Multi-carrier modulations over sparse channels: communication, channel estimation, and radar sensing

UNIVERSIT ` A DEGLI STUDI DI PARMA DOTTORATO DI RICERCA IN “TECNOLOGIE DELL’INFORMAZIONE” CICLO XXXIII Multi-Carrier Modulations Over Sparse Channels: Communication, Channel Estimation, and Radar Sensing Coordinatore: Chiar.mo Prof. Marco Locatelli Tutore: Chiar.mo Prof. Giulio Colavolpe Chiar.mo Prof. Giuseppe Caire Dottorando: Lorenzo Gaudio Anni 2017/2020 a chi crede in me Contents Introduction 1 State of the art 5 1 Multi Carrier Modulations 9 1.1 The Communication Channel . . . . . . . . . . . . . . . . . . . 9 1.2 Orthogonal Frequency Division Multiplexing (OFDM) Modulation ................................ 11 1.2.1 Input Output Relation . . . . . . . . . . . . . . . . . . . 14 1.3 Orthogonal Time Frequency Space (OTFS) Modulation . . . . 16 1.3.1 System Model and Definitions . . . . . . . . . . . . . . . 16 1.3.2 Modulation and Transmission over the Channel . . . . . 18 1.3.3 Demodulation........................ 20 1.3.4 Special Case: Rectangular Waveforms . . . . . . . . . . 25 1.3.5 Considerations on matrix Ψ................ 27 1.3.6 Symbols shift within the Doppler-delay grid . . . . . . . 32 1.3.7 General Waveforms . . . . . . . . . . . . . . . . . . . . . 33 1.3.8 The Cross-Ambiguity Function . . . . . . . . . . . . . . 36 2 ML Methods for Radar Parameter Estimation 39 2.1 Joint State Sensing and Communication . . . . . . . . . . . . . 39 2.1.1 OFDM ........................... 39 i ii Contents 2.1.1.1 Maximum Likelihood Estimator . . . . . . . . 39 2.1.1.2 Crame´r-Rao lower bound (CRLB) . . . . . . . 41 2.1.2 OTFS ............................ 43 2.1.2.1 Maximum Likelihood Estimator . . . . . . . . 43 2.1.2.2 Cram´er-Rao Lower Bound (CRLB) . . . . . . 48 2.1.2.3 ML Waterfall Analysis for Single Path . . . . . 49 2.2 Radar Resolution and Multi-Target Detection . . . . . . . . . . 53 2.3 SimulationResults ......................... 55 2.3.1 Joint State Sensing and Communication . . . . . . . . . 55 2.3.2 Joint Radar and Communication Performance . . . . . . 57 2.3.3 Self-Interference . . . . . . . . . . . . . . . . . . . . . . 61 3 ML Radar Methods in MIMO Configurations 65 3.1 Introduction............................. 65 3.2 Physicalmodel ........................... 68 3.2.1 OTFS Input Output Relation . . . . . . . . . . . . . . . 70 3.2.2 Beamforming matrices . . . . . . . . . . . . . . . . . . . 73 3.3 Joint Detection and Parameters Estimation . . . . . . . . . . . 74 3.3.1 Successive Interference Cancellation (SIC) and Joint Target Detection and Parameters Estimation Algorithm . . 76 3.3.2 Reduced-Complexity Parameter Estimation . . . . . . . 79 3.3.3 Cram´er-Rao Lower Bound (CRLB) . . . . . . . . . . . . 79 3.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.4.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . 83 4 OTFS Detection 91 4.1 Introduction............................. 91 4.2 TheDetectors............................ 93 4.2.1 Proposed MP-based detector (“Matrix Galgorithm” — MPG)............................ 93 4.2.2 Another MP-based algorithm (“Matrix Ψalgorithm” — MPΨ)............................ 97 Contents iii 4.2.3 Linear block-wise MMSE equalization . . . . . . . . . . 98 4.3 Performance of Separated Detection and Decoding . . . . . . . 99 5 Channel Estimation 105 5.1 Introduction............................. 105 5.2 OFDM Modulation and the CS Algorithm . . . . . . . . . . . . 109 5.2.1 The LASSO Solver . . . . . . . . . . . . . . . . . . . . . 113 5.2.1.1 Complexity of the LASSO Solver and Step Size Refinement....................114 5.2.1.2 Soft-Thresholding Operator . . . . . . . . . . . 115 5.2.1.3 Nesterov’s Acceleration Factor . . . . . . . . . 117 5.2.2 PilotScheme ........................118 5.2.3 Received Samples Expression — Real and Approximated Channel Conditions . . . . . . . . . . . . . . . . . . . . 119 5.3 OTFS Modulation and the Proposed Estimation Algorithm . . 120 5.3.1 PilotScheme ........................122 5.3.2 Channel Estimation . . . . . . . . . . . . . . . . . . . . 124 5.4 Comparison in Terms of Pragramatic Capacity . . . . . . . . . 127 5.4.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . 132 5.5 Conclusions.............................137 Bibliography 145 List of Figures 1.1 Channeldomains........................... 10 1.2 OTFSsystemmodel ........................ 16 1.3 Dirichlet functions . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.4 Doppler-delay shift example . . . . . . . . . . . . . . . . . . . . 33 1.5 Two dimensional Dirichlet function . . . . . . . . . . . . . . . . 34 1.6 Cross-ambiguity function examples . . . . . . . . . . . . . . . . 38 2.1 Waterfall behavior . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.2 RMSE curves for range and velocity, and Gaussian capacity . . 58 2.3 RMSE curves for multi-path case . . . . . . . . . . . . . . . . . 60 2.4 RMSE self-interference curves . . . . . . . . . . . . . . . . . . . 62 3.1 Broadcasting and tracking scenarios . . . . . . . . . . . . . . . 68 3.2 Receiver beam configuration . . . . . . . . . . . . . . . . . . . . 69 3.3 Transmitter beamforming examples . . . . . . . . . . . . . . . . 73 3.4 RMSE of a single target and detection w.r.t. distance . . . . . 84 3.5 RMSE of two targets including detection . . . . . . . . . . . . . 86 3.6 Example of masking effect . . . . . . . . . . . . . . . . . . . . . 87 3.7 Tracking scenario with multi-beams . . . . . . . . . . . . . . . 87 3.8 RMSE performance of tracking phase . . . . . . . . . . . . . . 88 4.1 Factor graph for matrix G..................... 95 4.2 Factor graph for matrix Ψ..................... 97 v 2 Introduction tion loss typical of that frequencies. Under this context, against common and well known radar systems, able to efficiently detect and locate a target within the range-velocity plane, new joint radar and communications techniques are taking part of the current literature. These system are mainly focused on the transmission of useful information towards the targets, which are, thus, not only “passively” detected, and a single equipment is able to perform both operational modes, avoiding to split the functionalities between two distinct subsystems (with increased cost and complexity). Hence, there are mainly two approaches to solve the aforementioned problem. The first one considers the application of common radar waveforms, adapted to carry useful information with them. The second one, which is the one explored in this dissertation, takes into account typical communication waveforms (singleor multi-carrier), and, while communication tasks come naturally, the radar processing is performed with novel methods exploiting the knowledge of the transmitted information (known by both transmitter and receiver, if physically colocated), thus differs from a more direct, or “radar-like”, threshold analysis of the backscattered power from the target. The choice of the communication waveform is subject to a non trivial tradeoff. On one hand, the system aims the maximization of the communication achievable rate, i.e., the amount of information sent in a time-frequency window. On the other hand, radar tasks have to be performed with as much precision as possible, in order to correctly localize a target in all dimensions, i.e., range, velocity, and space (angular) location. Typically, pure radar tasks are performed with chirp-like pulses, i.e., short single-carrier impulses with large bandwidth, such that the total energy delivered towards the targets is compensated by the band occupation of the signal. Thus, the joint definition of a short pulse, together with large bandwidth, leads to a very precise localization of the target over the three aforementioned domains. However, the amount of (possible) useful information, impressed on top of such chirp, is poor. A solution to improve the communication rate is the use of multi-carrier digital waveforms, modulating information symbols not only in time domain (as single-carrier) but also in the frequency band, split in Introduction 3 many subcarriers each occupied by a different modulation symbol. However, limitations are linked to the definition of the symbol time and the subcarrier spacing, which are in a one-to-one relation, and a good localization is not only challenging, for instance, in terms of signal processing algorithms, but also definitely sub-optimal with respect to single-carrier solutions, but this is the cost to pay in order to bring communication features together with radar tasks. In conclusion, the current literature is moving towards the definition of new multi-carrier schemes able to break the limits, in terms on communication rate, imposed by classical radar waveforms, and the optimization of the tradeoff between the two different tasks is an open problem, whose optimal solutions have not been defined yet. The choice of the multi-carrier modulation for joint radar and communication falls into two distinct waveforms, i.e., orthogonal frequency-division multiplexing (OFDM) modulation and orthogonal time frequency space (OTFS) modulation. OFDM is the most popular multi-carrier modulation of recent years, widely studied and standardized in most of the current communication standards, including 5G. The motivation of this choice is simple: thanks to the application of a cyclic prefix between symbols, i.e., a guard interval to prevent inter-symbol interference, and under the assumption of absence of inter-carrier interference, which holds under reasonable amount of the Doppler effect and subcarrier spacing, the communication channel can be diagonalized and symbol-by-symbol detection performed. Clearly, the appealing simplicity of detection makes OFDM the best choice for modern digital communications. On the other hand, OTFS is a modulation waveform with two big differences with respect to its direct competitor. First, it does not necessitate the insertion of the cyclic prefix, achieving a better communications rate, i.e., more information is sent over a time-frequency window, but at the cost of a more complex detection approach, working blockwise and not symbol-by-symbol. Second, OTFS is not sensitive to delay and Doppler shifts, meaning that its performance is kept constant whatever the distance and the speed between transmitter and (target) receiver. This feature is very appealing for joint radar 4 Introduction and communication tasks, being the scenario very dynamical, with possible remarkable Doppler shifts and delays, increased by considering the round-trip time between radar transmitter and target. Based on the aforementioned analysis, in this dissertation we take care of a fair comparison between the two digital modulation formats, from the point of view of radar, parameter estimation, achievable communication rate, channel estimation, and more other tasks, to determine their positive and negative aspects, such that a system designer is able to choose the most suitable waveform for a given scenario or application. State of the Art By extending the introduction, some details are provided here, with the corresponding references to literature, but some other are left to the introductions of chapters. The 5G communication standard will bring some novelties to overcome outdated and old techniques [1]. By mostly focusing on multi-carrier modulation formats, in particular OFDM [2, 3] and OTFS [4, 5, 6], this dissertation provides a complete analysis and performance comparison under different scenarios, with the common denominator of the sparse description of the communication channel [7], whose characteristics depend on the surrounding environment. Under this context, typical channels are characterized by few reflectors with their relative line-of-sight and small number of additional multi-path components (ground reflection and some other reflections from, e.g., metal surfaces) [8, 9]. Motivated by emerging vehicular applications (V2X) [10], joint radar and communication systems have been studied, in such a configuration the two functions share the same physical resources [11, 12]. Thus, differently from pure radar tasks, which aim to detect targets with high resolution [12, 13], also an active communication, i.e., the transmission of useful data, is considered, such that both functionalities might work together to jointly improve the performance. Note that this differs from typical beacon-based initial acquisition of communication standards [14, 15, 16, 17, 18], where the alignment between transmitter and receiver is achieved through a sort of handshake, i.e., the two entities talk together to achieve the common task. Given 6 State of the art the appealing of joint radar and communication systems for future vehicle-toeverything (V2X) communications, the recent literature provides many different and detailed solutions [19, 20, 21, 11, 22, 23, 24, 25, 26], basically divided into two classes: information-embedded radar waveforms [11, 27, 24] and communication waveforms applied to radar detection and parameter estimation [11, 19, 21, 26, 28, 29]. Moreover, note that the jointly approach could break the limits imposed by separated, or resource sharing, methods [30]. This dissertation, as said before, studies the case of communication waveforms applied to radar, by reviewing some well-known signal processing for OFDM (see [26, 21] and references therein), while exploiting new methods for OTFS, whose baselines are shared by other works in the literature, but in different shapes (see, e.g., [28, 29, 15]). In order to demonstrate that the joint approach is superior (not always but given some system setups) with respect to the physical resources split to one or the other task, the comparison will also take into account typical radar waveforms [13, 20]. When a communication waveform is employed, the problems of radar detection and parameter estimation are based on the knowledge of the information transmitted and backscattered from a target, which results to be known if radar transmitter and receiver are colocated. Note that this is also possible thanks to full-duplex configurations [31, 32, 33], which limits the selfinterference of the system, which results, on the other hand, unable to properly work if this condition is not fulfilled. Thus, information symbols are treated as known in the conditioning of probability density functions during digital signal processing operations, as, e.g., in [15], rather than unknown as in typical detection problems [34]. Hence, the joint radar sensing and communication paradigm results similar to classical channel estimation schemes, because the final goal is equivalent, i.e., the characterization of the surrounding channel. Being the channel state information, i.e., the knowledge of the communication channel, required to perform coherent detection in any scenario, the literature treating its estimation is wide [34]. Generally, this information is accessed through symbols, i.e., pilots, known at both transmitter and receiver State of the art 7 [35, 36, 37]. It is straightforward to understand that, by taking into account a full block of symbols backscattered from a target, in a joint radar and communication scheme, all symbols take the role of pilots. Many techniques to solve the channel estimation problem are present in literature. By taking into account the sparse channel representation in the Doppler-delay domain (see, e.g., [7, 5]), for OFDM, these techniques might use concepts from compressed sensing literature, e.g., [38, 39, 40, 41, 42, 35, 43], while OTFS propose a variety of solutions, based on minimum-mean square error estimation, maximum likelihood, compressed sensing, and others [37, 44, 45, 46, 47, 48]. At last, by taking into account the additional spatial dimension, multi antenna systems are studied. The choice of considering multiple-input multipleoutput (MIMO) configurations for radar is fundamental [49]. In fact, other than opening to angle of arrival (or, equivalently, space) estimation of the target, it allows, through a careful design of the beam pattern [50], to separately track different objects or targets [51, 52], while improving the power delivered towards a direction, thanks to the additional antenna gain, which is very relevant in V2X radars [20] and opens to transmission over millimeter wave frequency bands [50]. By considering the problem of radar detection, a non trivial tradeoff appears with respect to the angular coverage of the beam pattern. On one hand, a wider angular sector coverage enables to detect potentially more targets simultaneously, if the received backscattered power is high enough. On the other hand, a more directional allocation of the power towards a narrower angular sector, grants a higher received signal-to-noise ratio, at the cost of a time-consuming search (as classical radar successively swapping adjacent regions, see, e.g.,[13]). Different solutions can be found in the literature (see, e.g., [52, 53, 15, 54]). Moreover, this dissertation will treat the problem of mismatch between the number of antennas and the number of radio frequency chains. In fact, by considering MIMO configurations over millimeter wave frequency bands, it is difficult to implement a fully digital beamforming, or, equivalently, to associate one radio frequency chain per antenna (including A/D conversion, modulation, and amplification) in a small form factor 8 State of the art and highly integrated technology over a large signal bandwidth. Therefore, for millimeter wave automotive applications, we study hybrid digital-analog beamforming schemes (see, e.g., [55, 56] and references therein), thus, without relying on optimal full-duplex configurations as generally done in literature [49, 52, 54]. More details about the state-of-the-art will be given within the following chapters. Chapter 1 Multi Carrier Modulations 1.1 The Communication Channel The communication channel describes how the transmitted signal is modified when traveling through the communication medium (e.g., an optical fiber, the air, a copper line, etc.). Different impairments and effects characterize each different scenario, and the associated channel is completely described by its channel impulse response (CIR). These effects, including, for instance, fading fluctuations, shadowing, delay, frequency shifts, phase noise, etc., are described by mathematical models, exploited during the algorithmic design of detectors, estimators, and any other digital signal processing (DSP) which could be performed by the communication receiver (Rx). The channel considered in this dissertation is time-frequency varying, i.e., its behavior changes with respect to (w.r.t.) the time instant and the carrier (or subcarrier) frequency considered. In order to simplify its treatment, the channel can be represented in distinct domains, each owning its different (but behaviorally equivalent) description function. In fact, clearly, the channel behavior must remain the same while its representation changes. In order to switch between different domains, a direct or inverse Fourier transform (i.e., F or F−1, respectively) has to be applied [7]. Fig. 1.1 shows all the domains and 9 10 Chapter 1. Multi Carrier Modulations γ(t, τ)H(t, f) Γ(ν, f)h(ν, τ) F F−1 F F−1 F−1 F F−1 F Figure 1.1: Channel domains. the relative Fourier transforms. In the top right position of Fig. 1.1 we find the time-frequency domain, with function H(t, f). Considering only direct Fourier transforms, i.e., F, we first move to the Doppler-frequency domain (Γ(ν, f)), then to the Doppler-delay domain (h(ν, τ)), and finally to the time-delay domain (γ(t, τ)). It is interesting to note that the channel description in the Doppler-delay domain relies in its physical representation, or geometry, which simplifies the overall mathematical analysis [4, 5, 7]. In fact, typically, only a small number of reflectors (or propagation paths) takes part of a channel, which is thus sparse and can be modeled with few parameters. Moreover, the geometry of the surrounding environment slowly changes in time (w.r.t. the frame duration), behavior which could be exploited during the algorithmic design. The sparse representation of the channel h(ν, τ) can be given as [7] h(ν, τ) = P−1 X p=0 hpδ(τ−τp)δ(ν−νp),(1.1) where Pis the number of propagation paths, hp,τp, and νprepresent the path gain, delay, and Doppler shift associated to the p-th path. The key point is that the channel behavior is discrete in the number of paths. In other words, a single symbol transmitted over the channel is shifted in the delay domain, i.e., is received with a delay of τp, and its frequency is shifted of νp(Doppler 1.2. Orthogonal Frequency Division Multiplexing (OFDM) Modulation 11 effect). An extension of the (1.1) taking into account MIMO antenna systems can be found in Chapter 3 and in [7]. 1.2 Orthogonal Frequency Division Multiplexing (OFDM) Modulation Before entering into the details of OFDM signal processing, for radar and communication purposes, we will briefly describe the basics of this modulation technique, to better understand things to come. This pretends to be an overview of mainly aspects which are relevant for our analysis, while a more in-depth description can be found in many different digital communication books (see, e.g., [2, 3]) and works in literature [2, 57, 58]. As the name suggests, OFDM is a multiplexing scheme which modulates data (information symbols) on distinct parallel orthogonal frequencies (see also the pioneering work [58]). In general, OFDM uses a certain number of subcarriers, to equally split the available bandwidth, and some time slots, which, together, identify an OFDM frame. The dimension of such frame depends on the particular application, which could aim at low-latency systems, i.e., smaller frames in time, or necessitates larger dimensions to cope and estimate unknown channel impairments. As a notation, a set of modulation symbols transmitted over different subcarriers at the same time is called OFDM symbol, while more OFDM symbols (in time) form the OFDM frame. The orthogonality of the frequency division is achieved by choosing a constant subcarrier spacing ∆f, generally defined as the inverse of the symbol duration T, i.e., ∆f= 1/T, in order to avoid data loss during filtering operations. Thus, by assuming a rectangular shaping pulse of duration Tto modulate constellation symbols at the transmitter (Tx) side, whose expression is 18 Chapter 1. Multi Carrier Modulations cross-ambiguity function (CAF) between the two pulses, useful for successive considerations, i.e., Cgtx,grx (t, f),Zg∗ rx(t0−t)gtx(t0)e−j2πft0dt0.(1.15) We adopted the definition of [13], while other expressions might be found in literature (with no significant changes on the final results and system behavior).  Consider a time-varying channel where the maximum delay and Doppler shift over all multipath components are given by τmax and νmax, respectively. The parameters Tand ∆fdetermine the maximum tolerable delay and Doppler, respectively, such that νmax <∆fand τmax < T. We will now look into a detailed derivation of the OTFS input-output relation, which is the base of any signal processing applied afterwords. 1.3.2 Modulation and Transmission over the Channel The OTFS Tx first maps symbols x[k, l] to samples X[n, m], from the Dopplerdelay domain to the time-frequency domain, according to grids Γ and Λ, using the ISFFT, i.e., X[n, m] = 1 √NM N−1 X k=0 M−1 X l=0 x[k, l]ej2π(nk N−ml M),(1.16) for n= 0,...N −1, m = 0,...M −1.2Eq. (1.16) shows that each information symbol x[k, l], belonging to any complex constellation alphabet, is modulated by a two-dimensional basis function in the time-frequency domain, i.e., 2Note that, since the ISFFT is a Fourier transformation between two-dimensional domains, a normalization factor has to be taken into account. The normalization factor 1/(NM) could be at both direct and inverse transformation, with a square root, of just at one side, without the square root. 1.3. Orthogonal Time Frequency Space (OTFS) Modulation 19 exp j2πnk N−ml M. Next, the time-frequency modulator converts the samples X[n, m] to a continuous-time waveform s(t), by the use of the transmit (shaping) pulse gtx(t), i.e., s(t) = N−1 X n=0 M−1 X m=0 X[n, m]gtx(t−nT)ej2πm∆f(t−nT ).(1.17) Eq. (1.17) can be seen as a discrete Heisenberg transform parameterized by gtx(t) [5, 4]. The pulse s(t) is the product of the superposition of delay-andmodulate operations on the pulse waveform gtx(t), shifted in time and in frequency. Note that it is useful to express s(t) through an Heisenberg transform since the cascade of two Heisenberg transforms, one for the modulator and one for the channel, can be expressed as a unique function. Note that the equality ∆fT = 1 implies ej2πm∆f(t−nT )=ej2πm∆ft ,(1.18) which simplifies (1.17) leading to the equivalent signal model s(t) = N−1 X n=0 M−1 X m=0 X[n, m]gtx(t−nT)ej2πm∆ft ,(1.19) which can be found, e.g., in [26, Page 13, Equation (3.4)]. The signal s(t) is transmitted over the time-frequency varying channel with complex baseband CIR h(ν, τ) specified in (1.1). The received signal, neglecting for simplicity the noise, is r(t) = ZZ h(ν, τ)s(t−τ)ej2πνtdτdν , (1.20) which is a continuous Heisenberg transform parameterized in h(ν, τ). Note that, by substituting (1.1) into (1.20), the double integration is valid only where the two deltas are equal to one, simplifying in a single summation over p= 0, . . . , P −1. This substitution is done in subsequent calculus. 20 Chapter 1. Multi Carrier Modulations 1.3.3 Demodulation At the Rx, a matched filter computes the CAF (see (1.15)) in the following way Y(t, f) = Agrx,r (t, f) = Zg∗ rx t0−trt0e−j2πft0dt0.(1.21) By substituting (1.20) in (1.21), we obtain Y(t, f) = Zg∗ rx t0−tZZ h(ν, τ)st0−τej2πνt0dτdνe−j2πft0dt0, (1.22) and, by using (1.17) Y(t, f) = Zg∗ rx t0−t"ZZ N−1 X n0=0 M−1 X m0=0 h(ν, τ)Xn0, m0gtx t0−τ−n0T ej2πm0∆f(t0−τ−n0T)ej2πνt0dτdν#e−j2πft0dt0,(1.23) while, by reordering terms Y(t, f) = N−1 X n0=0 M−1 X m0=0 Xn0, m0"ZZ h(ν, τ)(Zg∗ rx t0−tgtx t0−τ−n0T ej2πm0∆f(t0−τ−n0T)ej2πνt0e−j2πft0dt0)dτdν#.(1.24) The matched filter output is obtained by sampling Y(t, f) as Y[n, m] = Y(t, f)t=nT,f=m∆f.(1.25) Now, by recalling the function inside the square brackets and sampling, we define Hn,m n0, m0=ZZ h(ν, τ)"Zg∗ rx t0−nTgtx t0−τ−n0T ej2πm0∆f(t0−τ−n0T)ej2π(ν−m∆f)t0dt0#dτdν . (1.26) 1.3. Orthogonal Time Frequency Space (OTFS) Modulation 21 By substituting t00 =t0−τ−n0T, we get Hn,m n0, m0=ZZ h(ν, τ)"Zg∗ rx t00 −n−n0T+τgtx t00 ej2πm0∆ft00 ej2π(ν−m∆f)(t00+n0T+τ)dt00#dτdν =ZZ "Zg∗ rx t00 −n−n0T+τgtx t00 e−j2π((m−m0)∆f−ν)t00 dt00#h(ν, τ)ej2π(ν−m∆f)(n0T+τ)dτdν =ZZ h(ν, τ)Agrx,gtx n−n0T−τ, m−m0∆f−ν ej2πνn0Tej2πντ e−j2πm∆fτ dτdν , (1.27) and, by considering the channel specified in (1.1), it becomes Hn,m n0, m0= P−1 X p=0 hpAgrx,gtx n−n0T−τp,m−m0∆f−νp ej2πνpn0Tej2πνpτpe−j2πm∆fτp.(1.28) It is straightforward to obtain the input-output relation of OTFS, given by Y[n, m] = N−1 X n0=0 M−1 X m0=0 Hn,m n0, m0Xn0, m0.(1.29) Note that Eq.(1.29) can be split to directly and separately show the parts 22 Chapter 1. Multi Carrier Modulations involved in (Doppler-delay) ISI and ICI, thus Y[n, m] = N−1 X n0=0 M−1 X m0=0 Hn,m n0, m0Xn0, m0 =Hn,m [n, m]X[n, m] + M−1 X m0=0 m06=m Hn,m n, m0Xn, m0 + N−1 X n0=0 n06=n M−1 X m0=0 Hn,m n0, m0Xn0, m0,(1.30) in which the first term indicates the current symbol X[n, m], with the associated channel response, the second term is the ICI, i.e., the total interference at different frequencies m06=mbut within the same time slot nof the current symbol X[n, m], and, at last, the third term is the ISI. Note that, at this point, the shape of the pulses is unknown, so there are possibly infinite (past and future) interfering terms. Proceeding further, starting from (1.30) and exploiting (1.16), we obtain Y[n, m] = N−1 X n0=0 M−1 X m0=0 Hn,m n0, m0Xn0, m0 = N−1 X n0=0 M−1 X m0=0 Hn,m n0, m0"N−1 X k0=0 M−1 X l0=0 x[k0, l0] √NM ej2πn0k0 N−m0l0 M#.(1.31) By applying the ISFFT (from now on, for the sake of brevity, we remove the summation subscripts and superscripts, which are, however, in accord to the 1.3. Orthogonal Time Frequency Space (OTFS) Modulation 23 aforementioned treatment) y[k, l] = X n,m X n0,m0X k0,l0 Hn,m n0, m0x[k0, l0] NM ej2πn0k0 N−m0l0 Me−j2π(nk N−ml M) =X k0,l0 x[k0, l0] NM  X n,m X n0,m0 Hn,m n0, m0ej2πn0k0 N−m0l0 Me−j2π(nk N−ml M)  =X k0X l0 x[k0, l0] NM hk,l k0, l0,(1.32) in which, by using the definition in (1.28), including the CIR in (1.1), we get hk,l k0, l0=X N,M X n0,M0 P−1 X p=0 hpAgrx,gtx n−n0T−τp,m−m0∆f−νp ej2πνpn0Tej2πνpτpe−j2πm∆fτpej2πn0k0 N−m0l0 Me−j2π(nk N−ml M).(1.33) Note that Agtx,grx n−n0T−τ, m−m0∆f−ν= Zg∗ rx t0−n−n0T+τgtx t0e−j2π[(m−m0)∆f−νp]t0dt0,(1.34) and since the received signal r(t) is sampled at time intervals t0= 1/(M∆f) or equivalently t0=T/M, we get Agtx,grx =T M i=∞ X i=−∞ g∗ rx iT M−n−n0T+τpgtx iT M e−j2π[(m−m0)∆f−νp]i M∆f.(1.35) 24 Chapter 1. Multi Carrier Modulations So, by reordering terms (note that ∆fT = 1) and defining h0 i=hiej2πντ hk,l k0, l0=T M P−1 X p=0 h0 p(X n e−j2π(k N)n"X n0 ej2πk0+νpNT Nn0 i=∞ X i=−∞ g∗ rx iT M−n−n0T+τpgtx iT M X m ej2π−∆fi M∆f+τp+l MmX m0 e−j2πl0 M−i Mm0ej2πνpi M∆f#) =T M P−1 X p=0 h0 p(X n e−j2π(k N)n"X n0 ej2πk0+νpNT Nn0 i=∞ X i=−∞ g∗ rx iT M−n−n0T+τpgtx iT M (X m ej2π(−i−M∆fτp+l)m M)(X m0 e−j2π(−i+l0)m0 M)ej2πνpi M∆f#). (1.36) At this point, it is useful to define the Dirichlet kernel function, that is Dir (φ, Z), Z−1 X z=0 ej2πφ z Z=ej2πφ −1 ej2πφ/Z −1=ejπφ ejπφ −e−jπφ ejπφ/Z ejπφ/Z −e−jπφ/Z =ejπφ(Z−1)/Z sin (πφ) sin (πφ/Z).(1.37) A plot of the function is given in Fig. 1.3. The value of the function is equal to Z(or −Zdepending if Zis even or odd) when φis a multiple of Z, and is equal to zero for all others integer values of φ. Moreover, note that the Dirichlet kernel has the following property sin (πφ) sin π Zφ=sin (π(φ+Z)) sin π Z(φ+Z).(1.38) which could be useful for successive analysis. 1.3. Orthogonal Time Frequency Space (OTFS) Modulation 25 -20 -15 -10 -5 0 5 10 15 20 -1 0 1 2 3 4 5 (a) Z= 5. -15 -10 -5 0 5 10 15 -6 -4 -2 0 2 4 6 (b) Z= 6. Figure 1.3: Example of Dirichlet functions Dir (φ, Z) = sin(πφ) sin(πφ/Z). Thus, the expression for y[k, l] becomes y[k, l] = X k0,l0 x[k0, l0] NM T M P−1 X p=0 h0 p(X n e−j2πk n NX n0"∞ X i=−∞ gtx iT M g∗ rx iT M−n−n0T+τpDir (l−i−τpM∆f, M) Dir l0−i, Mej2πνpi M∆f#ej2π(k0+νpNT )n0 N) =X k0,l0 x[k0, l0] NM T P−1 X p=0 h0 p(X nX n0 g∗ rx l0T M−n−n0T+τp gtx l0T MDir l−l0−τpM∆f, Me−j2πk n Nej2πνpl0 M∆f ej2π(k0+νpNT )n0 N).(1.39) 1.3.4 Special Case: Rectangular Waveforms As a practical special case, consider a rectangular waveform of duration Tand amplitude 1/√T, i.e., gtx(t) = grx(t) = rect(t), as defined in (1.2). If τmax < T, only the signal of the first preceding slot is involved in the ISI calculation, i.e., 26 Chapter 1. Multi Carrier Modulations n0=n−1. In this case g∗ rx pT M−T+τpgtx pT M=1 √T 1 √T=1 T,(1.40) for the values of pwhere the product of the two pulses is nonzero. Thus, for rectangular pulses, the sum Pn0takes into account only two terms, i.e., n0=n and n0=n−1. By using these results, starting from (1.39), we obtain yrect [k, l] = X k0,l0 x[k0, l0] NM P−1 X p=0 h0 p (X n e−j2π(k−k0−νpNT )n Nej2πνpl0 M∆f Dir l−l0−τpM∆f, M)+(X n e−j2π(k−k0−νpNT )n N Dir l−l0−τpM∆f, Mej2πνpl0 M∆fe−j2π(k0+νpNT ) N)! =X k0,l0 x[k0, l0] NM P−1 X p=0 h0 pDir νpNT −k+k0, Nej2πνpl0 M∆f Dir l−l0−τpM∆f, M1 + e−j2πk0+νpNT N.(1.41) However, note that, having considered the multiplication between the two rectangular pulses gtx(t) and grx(t), the l0involved in the two terms inside the curly brackets is different, since it considers the pulses overlap within the interval [0, M −1−dτp/(T/M)e] and [M−1−bτp/(T/M)c, M −1], where d·e and b·c indicates the nearest upper and lower integer, respectively. We finally get y[k, l] = P−1 X p=0 h0 pX k0 Dir νpNT −k+k0, NX l0 Dir l−l0−τpM∆f, M ej2πνpl0 M∆fx[k0, l0] NM ×   1 if l0∈l0 ISI e−j2πνpT+k0 Nif l0∈l0 ICI ,(1.42) 1.3. Orthogonal Time Frequency Space (OTFS) Modulation 27 where we used    l0 ISI ,n0, M −1−d τp T/M eo l0 ICI ,nM−1−b τp T/M c, M −1o.(1.43) Thus, the channel matrix expression for the p-th path becomes Ψp k,k0l, l0=1 NM Dir νpNT −k+k0, NDir l−l0−τpM∆f, M ej2πνpl0 M∆f×   1l0∈l0 ICI e−j2πνpT+k0 Nl0∈l0 ISI .(1.44) The input output relation becomes y[k, l] = X k0,l0 P−1 X p=0 h0 pΨp k,k0l, l0xk0, l0,(1.45) which can be represented in matrix form as y=  P−1 X p=0 h0 pΨp x,(1.46) with yand xvectors of dimension NM ×1 obtained by stacking the received samples and information symbols, respectively, and Ψpmatrix of dimension MN ×MN, whose {k, k0, l, l0}element is defined in (1.44). 1.3.5 Considerations on matrix Ψ We now consider some special cases of matrix Ψp, for any single path p.  Zero delay — Zero Doppler:τp=νp= 0. Ψp k,k0l, l0=   1 if (l=l0, k =k0) 0 else .(1.47) When both delay and Doppler are zero, Ψpresults to be the identity matrix. The channel does not modify the transmitted symbols, the received signal is simply y=x+ noise. 34 Chapter 1. Multi Carrier Modulations −5 0 −4−2024 0 0.5 1 |A| Figure 1.5: Two dimensions visual example of Dirichlet functions in both domains, where |A|is the normalized amplitude value, in accord to the heatmap of Fig. 1.4. support. By starting from (1.39) and defining q,(n−n0), we get y[k, l] = T NM X k0,l0 xk0, l0P−1 X p=0 h0 p(X n e−j2πk n NX n0 ej2π(k0+νpNT )n0 Nej2πνpl0 M∆f g∗ rx l0T M−n−n0T+τpgtx l0T MDir l−l0−τpM∆f, M) =X k0,l0 x[k0, l0]T NM P−1 X p=0 h0 p(X n e−j2πk n N ∞ X q=−∞ ej2π(k0+νpNT )n−q Nej2πνpl0 M∆f g∗ rx l0T M−qT +τpgtx l0T MDir l−l0−τpM∆f, M) =x[k0, l0]T NM X k0,l0 P−1 X p=0 h0 pgtx l0T MDir l−l0−τpM∆f, Mej2πνpl0 M∆f X n ej2π(k0+νpNT −k)n N!∞ X q=−∞ e−j2π(k0+νpNT )q Ng∗ rx l0T M−qT +τp. (1.60) 1.3. Orthogonal Time Frequency Space (OTFS) Modulation 35 Note that only the term in brackets depends on n, and it is a Dirichlet function X n ej2π(k0+νpNT −k)n N,Dir νpNT −k+k0, N.(1.61) Hence y[k, l] = X k0,l0 x[k0, l0]T NM P−1 X p=0 h0 pDir l−l0−τpM∆f, MDir νpNT −k+k0, N gtx l0T M(∞ X q=−∞ e−j2π(k0+νpNT )q Ng∗ rx l0T M−qT +τp) ej2πνpl0 M∆f.(1.62) The term under curly brackets takes into account interfering pulses shifted of multiples of Tw.r.t. the summation index q. Depending on the support of the pulses, i.e., when the energy is above a certain threshold if the pulse support is infinite, only a fixed number of interferes appears (to left and to right), indicated by Ip. The summation over qcan be thus limited to [−Ip, Ip]. We finally get y[k, l] = X k0,l0 x[k0, l0]T NM P−1 X p=0 h0 pDir l−l0−τpM∆f, MDir νpNT −k+k0, N ej2πνpl0 M∆fgtx l0T M(Ip X q=−Ip e−j2π(k0+νpNT )q Ng∗ rx l0T M−qT +τp). (1.63) Note that a pulse not satisfying the Nyquist condition, i.e., flat frequency representation, changes the noise properties. The noise associated to the received samples is not white anymore, and changes within the pulse definition. This fact must be considered in the mathematical and simulation model. What is the role of the shaping pulses in the construction of the channel matrix Ψ? The structure of Ψis dominated by the Dirichlet function values, w.r.t. the integer/fractional delay and Doppler shifts and the indices l, l0, k, k0. 36 Chapter 1. Multi Carrier Modulations The ISI and ICI effects caused by the shaping pulses add to the Dirichlet behavior, but how? One can think that optimized “well-known” pulses having limited CAF in time-frequency domain should be adopted [60], but the effect in the dual Doppler-delay domain remains not clear. In fact, the Doppler-delay ISI and ICI weakly depend on the adopted shaping pulse, and are dominated by the Dirichlet functions, whose expressions appear from the particular transformations performed by modulator and demodulator, and not from the choice of the transmitted and received pulses (see Sec. 1.3.1). For these reasons, once common well-confined time-frequency pulses are adopted [60], it is not guaranteed to achieve good performance also in the Doppler-delay domain. This fact is also confirmed in [61]. The conclusion could be that, whatever the chosen pulse (also different between transmitter and Rx), there are no performance guarantees. For completeness, the next section will present some known pulses and the associated CAFs, together with final considerations on the adopted pulses. 1.3.8 The Cross-Ambiguity Function In radar scenarios, the CAF is a two-dimensional function of delay and Doppler showing the distortion of a returned pulse at the Rx matched filter due to delay and Doppler shift of the moving target (see, e.g., [13, 60]). The ambiguity function description is only determined by the properties of the transmitted pulse and the matched filter (received pulse). By taking into account the definition of CAF in (1.15) [13], recalled here for convenience Cgtx,grx (t, f),Zg∗ rx(t0−t)gtx(t0)e−j2πft0dt0,(1.64) we show its behavior when different pulses gtx(t) and gtx(t) are used. Fig. 1.6 shows that different pulses achieve distinct performance in terms of spreading in time and in frequency of the CAF. For instance, as shown in Fig. 1.6 (a), since rectangular pulses have infinite support in the frequency domain, the projection of the CAF on the frequency plane slowly decays and 1.3. Orthogonal Time Frequency Space (OTFS) Modulation 37 necessitates some time to reach the zero “floor”, while values near the peak point have remarkable magnitude. A different behavior is shown when two Gaussian pulses are adopted (Fig. 1.6 (b)). In fact, having Gaussian pulses finite support in the frequency domain, they exhibit a rapid decay to zero around peak point. A similar consideration occurs in the time domain, which is confirmed by looking at the projection on the time plane. At last, compare Fig. 1.6 (b) and 1.6 (c) to have an idea on how Gaussian pulses with different variances change the behavior of the projections over the time and frequency planes. The definition of good shaping pulses, with the corresponding CAF, is a problem of primary interest in typical radar systems. For more details, for instance, refer to the analysis carried on in [60], suggesting the use of pulses well localized in the time-frequency domain. Regarding OTFS, pioneering works [4, 5] are based on the assumption of ideal bi-orthogonal pulses satisfying perfect interference properties, i.e., resulting to be deltas in both time and frequency planes. However, these pulses cannot be created in real electronic circuits, and different solutions must be adopted. Thus, it is possible to find in literature many examples of OTFS modulation based on rectangular pulses [62, 46], whose CAF spreads in frequency, according to Fig. 1.6, but it easier to handle (mathematically speaking) within the derivation of the OTFS input-output relation. An in-depth numerical analysis comparing the performance of OTFS with different pulses (rectangular, root-raised cosine, Gaussian) has been carried on, resulting in similar performance, and thus not shown here for the sake of brevity. This fact has been also confirmed in [61]. Thus, we took advantage of the simplicity in terms of mathematical treating of rectangular pulses, as generally done in literature, by keeping in mind that a different treatment is possible, but does not lead to remarkable performance improvement, at least in the considered scenarios. 38 Chapter 1. Multi Carrier Modulations −1 0 1−10 −50510 0 0.5 1 t f |A| (a) Rectangular pulses, gtx(t) = grx(t). −2 0 −10 −50510 0 0.5 t f |A| (b) Gaussian pulses, σ2 1,gtx(t) = grx(t). −2 0 −10 −50510 0 0.2 0.4 t f |A| (c) Gaussian pulses, σ2 2> σ2 1,gtx(t)6= grx(t). Figure 1.6: Cross-ambiguity function for different pulses. Chapter 2 ML Methods for Radar Parameter Estimation 2.1 Joint State Sensing and Communication 2.1.1 OFDM 2.1.1.1 Maximum Likelihood Estimator Starting from the aforementioned analysis, by focusing for simplicity on a single-target case (P= 1), we neglect the p-path subscript for the following derivations based on OFDM. Since data symbols are known by the radar Rx (which could be colocated with the Tx (monostatic radar) or not (bistatic radar)), and the noise is i.i.d. Gaussian circularly symmetric, the radar Rx can shift the data symbol phase without changing the noise statistics. Therefore, the radar observation, including the noise and symbol-by-symbol phase rotation, can be written as zn,m =An,mhej2πnToνe−j2πm∆fτ +wn,m ,(2.1) where An,m =|xn,m|denotes the amplitude of the transmitted symbol and wn,m is the additive white Gaussian noise (AWGN) with zero mean and unit 39 40 Chapter 2. ML Methods for Radar Parameter Estimation variance. The maximum likelihood (ML) estimator of channel gain, range, and velocity for the observation model in (2.1) is obtained by generalizing the approach in [26, Chapter 3.3.3] to the case of arbitrarily amplitude Asymbols, with Abeing the matrix collecting the time-frequency entries {An,m}. For the set of parameters θ= (h, ν, τ), we wish to find the estimator minimizing the likelihood function l(z|θ,A) = X nX mzn,m−hAn,mej2π(νnTo−m∆fτ) 2 .(2.2) Assuming that the pair (ν, τ) is known, by letting the derivative of l(z|θ,A) w.r.t. hequal to zero, we obtain the estimation ˆ hof the complex channel gain h, which is ˆ h=Z(ν, τ) Pn,m A2 n,m ,(2.3) where the DFT/inverse discrete Fourier transform (IDFT) operation is defined as Z(ν, τ), M−1 X m=0 N−1 X n=0 zn,mAn,me−j2πνnToej2πm∆fτ ,(2.4) which is a two-dimensional periodogram. By plugging (2.3) into (2.2) and following similar steps as [13, Chapter 7.2.2]), the estimator of the remaining unknown parameters in given by (ˆν, ˆτ) = arg max (ν,τ)∈Γ0|Z(ν, τ)|2,(2.5) where we considered a discretized set Γ0of delay and Doppler frequency axes with step sizes 1/(M0∆f) and 1/(N0To), respectively, with N0≥Nand M0≥ M. Note that Γ0is in line with the definition in (1.13), but with greater granularity to achieve an higher accuracy within the estimation process. In summary, to compute the joint ML estimator of the set of unknown parameters (h, τ, ν) the following steps are done: 2.1. Joint State Sensing and Communication 41 1. Compute the DFT/IDFT output Z(ν, τ). This step which can be efficiently implemented by using fast Fourier transform (FFT)-based design. 2. Choose (ˆν, ˆτ) maximizing |Z(ν, τ)|2over Γ0, for some N0and M0(depending to the target accuracy level). 3. Let the channel gain be ˆ h=Z(ˆν, ˆτ)/Pn,m A2 n,m. Clearly, starting from the delay and Doppler estimations, it is possible to derive the corresponding range and velocity estimations, respectively given by ˆr= ˆτc/2 and ˆv= ˆνc/(2fc). 2.1.1.2 Crame´r-Rao lower bound (CRLB) It is well known that the Crame´r-Rao lower bound (CRLB) provides a theoretical lower bound on the variance of any estimator [3, 34]. The derivation of the CRLB depends on the particular system setting, but it is always based on a common denominator, i.e., the construction of the Fisher information matrix. Suppose we have only one path, i.e., P= 1, to simplify the notation. For the calculation of the CRLB consider a vector of unknown parameters θto be estimated. Given f(y|θ), which is the conditional distribution of the channel output ygiven the set of unknown parameters θ, if the regularity condition is satisfied Ey∂ ∂θln f(y|θ)=   Ey∂ ∂h ln f(y|θ) Ey∂ ∂τ ln f(y|θ) Ey∂ ∂ν ln f(y|θ)  =   0 0 0   ,(2.6) then, any unbiased estimator providing ˆ θhas covariance matrix Cˆ θ=Ehˆ θ−Ehˆ θiˆ θ−Ehˆ θi∗i,(2.7) which satisfies Cˆ θ−I(θ)−1≥0∀θ.(2.8) 42 Chapter 2. ML Methods for Radar Parameter Estimation The matrix I(θ) is the Fisher information matrix, whose (i, j) element is [I(θ)]i,j =−Ey∂2ln f(y|θ) ∂θi∂θj,(2.9) where indexes (i, j) select the unknown parameters within the vector θ. Moreover, (2.8) implies that diagonal elements of Cθdominates those of I(θ)−1, hence Var hˆ θii≥hI(θ)−1iii ≥1 [I(θ)]ii .(2.10) Consider the vector of unknown parameters θ= (α, ϕ, f, t), where α=|h|, ϕ=∠(h), f=Toν, and t= ∆fτ, from (2.1) we obtain zn,m =An,mαejϕej2πnf e−j2πmt+wn,m .(2.11) By letting sn,m =An,mαejϕej2πnf e−j2πmt, we derive the 4 ×4 Fisher information matrix defined as [I(θ,A)]i,j = 2PavgRe (X n,m ∂sn,m ∂θi∗∂sn,m ∂θj),(2.12) where Pavg takes into account any possible power constraint on transmitted symbols. We thus have ∂sn,m ∂α =An,mejϕe+j2πnf e−j2πmt(2.13a) ∂sn,m ∂ϕ =jαAn,mejϕe+j2πnf e−j2πmt(2.13b) ∂sn,m ∂f = (j2πn)αAn,mejϕe+j2πnf e−j2πmt(2.13c) ∂sn,m ∂t= (−j2πm)αAn,mejϕe+j2πnf e−j2πmt.(2.13d) For a model given in (2.11), the MMSE of f and tis lower bounded by σ2 ˆ f≥N0 2|h|2 ef(A) d(A)(2.14a) σ2 ˆ t≥N0 2|h|2 et(A) d(A),(2.14b) 2.1. Joint State Sensing and Communication 43 where ef(A), et(A), d(A) are given by ef(A) = (2π)2(X n,m (A2 n,m)·X n,m (m2A2 n,m)−hX n,m mA2 n,mi2),(2.15a) et(A) = (2π)2(X n,m (A2 n,m)·X n,m (n2A2 n,m)−hX n,m nA2 n,mi2),(2.15b) d(A) = (2π)4nX n,m (A2 n,m)hX n,m n2A2 n,mihX n,m m2A2 n,mi + 2hX n,m nA2 n,mihX n,m mA2 n,mihX n,m nmA2 n,mi−X n,m (A2 n,m)hX n,m nmA2 n,mi2 −hX n,m n2A2 n,mihX n,m mA2 n,mi2−hX n,m m2A2 n,mihX n,m nA2 n,mi2o.(2.15c) In the regime of large Mand N, the CRLB of fand tare given by σ2 ˆ f≥6 |h|2Pavg(2π)2MN(N2−1) ,(2.16a) σ2 ˆ t≥6 |h|2Pavg(2π)2MN(M2−1) .(2.16b) For a special case of constant envelope (An,m =pPavg for all n, m), the above expressions coincide with those in [26, Section 3.3]. 2.1.2 OTFS 2.1.2.1 Maximum Likelihood Estimator Based of the results of Chapter 1 and Sec. 1.3.1, the vectorized input-output relation is y= P−1 X p=0 hpΨp(τp, νp)x+w.(2.17) We wish to estimate the set of unknown parameters θ= (¯ h,¯ τ,¯ ν), with ¯ h= [h0, . . . , hP−1], ¯ τ= [τ0, . . . , τP−1], and ¯ ν= [ν0, . . . , νP−1], where the bar indicates the true channel parameter value. 50 Chapter 2. ML Methods for Radar Parameter Estimation maximization based estimator in this dissertation): namely, when the noise dominates the useful signal, the maxima of the likelihood functions tend to be randomly placed anywhere on the search grid. In the following, we analyze the waterfall effect, or transition, for the single path case P= 1, i.e., we search and study the region around the threshold SNR value where the rapid deterioration occurs. Moreover, simulations results show that the provided waterfall prediction for P= 1 is also very accurate for the multipath case P > 1.1This also confirms the evidence that the proposed approximated iterative ML estimation described in Algorithm 1 is effectively very good, and performs very close to the true ML.2 Following the reasoning of [64, 65], we define as “outlier” the event that a maximum of the likelihood function is randomly placed on the grid Γ, rather than within the cluster of points surrounding the true value (¯τ0,¯ν0). Let α∈ {τ, ν}be the unknown parameter to be estimated, i.e., τor νindifferently. By the law of total probability over the discretized grid Γ (where we calculate the ML estimator), we can express the estimation MSE as MSE = Eh(ˆα−¯α)2i=X i∈Γ Pr (ε(i)) (ˆαi−¯α)2 ≤X i∈Γ Pr (˜ε(i)) (ˆαi−¯α)2.(2.36) where ¯αis the true value of the parameter, ˆαis an estimation of such parameter, and Pr (ε(i)) denotes the probability of error of choosing αirather than ¯α. While evaluating Pr (ε(i)) may be extremely difficult, also because the true parameter ¯α∈R, we obtain an upper bound by considering pairwise error probabilities, i.e., replacing Pr (ε(i)) with the probability that the detector chooses αirather than ¯αwhen these are the only two alternatives. Since the 1This is also true because the MSE performance between the single path and the multipath case are very similar. 2Obviously, the CRLB for P= 1 yields also a lower bound for the case P > 1, in the case where we simply add more multipath components with independent statistics, and the presence of more unknown parameters does not generally help the estimation of each single target parameters. 2.1. Joint State Sensing and Communication 51 pairwise error event ˜ε(i) contains the true error event ε(i), it follows that the inequality of (2.36) provides an upper bound. Thus, the definition of the pairwise error probability is Pr (˜ε(i)) ,Pr nl(y|θi,x)> l y|¯ θ,xo,(2.37) where l(y|θi,x) and ly|¯ θ,xare the values obtained from the evaluation of the likelihood function at grid point i, with parameter θi, and when the set of true parameters ¯ θis taken into account, respectively. At this point, the problem is reduced to the computation of the pairwise error probabilities Pr (ε(i)), for i∈Γ, which can be derived as follows. We notice that, taking into account the useful signal appearing in (2.28) Pr (˜ε(i)) ≈Pr |xHΨH iy|2>|xH¯ ΨHy|2,(2.38) with ¯ Ψthe channel matrix computed with the true set of parameters, and where we exploit ΨH iΨi=I,∀i, for P= 1. Defining the jointly conditionally Gaussian random variables zi,xHΨH iy=xHΨH i¯ Ψx+xHΨH iw,(2.39) ¯z,xH¯ ΨHy=xHx+xH¯ ΨHw,(2.40) with first and second order moments    E [¯z] = kxk2,Var [¯z] = σ2 wkxk2 E [zi] = xHΨH i¯ Ψx,Var [zi] = σ2 wkxk2,(2.41) and Cov [¯z, zi] = σ2 wxH¯ ΨHΨix,(2.42) a good approximation for Pr (˜ε(i)) is given by [64] Pr (˜ε(i)) ≈1 2exp (−kxk4 2σ2 wNM )I0xHΨH i¯ Ψx·kxk2 2σ2 wNM ,(2.43) 52 Chapter 2. ML Methods for Radar Parameter Estimation −30 −28 −26 −24 −22 −20 −18 −16 −14 −12 −10 10−1 100 101 102 103 104 SNR RMSE CRLB Waterfall Estimation Random Estimation Figure 2.1: Waterfall behavior example w.r.t. other bounds. where I0is the modified Bessel function of the first kind of order 0. The MSE estimation can be thus computed by substituting (2.43) into (2.36), using the above definitions. The resulting (approximated) upper bound tends to be loose at low SNR, where the outliers are likely to occur and the associated error probabilities are high, but becomes more accurate moving towards the ML waterfall region, thus, when the outlier probability decreases. Furthermore, the MSE strictly depends on the grid resolution and may or may not reach the CRLB for increasing SNR depending on the systematic error incurred by the grid discretization. As a matter of fact, in our numerical results we took care of using a search grid fine enough such that the discretization systematic error is not visible in the explored range of SNRs, and, particularly, when the waterfall falls to the CRLB. Moreover, to contrast the looseness of the MSE upper bound obtained in (2.36) at low SNR, we define the trivial upper bound MSE ≤1 |Γ|X i∈Γ (ˆαi−¯α)2,(2.44) 2.2. Radar Resolution and Multi-Target Detection 53 that corresponds to estimate by choosing random grid points, i.e., completely disregarding the received signal. Then, putting together (2.36) (with the pairwise error probability approximation in (2.43)) and the above “random estimation” bound in (2.44), we finally obtain the approximated MSE upper bound as MSE .min (X i∈Γ Pr (˜ε(i)) (ˆαi−¯α)2,X i∈Γ (ˆαi−¯α)2 |Γ|),(2.45) whose behavior is shown in Fig. 2.1, together with the CRLB introduced hereafter. Thus, the ML performance of the estimator stands on the random estimation straight line for low SNR, follows the CRLB for high SNR, and the transition between these two extremes occurs around the waterfall estimation. Simulation results show that the aforementioned analysis is able to accurately predict the waterfall behavior of the ML estimator. 2.2 Radar Resolution and Multi-Target Detection Under the assumption of the point target model [17, 66], until now we have analyzed a single target scenario. Radar operations can be essentially performed with two different approaches. The first one, considers a radar periodically scanning angular sectors, as typical naval or airplane radars. In this context, by making the beam as directive as possible, i.e., assuming a very narrow beam and angular coverage, it is reasonable to assume the presence of just one target in any direction (or multiple targets sharing the same direction and assuming no blockage of the signal propagation). As an alternative, the second approach is based on a wider angular sector coverage. In such a case, since the target location is not a priori known, a joint estimation of Doppler, delay, and angle of arrival (AoA) should be performed at the radar Rx. In both approaches, the radar range and velocity resolutions acquire a central role, providing the minimum distance and velocity at which two different targets can be separately detected, i.e., such that they are not seen as a single entity. 54 Chapter 2. ML Methods for Radar Parameter Estimation Consider the transmitted symbols to be arranged within the Doppler-delay grid in (1.13). The definition of N,M,B(bandwidth), T, and fc(carrier frequency) is very important. By following the continuous time signal expression in (1.17), symbols are spaced by T(seconds) in time and ∆f(Hertz) in frequency. The radar resolutions are computed in the following. Starting from the definition of Doppler shift we get ν,2vfc c⇒v,νc 2fc ,(2.46) and since the minimum Doppler step is ∆f/N, the velocity resolution results to be νstep =∆f N⇒vres ,νstepc 2Nfc =∆fc 2Nfc .(2.47) Equivalently, starting from the definition of delay we get τ,2r c⇒r,τc 2,(2.48) and since the minimum delay step is T/M, the range resolution is τstep =T M⇒rres ,τstepc 2M=Tc 2M.(2.49) If we impose, as generally done in multi-carrier systems, the equality ∆f, 1/T, together with B=M∆f, previous formulas become    rres =Tc/2 = c/(2B) vres = ∆fc/(2f) = Bc/(2NMfc) .(2.50) This analysis provides the limits of a radar system based on a multi-carrier communication waveform. Even if the range resolution strictly depends on the bandwidth B, and this is valid for any radar system, the only way to reduce the velocity resolution, having imposed ∆f= 1/T, is to increase the dimension of the transmitted block of data, through Nand M. However, this translates in increasing the total signal duration and decreasing the subcarriers spacing, which could lead to an increase of the computational complexity (if 2.3. Simulation Results 55 blockwise operations have to be computed), or increased interference effects (for instance for OFDM). This is the cost of the proposed joint radar parameter estimation and communication setup using OFDM and/or OTFS, which allows the simultaneous transmission of useful information and radar sensing without any tradeoff, but requires complex signal processing operations (to contrast the block dimension or the interference). On the other hand, one can bypass the aforementioned problem by considering the additional spatial dimension, i.e, with a multi antenna system, such that distinct targets can be identified in three different domains: range (delay), velocity (Doppler), and angular position. As we will see, our proposed method is able to correctly detect targets which are separable in at least one domain out of three (delay, Doppler, and angle). 2.3 Simulation Results 2.3.1 Joint State Sensing and Communication The radar (backscattered) and forward communication SNRs are defined as SNRrad =λ2σrcsG2 (4π)3r4 Pavg σ2 w ,(2.51) SNRcom =λ2G2 (4π)2r2 Pavg σ2 w ,(2.52) respectively, where λ=c/fcis the wavelength, σrcs is the radar cross-section in m2,Gis the antenna gain, and ris the distance between Tx and Rx. In the case of multipath, we fix SNRcom to be the SNR of the line of sight (LoS) component, and we add multipath components with progressively lower SNRs, such that the sum SNR of the channel increases with the number of paths P. This corresponds to the physically meaningful case that a richer propagation environment conveys more signal power. Table 2.1 summarizes the relevant simulation parameters inspired by the automotive communication standard 56 Chapter 2. ML Methods for Radar Parameter Estimation Table 2.1: Simulation parameters fc= 5.89 GHz M= 64 B= 10 MHz N= 50 ∆f=B/M = 156.25 kHz T= 1/∆f= 6.4µs σrcs = 1 m2G= 100 r= 20 m v= 80 km/h IEEE 802.11p [21], where rand vdenote the target range and velocity taken into account.3 By briefly reviewing OFDM modulation, we introduce a widely used radar waveform known as frequency modulated continuous wave (FMCW) [13, Chapter 4.6], in order to understand the gain of the proposed methods against well-known and currently used approaches for radar vehicular applications. For both OFDM and FMCW we consider a symbol length of To=TGI +T, i.e., including the possible presence of a guard interval denoted by TGI, generally longer than the maximum path delay τmax. In OFDM the guard interval results to be the CP, extensively discussed in Sec. 1.2. In FMCW, the radar transmitter sends a sequence of identical “chirp” pulses of duration T, each followed by a guard interval of length TGI to avoid inter-pulse interference. By denoting with φ(t)=(fc+B 2Tt)tthe phase at time t, the transmit signal is given by s(t) = N−1 X i=0 ej2πφ(t−iT0)rect t−iT0 T,(2.53) where Nis the number of consecutive pulses. Transmitting (2.53) over the channel in (1.1), we get the received signal r(t), whose expression, by neglecting 3Note that, if the conditions τmax < T and νmax <∆fare satisfied, the numerical results are independent of the choice of rand v. 2.3. Simulation Results 57 the noise, is given in (1.20), but with the current s(t). After some algebra, it is easy to show that the product of the received and the transmit signals gives y(t) = r(t)s∗(t) = P−1 X p=0 hpej2πνpte−j2πfcτp N−1 X i=0 e−j2πfb,p(t−iT0−τp/2) ,(2.54) where fb,p =B Tτpdenotes the so-called beating frequency of path p. The receiver samples y(t) every T/M for each pulse, i.e., for t=iTo+lT Mwhere i denotes the pulse index and ldenotes the sample index. By letting L=M+C denotes the number of samples per pulse, the sampled received signal can be rewritten as y[i, l] = yiT0+l fs=X p hpej2π(fb,p+νp)l fsej2πνpiT0,(2.55) for i= 0, . . . , N −1 and l= 0, . . . , L −1, where hpabsorbs a constant phase term independent of the indices i, l. The estimation of the 2Punknown parameters {τp, νp}is thus obtained by selecting the peak of the range-Doppler map found by applying a twodimensional DFT to the noisy samples in (2.55) as proposed in [26, 20]. The range and velocity of the target are obtained by the estimates of {τ0, ν0}. 2.3.2 Joint Radar and Communication Performance The first two subfigures of Fig. 2.2 show the velocity and range estimation root MSE (RMSE) versus SNRrad for a pure LoS channel (P= 1) and for OTFS, OFDM, and FMCW. We notice that both digital modulation formats provide as accurate radar performance as FMCW, while transmitting useful information to any possible Rx. In addition, the third subfigure of Fig. 2.2 shows the achievable rate of OFDM and OTFS, for simplicity of derivation, when Gaussian independent and independent and identically distributed (i.i.d.) symbols CGauss are sent through the channel. This provides an achievable rate in the case of joint detection and decoding, with unconstrained complexity, as a function of SNRcom, 58 Chapter 2. ML Methods for Radar Parameter Estimation −40 −35 −30 −25 −20 −15 −10 −5 0 5 10 10−2 10−1 100 101 102 103 SNRrad [dB] RMSE[ˆv] / [m/s] OTFS OTFS CRLB OFDM FMCW OTFSWaterfall Estimation OTFSRandom Estimation −40 −35 −30 −25 −20 −15 −10 −5 0 5 10 10−2 10−1 100 101 102 103 SNRrad [dB] RMSE[ˆr] / [m] OTFS OTFS CRLB OFDM FMCW OTFSWaterfall Estimation OTFSRandom Estimation 0 5 10 15 20 25 30 35 40 45 50 0 2 4 6 8 10 12 14 16 SNRcom [dB] CGauss / [bits/s/Hz] COTFS Gauss COFDM Gauss Figure 2.2: From top to bottom: the RMSE of the target velocity estimation ˆvvs SNRrad, the RMSE of the target range estimation ˆrvs SNRrad, and the Gaussian capacity CGauss vs. SNRcom, for the curves indicated in the different legends. 2.3. Simulation Results 59 giving a qualitative idea of performance curves. A more accurate model should consider only detection, i.e., separated from decoding, calculating the achievable communication rate in terms of pragmatic capacity, as extensively done in Chapter 4. Given the fact that we can model the block input-output relation of OTFS as a MIMO channel, the mutual information with Gaussian inputs and perfect channel state information (CSI) at the receiver is given by [67] COTFS Gauss =NT NT +TGI 1 NM log2det I+ SNRcomΨΨH.(2.56) A similar expression for OFDM, owing to the fact that the channel matrix in OFDM is diagonal, with consequent symbol-by-symbol detection, yields COFDM Gauss =T T+TGI log2(1 + SNRcom).(2.57) The factors NT/(NT +TGI) and T/(T+TGI) for OTFS and OFDM, respectively, are introduced in order to take into account the (possible) insertion of a guard interval. In OTFS, a guard interval of duration TGI is inserted at the end of each frame, whose duration is TOTFS f=NT, comprising Nconsecutive symbols in the time domain (see Chapter 1). In contrast, in OFDM, the guard interval takes the form of a CP, inserted for each OFDM symbol of duration T. The frame size is thus TOFDM f=NTo=N(T+Tcp). It is clear that for practical values of the OTFS frame length N, the overhead paid by OTFS is much less than the CP overhead paid by OFDM (in these results we used TGI =T/4, which is typical in the IEEE 802.11 family of standards). On the other hand, the larger overhead incurred by OFDM yields a particularly simple receiver structure, allowing symbol-by-symbol detection thanks to the diagonalization of the channel. In contrast, OTFS requires blockwise detection over the whole frame, which can be very computationally intensive, especially for large Nand M. This is why the proposed soft-output symbol detector presented in Section 4.2.1 is of particular interest. Next, we present a second set of results where we consider only OTFS but in the presence of a multipath channels (P > 1, up to P= 4), showing 66 Chapter 3. ML Radar Methods in MIMO Configurations on a detected target, maximizing the BF gain and the received SNR, shall be used to minimize “multi-target” interference during a possible tracking and communication phase [49, 13, 70]. Clearly, within the target detection phase, a non-trivial tradeoff appears. On one hand, wider angular sectors coverage enable the detection of multiple targets if the received backscattered power is high enough. On the other hand, a more directional BF grants a higher received SNR, and thus longer detectable target distance, at the cost of a time-consuming search over narrower angular sectors (as classical radar successively swapping adjacent regions, see, e.g., [13]). Different solutions to the aforementioned problem can be found in literature (see, e.g., [52, 53, 15, 54]). As an extension of [71] and of the concepts introduced in the previous parts, this chapter studies the joint target detection and parameter estimation problem with a mono-static MIMO radar adopting OTFS, and extends the literature results on MIMO configurations (see, e.g., [44, 72]). The use of communication waveforms for MIMO radar has been motivated by the joint radar and communication paradigm (as showed in Sec. 2.1), where two functions are implemented by sharing the same resources and the same waveform (see e.g. [24, 27, 25] and references therein). Contrary to the existing works on radar sensing using OTFS [46, 71, 28], this use case considers a MIMO radar under a practical mmWave system architecture, such that the number of radio frequency (RF) chains (Nrf) is much smaller than the number of antennas (Na). In fact, the difficulties related to the implementation of a fully digital BF, or, equivalently, associate one RF chain per antenna (including A/D conversion, modulation, and amplification) in a small form factor and highly integrated technology over a large signal bandwidth, are well known. Therefore, focusing on mmWave automotive applications, we consider hybrid digital-analog (HDA) BF schemes as typically considered in literature (see, e.g., [55, 56] and references therein). We will study two different scenarios, exploring the aforementioned BF tradeoff. The first scenario considers a Tx BF design with beam covering a wide angular sector, to jointly perform target detection, parameter estimation, and multicasting of a common message to 3.1. Introduction 67 all possible active users (see Fig. 3.1a). A possible application of this model is, for instance, a base station mounted near a highway able to collect traffic information through radar capabilities while communicating its knowledge to active Rx.1Assuming the communication phase established, i.e., detection already performed, the second scenario considers a Tx BF with directed narrow beams, such that individual information streams are sent to the detected users, or groups of users, as depicted in Fig. 3.1b. It is important to stress the fact that the radar Rx uses a wide beam pattern consisting of Nrf beams, as illustrated in Fig. 3.2, in order to obtain a meaningful vector observation, necessary for AoA estimation, regardless of the operating phase. This is in a sharp contrast to the hybrid beam alignment considered in a typical communication system, where the Rx also applies BF and obtains a scalar observation precluding the estimation of the AoA (see, e.g., [55, 50] and references therein). Under this setup, we propose an efficient ML-based scheme combined with HDA BF to jointly perform target detection and parameter estimation. More precisely, our scheme first performs target detection and super-resolution estimation of delay, Doppler shift, and AoA using a wide angular beam along which a single data stream is sent. Then, once the targets are detected, the subsequent tracking phase performs the parameter estimation using multiple narrow beams along which multiple data streams can be sent. Our numerical results demonstrate that the proposed scheme is able to reliably detect multiple targets while essentially achieving the CRLB for radar parameter estimation. Furthermore we investigate various scenarios of near-far effects of targets, showing that a successive interference cancellation (SIC) mechanism is able to efficiently remove the masking effect between targets located at different ranges from the radar, and we provide an in-depth analysis of the two scenarios of interest, showing their limits and advantages. 1It is clear that, getting rid of the message sent, radar tasks can be performed by using any well known radar waveform [13], and the use of a digital communication format is pointless (the transmission of information could start in a second communication phase, e.g., within a time division protocol). 68 Chapter 3. ML Radar Methods in MIMO Configurations Tx / Radar Rx Communication Rx / Radar Target Tx / Radar Rx Wide Angular Sector / GTX Wide Angular Sector / GTX (a) Detection phase Tx / Radar Rx Communication Rx / Radar Target Target BF / GTX Target BF / GTX Tx / Radar Rx Target BF / GTX Target BF / GTX (b) Tracking phase Figure 3.1: Two scenarios with two different Tx beam patterns. In (a), a Tx (a base station or a car) broadcasts a common message exploring a wide angular sector. In (b), we consider directional BF towards the detected targets. The Rx always makes use of a wide beam within the angular sector of interest. We remark here the used notation. (·)Tdenotes the transpose operation. (·)Hdenotes the Hermitian (conjugate and transpose) operation. Operator |·| denotes the absolute value |x|if x∈R, or the cardinality (number of elements) of a discrete set, i.e., |F|, if Fis a discrete set. 3.2 Physical model We consider a joint radar detection and parameter estimation system operating over a channel bandwidth Band at the carrier frequency fc. The Tx is equipped with a mono-static MIMO radar with Naantennas and Nrf RF 3.2. Physical model 69 Radar Rx Nrf Beams Wide Angular Sector Figure 3.2: Beam configuration at the radar Rx for the two scenarios depicted in Fig. 3.1. Nrf beams cover a wide angular sector. chains, operating in full-duplex mode.2The radar Rx (colocated with the Tx, i.e., mono-static assumption) processes the backscattered signal to identify the presence of targets within the beam, while estimating parameters of interest such as range, velocity, and AoA. A point target model is taken into account, such that each target can be represented through its LoS path only [17, 21, 15]. By letting the steering angle φ∈[−π 2,π 2], by considering an antenna array with λ/2 spacing (where λis the wavelength), the Tx and Rx arrays are given by a(φ) and b(φ) respectively, where a(φ)=(a1(φ), . . . , aNa(φ))T∈CNa, denotes the uniform linear array response vector of the radar Rx with an(φ) = ej(n−1)πsin(φ), n = 1, . . . , Na,(3.1) and bn(φ) = an(φ). In fact, given the mono-static radar, the same steering angle φis both at radar Tx and Rx, thus, vectors aand bresult to be equal. The channel is modeled as an extension of the P-tap time-frequency selective 2Full-duplex operations can be achieved with sufficient isolation between the Tx and the (radar) detector and possibly interference analog pre-cancellation in order to prevent the (radar) detector saturation [31, 73, 32]. 70 Chapter 3. ML Radar Methods in MIMO Configurations channel in (1.1) including the addition spatial dimension, which is given by [7] H(t, τ) = P−1 X p=0 hpb(φp)aH(φp)δ(τ−τp)ej2πνpt,(3.2) whose dimension is Na×Na, where Pis the number of targets, hpis a complex channel gain including the pathloss, νp=2vpfc cand τp=2rp care the round-trip Doppler shift and delay, while φpdenotes the AoA, each corresponding to the p-th target, respectively. 3.2.1 OTFS Input Output Relation We consider OTFS with Msubcarriers of bandwidth ∆feach, such that the total frequency band is given by B=M∆f. As already written in many previous chapters, Tdenotes the symbol time, and the OTFS frame duration is NT. The Doppler-delay dimensions, according to (1.13), are Nand M, while the equation T∆f= 1 is always valid (see Chapter 1 and Sec. 1.3 for more details). In order to consider the aforementioned different operational modes, i.e., detection and tracking phases, we let Nsdenote the number of data streams to be sent in each time-frequency slot, such that Ns= 1 corresponds to the multicasting of a single data stream (in a possible detection phase where a base station wants to share a common information while detecting active targets) and Ns≤Nrf corresponds to the broadcasting of individual data streams (towards the active targets already detected in the previous phase). Following the standard derivation of the input-output relation of OTFS (see Sec. 1.3), here we extend the equations to also consider the additional spatial dimension. Let us deal with Ns-dimensional data symbols {xk,l}, for k= 0, . . . , N−1, l= 0, . . . , M−1, belonging to any constellation, and arranged in the N×Mtwo-dimensional Doppler-delay grid Γ = {(k/NT, l/M∆f)}in (1.13). The Tx first applies the ISFFT to convert data symbols {xk,l}into a Ns×1 block {X[n, m]}in the time-frequency domain X[n, m] = N−1 X k=0 M−1 X l=0 xk,lej2π(nk N−ml M),(3.3) 3.2. Physical model 71 for n= 0, . . . , N −1, m= 0, . . . , M −1, satisfying the average power constraint E[X[n, m]HX[n, m]] = Pavg/(NMNa)INa, where INsdenotes an identity matrix of dimension Nsand Pavg some total maximum power allowed by the system. After assigning Nsstreams to Nrf RF chains through a mapping matrix V∈ CNrf ×Ns(since generally Ns≤Nrf, thus a mapping is necessary), the Tx generates the Nrf-dimensional continuous-time signal s(t) = V N−1 X n=0 M−1 X m=0 X[n, m]gtx(t−nT)ej2πm∆f(t−nT ),(3.4) in which gtx(t) is the shaping pulse applied in transmission (see Chapter 1 for more details and Sec. 1.3.8). Since the number of RF chains is typically much smaller than the number of antennas, different types of HDA architectures between RF chains and antennas have been considered in the literature (see e.g. [55]). In this paper, we focus on the fully-connected HDA scheme of [55]. For any HDA architecture, the Tx applies the hybrid BF matrix denoted by F∈CNa×Nrf that captures both baseband and RF analog BF (see [51, 54]), while the Rx sees the received signal of a reduced dimension through a projection matrix denoted by U∈CNrf ×Na. By imposing tr(FVVHFH) = Na, the total power constraint Pavg is satisfied. In other words, the Rx cannot access to each antenna element, but obtains only a projection of the received signal. By transmitting the signal (3.4) over the channel (3.2), the Nrf-dimensional continuous-time received signal is given by r(t) = P−1 X p=0 hpUb(φp)aH(φp)Fs(t−τp)ej2πνpt,(3.5) where we omitted the noise for simplicity. It is interesting here to compare these equations with the single spatial dimensional ones presented in Sec. 1.3. The output of the Rx filter-bank adopting a generic receive shaping pulse 72 Chapter 3. ML Radar Methods in MIMO Configurations grx(t) is y(t, f) = Zr(t0)g∗ rx(t0−t)e−j2πft0dt0 =Zt0 g∗ rx(t0−t) P−1 X p=0 hpUb(φp)aH(φp)Fs(t0−τp)ej2πνpt0e−j2πft0dt0 =X p,n0,m0 hpUb(φp)aH(φp)FVX[n0, m0]Zt0 g∗ rx(t0−t)gtx(t0−τp−n0T) ej2πm0∆f(t0−τp−n0T)ej2π(νp−f)t0dt0.(3.6) By sampling at t=nT and f=m∆f, we obtain y[n, m] = y(t, f)|t=nT,f=m∆f= N−1 X n0=0 M−1 X m0=0 hn,m[n0, m0],(3.7) where the time-frequency domain input-output relation hn,m[n0, m0] is hn,m[n0, m0] = P−1 X p=0 h0 pUb(φp)aH(φp)FVX[n0, m0]ej2πn0T νp Cgtx,grx ((n−n0)T−τp,(m−m0)∆f−νp)e−j2πm∆fτp,(3.8) where Cu,v(τ, ν) denotes the CAF, we let h0 p=hpej2πνpτp, and imposed the term e−j2πmn0∆fT = 1, ∀n0, m, under the hypothesis T∆f= 1. Since each Xi[n, m] is generated via ISFFT, the received signal in the delay-Doppler domain is obtained by the application of the SFFT y[k, l] = X n,m y[n, m] NM ej2π(ml M−nk N)=X k0,l0 gk,k0l, l0,(3.9) where the ISI coefficient of the Doppler-delay pair [k0, l0] seen by sample [k, l] is given by gk,k0l, l0=X p h0 pUb(φp)aH(φp)FVxk0,l0Ψp k,k0[l, l0],(3.10) 3.2. Physical model 73 −60−50−40−30−20−10 0 10 20 30 40 50 60 −50 −40 −30 −20 −10 0 10 20 Angle BF Gain [dB] Coverage 10◦ Coverage 20◦ Coverage 30◦ Figure 3.3: BF design for different angular coverage. Clearly, wider the beam, less the BF gain. with Ψp k,k0[l, l0] defined as Ψp k,k0[l, l0] = X n,n0,m,m0 Cgrx,gtx ((n−n0)T−τp,(m−m0)∆f−νp) NM ej2πn0T νp e−j2πm∆fτpej2πn0k0 N−m0l0 Me−j2π(nk N−ml M),(3.11) while simplified version of Ψp k,k0[l, l0] obtained by approximating the CAF can be found in Sec. 1.3. 3.2.2 Beamforming matrices The design of the BF matrix Fat the radar Tx depends on the operating phase, and anyway Fis chosen such that Tx and Rx are aligned towards the same wide angular sector. Following [51, Section III.C], we construct F∈CNa×Nrf to cover a wide angular sector [−θ, θ] as follows. By representing this angular sector by a discrete set of Nrf angles, denoted by Θ = {±(θ/(2Nrf) + kθ/Nrf)}, 74 Chapter 3. ML Radar Methods in MIMO Configurations for k= 0, . . . , Nrf/2−1, each column of F= [f1,...,fNrf ] takes the form fi=a(θi) |a(θi)|, i = 1, . . . , Nrf ,(3.12) where a(θi) is defined in (3.1) and taking into account a suitable normalization. An example on how the BF beams, w.r.t. the associated gain, look like is shown in Fig. 3.3, for different angular coverage. During the target tracking phase, we form multiple narrow beams corresponding to the estimated AoA of the detected targets. This is illustrated with red and blue beams, corresponding to two different AoA, in Fig. 3.1b. Assuming that Ptargets are detected and their respective AoA are estimated, we construct Fby replacing θiby ˆ φpfor the first Pcolumns in (3.12) [51, Section III.B]. In such a case, the BF gain towards a single target is approximately ≃Na, while decreases for each other considered target, i.e., ≃Na/P. Contrary to the transmit beamforming matrix, the reduction matrix Uat the radar Rx remains the same for both detection and tracking phases. Namely, we set U=FH, where each column is given in (3.12). This is illustrated in Fig. 3.2. This choice of an isotropic receive beam enables to obtain a multidimensional signal for AoA estimation in both detection and tracking phases. 3.3 Joint Detection and Parameters Estimation We wish to estimate the set of four parameters θ={h0 p, φp, τp, νp}∈TP, with T=C×R×R×R. By defining Gp(τp, νp, φp),(Ub (φp)aH(φp)FV)⊗Ψp,(3.13) where ⊗is the Kronecker product,3as the NrfNM ×NsNM matrix obtained by multiplying Ψpby a different coefficient of (Ub (φp)aH(φp)FV). Thus, by stacking Xinto a NsNM-dimensional vector xand defining an output vector 3Note that AX×Y⊗BZ×K=CXZ×Y K . 3.3. Joint Detection and Parameters Estimation 75 yof dimension NMNrf ×1, the received signal in the presence of noise is y= P−1 X p=0 h0 pGp(τp, νp, φp)x+w,(3.14) where wdenotes the AWGN vector with independent and identically distributed entries of zero mean and variance σ2 w. The problem consists, first, of the detection of the Ptargets (in case of a first acquisition phase), together with the estimation of the associated 4Pparameters (complex channel coefficient, Doppler, delay, and angle) from the NrfMN-dimensional received signal. To this end, we define the ML function as l(y|θ,x) =  y−X p h0 pGpx 2 , =yHy−yHX p h0 pGx −X p h0∗ pxHGHy +xH X p h0 pG!H X q h0 qGq!x,(3.15) where we use the short hand notation Gp,G(τp, νp, φp). Note the similarities and differences w.r.t. (2.27). The ML solution is given by ˆ θ= arg min θ∈T P l(y|θ,x).(3.16) For a fixed set of {φp, τp, νp}, the ML estimator of {h0 p}is given by solving the following set of equations xHGH p  P−1 X q=0 h0 qGq x=xHGH py, p = 0, . . . , P −1.(3.17) By plugging (5.34) into (5.32), it readily follows that minimizing l(y|θ,x) reduces to maximize the following function l2(y|θ,x) = X p h0 pyHGpx=X p Sp(τp, νp, φp)−Ip({h0 q}q6=p,θ),(3.18) 82 Chapter 3. ML Radar Methods in MIMO Configurations While two distinct targets in the angle domain can be identified if the angular resolution meets some conditions (depending on the number of antennas, the angular distance between the two targets, and the antenna array properties) [13]. The velocity and the range resolution is determined by the system parameters in Table 3.1 and given by vres =Bc 2NMfc [m/s] , rres =c 2B[m] .(3.29) In order to get a reasonable range resolution, e.g., <1 [m], a large bandwidth has to be considered.7Since the velocity resolution is directly proportional to B, for a fixed fc, the only way to obtain lower values is to increase the block size NM, leading to a remarkable increase in computational complexity, which could be not affordable. For this reason, we set the system parameters by focusing on a reasonable range resolution (and theoretical maximum range) under a feasible computational complexity. Note that the maximum range could not be achieved if the backscattered power is below the noise floor. However, the chosen system setup leads to an unavoidable very large velocity resolution. Under the aforementioned assumptions, taken at the beginning of Section 5.4, the problem of targets identifiability appears only in the angular domain. However, this only happens at mmWave, thus, range and velocity resolutions are reported here for completeness, since the proposed scheme could target lower frequencies, where the aforementioned assumption might not be satisfied. Remark: The parameter estimation performance of the proposed MLbased algorithm, in particular range and velocity estimation, strictly depends on the dimension of the block of data sent, i.e., the product N·M. Thus, the system parameters of Tab. 3.1 can be easily tuned to achieve the desired levels of radar resolutions (modifying the bandwidth), acquisition time (based on the length of the OTFS frame in time), maximum range, etc. Clearly, the CRLB changes accordingly. Moreover, note that this is also possible thanks to 7Note that a tradeoff appears. Larger bandwidths mean more precise resolution, but lower theoretical maximum range (with the same N×Mgrid). We remark that our algorithm is completely independent of these choices. 3.4. Numerical Results 83 OTFS modulation, which is not sensitive to Doppler and delay effects. Remark: The (radar) range and velocity resolution in (3.29) indicates the minimum necessary targets spacing, in one of the two domains, such that both of them are distinguishable at the radar Rx. This is not linked to the performance of our ML-based detector, which is able to accurately estimate the parameters far beyond the resolution in (3.29). Thus, there is a huge difference between targets identifiability and estimation performance. 3.4.1 Simulation Results Fig. 3.4 shows the radar performance in terms of probability of detection (PD), range/velocity/AoA estimation during the detection phase (Fig. 3.1a). When more than one target is considered within the simulation scenario, the PD Pd is averaged w.r.t. all Ptargets, i.e., Pd=PP−1 p=0 Pd(p) P,(3.30) where Pd(p) denoted the PD of the p-th target. First, note that, by considering an angular coverage of 10 degrees (blue line), the maximum range to correctly identify a target, limited by the pathloss and thus different from the theoretical limit indicated in Table 3.1, is about 110 m. For any distance between Tx and target, the estimation performance of radar parameters of interest (range, velocity, and AoA) follows the corresponding CRLB. More in details, note that at the limit range of 110 m, the RMSE for range, velocity, and angle are respectively, ≃4·10−2[m], ≃1.6·101[m/s], ≃4·10−2[degree]. As expected, given the system parameters, the velocity RMSE is quite poor, while the other estimation performances are satisfactory. However, a proper BF design towards targets, in a subsequent tracking phase, could improve the estimation performance maximizing the received SNR, as showed in next results. As seen from Fig. 3.4, by increasing the angle sector from 10◦to 30◦, the backscattered power gets smaller (less BF gain), hence the maximum range significantly decreases. There exists a non-trivial tradeoff 84 Chapter 3. ML Radar Methods in MIMO Configurations 10 20 30 40 50 60 70 80 90 100110120130 140 150 0 0.2 0.4 0.6 0.8 1 Range [m] Pd Sector 10◦ Sector 20◦ Sector 30◦ 10 20 30 40 50 60 70 80 90 100110120130 140 150 10−4 10−3 10−2 10−1 100 Sector 10◦20◦30◦ r[m] CRLB Range [m] RMSE[ˆr] / [m] 10 20 30 40 50 60 70 80 90 100110120130 140 150 10−1 100 101 102 Sector 10◦20◦30◦ v[m/s] CRLB Range [m] RMSE[ˆv] / [m/s] 10 20 30 40 50 60 70 80 90 100110120130 140 150 10−4 10−3 10−2 10−1 100 Sector 10◦20◦30◦ ϕ[deg] CRLB Range [m] RMSE[ˆ ϕ] / [deg] Figure 3.4: Detection phase. Single target at a different distance within the illuminated angular sector of specified coverage. RMSE performance with associated CRLB and detection performance. Na= 128. 3.4. Numerical Results 85 between the width of beams and radar performance. Wider angular sectors allow to explore the environment in less time, but with limited maximum range, while narrower sectors maximize the received power and the maximum target range, at the cost of a time consuming beam sweeping search. Clearly, RMSE performance can not be computed if the PD is equal to 0, i.e., the target is not detected, thus RMSE curves may stop at certain ranges, as visible in Fig. 3.4. Fig. 3.5 shows the performance of the SIC technique presented in Algorithm 2 during the detection phase in the scenario depicted in Fig. 3.6. Namely, the Tx wishes to detect two targets, one at fixed distance of 10 [m], the other moving w.r.t. the x-axis, i.e., from 20 to 150 [m] (see Fig. 3.6). SIC is necessary because the closer target (black car) will mask the further target (blue car) so that the latter cannot be detected. First, the first plot of Fig. 3.5, referred to the PD, shows that, when the moving target is located at ranges greater than 90 [m], corresponding to the relative range beyond 80 [m], the masking effect is not removed efficiently by the SIC technique (the residual interference is remarkable), and the target at longer distance is not detected correctly. In fact, at the extreme point, the curve flats to Pd= 0.5, because only one target out of two (clearly, the closest to the radar Rx, i.e., the one fixed at 10 [m]) is correctly detected. As clearly visible, the performance in terms of RMSE, which considers in this case the estimation performance averaged w.r.t. the detected targets (note that the target located at 10 [m] is always detected correctly), slightly changes while considering one or two targets, as a confirmation of the effectiveness of the proposed algorithm. Note that the blue curves of Fig. 3.5 correspond to the blue ones of Fig. 3.4. Now we consider the tracking phase corresponding to Fig. 3.1b. The scenario takes into account one Tx and three targets within an angular sector of 10◦, as shown in Fig. 3.7. Fig. 3.8 shows the RMSE performance of the reference target (black car), in the presence of other two targets (blue cars), Note that distance, velocity, and angular position of all three targets are randomly chosen at every Monte Carlo iteration, in such a way the complete masking 86 Chapter 3. ML Radar Methods in MIMO Configurations 20 30 40 50 60 70 80 90 100 110 120 130 140 150 0 0.2 0.4 0.6 0.8 1 Range [m] Pd 1 Target 2 Targets 20 30 40 50 60 70 80 90 100 110 120 130 140 150 10−3 10−2 10−1 N. Targets 1 2 r[m] CRLB Range [m] RMSE[ˆr] / [m] 20 30 40 50 60 70 80 90 100 110 120 130 140 150 10−1 100 101 102 N. Targets 1 2 v[m/s] CRLB Range [m] RMSE[ˆv] / [m/s] 20 30 40 50 60 70 80 90 100 110 120 130 140 150 10−3 10−2 10−1 N. Targets 1 2 ϕ[deg] CRLB Range [m] RMSE[ˆ ϕ] / [deg] Figure 3.5: Detection phase. Two target, one located at 10 [m] and the other moving at a different distance (x-axis) within the illuminated 10◦angular sector as shown in Fig. 3.6. RMSE performance with associated CRLB. Detection performance. Na= 128. 3.4. Numerical Results 87 Fixed Target Moving Target Tx / Radar Rx Figure 3.6: Example of scenario depicted in Fig. 3.5. The fixed target (in black) masks the moving target, in blue, which changes its location within the illuminated angular sector. Reference Target Tx / Radar Rx Interference Target Interference Target Rx BF Pattern Figure 3.7: Example of scenario depicted in Fig. 3.8. The goal is to correctly estimate the parameters of the reference target (in black), while interference targets (in blue) lay within the same Rx BF pattern (depicted in Fig. 3.2 and with the black shape in this figure). 88 Chapter 3. ML Radar Methods in MIMO Configurations 50 100 150 200 250 300 350 400 10−4 10−3 10−2 10−1 100 101 102 NaSim CRLB 16 32 64 128 Range [m] RMSE[ˆr] / [m] 50 100 150 200 250 300 350 400 10−1 100 101 102 103 NaSim CRLB 16 32 64 128 Range [m] RMSE[ˆv] / [m/s] 50 100 150 200 250 300 350 400 10−4 10−3 10−2 10−1 100 101 NaSim CRLB εBW 16 32 64 128 Range [m] RMSE[ˆ φ] / [degree◦] Figure 3.8: Tracking scenario. Tx BF distinct beams towards three different targets. 3.4. Numerical Results 89 effect presented in Fig. 3.5 does not occur, and an average of RMSE results is finally computed. From Fig. 3.8, we observe that the RMSE critically depends on the number of antennas. This is because the BF gain grows proportionally with the number of antennas and increases the backscattered signal power. Moreover, note that a (reversed) “waterfall” behavior is shown for range and velocity estimation. This is because, even if the presence of the target is given for granted, low SNR values might still lead to a large error during the Dopplerdelay ML maximization (see Algorithm 2). The waterfall behavior is typical of ML estimators and has been extensively analyzed in Sec. 2.1.2.3. Also note that the AoA RMSE performance is upper limited by the 3-dB beamwidth of the beam pattern (see, e.g., [51, 76]). In fact, supposing that the target position lies within the 3-dB beamwidth, also the initial AoA estimation (the upper and lower limit of matrix Ω in (3.21)) is limited to that width. As a consequence, the RMSE does not exceed a systematic error, indicated as εBW, calculated by averaging over random AoA estimation realizations within the range of possibilities, i.e., between the upper and lower limit set by the 3-dB beamwidth of the beam pattern, equivalently of the analysis related to (2.44), which can be also applied to range and velocity estimations. Chapter 4 OTFS Detection 4.1 Introduction We will now focus on the OTFS data detection at the Rx side. By considering the communication-oriented channel model in (1.1), i.e., taking into account one-way Doppler and delay shift (modeling a forward communication channel, against the radar two-way scenario), in line with most of the current literature on OTFS detection (e.g., [62, 4]) we assume perfect CSI at the Rx, i.e., the knowledge of channel matrix Ψ. The acquisition of such CSI is a distinct problem of independent interest, that will be analyzed in 5 and has been studied in literature in recent years (see, e.g., [37, 44, 45]). In this chapter, we consider separate detection and decoding, where the Rx consists of the concatenation of a soft-output symbol detector, producing soft-estimates of the coded symbols x, and a (separate) decoder that takes such estimates as the output of a virtual channel that incorporates also the detector. Thus, we do not consider “turbo equalization” schemes, in which detection and decoding are jointly performed through successive iterations involving feedback loops (as, e.g., in [77]). This choice is justified by the fact that the presence of a forward error correction scheme, together with the specific definition of the channel model, could obscure the real performance of the de91 98 Chapter 4. OTFS Detection discrete variables taking values in C. Therefore, it has a complexity equal to |C|di−1, where |·|indicates the cardinality of the set, which may be prohibitively large for large constellations and, above all, it depends on the sparsity of the channel. Therefore, the exact application of the SPA computation rules to the FG obtained directly from the matrix Ψis highly impractical. For this reason, the authors of [62] propose to use a Gaussian approximation of the interfering symbols in the computation at nodes Qi(x), which effectively relies to a soft interference cancellation approach, as already widely used in turbo equalization and in the context of multiuser detection in [39, 85]. The details of the resulting MP algorithm can be found in [62]. It should be also mentioned that, for the sake of simplicity and in order to increase the sparsity of the FG in Fig. 4.2, the detector proposed in [62] constructs the nominal matrix Ψby rounding the delay shifts to integers on receiver sampling grid. Under this condition, the channel matrix is very sparse since many coefficients corresponding to sampling at non-integer delay shifts are identically to zero, and the number of connections for each node is reduced, while preserving the length-4 cycles problem, unavoidable with this approach. Nevertheless, since the assumption is generally not satisfied by realworld channels, such approximation of the channel matrix results in neglecting a significant component of the ISI. We shall verify that when such integer delay shift rounding is applied to the construction of the nominal matrix Ψused by the detector but the actual delay shifts have a fractional component (as it is always the case in practice), the mismatch yields a significant performance degradation. This shows that neglecting the fractional part of delays and Doppler shifts, as routinely done in the literature of OTFS, may be indeed quite misleading. 4.2.3 Linear block-wise MMSE equalization As a further term of comparison we consider also the standard linear MMSE block equalizer, applied to the channel model (2.17). In this case, the softoutput is simply the linear MMSE estimate of symbols xfrom the observation 4.3. Performance of Separated Detection and Decoding 99 y, given by ˆ xLMMSE =ΨHΨΨH+σ2 wI−1y.(4.13) The complexity of this approach is proportional to O(NM)3, so it becomes quickly unfeasible for typical values of Nand M. A low-complexity (mismatched) linear minimum mean square error (LMMSE) approach was recently proposed in [47], relying on the cyclic properties of channel matrix Ψunder perfect bi-orthogonality of shaping pulses gtx(t) and grx(t), together with the assumption of on-grid Doppler and delay shifts (with Doppler-delay grid Γ). In real channel conditions, i.e., in the presence of practical rectangular pulses and non-integer delay and Doppler shifts, the performance of this approach visibly degrades. 4.3 Performance of Separated Detection and Decoding As already mentioned, we characterize the performance of separated detection and decoding schemes in terms of pragmatic capacity, i.e., the mutual information of the virtual channel with input the constellation symbols, assumed with uniform probability, and output provided by the soft-output of the detector. This mutual information provides an achievable rate for separated detection and decoding, for any given detection scheme [78, 79]. Let us consider a sequence of symbols {xk}belonging to the signal constellation C, and let Vk(xk) denotes the detector soft-output. In the case of MP-based detectors, Vk(xk) is given in the form of a posterior probability distribution on xk∈ C, while in the case of linear equalizers (e.g., the linear MMSE estimator in (4.13)), this is given as the noisy estimate ˆxkwhich is treated as the output of a (virtual) AWGN channel. In any case, the pragmatic capacity is simply defined as the symbol-by-symbol mutual information IPC (xk;V(xk)). When V(xk) takes on the form of a posterior probability dis- 100 Chapter 4. OTFS Detection tribution, this can be easily calculated as IPC(xk;V(xk)) ,H(xk)−H(xk|V(xk)) = log2C − X xk∈C V(xk) log2 1 V(xk),(4.14) and, by considering a Monte Carlo simulation over Kdifferent realizations IPC = K−1 X k=0 H(xk)−H(xk|V(xk)) = log2C − 1 K K−1 X k=0   X xk∈C V(xk) log2 1 V(xk)   .(4.15) In the case of linear equalization (i.e., V(xk) = ˆxk), the pragmatic capacity is simply given by the symmetric capacity (i.e., with symbols used with uniform probability) of the signal constellation C, for an AWGN channel with SNR equal to the output signal-to-interference noise ratio (SINR) of the equalizer [67]. For the sake of comparison, we also show the symmetric capacity of the signal constellation in an AWGN channel with SNR equal to SNRcom (i.e., the SNR of the LoS path), denoted by Csym AWGN, and the mutual information with Gaussian inputs COTFS Gauss , for the case P= 1. Fig. 4.3 and Fig. 4.4 show the performance of the various methods for OTFS soft-output detection for a 16-quadrature amplitude modulation (QAM) and system parameters listed in Tab. 2.1. Fig. 4.3 shows the results for the (unrealistic) case where the channel Doppler shifts and delays are exactly on the discrete Doppler-delay grid Γ used by the receiver sampling. In contrast, Fig. 4.4 shows the results when the actual channel has arbitrary Doppler and delay shifts (with a random uniformly distributed fractional part), while certain algorithms still assume such integer grid when constructing the nominal channel matrix Ψused by the detector (as advocated for example in [62, 47]). We notice that the proposed MP-based approach based on the definition of matrix Goutperforms the one in [62] in both scenarios. In particular, it 4.3. Performance of Separated Detection and Decoding 101 −4−2 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 3 3.5 4 SNRcom [dB] Pragmatic Capacity — IPC hbits symboli 16-QAM P= 1 P= 2 P= 3 P= 4 MPG MPΨ LMMSE LMMSELC Csym AWGN COTFS Gauss Figure 4.3: Symbols detection performance in terms of pragmatic capacity for 16-QAM modulation. The curves show the behavior of matrix Gbased MP algorithm and MP algorithm of [62] under approximated channel conditions, i.e., with delay and Doppler shifts on the Doppler-delay grid, for a multipath channel with different number of components P. 102 Chapter 4. OTFS Detection −4−2 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 3 3.5 4 SNRcom [dB] Pragmatic Capacity — IPC hbits symboli 16-QAM P= 1 P= 2 P= 3 P= 4 MPG MPΨ LMMSE LMMSELC Csym AWGN COTFS Gauss Figure 4.4: Symbols detection performance in terms of pragmatic capacity for 16-QAM modulation. The curves show the behavior of matrix Gbased MP algorithm and MP algorithm of [62] under real channel conditions, i.e., with delay and Doppler shifts not on the Doppler-delay grid, for a multipath channel with different number of components P. 4.3. Performance of Separated Detection and Decoding 103 suffers from almost no degradation due to the non-integer Doppler and delay shifts, unlike the method of [62], which has been derived based on these strong assumptions. The LMMSE equalizer (4.13) with full complexity yields very good performance, paying only a small SNR penalty w.r.t. the proposed MP-based scheme for P= 1,2, and outperforming the MP-based scheme for richer scattering P= 3,4. However, as said before, its complexity is cubic in the frame dimension NM, which is unaffordable in practical implementations. Unfortunately, the low-complexity LMMSE estimator (curves indicated by LMMSELC in the figures), proposed in [47] and exploiting a specific structure of the channel matrix Ψ, which requires doubly block circulant feature of the OTFS channel matrix, is satisfied only when gtx(t) and grx(t) are biorthogonal. Under such condition, as demonstrated in [47], this scheme coincides with the MMSE block equalizer, but without the need of a large matrix inversion. However, by adopting physically realizable and realistic rectangular pulses, which clearly do not satisfy the bi-orthogonal condition the doubly block circulant feature is lost. Thus, our simulations show that the approach [47] is not competitive when applied to a channel model using rectangular pulses, as intuitively expected. It should be noticed that bi-orthogonality for pulses with time-frequency product equal to 1 is mathematically impossible [84], and relaying on such assumption may be very misleading. Note that we only treat a 16-QAM modulation. However, the results are very clear and plots with different modulation formats would have only been a confirmation of what we stated above. Chapter 5 Channel Estimation 5.1 Introduction In a given communication scenario, the CSI, i.e., the knowledge of the communication channel, is required at the Rx to perform coherent detection [34], i.e., to correctly demodulate the transmitted symbols based on the (known) channel transformation. The most common approach to acquire the CSI is through the transmission of known symbols, usually called pilots [34]. Generally, these pilots are arranged within the block of information symbols, following a chosen fixed pattern known to both Tx and Rx (see e.g, [35, 36, 86]), in such a way when a complete block is received, the Rx is able to instantly perform coherent detection, for instance, without waiting for some other side information from another source.1It is also well-known that, subject to the meaningful and widely used assumption of block fading (i.e., the propagation channel remains constant over blocks of consecutive time-domain symbols, while it may change 1In some communication scenarios (or standard) the Tx could send an entire block of pilot symbols, i.e., without information data, followed by blocks with no pilots. Under the assumption of slowly varying channel statistics, the symbol detection is based on the estimation made from the first block. This case has not bee considered, by focusing on a per block channel estimation, with associated benefits and losses. 105 106 Chapter 5. Channel Estimation independently from block to block), pilot-aided schemes are indeed nearly information-theoretically optimal in terms of the capacity scaling in the high spectral efficiency / high SNR regime (see, e.g., [87, 88, 89, 90]). Clearly, depending on the particular application and communication scenario, both the pilot pattern and the channel estimation algorithm should be optimized. As typical electronic systems are affected by thermal noise, the estimated CSI is not perfect but afflicted by an estimation error, whose magnitude depends on different parameters, e.g., the channel SNR, the number of pilots per block, the estimation algorithm (with related convergence speed, complexity, accuracy). The estimated communication channel expression is thus used to perform coherent detection at the Rx side. By taking the achievable communication rate as the most relevant and meaningful performance metric, which is a measure of the amount of useful information sent in a block of symbols, a tradeoff appears between the number of pilots per block, dedicated to the CSI estimation, and the number of information-bearing symbols. The optimization of this tradeoff is generally not trivial, depending on the modulation format and on the channel propagation characteristics, and its optimization is usually performed by trials, given the difficulty to find closed-form optimization criteria, as done in our successive analysis. Given the aforementioned system setup and related problems and challenges, we analyse the symbol detection performance in terms of pragmatic capacity, i.e., the achievable rate of the channel induced by the signal constellation and the detector soft-output [78, 79], of OTFS and OFDM, both designed to handle time-frequency selective channels as (1.1). This is equivalent to what is done in Chapter 4, recalled here for the sake of completeness. In general, a soft-output detector at the Rx side produces an estimate of the posterior probability of the transmitted symbols given the received signal block (pilots and data). This estimated posterior probability (e.g., in the form of log-likelihood ratios) is then passed to a decoder, that treats the sequence of soft-output symbols as the output of a virtual channel. The pragmatic capacity is the capacity of such virtual channel, with discrete input represented by the 5.1. Introduction 107 modulation symbols, and soft-output generated by the detector. Hence, the pragmatic capacity is representative of the achievable rate under the assumption of separated detection and decoding, i.e., without “turbo” reprocessing of the channel output once the decoder output is available (see, e.g., [91, 92] and references therein). In practice, iterative “turbo” detection is very hard to implement since often the detector is implemented in hardware (e.g., in an integrated circuit) and the decoder is implemented in software, and maybe even in a different location (as for example in the so-called 7.2 split between hardware and software, enabling cloud-based processing of the signals from remote radio heads [93]). For this reason, we believe that pragmatic capacity for separated detection and decoding is a very meaningful performance metric to compare different modulation formats and the associated pilot schemes and soft-output detectors. In order to make a fair comparison between the two modulation formats in terms of achievable communication rate, the pilot overhead necessary to achieve a satisfactory estimation of the CSI has been taken into account. Since the loss (no useful information is transmitted) associated to the presence of pilots cannot be neglected, the achievable communication rate inevitably deviates from the corresponding upper bound, depicted by the AWGN symmetric capacity. Moreover, together with the pilot overhead, we also consider the (likely) presence of a guard interval (GI) or CP, which additionally reduces the achievable rate. A per-block GI is used in OTFS to avoid inter-block interference (IBI) [61], while a CP is used for every OFDM symbol to avoid ISI and to make symbols orthogonal, thanks to the diagonalization of the channel matrix, under the assumption of low ICI, condition which is not hard to be satisfied. Here a first significant difference between the two modulation formats evidently appears. In fact, especially when the channel delay is significant, the CP length tends to be a large fraction (e.g., 25%) of the symbol time, leading to a remarkable loss in terms of capacity. On the other hand, OTFS does not need a per symbol separation, but this comes at the cost of a non-negligible increase in signal processing complexity caused by the remarkable Doppler-delay 114 Chapter 5. Channel Estimation We used as stopping criterion the maximum number of iterations. Note that the shrinking operation is allowed because zero entries of vector ˆ hat iteration icannot assume a value 6= 0 at iteration i0> i [99]. From a complexity point of view, the first iterations are the most costly, while the algorithm can run >106times keeping the complexity almost constant and the computational time linear (when the number of iterations is large enough, i.e., far away from starting costly ones). 5.2.1.1 Complexity of the LASSO Solver and Step Size Refinement The sensing matrix Dis composed of Gcolumns of length NM. While the dimensions Nand Mdepends on system settings and can be somehow controlled or tuned, the dimension Gtakes into account the estimation precision, or granularity, of the searching grid Γ. Hence, the larger the dimension G, the more reliable the result. Using some examples:  If Γ is equivalent to the Doppler-delay grid (delay and Doppler shifts integer multiple of the grid), G=NM and Dis a NM ×NM matrix. Blockwise operations adopted by any LASSO solver are feasible in this framework.  If the step size for both Doppler and delay axis is a fraction 1/ρ of the Doppler-delay grid step, G≃ONM ·ρ2and Dis approximately a NM ×NM ·ρ2matrix. Clearly, increasing the granularity of the grid quickly increases the complexity. In order to overcome the complexity induced by searching grid with fine granularity, it is possible to refine the step size in successive phases, rather than directly defining a low fractional value for the entire grid. The proposed refinement scheme is illustrated in Algorithm 3. During the Peak Selection step, if the number of multipath components Pis not available at the receiver, instead select all local maxima or peaks (of the groups of estimates) whose magnitude is above a certain threshold (to be defined). 5.2. OFDM Modulation and the CS Algorithm 115 Algorithm 3: Refinement of the Granularity Result: Fine estimation ˆ hfor LASSO problem (5.14). Coarse Estimation: For any LASSO solver, get a first coarse estimation ˆ hsuch that the searching grid Γ is equivalent to the Doppler-delay grid (i.e., delay and Doppler shifts integer multiple of the grid). In this case G=NM and Dis a NM ×NM matrix; For Iteration i= 1,2, . . . do •Peak Selection: Select the first Plocal maxima of ˆ h; •Step Refinement Around Maxima: Build a new sensing matrix based on an extension of matrix Dsuch that the step size around the peaks is decreased (i.e., the granularity and the precision are increased); •Finer Estimation: For any LASSO solver, get a finer estimation ˆ h. End Note that during the Coarse Estimation step, i.e., when Dis an NM × NM matrix, it is possible to adopt the approach proposed in [100] to solve the LASSO minimization in (5.14). The algorithm of [100], benefiting of the hierarchical structure of vector h, is able to provide a first coarse and reliable estimation optimizing the computational complexity. However, if htakes offgrid values, the approach of [100] becomes inappropriate, as confirmed by the presented simulation results. For this reason, after a Coarse Estimation, i.e., within the Iteration step, another LASSO solver must be chosen to obtain the best performance in terms of channel estimation. 5.2.1.2 Soft-Thresholding Operator In order to solve the LASSO minimization problem, a soft thresholding operator is used in (5.17). The choice of this function is justified in the following. 116 Chapter 5. Channel Estimation By starting from the input-output relation in matrix form y=xSDh +z,Ah +z,(5.19) the residual sum of squares (RSS) for the LASSO solver is given by RSS (h) = 1 2 NM−1 X i=0  yi− G−1 X j=0 hjAi,j  2 +λ G−1 X j=0 |hj|.(5.20) By taking the derivative of the first and second terms of the RSS w.r.t. θkwe get ∂ ∂hk   1 2 NM−1 X i=0  yi− G−1 X j=0 hjAi,j  2   =− NM−1 X i=0 Ai,k    yi− G−1 X j=0 j6=k hjAi,j   +hk NM−1 X i=0 Ai,k ,−ρk+hkηk,(5.21) ∂ ∂hk   λ G−1 X j=0 |hj|   =         −λif hk<0 [−λ, λ] if hk= 0 λif hk>0 ,(5.22) in which the derivative at hk= 0 can assume two different values. By putting things together ∂RSS (h) ∂hk =−ρk+hkηk+∂ ∂hk λ|hk|,(5.23) and, by setting the derivative equal to zero (since we are trying to minimize the LASSO cost function) ∂RSS (h) ∂hk =0=         −ρk+hkηk−λif hk<0 [−ρk−λ, −ρk+λ] if hk= 0 −ρk+hkηk−λif hk>0 .(5.24) 5.2. OFDM Modulation and the CS Algorithm 117 We must ensure that the closed interval [−ρk−λ, −ρk+λ] contains the zero such that hkis a global minimum, i.e., 0∈[−ρk−λ, −ρk+λ]⇒ −λ≤ρk≤λ . (5.25) Thus, finally ψst (hk, λ) =          (ρk+λ)/ηkif ρk<−λ 0 if −λ≤ρk≤λ (ρk−λ)/ηkif ρk> λ ,(5.26) which is the soft thresholding operator ψst (hk, λ) with normalization constant 1/ηk. 5.2.1.3 Nesterov’s Acceleration Factor The Nesterov’s acceleration factor governs the dependency between two successive estimations, while remarkably reducing the convergence time of the algorithm [97]. The choice of the optimization coefficient αiis not mandatory. By following the pioneering work [98], which inspired many other works, e.g., [96], αican be recursively defined as ξi+i=1 + q4ξ2 i+ 1 2,(5.27) αi=ξi−1 ξi+i ,(5.28) with ξ0= 1. Another choice simply based on the iteration index iis αi= (i−1)/(i+ 2) [97, 98]. Both solutions describe a curve growing from an initial value (“far” from 1) up to 1. The associated plots, with their similar behaviors, can be seen in Fig. 5.1. The most conservative choice is αi= 1, for which the dependency from the previous estimated values is maximized, while the less conservative choice is αi= 0, completely forgetting the previous estimated values. Intuitively, a small value of αiis preferable for the first noisy iterations, 118 Chapter 5. Channel Estimation 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 i αi αi= (i−1) /(i+ 2) αi= (ξi−1) /ξi+i Figure 5.1: The evolution of αiover iterations ifor different approaches. while a parameter αinear to one should be chosen when the reliability of the estimation increases. 5.2.2 Pilot Scheme The optimization of a deterministic sensing matrix for CS configurations, such as LASSO, is up to now one of the most studied open problems in CS theory. In fact, the typical performance guarantees of CS require properties such as the restricted isometry property [101, 38], for which explicit constructions are not available and even checking the property for a given randomly generated matrix is exponentially complex [102]. On the other hand, ensembles of randomly generated matrices have the property of satisfying these properties with high probability [38]. Hence, here we resort to a pseudo-random pilot placement on the 2-dimensional time-frequency grid of transmitted symbols. Simulation results have shown that such random placement achieves with high probability the best performance w.r.t. regular “lattice” placements (e.g., equally spaced combinations of subcarriers or time slots), as usually specified in wireless standards [36]. An example of a random pilot scheme is depicted in Fig. 5.2. Moreover, generally, distinct configurations of a fixed number of 5.2. OFDM Modulation and the CS Algorithm 119 Figure 5.2: Example of a random pilot scheme for OFDM modulation. pilots, randomly placed within the 2-dimensional grid, provides similar performance in terms of channel estimation. If pilots are not placed randomly but follow some periodic pattern, the algorithm for solving the LASSO produces far inferior results. This behavior is caused by the periodic sampling of a random Fourier matrix (i.e., Hor Dhsp). This is the reason why commonly used pilot schemes (see, e.g., [36] and references therein), generally structured or periodic, are not suitable for the CS-based estimation of OFDM systems (assuming that the OFDM channel is represented by a Fourier matrix). Overall, the aim it to maximize the overall achievable rate under random pilot placement. Hence, we can optimize the number of pilots per block to seek the optimal tradeoff between CSI estimation quality and pilot overhead (see (5.41) in the following and numerical results in Sec. 5.4). 5.2.3 Received Samples Expression — Real and Approximated Channel Conditions Without entering into details of the complete input-output derivation of a CP OFDM system which can be found in Section 1.2, we only provide the received 120 Chapter 5. Channel Estimation sample expression, which is y[n, m] = 1 M P−1 X p=0 hpej2πνpnT M−1 X m0=0 xn, m0e−j2πm0∆fτp M−1 X i=0 ej2πi M νp ∆fej2πi(m0−m) M (5.29) ≈1 M P−1 X p=0 hpej2πνpnT M−1 X m0=0 xn, m0e−j2πm0∆fτp M−1 X i=0 ej2πi(m0−m) M = P−1 X p=0 hpej2πνpnT e−j2πτpm∆fx[n, m].(5.30) By considering real and approximated channel conditions, the received samples at time instant nand subcarrier mare respectively given by (5.29) and (5.30), in which the ICI-free approximation follows the assumption νmax/∆f1, and the last equality follows by using the orthogonality property. Note that the expression (5.30) is equivalent to (5.6), meaning that the ICI-free assumption has been taken into account for the algorithmic design. When comparing OTFS and OFDM, the channel model in (5.30) is considered, such that the absence of ICI assumption holds. However, by focusing the attention on OFDM only, the performance comparison is extended to the case of real channel conditions, i.e., with input-output relation (5.29), showing the performance degradation when the ICI-free assumption is far to be satisfied. Clearly, a similar analysis for OTFS is meaningless, being the waveform not sensitive to the magnitude of delay and Doppler shifts. 5.3 OTFS Modulation and the Proposed Estimation Algorithm By neglecting, but keeping in mind, the complete derivation of the OTFS input-output relation, which can be found in Chapter 1, we can proceed further describing the channel estimation procedure, which is inspired by [86]. The input-output relation expressed in matrix, shown the first time in (1.44), is 5.3. OTFS Modulation and the Proposed Estimation Algorithm121 reported here for convenience y=  P−1 X p=0 hpΨp x+z,(5.31) where zdenotes the AWGN with zero mean and covariance matrix σ2INM . Note that Ψpimplicitly takes into account a Doppler-delay pair (τp, νp), i.e., Ψp,Ψp(τp, νp). Without entering into mathematical details which can be found in Sec. 1.3, the effect of the channel to symbols arranged in blocks is shortly described in the following. Consider a block composed by all zero magnitude symbols but one nonzero, having enough energy to be well distinguishable, w.r.t. to the noise floor, and positioned anywhere within the block. Note that the position of the symbol does not influence the result, since the channel shift effect is circular within the block, as proved by the results of Sec. 1.3. This block is thus transmitted over the time-frequency selective channel in (5.3). At the Rx, most of the energy concentrates in a point of the block (or, more precisely, a point per multipath component), while dissipating to the surrounding positions, according to Fig. 1.4, where and example of blocks of transmitted symbols and received samples blocks are depicted. The intuitive estimation of the pairs (τp, νp), for each multipath component, follows by searching the peaks within the received samples grid, successively associated to an estimate (ˆτp,ˆνp) (as suggested in [86]). However, this intuitive estimation procedure is only able to provide the integer parts of the Doppler and delay shifts, associated to the Doppler-delay grid point, where a peak is detected, collecting enough energy. The fractional parts of delay and Doppler shifts are, instead, associated to the dissipation of the energy around the peak points (see Fig. 1.4), and have to be treated and analyzed separately. The approximation of the channel behavior to integer Doppler and delay shifts, as done in [86], allows this intuitive estimation procedure to work correctly, but only under such non-realistic channel conditions. Thus, based on the ML estimator proposed in Sec. 1.3, the idea of [86] is extended, and a 122 Chapter 5. Channel Estimation 25 50 32 64 0 10 20 NM  Figure 5.3: Example of a pilot scheme for OTFS modulation. Within the block of dimension N×Mthe centered pilot with high energy εis well distinguishable, and surrounded first by zero pilots (the hollow zone) and after by information symbols (here, for convenience, with unit energy). reliable estimation algorithm working under realistic channels is provided. 5.3.1 Pilot Scheme A block of N×Mtransmitted symbols contains both information bearing symbols and pilots. The arrangement of pilots consists of a rectangular region placed within the block (not necessarily in the middle, since the channel effect is circular) containing two types of symbols (see Fig. 5.3):  Zero Pilots: Placed between information symbols and non-zero pilots to guarantee as less interference as possible between them. The Dirichlet kernel functions appearing in the OTFS input-output relation (see Sec. 1.3 and in particular (1.44)) are rapidly decreasing non-negative functions (see Fig. 1.3 and 1.5), hence, perfect orthogonality between information symbols and pilots cannot be achieved, but, at least, the Doppler-delay ISI can be reduced. 5.3. OTFS Modulation and the Proposed Estimation Algorithm123 25 50 32 64 0 2 4 N M  Figure 5.4: Example of a pilot scheme for OTFS modulation. Within the block of dimension N×Mthe plateau of pilots with high energy εis well distinguishable, and surrounded first by zero pilots (the hollow zone) and after by information symbols (here, for convenience, with unit energy).  Peak Pilot: A single pilot symbol with high energy, collecting the energy of all zero pilots, is placed at the grid center. Its shifts in the Dopplerdelay grid are used to provide the initial coarse estimation of the Dopplerdelay pairs, which results to be fast and simple. Given this pilot arrangement, the number of pilot symbols has to be optimized to match the optimal performance-overhead tradeoff, while keeping constant the total block energy. Given some particular system setups, such as communication including non-linear amplifier not constant channel behaviours over block symbols in time, basing the estimation algorithm on just one symbol (the peak pilot) could not be a safe approach. In such cases, the pilot scheme can be extended, for instance, by considering a rectangular region composed of zero pilots and a plateau on non-zero pilots (see Fig. 5.4), instead of just the peak pilot. We would like to stress the fact that the successive algorithm design is not only suitable for the aforementioned pilot configuration, but is able to provide reli- 130 Chapter 5. Channel Estimation of pragmatic capacity. For instance, by considering a CP of length T/4, being Tthe symbol time, the loss is T T+T/4=T 5T/4=4 5= 0.8 = 20% .(5.42) This means that, with a modulation of cardinality C, while the maximum achievable rate is log2Cbits/symbol, OFDM saturates at 0.8·log2Cbits/symbol. The overall loss takes into account both the pilot overhead and the presence of a CP and/or GI (also of length T/4, for consistency). Another important aspect is the definition of the number of pilots |P| w.r.t. the ambient dimension G. Many features are influenced by this choice. Consider first OFDM modulation. It is well known in the CS literature that the minimum number of pilots (or measurements, from CS literature) to recover a sparse signal is given by the logarithmic scaling factor [105] |P| ≥ Plog G P,(5.43) where Phere represent the number of non-zero components of the vector to be estimated. Thus, given a multipath channel with Ppaths and a sensing matrix of dimension G(defined in (5.7)), the (asymptotically) minimum number of pilot symbols necessary to solve the minimization problem in (5.14) is given by (5.43). Due to the logarithmic scaling, the function is slowly increasing, even if the granularity of the sensing matrix and its dimension grows (see Section 5.2.1.1). However, while (5.43) provides a lower bound on the dimension, the optimal performance might be achieved for a number of pilots different from the minimum. For this reason, if the number of pilots used for a given setup (i.e., for fixed Pand G) is well above the lower limit, we can state that while increasing the dimension G, the optimal number of pilots slightly changes. Hence, the pilot loss in (5.41) tends to zero, while increasing the ambient dimension G, which is directly linked to the block size NM (assuming, for simplicity, on-grid paths shifts and neglecting the refinement solution depicted in Alg. 3), and OFDM benefits from large blocks. Note that this is not a precise quantitative analysis but it just gives a qualitative idea or intuition 5.4. Comparison in Terms of Pragramatic Capacity 131 on how large the number of pilots per block should be, knowing that the aforementioned CS-based conditions are always given up to constant factors that depend on the specific problem, SNR, shape of the sensing matrix, and other variables. For OTFS modulation, the pilot scheme presented in Sec. 5.3.1 and its estimation algorithm are independent of the block dimension, since the shift of the peak pilot, and so the rough estimation, is only associated to the maximum Doppler and delay of the channel defined in (5.2). Hence, it follows that the pilot overhead tends to zero while increasing the block dimension, hence, as for OFDM, also OTFS benefits from large blocks. Moreover, if the dimension of the block increases, one can set more pilots to zero to raise the power of the peak pilot, allowing the detection of low power scattering components in the threshold-based approach (as explained at the beginning of Sec. 5.4). However, limits on the block dimension comes, first, from important restrictions on OTFS detection computational complexity (as seen in Chapter 4 and [46]), and then from the block fading assumption, which breaks down if the block becomes too large (see Chapter 1). Thus, realistic block sizes have to be considered in both directions. Moreover, as anticipated in Sec. 5.2, in order to restrict to the classical low-complexity symbol-by-symbol MMSE estimation for OFDM we have neglected the ICI. As already seen in (5.29), the ICI depends on the ratio between the subcarrier spacing ∆fand the maximum Doppler shift introduced by the channel. In order to have negligible ICI the necessary condition is ∆fνmax, or, equivalently, νmax/∆f1. Since ∆f=B/M, with Btotal bandwidth, the condition may not be satisfied when the number of subcarriers Mbecomes too large, even for moderate Doppler. Here, we insist on neglecting ICI and consider the range of system parameters for which this assumption is indeed virtually exact. Furthermore, we notice that while OFDM incurs in this additional limitation, OTFS remains not sensitive to the Doppler shift. 132 Chapter 5. Channel Estimation Table 5.1: System parameters fc= 5.89 [GHz] M= 64 B= 10 [MHz] N= 50 ∆f=B/M = 156.25 [kHz] T= 1/∆f= 6.4 [µs] 5.4.1 Simulation Results In the following figures, we plot the pragmatic capacity vs. SNR for OTFS and OFDM modulations with quadrature phase-shift keying (QPSK) modulated symbols, for a time-frequency multipath channel with Pcomponents affected by AWGN, and under non-perfect CSI. The channel estimation has been performed with the pilot schemes and the algorithms of Sec. 5.2 for OFDM and Sec. 5.3 for OTFS. As a reference benchmark, we plot the AWGN (symmetric) capacity Csym AWGN for QPSK modulation, which results to be not achievable due to the pilot overhead. The system parameters are listed in Table 5.1. Fig. 5.5 shows the performance of OFDM for a different number of pilot symbols. For the case P= 1, it easy to note that the performance slightly changes for different pilot overheads, whose percentage is indicated in the legend. On the other hand, as suggested by (5.43) and associated discussion, if the number of non-zero components to be estimated increases, i.e., with P > 1, the channel estimation algorithm needs more pilots to work efficiently. Given these results, from now on, we will consider a pilot overhead of 3.125%, achieving, in our setup, the best tradeoff between estimation accuracy and achievable pragmatic capacity (for any number of scattering components). Fig. 5.6 shows the performance of OTFS with different detection algorithms. The MP soft-output detection approach of [46] is able to almost achieve the AWGN capacity under non-perfect CSI for a low number of scattering components, together with a remarkable reduction of the computational complexity [46]. However, in line with the results of [46] and Chapter 4, the detec- 5.4. Comparison in Terms of Pragramatic Capacity 133 −5 0 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 P= 1 SNR [dB] IPC hbits symboli 0.625% 1.25% 1.875% 2.5% 3.125% 3.75% 4.375% Csym AWGN −5 0 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 SNR [dB] 0.625% 1.25% 1.875% 2.5% 3.125% 3.75% 4.375% Csym AWGN P= 2 −5 0 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 SNR [dB] 0.625% 1.25% 1.875% 2.5% 3.125% 3.75% 4.375% Csym AWGN P= 3 −5 0 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 SNR [dB] 0.625% 1.25% 1.875% 2.5% 3.125% 3.75% 4.375% Csym AWGN P= 4 Figure 5.5: Pragmatic Capacity vs. SNR for OFDM modulation with multipath components Pand different pilot overhead (whose percentage is indicated in the legend). 134 Chapter 5. Channel Estimation −5 0 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 P= 1 SNR [dB] IPC hbits symboli LMMSE / 5.28% G/ 5.28% Csym AWGN −5 0 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 P= 2 SNR [dB] LMMSE / 5.28% G/ 5.28% Csym AWGN −50 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 P= 3 SNR [dB] LMMSE / 5.28% G/ 5.28% Csym AWGN −5 0 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 P= 4 SNR [dB] LMMSE / 5.28% G/ 5.28% Csym AWGN Figure 5.6: Pragmatic Capacity vs. SNR for OTFS modulation with multipath components P, different detection algorithms, i.e., LMMSE and MP approach of [46], for a fixed pilot overhead, in percentage 5.28%. 5.4. Comparison in Terms of Pragramatic Capacity 135 tor performance decreases with increasing multipath. Note that the small loss w.r.t. Csym AWGN under non-perfect CSI is an indicator of the performance of the channel estimation algorithm, which results to be very accurate (otherwise the curve would have deviated from the benchmark). In light of these results, we see that there is no reason to adopt the LMMSE estimator for OTFS, which results in higher complexity and worse performance (see Chapter 4 and [46] for a more detailed analysis). Hence, from now on, for the comparison with OFDM modulation, we consider the MP “Matrix-G” approach of Chapter 4. In Fig. 5.7, we plot the pragmatic capacity vs. SNR for OFDM and OTFS under the configurations mentioned above. First of all, it is possible to note that performance slightly decrease while increasing the number of multipath components, proving the robustness of the pilot schemes and the algorithms proposed. The pilot overhead has been chosen ad hoc for both modulations. For OTFS, as said in Sec. 5.3.1, the peak centered pilot collects all the energy of surrounding zero pilots, and the percentage of overhead (indicated in the figures) is also an indicator of the peak pilot energy. For OFDM, as pointed out before, the optimal tradeoff between estimation performance and pilot overhead has been found by brute-force search over a suitable set of possibilities (some of them are visible in Fig. 5.5). While the performance of the two modulations is similar, the presence of a per symbol CP for OFDM remarkably deteriorates the pragmatic capacity, while a per block GI for OTFS introduces a negligible loss. In Fig. 5.8, we plot the pragmatic capacity of OFDM for a fixed value of SNR, i.e., 18 dB, while changing the ratio between the maximum Doppler shift and the subcarrier spacing, i.e., νmax/∆f, taking into account a different number of subcarriers M(N= 50 for all cases). In this case, the received samples are obtained by considering a real channel taking into account the ICI, i.e., (5.29), while the channel estimation works under the hypothesis of an ideal interference-free channel. Intuitively, the performance degrades when the ICI becomes significant. Note that the estimation performance of the LASSO solver is independent of the number of subcarrier Mand, for this reason, 136 Chapter 5. Channel Estimation −5 0 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 P= 1 SNR [dB] IPC hbits symboli −5 0 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 P= 2 SNR [dB] −50 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 P= 3 SNR [dB] IPC hbits symboli −5 0 5 10 15 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 P= 4 SNR [dB] MOD OFDM OTFS (G)Csym AWGN Pilot Overhead 3.125% 5.28% - OFDM CP OTFS GI (G) Figure 5.7: The curves take into account the loss related to the presence of pilots in the block of N×Msymbols. The OFDM CP curve includes the CP overhead which is 0.25 of the symbol time, while the OTFS GI curve includes a GI for the entire block. A QPSK modulation is used. The legend indicates the percentage of pilot symbols. 5.5. Conclusions 137 0 0.05 0.1 0.15 0.2 0.25 0 0.5 1 1.5 2 νmax/∆f IPC hbits symboli M= 64 M= 128 M= 256 M= 512 Csym AWGN Figure 5.8: Pragmatic Capacity vs. νmax/∆fratio for OFDM modulation with P= 1, at SNR = 18 dB. whatever the choice of νmax and M, the performance of OFDM depends only on their ratio. Fig. 5.8 shows that the pragmatic capacity performance starts decreasing significantly for νmax/∆f≃0.15. Almost the same behavior is shown for different number of subcarriers M(not reported here for the sake of space limitation), except for the percentage of pilot loss due to different block dimensions, supporting what stated above. However, as pointed out in Table 5.2, while the performance is almost constant, the maximum tolerable Doppler (or velocity), inversely proportional to M, is not. For these reasons, as expected, OFDM is not independent of the block dimension and the system has to be defined properly to operate in the range where the ICI is negligible. 5.5 Conclusions We carried out a fair comparison between OFDM and OTFS modulation formats in terms of maximum achievable rate for practical separated detection and decoding, quantified by the pragmatic capacity measured at the soft- 138 Chapter 5. Channel Estimation νmax/∆f 0.1 0.15 0.2 M 64 1431 [km/h] 2147 [km/h] 2863 [km/h] 128 715 [km/h] 1073 [km/h] 1431 [km/h] 256 357 [km/h] 536 [km/h] 715 [km/h] 512 178 [km/h] 268 [km/h] 357 [km/h] Table 5.2: Maximum supportable velocity w.r.t. the ratio νmax/∆ffor OFDM modulation with subcarrier spacing ∆f=B/M and carrier frequency fc= 5.89 GHz. The velocity is given by v=c·νmax/(2fc). detector output. We considered two pilot schemes and channel estimation algorithms each one specifically suited for the given modulation scheme. Both pilot and CSI estimation schemes are able to achieve very good performance (near genieaided) under time-varying communication channel in the sparsity regime of a small number of number of multipath components. This conclusion is fully supported by numerical results, where simulation curves achieve the theoretical benchmark under non-perfect CSI, proving the quality of the proposed approaches. OTFS achieves a better communication rate mainly because of the presence of a per block guard interval rather than a per symbol cyclic prefix as in OFDM. This of course comes at the cost of a more complex channel estimation scheme, working on large block-wise operations. In terms of soft-output data detection, the use of the message passing soft-output detector of Chapter 4 yields constant per symbol complexity for OTFS, which is the same scaling law of symbol-by-symbol MMSE detection for OFDM. Although we do not claim that the complexity of the two detectors 5.5. Conclusions 139 is identical, in fact the actual complexity differ for some implementation-based constant. Finally, we can observe that OTFS is indeed very insensitive to the magnitude of the Doppler shifts, while the performance of OFDM degrades significantly even under small-to-moderate Doppler values if the number of subcarriers increases. Therefore, OTFS is effectively a good candidate for highmobility systems in rural environments (e.g., high speed trains [106]) or aerial environments (e.g., UAVs [107]), where Doppler shifts may be large, and the propagation channel contains typically the line-of-sight and a few other reflection components (e.g., ground reflection, hills, large buildings), and it is therefore sparse in the Doppler-delay domain.