Doppler-Supervised Channel Charting

Doppler-Supervised Channel Charting Anil Kumar1, Yamil Vindas Yassine2, and Maxime Guillaud1 1Inria, INSA Lyon, CITI Laboratory UR3720, Villeurbanne, France Email: {anil.kumar, maxime.guillaud}@inria.fr 2Data Science for Digital Health group, University of Geneva, Switzerland Email: [email protected] Abstract—Channel charting is an integrated sensing and communication technique where passively collected channel state information (CSI) at the base-station is used for obtaining a lowdimensional representation of the distribution of all observed CSI samples, referred to as a channel chart. The usage of the channel chart for various applications depends highly on the charting accuracy, which is typically defined by its similarity to the users’ geometric positions; an accurate channel chart can be used for a variety of applications like beam selection, beam tracking, radio maps, pilot assignment, signal strength prediction etc. Thus, improving the charting performance is of utmost importance. In this paper, we propose to exploit the delayDoppler based multipath CSI which occurs naturally in modern techniques such as orthogonal time frequency space modulation as a form of supervision to produce channel charts that more closely match the geometric ground truth of user positions. We use a combination of classical triplet based loss function along with a novel Doppler based loss function to train a deep neural network which provides an efficient charting function. The charting performance for a variety of user trajectories demonstrates the effectiveness of our proposed approach. I. INTRODUCTION Channel charting (CC) has emerged as an integrated sensing and communication technique which singularly does not require estimating the geometric propagation parameters associated to the mobile users [1]. CC maps the high dimensional channel state information (CSI) to a low dimensional space called channel chart. CC works on the principle of neighbourhood preservation such that similar CSI samples are mapped close to each other on the channel chart. Channel charts can be obtained through the application of dimensionality reduction to well-chosen CSI-based features [2]. An alternative solution is to obtain the channel charting function as the solution to a proxy problem of learning a distance between CSI samples. This contrastive approach exploits the fact that CSI samples are acquired sequentially in communication systems; this gives rise to a triplet-based approach [3]. Once the charting function is efficiently learned, multiple users can be mapped onto the chart based on their CSI, and their trajectories can be tracked in real time on the chart as CSI is regularly collected. Channel charts, thus produced, can be used for various applications, such as pilot assignment based on the chart coordinates [4], This work was supported by ANR (grant ANR-23-CHR4-0001-01) under the CHIST-ERA project CHASER (CHIST-ERA-22-WAI-01), and by Horizon Europe SNS project INSTINCT (grant 101139161). It was performed using HPC resources from GENCI-IDRIS (Grant 2023-AD010614930). best beam prediction [5] and beam tracking in millimeter wave (mmWave) systems [6], radio mapping [7], signal to noise ratio (SNR) prediction [8] to name a few. Current state-of-the-art signalling techniques such as Orthogonal Frequency Division Multiplexing (OFDM) are known to degrade in terms of performance and data rate in high mobility scenarios [9], [10]. With the advent of technologies such as autonomous vehicles, drone-assisted communications among others, we expect high mobility scenarios to be more frequently encountered. A recently proposed alternative to OFDM called Orthogonal Time Frequency (OTFS) modulation, specifically caters to high-mobility scenarios [11], [12], [13]. In OTFS modulation, data symbols are modulated on a two-dimensional delayDoppler grid and transmitted over the radio channel. In that context, CSI is represented by the multipath components (MPCs) of the propagation channel. Each CSI sample thus consists of several MPCs each having its own gain, delay and Doppler shift. Our focus is on utilizing this multipath CSI to enhance the fidelity of the obtained channel chart with respect to the geometric ground truth. Note that the proposed approach is applicable beyond the realm of delay-Doppler modulations, since the considered MPC-based CSI can also be obtained in MIMO-OFDM and mmWave scenarios [14], [15]. A. Related work and Contribution User velocity based channel charting is considered in [16], [17], [18] where velocity information is used in learning the charting function. However, velocity is estimated in these works either by using additional sensors at the users or by additional reference receivers. Also, user ground truth locations are used for learning the chart. In this work, we propose to leverage the Doppler shift information of the MPC-based CSI to make inferences about instantaneous user velocity, and to integrate this information as a form of supervision in the channel charting process. Thus, our method does not require any additional sensor/reference receiver or access to the ground truth user locations. More precisely, our contributions are summarised below: •We introduce a method to obtain a lower bound on the user velocity based on MPC-based CSI. •We propose a heuristic approach to integrate the lower bound on the velocity over sequences of CSI samples and supervise a triplet-based channel charting process based Vk BS Vk-1 θkl Pk scatterer scatterer (a) A segment of user trajectory showing MPCs from point Pk. Vk BS Vk-1 θkl Pk Pi Pj Pk+1 (b) Segment of trajectory between points Pi and Pj. (c) A street/building map used in Sionna showing a user trajectory. Fig. 1: A typical user trajectory segment is shown. Points Pi,Pk,Pjdenote the locations on the trajectory from where CSI samples xi,xk,xjwere obtained respectively. on the inferred Euclidean distance in the ground truth geometric model. •The performance of the MPC-enhanced charting approach is evaluated for realistic synthetic user trajectories generated using NVIDIA’s Sionna ray-tracer [19] (an example shown in Figure 1(c)). The improvement in charting performance, in comparison to the classical triplet-based charting, validates the proposed heuristic and demonstrates the effectiveness of our method. II. SYSTEM FRAMEWORK Consider a single antenna mobile user communicating with an M–antenna base station (BS). As the MPC-based uplink CSI is estimated at the BS regularly, we consider a dataset of timestamp-ordered CSI samples, denoted by X= {x1, ..., xN}, with Doppler shifts measurements available for each MPC of every CSI sample. Let Nkdenote the number of MPCs for sample xk, and Fkl represent the Doppler shift of the lth MPC for that sample. Additionally, denote by Pk the true geographical position of the user at the time the CSI sample xkwas acquired. Let θkl denote the angle which the lth MPC makes with direction of the user velocity. This scenario is depicted in Figure 1(a). The Doppler measurement Fkl for the lth path in CSI sample xkis linked to the user velocity at point k, denoted by vk, by Fklλ=vkcos(θkl),(1) where λ:= c fis the carrier wavelength, c= 3×108m/s is the electromagnetic wave propagation speed, and fis the carrier frequency. A. User Speed Estimate using Doppler Taking the absolute value in eq. (1) yields |vk|.|cos(θkl)|= λ|Fkl|. Notice here that θkl is typically unknown at the BS, however since |cos(θkl)| ≤ 1we can obtain a lower bound on the user speed |vk| ≥ λ|Fkl|for each component l; the quantity λ|Fkl|can be computed at the BS directly from the MPC-based CSI. The respective lower bounds corresponding to each path can be combined as |vk| ≥ v∗ k≜max l=1,2,...,Nk λ|Fkl|.(2) to increase the tightness of the bound. For a large number of MPCs (large Nk) in a rich scattering environment, it is likely that at least one of the MPC is approximately collinear with the direction of movement, hence maxlcos(θkl)≈1; in this case, v∗ kcan provide a tight lower bound on the user speed. For demonstrating the effectiveness of this approach, we consider a scenario shown in Figure 2(a) where the user follows a nearly circular trajectory. The simulated velocity along the trajectory is approximately 1m/s, with 10 stops of varying duration during which the velocity is zero. This scenario is simulated in NVIDIA’s SIONNA software [19] which provides the MPC-based CSI for all points on the trajectory. The true user velocity and the estimated velocity are shown in Figure 2(b) for the trajectory in Figure 2(a). It can be observed that the estimate closely follows the true user speed. III. BOUNDS ON THE EUCLIDEAN DISTANCE We now turn our attention to the problem of leveraging the previous results for the purpose of making inference about the Euclidean distance between distinct points in the user trajectory; this is because the objective function of the tripletbased channel charting approach involves pairwise distances [3]. We thus consider a sequence of successive samples indexed from ito j. We have the classical triangle inequality for the Euclidean distance ∥Pi−Pj∥between samples iand j(see Figure 1(b)): ∥Pi−Pj∥ ≤ j−1 X k=i ∥Pk−Pk+1∥,(3) (a) A simple circular trajectory simulated in Sionna. (b) True user speed vs. its lower bound provided by (2). (c) True Euclidean distance vs. the estimated distance. Fig. 2: Comparison of true user velocity with its lower bound and that of true Euclidean distance with the estimated distance for a simple, nearly circular, user trajectory simulated in Sionna. Let us assume that ∥Pk−Pk+1∥=|vk|(tk+1 −tk)since for small time steps, the average speed between samples kand k+ 1 is close to the instantaneous speed at k; this yields ∥Pi−Pj∥ ≤ j−1 X k=i |vk|(tk+1 −tk).(4) Note that the inequalities in eqs. (2) and (4) cannot be composed. However, note that if the trajectory is nearly a straight line (between Piand Pj), the inequality in (4) becomes an equality (approximately) resulting in ∥Pi−Pj∥ ≈ j−1 X k=i |vk|(tk+1 −tk)≥ j−1 X k=i v∗ k(tk+1 −tk).(5) In particular, this is the case when |tj−ti|is small since Newtonian physics dictate that the velocity cannot change arbitrarily fast, and can be considered constant over short duration. Intuitively, eq. (5) indicates that if samples iand jare close, the Euclidean distance between points iand j can be lower bounded with a quantity which can be related to geometric parameters (note that if no Doppler shift can be observed in any of the MPCs, the lower bound is trivially zero). We refer to the rightmost term in eq. (5) as the estimated distance,dij ≜Pj−1 k=iv∗ k(tk+1 −tk). A comparison of the true Euclidean distance with the estimated distance is depicted in Figure 2(c) where it can be observed that the estimate provides a lower bound on the Euclidean distance between points iand jwhen ∆Tij(≜|ti−tj|)is small. Coordinates of each point in this figure denote the true Euclidean distance on y-axis and estimated distance on the x-axis. For the points lying above the thin blue line (which has a unit slope), Euclidean distance is larger than the estimate and vice-versa for the points lying below the blue line. It can be observed that for a relatively large ∆Tij , the trajectory from Fig. 2(a) significantly deviates from a rectilinear path and the lower bound in (5) does not hold. Now, we analyze the relation of the Euclidean distance (∥Pi−Pj∥) with the estimated distance for a more complex trajectory shown in Figure 3(a). Observe that the travel segments on this complex path deviate significantly from a straight line even for small ∆Tij, thereby forcing the estimate to provide a lower bound on the Euclidean distance ∥Pi−Pj∥ only for very small ∆Tij. It can be observed from Figure 3(b) that the estimate provides a lower bound on the Euclidean distance ∥Pi−Pj∥for a very small ∆Tij. The important point to note here is that for a sufficiently small ∆Tij , a lower bound on the Euclidean distance can be obtained for simple as well as complex trajectories, as depicted by the samples in red in Figures 2(c) and 3(b)). This point is encapsulated in the following remark. Remark 1. For a sufficiently small ∆Tij (say ∆Tij ≤Tth), the estimated distance dij provides a lower bound on the true Euclidean distance ∥Pi−Pj∥for almost all user trajectories (see Figure 3(c) as a validation). In the next section, we show how to leverage this observation to include a weak Doppler-based supervision to the triplet-based CC objective function. IV. LOSS FUNCTION In this section, we will try to incorporate our observations from Section III into formulating an objective function to train a deep neural network (NN) which, after training, can serve as a charting function that maps a CSI sample to a point on the chart. As we have obtained a lower bound based on Doppler shifts for a pair of CSI samples, we can combine the triplet based loss function, which works quite well in practice [3], with a new loss function that satisfies our findings in Remark 1. Let fbe a charting function which maps a CSI sample to a point in the lower dimensional chart. In NN-based channel charting, fis learnt through a well-chosen loss function. A. Triplet based Loss The main idea behind this approach introduced in [3] is that the CSI sampling process does not yield independent and identically distributed (i.i.d.) samples. In fact, successive samples can be expected to be similar, hence sampling timestamps can be used to form a contrastive loss function. Specifically, (a) A complex user trajectory simulated in Sionna. (b) True Euclidean distance v/s the estimated distance. (c) Percentage of pairs with |ti−tj| ≤ Tth satisfying ||Pi−Pj|| ≥ dij . Fig. 3: True Euclidean distance v/s the estimated distance for a complex trajectory. Observe that for small Tth, estimated distance provides a lower bound on the true Euclidean distance as stated in Remark 1. two samples that are separated by a small time difference are expected to be closer on the chart than the two samples that are separated by a relatively larger time difference. Based on this, let us consider a triplet of samples, say xi,xj,xmsuch that |ti−tj| ≤ T′ th while |ti−tm|> T ′ th and T′ th is a suitably chosen time difference threshold. Here, xiis called the anchor while xjand xmare called positive and negative samples respectively. The triplet loss function is given by Lt= max(||f(xi)−f(xj)||−||f(xi)−f(xm)||+M, 0),(6) where ||.|| denotes the l2norm and M > 0is a margin hyperparameter. The loss function is penalized by all triplets for which the inequality ||f(xi)−f(xj)|| ≤ ||f(xi)−f(xm)|| is violated by at least M. B. Doppler based Loss We will use the conclusion in Remark 1, which states that if ∆Tij ≤Tth,||Pi−Pj|| ≥ Pj−1 k=i|tk+1 −tk|v∗ k, to control the distance on the chart i.e. ||f(xi)−f(xj)|| ≥ dij (recall that dij =Pj−1 k=i|tk+1 −tk|v∗ k). Thus, the Doppler loss function can be formed as Ld= max(−||f(xi)−f(xj)|| +dij,0).(7) C. Combined Loss Function Finally, we combine the triplet and Doppler loss functions and formulate the final loss function which is used during the training to learn f. The combined loss function is given by, L=w1Lt+w2Ld,(8) where w1>0and w2>0are the weights or preferences given to triplet and Doppler loss functions respectively. V. EXPERIMENTAL SETUP A. Dataset Multipath CSI is obtained for the two scenarios of simple and complex trajectories depicted in Figs. 2(a) and 3(a) respectively. These two scenarios are simulated in Sionna and multipath CSI is obtained every 100 ms. The model described above is trained and tested for two similar (both simple or both complex) trajectories. Loss functions in (6) and (8) are used for optimizing the parameters of the model during the training. B. Experiments The fully connected multi-layer NN architecture described in [20] was optimized using the proposed loss function. We put a lot more weight on Doppler loss function compared to that on the triplet loss by taking w1= 0.05 and w2= 0.95. In addition to that, we perform a pure triplet based experiment by taking w1= 1 and w2= 0 and compare the performance of our Doppler assisted method with this baseline approach. We use the following hyper-parameters throughout our experiments: Tth =T′ th = 2 s and M= 1. The training and testing results for simple and complex trajectories are shown in Figure 4 and Figure 5 respectively for both experiments. VI. RESULTS AND DISCUSSION A comparison of the charting performance of the triplet based learning method and that of the proposed Doppler assisted learning method for a simple and nearly circular trajectory is depicted in Figure 4. It is easy to observe that Doppler assisted method better capture the trajectory shape and consistency. Similarly, performance of these two methods for a complex trajectory is compared in Figure 5. It can be easily noticed from the figures that the Doppler assisted method better captures the twists and bends of the trajectory and produces a chart that nearly follows the true user trajectory. In addition to the advantages demonstrated through the depiction of the charting performance, we compute the performance metrics of continuity, trustworthiness and Rajski’s distance for both methods. These metrics with their significance and applicability are explained in detail in [3]. We compute these metrics for both the methods in case of simple and complex trajectories as listed in Table I. It can be observed from the table that most of the metrics have improved for Doppler assisted learning compared to the classical triplet based learning. Performance comparison demonstrated through metric comparison in Table I and charted trajectories in Figure 4 and Figure 5 clearly indicates the superiority of the proposed (a) Ground truth training trajectory. (b) Chart obtained by triplet learning. (c) Charting by Doppler assisted learning. (d) Ground truth testing trajectory. (e) Chart obtained by triplet learning. (f) Charting by Doppler assisted learning. Fig. 4: Charting performance for baseline triplet and Doppler assisted methods for a simple trajectory. Here, (a) and (d) show the simulated ground truth user trajectories for training and testing respectively, (b) and (e) show the charting performance for the baseline triplet learning, and (c) and (f) show the charting results for Doppler assisted learning approach. Different parts of the ground truth trajectories and their corresponding charts are shown by different colors. TABLE I: Performance metrics table. TW stands for trustworthiness, Cont denotes continuity and RD denotes Rajski’s distance. ↑shows that greater quantities denote better results and vice-versa. Trajectory/Method TW(↑) Cont(↑) RD(↓) Simple/Triplet 95.62 96.32 0.836 Simple/Doppler-assisted 95.87 96.116 0.8358 Complex/Triplet 93.515 92.09 0.886 Complex/Doppler-assisted 95.517 94.36 0.847 Doppler assisted learning approach. It is also important to note that we have used weights w1= 0.05 and w2= 0.95 indicating that the training relies heavily on the Doppler loss function given in (7) in case of Doppler assisted learning (notice that w2= 1 −w1and a search over w1∈[0,1] was used to optimize the performance). VII. CONCLUSION We have proposed a novel Doppler assisted approach for channel charting that uses multipath CSI. Multipath CSI is naturally estimated in delay-Doppler communications such as OTFS modulation. Our results demonstrate the superior charting performance of the proposed method as compared to the classical triplet based learning. 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