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Reliable, Secure, and Spectrally Efficient ISAC using Distributed Multiuser MIMO and Non-Orthogonal Waveform Tongyang Xu, Christos Masouros, and Izzat Darwazeh Department of Electronic and Electrical Engineering, University College London, London, UK Email: [email protected], [email protected], [email protected] Abstract—This paper presents a hardware-validated study of integrated sensing and communication (ISAC) using a distributed multiple-input multiple -output (MIMO) architecture, with a focus on enhancing reliability and security in dynamic wireless environments. We compare centralized and distributed ISAC systems under varying conditions, including signal blockage, and demonstrate that distributed MIMO offers superior resilience by maintaining stable communication performance and could enable real-time environmental sensing through channel state information (CSI) variations. To address security challenges arising from shared sensing and communication waveforms, we implement a waveform-defined security (WDS) technique based on spectrally efficient frequency division multiplexing (SEFDM). Experimental results show that while the legitimate user, equipped with prior knowledge of the waveform parameters, can successfully decode signals, an eavesdropper experiences severe degradation, with an error vector magnitude (EVM) penalty of over 22 dB. In addition, employing the non-orthogonal signal waveform enables over 10% spectral efficiency improvement. These findings highlight the potential of non-orthogonal waveform design and distributed MIMO architectures in achieving reliable, secure, and spectrally efficient ISAC. Index Terms—Waveform, ISAC, OFDM, SEFDM, nonorthogonal, secure, distributed, MIMO, over-the-air, softwaredefined radio (SDR). I. INTRODUCTION Integrated sensing and communication (ISAC) [1] has become a fundamental enabler for next-generation wireless networks, enabling joint communication and sensing through the shared use of hardware and spectrum resources. A key component of this paradigm is multi-user multipleinput multiple-output (MU-MIMO), which can be realized through two system architectures in ISAC: centralized MUMIMO [2] and distributed MU-MIMO [3]. In traditional dual-functional centralized MU-MIMO ISAC [4], sensing and communication tasks are coordinated by a central base station (BS). This structure offers efficient global resource management and joint signal processing, often resulting in high spectral efficiency and sensing accuracy. However, it also introduces a single point of failure and raises concerns regarding data security. Signal blockage poses a significant challenge to the reliability of centralized MU-MIMO ISAC systems, which relies heavily on lineof-sight (LoS) path to ensure high-quality beamforming signal transmission and accurate environmental sensing. As demonstrated in our previous experiment [5], the dualfunctional ISAC successfully integrates both communications and sensing. However, the presence of physical obstacles between users and BSs such as buildings, vehicles, trees, or even pedestrians can lead to severe signal attenuation or complete blockage. This degradation is especially critical in higher-frequency bands (e.g., mmWave), where signals are more susceptible to absorption and diffraction losses. When a user’s signal path is blocked, the centralized MU-MIMO experiences a significant decline in received signal power, which directly reduces the communication quality, leading to increased latency, reduced throughput, or link failures. Simultaneously, the sensing capability is compromised, as blocked or weak signal reflections degrade the spatial resolution and accuracy of target detection. Overall, signal blockage introduces critical limitations to centralized MU-MIMO ISAC systems, affecting both communication reliability and sensing accuracy. Distributed MU-MIMO ISAC systems offer a promising approach to addressing the impact of signal blockage. In distributed ISAC, geographically distributed antenna units or access points (APs), connected to a central processing unit (CPU) or control center, are deployed across the coverage area to jointly perform communication and sensing tasks. This spatially diverse deployment provides rich angular coverage and enhances the probability that at least some antennas maintain an unobstructed LoS paths to users and targets, even in the presence of dynamic blockages. When a user’s signal is blocked from one or more antennas, other nearby antennas can still provide connectivity and contribute to communication and sensing, ensuring robustness through spatial diversity. Unlike centralized MIMO ISAC, which suffers significantly from blockage at the BS, distributed systems dynamically reallocate communication and sensing loads across available antennas, thus maintaining system functionality and performance. While distributed MUMIMO ISAC systems enhance reliability and robustness against signal blockage, they also introduce new security vulnerabilities. In these systems, the widespread deployment of remote radio units or APs creates a large and distributed attack network. These remote antennas may be physically compromised due to the broadcast nature of radio signals, especially in untrusted or public deployment scenarios. An attacker could exploit unsecured APs to eavesdrop on user transmissions, thereby undermining the confidentiality of the ISAC service. To address these challenges, this work explores the use of a non-orthogonal signal waveform to enhance ISAC
security. Specifically, we leverage the intentional creation of inter carrier interference (ICI), which disrupts the eavesdropper’s ability to accurately decode the signal, while the authorized user, equipped with knowledge of the waveform design, can still successfully recover the intended information. This approach introduces a novel form of physical-layer security that is especially advantageous for distributed MU-MIMO ISAC systems operating in adversarial or untrusted environments, where conventional beamforming techniques are impractical or ineffective. II. CENTRALIZED MIMO BEAMPATTERN-BASED ISAC A. Communication Transmission Model In a centralized multi-user MIMO transmission system, where antennas are co-located at a central base station, the received signal at the user can be expressed as Y=H˜ X+W,(1) where H∈CK×Mdenotes the MIMO channel matrix between the Mtransmit antennas and Kusers, ˜ X= [˜x1,˜x2, ..., ˜xM]∈CM×Qis the precoded transmission symbol matrix with Qtime samples per data stream, and Wrepresents additive white Gaussian noise (AWGN). The signal vector ˜xmcorresponds to the precoded signal from the m-th antenna. To simplify, (1) can be rewritten as: Y=X+W+H˜ X−X |{z } MUI ,(2) where X= [x1,x2, ..., xK]T∈CK×Qdenotes the ideal symbol matrix intended for the Kusers, and the term H˜ X− Xaccounts for multi-user interference (MUI). The severity of MUI is quantified by its total power: PMUI = H˜ X−X 2 F,(3) where ∥·∥2 Fdenotes the Frobenius norm. The goal of precoding design is to minimize PMUI by optimizing ˜ X. B. MIMO Sensing Model MIMO based radar sensing waveform design generally falls into two categories: orthogonal waveforms, which produce an omnidirectional beampattern suitable for detecting unknown targets, and directional waveforms, which focus energy in specific directions to track known targets more effectively [6]. The resulting sensing beampattern is determined by the spatial covariance matrix of the transmit signal matrix ˜ X[7]. 1) Omnidirectional Beampattern: To synthesize an omnidirectional beampattern, the signal matrix ˜ Xmust be orthogonal. This ensures that its spatial covariance matrix is proportional to the identity matrix. The corresponding optimization problem can be formulated as: min ˜ X H˜ X−X 2 F(4) s.t. 1 Q ˜ X˜ XH=PT MIM,(4a) where PTdenotes the total transmit power, and IMis the M×Midentity matrix. The constraint enforces equal power allocation and orthogonality across the transmit antennas. 2) Directional Beampattern: To form a directional beampattern, the spatial covariance matrix must match a desired positive-definite matrix Rd. This design allows the transmitted energy to be steered toward specific spatial directions. The optimization problem is given by: min ˜ X H˜ X−X 2 F(5) s.t. 1 Q ˜ X˜ XH=Rd.(5a) C. Trade-off Between Communication and Sensing In conventional centralized MIMO ISAC systems, achieving a balance between communication and sensing performance requires the introduction of a trade-off factor γ∈ [0,1]. Let Xddenote the desired radar sensing signal. The joint optimization problem for dual-functional transmission can then be formulated as: min ˜ X γ H˜ X−X 2 F+ (1 −γ) ˜ X−Xd 2 F(6) s.t. 1 Q ˜ X 2 F=PT.(6a) This formulation enables flexible prioritization between communication and sensing objectives. Specifically, when γ= 0, the problem reduces to a pure sensing design. Conversely, setting γ= 1 transforms the problem into a pure communication optimization. By tuning γwithin the range [0,1], the system can adaptively allocate resources to meet varying requirements, achieving a controllable tradeoff between communication and sensing. D. Experiment Validation for Centralized MU-MIMO ISAC We design an experiment to evaluate the limitations of centralized MU-MIMO ISAC systems. As illustrated in Fig. 1(a), six omnidirectional antennas spaced at half-wavelength intervals are deployed to synthesize a dual-functional ISAC beampattern. To simplify the setup, the experiment involves two users. Six NI USRP-RIO 2953R software-defined radio (SDR) [8] [9] are used to emulate the transmitter and user devices. Each SDR is equipped with two independent RF chains, one for transmission and one for reception. The system operates in the 2.4 GHz unlicensed band, with each antenna connected to a dedicated RF chain. This configuration allows for flexible beamforming and signal processing. The signal frame structure follows the design in [10], and time-orthogonal pilot symbols [10] are employed to avoid inter-antenna interference and enable accurate channel estimation. Synchronization among the six USRPs is achieved using a CDA-2990 OctoClock module, which distributes a common 10 MHz reference and pulseper-second (PPS) signals to all devices. LabVIEW on a control host generates raw digital baseband signals, which are transmitted to a CPS-8910 PCIe switch box via an NI MXI-Express Gen 2 ×8 cable. The switch box then routes six parallel data streams to the USRPs at up to 3.2 Gb/s. The experiment targets indoor scenarios, with the transmitter and users placed approximately 2 meters apart. Two scenarios are tested, as shown in Fig. 1(b)(c). In the first scenario (Fig. 1(b)), both users have a direct LoS to the
Fig. 1. Centralized Multiuser-MIMO testbed setup. (a) Transmitter side. (b) Receiver side with users Rx-1 and Rx-2 without metal block. (c) Receiver side with users Rx-1 and Rx-2 with metal block. Fig. 2. Measured results for centralized Multiuser-MIMO. (a) Spectral response at Rx-1 without metal block. (b) Time-domain signal pattern at Rx-1 without metal block. (c) Time-domain signal pattern at Rx-2 without metal block. (d) Spectral response at Rx1 with metal block. (e) Time-domain signal pattern at Rx-1 with metal block. (f) Time-domain signal pattern at Rx-2 with metal block. transmitter. In the second (Fig. 1(c)), a metal plate is positioned between the transmitter and Rx-1, introducing a non-line-of-sight (NLoS) condition due to signal blockage. Spectral and time-domain measurements are taken to assess performance. As seen in Fig. 2(a)(d), Rx-1 experiences approximately 20 dB of signal power attenuation due to the metal obstruction. This degradation is further evident in the time-domain signal (Fig. 2(b)(e)), where Rx-1’s received power is significantly reduced. In contrast, Rx-2 remains unaffected, as shown in Fig. 2(c)(f). These results highlight a critical vulnerability of cenFig. 3. Distributed Multiuser-MIMO testbed setup. (a) Transmitter distributed antennas and users Rx-1 and Rx-2 without metal block. (b) Transmitter distributed antennas and users Rx-1 and Rx-2 with metal block. Fig. 4. Measured results for distributed Multiuser-MIMO testbed setup. (a) Spectral response at Rx-1 without metal block. (b) Timedomain signal pattern at Rx-1 without metal block. (c) Timedomain signal pattern at Rx-2 without metal block. (d) Spectral response at Rx-1 with metal block. (e) Time-domain signal pattern at Rx-1 with metal block. (f) Time-domain signal pattern at Rx-2 with metal block. tralized MIMO systems: performance heavily depends on maintaining an unobstructed signal path. When the dominant beamformed direction is blocked, the system suffers substantial performance degradation, indicating the need for more resilient ISAC architectures. III. DISTRIBUTED MIMO CSI-BASED ISAC To overcome the signal blockage limitations of centralized MIMO systems, a distributed MIMO architecture is proposed. In contrast to centralized MIMO ISAC sys-
tems, which depend on directional spatial beampatterns for sensing and communications, distributed MIMO enables a coexistence-based sensing and communication framework. This approach leverages channel state information (CSI) from different antennas to sense and communicate. A. Principle of Co-Existence ISAC The signal patterns shown in Fig. 2(b)(c)(e)(f) indicate that the first six segments correspond to pilot signals transmitted from six Tx antennas. These are followed by a data segment used to evaluate communication performance. In our MU-MIMO system, pilot and data signals coexist in the time domain, enabling interference-free CSI acquisition for monitoring channel variations. The estimated CSI not only facilitates data equalization but also reflects environmental changes through variations in CSI power. Furthermore, the pilot pattern is independently designed from the data pattern, ensuring no mutual interference between them. 1) Multiuser Communications: The proposed distributed MIMO framework retains the multiuser communication capability that is fundamental to ISAC systems. 2) CSI Monitoring for Sensing: By enabling continuous CSI tracking, the CSI variations allows the system to detect and track changes in user positions or movements. Channel estimation relies on time-orthogonal pilots, where non-overlapping symbols offer more accurate results. In the context of ISAC, these pilots serve dual purposes via enabling both communication and environmental sensing. Considering a six-antenna setup serving two users, the MIMO channel matrix is defined as H=h11 h12 h13 h14 h15 h16 h21 h22 h23 h24 h25 h26,(7) where hmn denotes the channel coefficient from the nth transmit antenna to the mth user. To eliminate spatial interference, time-orthogonal pilot symbols are transmitted sequentially, only one antenna is active per time slot. The pilot matrix is defined as: P= p100000 0p20 0 0 0 0 0 p3000 0 0 0 p40 0 0 0 0 0 p50 00000p6 .(8) The corresponding received signal matrix is: Y=y11 y12 y13 y14 y15 y16 y21 y22 y23 y24 y25 y26=HP +Z,(9) where ymn represents the symbol received by the mth user at the nth time period and Zdenotes the AWGN. The estimated channel matrix ˆ His then computed as: ˆ H=ˆ h11 ˆ h12 ˆ h13 ˆ h14 ˆ h15 ˆ h16 ˆ h21 ˆ h22 ˆ h23 ˆ h24 ˆ h25 ˆ h26 =y11/p1y12/p2y13/p3y14/p4y15/p5y16/p6 y21/p1y22/p2y23/p3y24/p4y25/p5y26/p6. (10) This estimated CSI matrix enables real-time tracking of environmental changes, forming the basis for sensing in distributed MIMO ISAC systems. B. Experiment Validation for Distributed MU-MIMO ISAC Building upon the ISAC testbed from [5], we enhance the system by enabling real-time CSI monitoring without requiring additional sensing hardware. While maintaining the core hardware from Section II-D, we adopt a distributed MIMO architecture where each spatially separated antenna connects to its own USRP. This setup improves spatial diversity and robustness in dynamic environments. Operating in the 2.4 GHz band, the system uses a CDA-2990 OctoClock for synchronization and a CPS-8910 PCIe switch box for high-speed data transmission and CSI processing. To evaluate system performance, we test two scenarios: one with clear LoS path between the transmitter and both receivers, and another where a metal board is placed between the transmitter and Rx-1 to simulate signal blockage, as shown in Fig. 3. In the LoS scenario, each distributed antenna maintains direct links with both users, providing a performance baseline. In the blocked scenario, we assess the system’s ability to maintain functionality under obstruction. As illustrated in Fig. 4(a)(d), the received power at Rx1 remains stable around -60 dBm despite the blockage, indicating the robustness of the distributed MIMO setup against localized interference. Furthermore, changes in the pilot signal patterns (Fig. 4(b)(c)(e)(f)) reflect distinct environmental conditions, enabling real-time sensing through CSI variations. These results demonstrate the system’s capability to maintain communication and detect environmental changes without external sensors, highlighting the advantages of distributed MIMO in dynamic ISAC applications. IV. EXPERIMENTAL RESULTS FOR CENTRALIZED AND DISTRIBUTED MU-MIMO ISAC Fig. 5 presents measured results comparing centralized and distributed MU-MIMO ISAC systems. In traditional MU-MIMO, the focus is solely on optimizing communication. As shown in Fig. 5(b), this yields excellent communication quality, with an error vector magnitude (EVM) of -29.2 dB. However, the corresponding radar sensing performance in Fig. 5(a) is poor, lacking a clear directional or an omnidirectional beampattern. In dual-functional ISAC systems, as illustrated in Fig. 5(c), a directional sensing beampattern is generated for accurate target detection. To balance sensing and communication, we apply a trade-off factor of γ= 0.9, enabling simultaneous functionality. However, this trade-off leads to a decline in communication performance, with the EVM dropping to -12.7 dB, as shown in Fig. 5(d), compared to -29.2 dB in Fig. 5(b). When a metal board is introduced to block the signal path, communication quality in the centralized ISAC system degrades greatly. As shown in Fig. 5(f), the EVM drops to -5.6 dB, indicating significant degradation. However, the radar sensing beampattern remains stable despite the obstruction, as seen in Fig. 5(e), demonstrating that sensing performance is largely unaffected by signal blockages. In contrast, the distributed MIMO system yields different outcomes. Rather than forming a directional beam, it leverages power variations in pilot signals from spatially
Fig. 5. Measured ISAC results in centralized and distributed multiuser-MIMO. (a) Sensing in centralized multiuser-MIMO without block. (b) Constellation performance in centralized multiuser-MIMO without block. (c) Sensing in centralized multiuser-MIMO ISAC without block. (d) Constellation performance in centralized multiuser-MIMO ISAC without block. (e) Sensing in centralized multiuserMIMO ISAC with block. (f) Constellation performance in centralized multiuser-MIMO ISAC with block. (g) Sensing at Rx-1 without block. (h) Sensing at Rx-2 without block. (i) Sensing at Rx-1 with block. (j) Sensing at Rx-2 with block. (k) Constellation performance without block. (l) Constellation performance with block. separated antennas to detect environmental changes. Figures 5(g)–(j) reveal that pilot signal strength varies between Rx1 and Rx-2 due to their distinct locations. When the metal board is added, these variations further reflect the change in surrounding environment. Crucially, the distributed MIMO system maintains consistent communication performance even in the presence of obstructions. As illustrated in Figs. 5(k,l), the EVM remains stable at -29.6 dB and -29.8 dB, respectively, regardless of the blockage. This highlights a major benefit of distributed MIMO: strong resilience to localized signal blockage while preserving reliable communication and environmental sensing through CSI variations without the tradeoffs in centralized MIMO ISAC systems. V. WAVEFORM-DEFINED SECURITY FRAMEWORK TO PROTECT ISAC The main challenge in ISAC security arises from the shared waveform used for both communication and sensing, which can lead to data leakage. Traditional solutions mitigate this by using narrow-beam techniques to spatially separate legitimate users and eavesdroppers, or by leveraging constructive interference precoding to strengthen signals at the intended receiver while causing destructive interference at potential eavesdroppers [11]. However, these methods rely on accurate CSI, which may not be available in dynamic or harsh environments. Moreover, when an eavesdropper is close to a legitimate user, spatial separation through precoding becomes physically infeasible. To address these limitations, a waveform-defined security (WDS) framework [12] was proposed and experimentally validated in a WiFibased communication testbed. WDS introduces intentional ICI using a non-orthogonal waveform, spectrally efficient frequency division multiplexing (SEFDM). Its signal is mathematically given by Xk=1 √Q N−1 X n=0 snexp j2πnkα Q,(11) where Xkis the kth time sample, Nis the number of sub-carriers, and Q=ρN is the total number of time samples with oversampling factor ρ. The factor α= ∆f·T represents the bandwidth compression factor where ∆f is the sub-carrier spacing and Tis the symbol duration. When α=1, the waveform becomes standard OFDM with orthogonal sub-carriers. For α <1, sub-carriers are packed more tightly, leading to a non-orthogonal waveform known as SEFDM. The increased spectral efficiency gain compared to OFDM is given by: η= ( 1 α−1) ×100.(12) As illustrated in Fig. 6, OFDM maintains orthogonality between sub-carriers, while SEFDM introduces ICI due to non-orthogonal sub-carrier overlap. Without precise knowledge of αand signal detection algorithms, an eavesdropper is unable to correctly demodulate the signal, even if legitimate user signals are perfectly captured. In the experiment detailed in Fig. 3, two users (Rx1, Rx-2) are placed surrounded by six distributed transmitter antennas. To evaluate communication security here, we assume Rx-1 is the legitimate user and Rx-2 is an eavesdropper. Based on the measured results in Fig. 5, when OFDM signal is applied, the communication part shows
Fig. 6. Secure ISAC principle and experiment. (a) Insecure OFDM signal. (b) Secure non-orthogonal signal. (c) The legitimate user constellation performance after using the non-orthogonal signal. (d) The eavesdropper constellation performance after using the nonorthogonal signal. perfect constellation patterns, indicating simple but insecure signal recovery. Under the non-orthogonal signal with α= 0.9, the legitimate user, equipped with knowledge of the compression factor αand the signal detection algorithms [13], can successfully recover the transmitted signal, as evidenced by the clean constellation in Fig. 6(c). In contrast, the eavesdropper, lacking this information, experiences a significant degradation in communication quality, with the EVM deteriorating by approximately 22 dB. These results demonstrate that employing a non-orthogonal waveform design can effectively enhance the security of ISAC systems, even in the absence of accurate CSI. In addition, employing the non-orthogonal signal waveform can improve the spectral efficiency by (1/0.9-1)=11.1%. VI. CONCLUSION This work demonstrates the practical benefits of a distributed MIMO ISAC testbed in achieving reliable and secure wireless communication and sensing. Experimental results show that the proposed distributed MIMO ISAC architecture maintains consistent communication quality, with EVM values remaining around –30dB even under signal blockage, while centralized MIMO suffers degradation up to –5.6 dB. The system also enables environmental monitoring through CSI variations without additional sensing hardware. Furthermore, by integrating a waveform-defined security (WDS) scheme using a non-orthogonal SEFDM waveform with a bandwidth compression factor α < 1, the system achieves a spectral efficiency improvement of up to 11.1% (when α= 0.9) compared to OFDM. Importantly, this design results in a 22 dB EVM degradation at the eavesdropper, while the legitimate user maintains perfect signal recovery. These findings validate that combining distributed MIMO ISAC with non-orthogonal waveform design jointly enhances reliability, security, and spectral efficiency in complex wireless environments. VII. ACKNOWLEDGEMENT This work was supported in part by the UK Engineering and Physical Sciences Research Council (EPSRC) under Grant EP/Y000315/1, and in part by HORIZON-SNS2023: 6G-Multiband Wireless and Optical Signalling for Integrated Communications, Sensing and Localisation (6GMUSICAL) REFERENCES [1] International Telecommunication Union, “IMT towards 2030 and beyond,” https://www.itu.int/en/ITU-R/study-groups/rsg5/rwp5d/imt2030/Pages/default.aspx, June 2023. [2] E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “Massive MIMO for next generation wireless systems,” IEEE Communications Magazine, vol. 52, no. 2, pp. 186–195, 2014. [3] E. Bjornson and L. Sanguinetti, “Making cell-free massive MIMO competitive with MMSE processing and centralized implementation,” IEEE Transactions on Wireless Communications, vol. 19, no. 1, pp. 77–90, 2020. [4] F. Liu, L. Zhou, C. Masouros, A. Li, W. Luo, and A. Petropulu, “Toward dual-functional radar-communication systems: Optimal waveform design,” IEEE Transactions on Signal Processing, vol. 66, no. 16, pp. 4264–4279, 2018. [5] T. Xu, F. Liu, C. Masouros, and I. Darwazeh, “An experimental proof of concept for integrated sensing and communications waveform design,” IEEE Open Journal of the Communications Society, vol. 3, pp. 1643–1655, 2022. [6] P. Stoica, J. Li, and Y. Xie, “On probing signal design for MIMO radar,” IEEE Transactions on Signal Processing, vol. 55, no. 8, pp. 4151–4161, 2007. [7] D. R. Fuhrmann and G. San Antonio, “Transmit beamforming for MIMO radar systems using signal cross-correlation,” IEEE Transactions on Aerospace and Electronic Systems, vol. 44, no. 1, pp. 171–186, 2008. [8] National Instruments, “Overview of the NI USRP-RIO software defined radio,” http://www.ni.com/white-paper/52119/en/, Apr. 2018. [9] ——, “USRP-RIO 2953R, Software Defined Radio Reconfigurable Device,” http://sine.ni.com/nips/cds/view/p/lang/en/nid/ 213005, Jan. 2020. [10] T. Xu, C. Masouros, and I. Darwazeh, “Waveform and space precoding for next generation downlink narrowband IoT,” IEEE Internet of Things Journal, vol. 6, no. 3, pp. 5097–5107, Jun. 2019. [11] N. Su, F. Liu, Z. Wei, Y.-F. Liu, and C. Masouros, “Secure dualfunctional radar-communication transmission: Exploiting interference for resilience against target eavesdropping,” IEEE Transactions on Wireless Communications, vol. 21, no. 9, pp. 7238–7252, 2022. [12] T. Xu, “Waveform-defined security: A low-cost framework for secure communications,” IEEE Internet of Things Journal, vol. 9, no. 13, pp. 10 652–10 667, July 2022. [13] T. Xu, R. C. Grammenos, F. Marvasti, and I. Darwazeh, “An improved fixed sphere decoder employing soft decision for the detection of nonorthogonal signals,” IEEE Communications Letters, vol. 17, no. 10, pp. 1964–1967, Oct. 2013.