Scatterer Identification by Atomic Norm Minimization in Vehicular mm-Wave Propagation Channels

Sparse scatterer identification with atomic norm minimization (ANM) techniques in the delay-Doppler domain is investigated for a vehicle-to-infrastructure millimeter wave propagation channel. First, a two-dimensional ANM is formulated for jointly estimating the time-delays and Doppler frequencies associated with individual multipath components (MPCs) from short-time Fourier transformed measurements. The two-dimensional ANM is formulated as a semi-definite program and promotes sparsity in the delay-Doppler domain. The numerical complexity of the two-dimensional ANM limits the problem size which results in processing limitations on the time-frequency sample matrix size. Subsequently, a decoupled form of ANM is used together with a matrix pencil, allowing a larger sample matrix size. Simulations show that spatial clusters of a point-scatterers with small cluster spread are suitable to model specular reflection which result in significant MPCs and the successful extraction of their delay-Doppler parameters. The decoupled ANM is applied to vehicle-to-infrastructure channel sounder measurements in a sub-urban street in Vienna at 62 GHz. The obtained results show that the decoupled ANM successfully extracts the delay-Doppler parameters in high resolution for the channel’s significant MPCs.

the millimeter-wave spectrum enable high-throughput sensor 22 The associate editor coordinating the review of this manuscript and approving it for publication was Julien Sarrazin . data sharing among vehicles without consuming the limited 23 sub-6 GHz resources for intelligent transport systems. 24 Many vehicular wireless communication channels have 25 short stationarity times and limited stationarity band-26 widths [3], [4]. They are adequately modeled as randomly 27 time-varying due to significant movements of receiver, 28 transmitter, and interacting objects in the propagation 29 environment, commonly known as scatterers at high-level 30 characterization. The signal travels from transmitter (Tx) to 31 receiver (Rx) via multiple propagation paths, where each path 32 is a multipath component (MPC) [5]. Any movements cause 33 ing pursuit, basis pursuit, and LASSO, where an overcom-90 plete basis or dictionary is essential for sparse recovery 91 [19,Ch. 13]. Sparse methods also provide valuable tools in 92 the noisy non-compressed sensing case for signal approx-93 imations [20], denoising [16], even for long-tailed data 94 statistics [21]. Sparse channel estimation for time-varying 95 channels is considered in [14] and [20]. Sparse approxima-96 tions of vehicle-to-vehicle measurements with c-LASSO and 97 oversampled discrete Fourier transform (DFT) basis is pre-98 sented in [22]. The sparsity of wireless channels is often based 99 solely on intuitive analysis and the lack of proper measures is 100 discussed in [23] and [24] where a combination of indicators 101 is employed to estimate channel sparsity. 102 State of the art wireless systems employ a probing sig-103 nal to aid in channel estimation. The probing signal has 104 finite bandwidth and snapshot period which leads to a fun-105 damental uncertainty in the delay-Doppler domain [6]. Prior 106 knowledge about the delay-Doppler characteristics of the 107 propagation channel enables optimal design of such prob-108 ing signal. Imposing a suitable sparsity constraint as prior 109 information on the propagation channel enhances its recov-110 ery with super-resolution capability in the delay-Doppler 111 domain [15], [25]. High-resolution delay-Doppler estimates 112 enable improved cyclic prefix OFDM (CP-OFDM) channel 113 estimates when only a few significant MPCs are present [26]. 114 Furthermore, inferred propagation scenario geometry from 115 delay-Doppler estimates aid in target tracking [27] and 116 antenna beam alignment [28]. 117 As the resolution of individual MPCs in delay becomes 118 feasible due to increased bandwidth, delay estimation uti-119 lizes techniques similar to direction of arrival (DOA) esti-120 mation in array processing [29]. A gridless estimation of the 121 DOA of sources from a sensor array via ANM is possible 122 with an additional sparsity constraint [30]. An extension to 123 non-uniform linear arrays via irregular Vandermonde decom-124 position is shown in [31]. The application of ANM for real 125 array measurements is shown in [32]. DOA estimation with 126 deep learning methods from single delay-Doppler snapshots 127 in automotive radar is investigated in [33]. 128 Delay-Doppler estimation from passive CP-OFDM radar 129 signals is shown in [34] with basis pursuit. The authors 130 of [18] and [35] employ the ANM for simulated data where 131 model mismatch from varying delays is ignored. In contrast, 132 we demonstrate delay-Doppler estimation with ANM on real 133 data and simulation data. We think the employed simulation 134 model based on spatial clusters of point-scatterers is a rele-135 vant scenario for vehicular mm-Wave communication.  ∂t where f c is the center frequency and shows 218 the dependence between both parameters τ s (t) and ν s (t). 219 We assume stationarity within a short time interval T stat , 220 where c s (t) ≈ c s , τ s (t) ≈ τ s , and ν s (t) ≈ ν s . This allows 221 a simple spreading functionS τ,ν (τ, ν) = S s=1 = c s δ(τ − 222 τ s )δ(ν − ν s ) such that (1) simplifies to the noise-free sparse 223 time-varying transfer function It is common to specialize the general model in ( where we treat each time-frequency region t b independent 274 and write only Y for ease of notation. with delay index l , delay resolution τ = 1 B , signal band-291 width B, Doppler index k , number of time snapshots N ν , 292 number of frequency samples N τ , and Doppler resolution 293 ν = 1 N ν T R . The DFT relation in (8) implies, that the 294 channel is two-dimensional periodic, which is most likely 295 not the case. Similar to S τ,ν (τ, ν), the smoothed spreading 296 function S τ,ν [l , k ] describes the dispersion of the signal in 297 the delay-Doppler domain and was found to have an approx-298 imate sparse support in vehicular mm-Wave scenarios due to 299 employed directive antennas [38] and sufficient bandwidth 300 for resolving arrivals in time [20].

301
Finite symbol lengths, limited bandwidth, employed pulse-302 shaping, and point-scatterer model mismatch lead to practi-303 cal limits of a sparse representation of the spreading func-304 tion [14]. Already for point-scatterers, grid-mismatch of the 305 sampled wireless channel manifests itself as DFT leakage 306 since their discrete delay and Doppler naturally are not 307 exactly located on DFT bins.

308
A sparse representation of the approximate sparse spread-309 ing function is nonetheless desirable for simpler descrip-310 tion of the propagation channel. A structured sparse channel 311 model is advantageous in developing simpler channel track-312 ing algorithms [20]. 313 While the channel needs to fulfill the underspread property, 314 the choice of sampling parameters for pilot-based channel 315 sounding method in (6)   The linear signal model for an observed two-dimensional 340 with unobserved two-dimensional complex exponential sig-341 nal X ∈ C N ν ×N τ as in (11) and additive noise Z ∈ C N ν ×N τ .

342
The framework of atomic norm minimization helps in finding The continuous parametersτ andν are restricted to the unam-  2 · vec(X) A ≈ min.

366
where T 2 (u) ∈ C N τ N ν ×N τ N ν is a block-Toeplitz matrix with 367 Toeplitz blocks, further described together with sequence u 368 in Section II-C2, w ∈ R + is a free optimization variable,

376
The underlying atoms of (11), i.e. a ν (ν)a τ (τ ) T from esti-377 matesτ andν, are retrieved only after evaluating a dual 378 polynomial of the dual problem, finding the roots of the 379 dual polynomial, or from Vandermonde decomposition [40] 380 of T 2 (u). For the last two cases, there is no need for model 381 selection [15]. For Vandermonde decomposition, the number 382 of atoms S corresponds to the rank of T 2 (u) [40].

383
After scaling of the normalized estimates according to 384 the system parameters (see Sec. II-B), we derive the 385 non-normalized estimatesτ =ˆτ f andν =ˆν T R , which we 386 don't explicitly state from here on.

388
In the presence of noise Z, atomic norm minimization is 389 accompanied with a denoising term [16] where a trade-off 390 between measurement reconstruction of the observed sample 391 matrix Y and sparsity is set with a regularization parameter 392 µ which trades sparsity for data reconstruction. The ANM 393 problem in (14)   where X − Y F measures quality of reconstruction and the 399 remaining terms promote sparsity of the optimal solution. 400 Although no model order selection is needed, the choice of 401 regularizer µ depends on both the noise model and noise 402 level as shown for the one-dimensional case in [16] and 403 empirically set to for the two-dimensional case [18] for complex Gaussian 406 matrix Z with independent and identically distributed (i.i.d.) 407 entries Z ij ∼ CN (0, σ 2 Z ). In the following, we refer to ANM 408 with soft thresholding in (15) solely as ANM, as our focus is 409 on noisy observations.  T 2 (u)

433
where u k = (u) n ν ,n τ =k is a subsequence of u and each block th off-diagonal and zeros elsewhere. Note that, K . Due to the Hermitian property,

443
The parameterization of T 2 (u) through the sequence u  (13), the atomic norm in matrix formulation is used. 455 Taking N ν sequential snapshots equidistant in time of a signal 456 with one-dimensional complex exponentials = c s a ν (ν s ) and stationary τ s . This 461 is applicable to the sample matrix Y and model matrix X and 462 thus interpreted as a two-dimensional snapshot.

463
For the number of paths S ≤ min (N ν , N τ ), the solution for 464 the D-ANM problem is found with the SDP [41, eq. (28)]  (15). 473 Furthermore, for D-ANM we employ the same empirical 474 value µ e (16) as for ANM.

475
Note the reduced size of the PSD constraint in (22) is 476 . The model 478 order S depends on rank(T 1 (u τ )), rank(T 1 (u ν )), and on over-479 lapping delay or Doppler of paths [41]. We use a matrix 480 pencil method [40, eq. (33)] for Vandermonde decomposition 481 and parameter recovery where the poles of the characteris-482 tic polynomial of T 1 are found as a solution of a general-483 ized eigenvalue problem [42, eq. (13)]. An additional pairing 484 step is necessary where we try all possible combinations of 485 (τ i ,ν s ), i = 1, . . . , S τ , s = 1, . . . , S ν and keep only the pairs 486 with strongest contribution |a ν (ν s ) HX a τ (τ i ) * |, where eachτ i 487 andν s appear only once. This approach is similar to finding 488 the main peaks in the two-dimensional conventional beam-489 former (CBF). Estimates of S τ and S ν can be based on the 490 significant eigenvalues of T 1 (u τ ) and T 1 (u ν ), respectively. The goal is to estimate delay and Doppler of the significant 500 MPCs. As there is no ground truth other than the measured 501 data, we construct a sparse representation in (9) based on 502 two-dimensional complex exponentials X (11) and assess its 503 similarity with the measured propagation channel based on 504 the sample matrix Y (7). Based on Y in the SDP (15), we 505 retrieve a block-Toeplitz matrix with Toeplitz blocks T 2 (u) 506 andX, a denoised version of the sample matrix. We assume 507 T 2 (u) to have rank(T 2 (u)) = R < min (N τ , N ν ) such  To retrieve the still missing complex-valued path ampli-

522
An estimate for the discrete MPC model in (9), limited in 523 time-frequency as in (11), is then The sparse representation of the matrix is now described by 526 the parameter set {ĉ s ,τ s ,ν s | s = 1, . . . , S} only. We define 527 the normalized approximation error (NAE) as  The capability of identifying MPCs depends on employed 558 antennas, received signal strength, noise, and the resolution 559 limit in the delay-Doppler domain, among others. Frequency 560 and time spacing parameters f and T R of the channel 561 sounder define the delay-Doppler grid in this dual domain. 562 Parameter estimation performance from spectral analysis of 563 measurements in terms of resolution depends on the spectral 564 properties of the employed frequency and time windows. 565 The observed channel spectrum is then the convolution of 566 the received signal and a window. For small window length, 567 interpolation of the grid can lead to finer resolution when 568 searching for the maximum peak of the spectrum.

569
This approach does not necessarily lead to better estimates 570 for more than one closely located sources as their spectral 571 peaks combine and become unresolvable. In array processing, 572 the Rayleigh resolution limit describes the ability to resolve 573 two plane waves of same magnitude impinging on a sensor 574 array [29, p. 48]. Subspace methods like MUSIC and ESPRIT 575 overcome the Rayleigh resolution limit [15], but they need 576 multiple snapshots of the same time-frequency region which 577 are not available for the vehicular setting.

578
Increasing the number of samples in time and frequency 579 to increase resolution is desirable. For a fixed bandwidth and 580 given maximum Doppler shift, sufficient subcarrier spacing 581 f is guaranteed by ensuring f ν max . The multicarrier 582 system's employed rectangular transmit and receive filters 583 (see Sec. II-B) implicitly set a frequency window. Therefore, 584 power from one delay tap leaks to neighboring taps only with 585 quadratic decay [14, eq. (22)]. The Rayleigh resolution limit 586 in the delay Domain is τ = 1 N τ f .

587
The length of the time window is lower bounded due 588 to T R ≥ T and upper bounded by the approximation 589 error through stationary delays τ s (t) ≈ τ s . Similarly to the 590 frequency window, a rectangular time window leads to a 591 quadratic power decay for the spectral window in the Doppler 592 domain [48, eq. (21c)]. The Rayleigh resolution limit in the 593 Doppler domain is ν = 1 N ν T R for the rectangular time 594 window. Decreasing the number of snapshots in time N ν 595 at constant T R , the decreasing Doppler resolution suffers 596 additionally from increased spectral leakage due to the rect-597 angular window and thus employing a time window with a 598 trade-off between resolution and leakage is desirable.

599
For successful separation of components and estimation of 600 their parameters ν i and τ i with ANM a sufficient condition 601  The frequency domain channel model is defined in terms 645 of the band-limited, complex valued base-band formulation 646 is the subcarrier frequency. We approximate the time-varying 659 delays for 0 ≤ kT R < T obs as and insert it into (28) and (29)

678
We model the path amplitude for the LOS component as  We obtain time-frequency snapshots of the propagation 712 channelĤ [t b ; k, l] (6) at a snapshot period of T R = 125 µs, 713 limited to T obs = N ν T R , and centered at time t b and frequency 714 f c . For each time-frequency snapshot t b , there is a correspond-715 ing sample matrix Y (7). 716 We assume wide-sense stationary uncorrelated scatterers 717 (WSSUS) approximately for the channel, locally within T obs . 718 An estimate of the non-stationary spectral process of the 719 channel, known as the LSFĈ[t b ; l , k ], is calculated with the 720 local multitaper spectral estimates [52], [53] 721 where D i [k, l] with i = 1, . . . , I are suitable time-frequency 726 windows, i.e. the data tapers. Employing a single data taper 727 reduces estimation bias due to leakage. However, it also 728 reduces the effective sample size where the multitaper method 729 counteracts the increase in estimation variance [52].

730
Generally, the LSF describes the energy shift in the 731 time-frequency domain locally at specific time and frequency. 732 Delay and Doppler power profiles are derived from the LSF 733 for a suitable local description of the energy shift in only 734 time or frequency, respectively. We estimate the delay power 735 profiles as the expectation of the LSF estimate (37) with 736 respect to Doppler [13] and the Doppler power profiles as the expectation with 739 respect to delay We have dropped the dependency on frequency for the LSF 742 and the power profiles, where we assume stationarity in the 743 Long observation times can lead to problems for practical 780 implementations in the vehicular scenario, e.g. when the 781 stationarity time of the channel is exceeded or when memory 782 depth of the receiver system is limited. Therefore, a sample 783 reduction is often necessary. Reducing the number of channel 784 snapshots to N ν = 16, i.e. T obs = 2 ms, reduces the resolution 785 in the Doppler domain by a factor 512/16=32. When selecting 786 only every second subcarrier such that N τ = 77 and f = 787 2 MHz, the occupied bandwidth remains similar, thus also the 788 delay resolution is similar. However, reducing the subcarriers 789 by two reduces τ max by two, which is an issue if excess delays 790 of MPCs exceed τ max .

805
The model (9) has its limitations for joint delay-Doppler 806 estimation with vectorized ANM (15). A potential concern 807 is the independence assumption between delay and Doppler 808 shift in the approximations. In the continuous model (2)

844
The simulation model in [35] is based on (9) and thus 845 neglects these model mismatch issues.  To prevent aliasing in delay, N τ ≥ τ max B = 0.3 × 160 = 870 48. Each cluster is attenuated by −g s = 6 dB relative to the 871 LOS component. 872 We generate multiple LTV realizations X SC of the model 873 in Sec. III-B and add complex Gaussian noise Z ∈ C N ν ×N τ , 874 with i.i.d. entries Z ij ∼ CN (0, σ 2 Z ). The velocity vector and 875 scatterer positions are constant during one realization X SC . 876 We define the array signal-to-noise ratio (SNR) as SNR = 877 10 log 10 X SC 2 F / N τ N ν σ 2 Z . We set SNR = 30 dB and 878 normalize the sample matrix Y = X SC + Z such that Y 2 F = 879 N τ N ν . The solution of the SDP in (15) is a denoised matrix 880 X and a generated block-Toeplitz matrix T 2 (u), where each 881 block is a Toeplitz matrix itself. The regularizer µ defines the 882 trade-off between sparsity of the model structure and recon-883 struction error of the simulation data with added noise and 884 model mismatch. A Vandermonde decomposition of T 2 (u) 885 with the MAPP method [40] recovers delay-Doppler pairs 886 (τ s ,ν s ). The number of pairs is known for the simulation and 887 limited to S = 3.

888
Next we employ D-ANM to estimate delay-Doppler pairs. 889 The sample matrix Y for D-ANM is normalized as in the 890 ANM case. The solution of the SDP in (22) is a denoised 891 matrixX and two Toeplitz matrices T 1 (u τ ) and T 1 (u ν ). 892 For recovering the parameters from T 1 (u τ ) and T 1 (u ν ), the 893 number of estimated parameters S (significant MPCs) cor-894 responds to the number of significant eigenvalues of the 895 Toeplitz matrices. We set S τ = min(3, N λ(τ ) ) and S ν = 896 min(3, N λ(ν) ), where N λ(ν) and N λ(τ ) are the number of signif-897 icant eigenvalues of T 1 (u τ ) and T 1 (u ν ) with magnitude not 898  the ground truth in Fig. 6(a), the estimated parameters for 930 ANM do only partially agree with the true cluster parameters. 931 Therefore, we consider the vectorized ANM to be of little use 932 for the employed model. Furthermore, the high computational 933 complexity of the SDP (15) makes it susceptible to numerical 934 and convergence issues [55]. 935 Therefore, we restrict the remaining discussion of simula-936 tions to D-ANM, the decoupled SDP in (22).  Fig. 6. The intensity of the denoising 941 effect depends on the regularizer µ as shown in Fig. 7(a) and 942 Fig. 7(b) (see also Fig. 6(c)). The empirical value µ = µ e (16) 943 as in Fig. 6(c) is a reasonable choice according to our observa-944 tions with the simulation model. An increase in cluster spread 945 from a s = 2 mm in Fig. 6 to 100 × a s = 0.2 m in Fig. 7(c) 946 still achieves good estimation of the cluster centers. At larger 947 cluster spreads 1000 × a s = 2 m in Fig. 7(d) the algorithm 948 fails to identify one of the clusters because the cluster spread 949 is too large. 950 We compare the estimation accuracy between estimated 951 and simulated model in terms of estimated delay-Doppler 952 pairs (τ k ,ν k ) with the cluster parameters (τ Cl(s) , ν Cl(s) ) in a 953 squared Euclidean distance  The cluster spreads do not influence the RMSE significantly 976 until spreads reach the order of the propagation distance due 977 to delay resolution, i.e. τ c 0 = 1.87 m, where the effect is 978 most dramatic for problem size (c). However, for problem 979 size (c), cluster spreads near a s = 0.4 m result in a very low 980 RMSE.

981
Next, we evaluate the applicability of the sparse representa-982 tionH (24) on the noiseless sampled time-varying frequency 983 transfer function with different previously defined problem 984 sizes (a)-(c). We denoteH with parameters from the geomet-985 ric model asH SC andH with estimated parameters through 986 D-ANM (22) asH DANM . The NAE( · ; X SC ), in (25), of the 987 matrix X SC ∈ C N ν ×N τ depending on the cluster spread is 988 shown in Fig. 9 forH SC andH DANM . 989 NAE(H SC ; X SC ) increases with cluster spread as the lim-990 ited number of parameters become insufficient in approxi-991 mating the MPCs caused by a cluster of point-scatterers as a 992 single MPC with increased delay-Doppler spread. In Fig. 9 993 we see two types of mismatch: On the one hand, for low 994 levels of cluster spread we see an error floor in (c) due 995 to violation of the WSSUS assumption because the Tx is 996 moving. On the other hand, for high cluster spreads starting at 997 10 −1 m, the Fourier basis does not provide a good expansion 998 of the channel. Furthermore, for very large spreads of point-999 scatterers, the clusters are less suitable in modeling specular 1000 reflection and more likely modeling diffuse scattering [50]. 1001 In the case of the sparse model with D-ANM estimated 1002 parameters, NAE(H DANM ; X SC ) start for low cluster spreads 1003 at medium to high NAE (>0.3) and approach NAE(H SC ; X SC ) 1004 with increased cluster spread, see Fig. 9. Although absolute 1005 delay drifts get larger with (c) longer observation times, 1006 the sparse representation from estimated parameters results 1007  Fig. 11 shows the S = 16 largest peaks of the LSF 1040 C[t b ; l , k ] (37) for time-frequency snapshots (+) and the 1041 recovered delay-Doppler pairs (τ s ,ν s ) (o) from T 2 (u) derived 1042 with ANM and µ = 0.5µ e , µ = µ e , and µ = 4µ e . 1043 At recording time t 1 , the largest LSF peak (60 ns, −1.8 kHz) 1044 corresponds to the strong LOS component where neighboring 1045 peaks with similar delay or Doppler are likely due to spectral 1046 leakage despite data tapering. Further LSF peaks with delay 1047 > 100 ns and Doppler > 0 kHz correspond to other scatterers. 1048 The delay-Doppler ANM estimates in Doppler domain for all 1049 shown µ suffer from a leakage effect similar to the leakage 1050 effect from spectral analysis for strong LOS. ANM identifies 1051 only one additional scatterer to the LOS. At recording time 1052 t 2 , less LSF peaks are in the neighborhood corresponding to 1053 the LOS with moderate power. ANM identifies now more 1054 scatterers for µ = 0.5µ e and µ = µ e additional to the LOS, 1055 where delay-Doppler estimates still suffer mildly from the 1056 leakage effect. For high µ = 4µ e , the ANM fails to identify 1057 most of the scatterers. 1058 We now calculate an equivalent LSF with the samples of 1059 the time-frequency region described by the ANM denoised 1060 matrixX, for a description in the delay-Doppler domain. 1061 The background color in Fig. 11 shows the LSFsĈ[t b ; l , k ] 1062 based on the sample matrices Y and denoisedX,respectively. 1063 Increasing µ shows an intensification of the denoising effect 1064 in the LSF, where the strong differences between the LSF 1065 from Y and the LSF from denoisedX at t 2 and at µ = 4µ e 1066 coincides with failing identification of scatterers in the later. 1067 Next, we evaluate the applicability of the sparse representa-1068 tionH (24) on the sample matrix Y defined in (7). We denote 1069 H with estimated parameters through ANM asH ANM . The 1070 model order is unknown for the measurement data and esti-1071 mation is limited to the previously set maximum value of 1072 16. We show the approximation quality with NAE( · ; Y) (25) 1073 for reconstructing Y with the sparse representationH ANM 1074 defined in (24) with recovered parameters {ĉ s ,τ s ,ν s | s = 1075 1, . . . , S}. The trade-off between sparsity and measurement 1076 reconstruction is shown in Fig. 12 where the trace-norm 1077 is Tr (T 2 (u)) /(N ν N τ ). The NAE(X; Y) through the ANM 1078 denoisedX serves as a comparison. NAE(X; Y) decreases 1079 with decreasing µ as the trace-norm has only insignificant 1080 weight for the objective function in (15). NAE(H ANM ; Y) 1081 through the estimateH ANM of the discrete MPCs model 1082 also decreases with decreasing µ , where S = 16 paths 1083 lead to NAE < 0.16 in both cases for µ = µ e . For a 1084 further decrease of µ , NAE(H ANM ; Y) saturates and does 1085 not decrease anymore. Reducing the number of paths to S = 1086 14 paths slightly increases the error for t 2 , but dramatically 1087 increases it for t 2 and thus leads to a bad approximation for 1088 t 1 . Note, NAEs > 0.6 are not shown.  pencil method from the recovered Toeplitz matrices T 1 (u τ ) 1099 and T 1 (u ν ), respectively. Toeplitz matrices T 1 (u ν ) and T 1 (u τ ) with magnitudes not 1114 smaller than 20 dB from the largest eigenvalue each. 1115 We pair the separate parameter estimates (see Sec. II-D) 1116 and get S delay-Doppler pairs (τ i ,ν s ) as shown in Fig. 14.

1127
Next, we evaluate the applicability of the sparse repre-1128 sentationH (24) on the sample matrix Y defined in (7). 1129 We denoteH with estimated parameters through D-ANM 1130 asH DANM . We vary the number of paths up to S and cal-1131 culate NAE(H DANM ; Y), in (25), for reconstructing Y with 1132 a sparse representationH defined in (24). The trade-off 1133 between sparsity and measurement reconstruction is shown in 1134 Fig. 15 where the trace-norm is [Tr (T 1 (u τ )) + Tr (T 1 (u ν ))] / 1135 √ N ν N τ . The NAE(X; Y) through the D-ANM denoisedX 1136 serves as a comparison. NAE(X; Y) decreases with decreas-1137 ing µ as the trace-norm has only insignificant weight for the 1138 objective function as in the ANM case. NAE(H DANM ; Y) also 1139 decreases with decreasing µ. However, there is a saturation 1140 effect which differs strongly between the two time regions. 1141 For 8 paths and µ = µ e , NAE < 0.2 in both cases. 1142 We now analyze a larger part of the car's trajec-1143 tory as shown in Fig. 2 (Fig. 4(a)) to τ [t 2 ]=22.2 ns and 1149 ν[t 2 ]=1.43 kHz (Fig. 4(b)).

1150
Further, the estimated signal-to-noise ratio SNR = 1151 10 log 10 Y 2 F / N τ N νσ 2 Z − 1 decreased, see Fig. 16. 1152 We employ the NAE to compareH DANM from estimated 1153 parameters with the sample matrix Y for different record-1154 ing times t b . NAE depends further on the number of paths 1155 utilized for the sparse representationH DANM . Fig. 16 shows 1156 the NAE(H DANM ; Y) over time at fixed µ µ e ratios, where the 1157 number of paths of the sparse representation is limited to 1158 VOLUME 10, 2022 frequency windows, respectively [53], [56]. The choice of the 1169 windows is a trade-off between bias, variance, and resolution. 1170 We need high spectral resolution of the largest peak posi- . This leads to a 1197 total number of I = 4 orthogonal windows, which is again 1198 lower than the empirical maximum to combat leakage. The 1199 spectral resolution reduces further to R τ = 2 NW f τ = 1200 32.2 ns in delay and R ν = 2 NW t ν = 250 Hz in Doppler 1201 due to the employed windows. Prior to the calculation of 1202 the moments, thresholding is applied to the delay (38) and 1203 Doppler power profiles (39) to exclude values 30 dB lower 1204 than the peak power or values lower than 5 dB above the 1205 estimated noise power of the corresponding delay or Doppler 1206 power profiles.  Elapsed time for solving the semi-definite programs (SDPs) of atomic norm minimization (ANM) for µ = µ e and decoupled ANM (D-ANM) for µ = µ e with utilized solver [60] and different problem size (N ν × N τ ).
The RMS delay spreads in Fig. 17(b) and RMS Doppler 1223 shift spreads in Fig. 17(d) depend heavily on the number of 1224 paths where for the maximum number of paths, there is only a 1225 slight deviation to the LSF of the measurement data. Despite 1226 lower spreads in Fig. 17(b) and Fig. 17(d) for the D-ANM 1227 case compared to the measurement data, their trends follow 1228 the measurement data closely and nearly as good asH peaks . 1229 With only a single path forH DANM andH peaks , the spreads 1230 reduce to a residual value close to the delay resolution τ 1231 and Doppler resolution ν, respectively.

1232
C. SDP SOLVER 1233 We solve the SDPs (15) and (22) with CVXPY [59] utilizing 1234 an alternating direction method of multipliers-based cone 1235 splitting solver [60]. It employs a first-order method which 1236 scales better to larger problems and can solve them with mod-1237 est accuracy more quickly than interior point methods [60]. 1238 However, the complexity for the SDPs grows quickly with 1239 problem size such that also first-order methods become 1240 insolvable even for the decoupled case (22).

1241
Solvers for SDPs with first-order and interior point meth-1242 ods work iteratively. Available complexity analysis for the 1243 later case estimate O(P 3 ) steps for each iteration [41], where 1244 P is the size of the PSD matrix constraint of (15) and (22) and 1245 complexity of first-order methods is usually smaller. At most 1246 O( √ P ln(1/ )) iterations are necessary for recovery precision 1247 , thus O(P 3.5 ln(1/ )) of overall time complexity [41]. P = 1248 N τ N ν + 1 for ANM in (15) and the following Vandermonde 1249 decomposition with MAPP of the Toeplitz block-Toeplitz 1250 matrix [40] has complexity O(P 2 R) for rank R [41]. P = 1251 N τ +N ν for D-ANM in (22) and the two separate matrix pencil 1252 method have complexity O(N 2 τ ) and O(N 2 ν ).

1253
The sample matrix Y as input to the solver is scaled such 1254 that Y 2 F = N τ N ν . For given N τ and N ν , the empirical value 1255 for the regularizer µ e depends only on σ Z , (See (16)), thus it 1256 is scaled correspondingly.

1257
The time for solving the SDP depends not only on the 1258 problem size but also on the data and the regularizer µ for 1259 (15) and µ for (22). Increasing values usually lead to slower 1260 convergence of the solver and longer processing time. For 1261 large values, e.g. larger than right-most column in Fig. 11, the 1262 SDP solver will likely not converge in our experiments. The 1263 mean elapsed processing time for the different algorithms and 1264 problem sizes of the measurement data is shown in Table 1 for 1265 µ = µ e and µ = µ e when run on a single core of an AMD 1266 EPYC 7302 processor.

1267
The processing time for ANM is already nearly an hour 1268 for a small problem size of N ν = 16 and N τ = 77. The processing time decreases dramatically with D-ANM, i.e. from hours to seconds for N ν = 16 and N τ = 77, such that it 1271 is possible to solve larger problems.