Sparse Factorization-based Detection of Off-the-Grid Moving targets using FMCW radars
Gilles Monnoyer de Galland, Thomas Feuillen, Luc Vandendorpe, Laurent Jacques
SSPARSE FACTORIZATION-BASED DETECTION OF OFF-THE-GRIDMOVING TARGETS USING FMCW RADARS
Gilles Monnoyer de Galland (cid:63) , Thomas Feuillen (cid:63) , Luc Vandendorpe (cid:63) , Laurent Jacques † (cid:63) ELEN. † INMA. ICTEAM, UCLouvain, Belgium
ABSTRACT
In this paper, we investigate the application of continuous sparse signal reconstruction algorithms for the estimation of theranges and speeds of multiple moving targets using an FMCW radar. Conventionally, to be reconstructed, continuous sparse signalsare approximated by a discrete representation. This discretization of the signal’s parameter domain leads to mismatches with theactual signal. While increasing the grid density mitigates these errors, it dramatically increases the algorithmic complexity of thereconstruction. To overcome this issue, we propose a fast greedy algorithm for off-the-grid detection of multiple moving targets.This algorithm extends existing continuous greedy algorithms to the framework of factorized sparse representations of the signals.This factorized representation is obtained from simplifications of the radar signal model which, up to a model mismatch, stronglyreduces the dimensionality of the problem. Monte-Carlo simulations of a K -band radar system validate the ability of our methodto produce more accurate estimations with less computation time than the on-the-grid methods and than methods based on non-factorized representations. Index Terms — Radar, sparsity, off-the-grid, continuous matching pursuit, factorization.
1. INTRODUCTION
Sparse signal processing grew in interest for radar applications as it enables using Compressive Sensing (CS) to reduce the amount ofacquired data required to detect the targets [1, 2, 3], or to achieve super-resolution [4, 5]. In this paper, we consider the estimation ofthe ranges and the radial speeds of several targets with a Frequency Modulated Continuous Wave (FMCW) mono-static radar emit-ting linear chirp modulated waveforms. The sampled measurement signal from this system is assumed to be a linear combination ofa few waveforms, each corresponding to the presence of one of the targets. These waveforms (or atoms) are taken from a continuousparametric dictionary, in which the measurement is represented. The detection of targets is thereby formulated as the reconstructionof a continuous sparse signal in the range-Doppler domain which models the observed scene. Traditional approaches for sparse sig-nal reconstruction capitalize on a discretization of the parameter domain and on the assumption that all the target parameters matchthe resulting grid [6]. Yet, reconstructions from such discretized models are affected by grid errors that limit both the precision andthe resolution of the algorithms [7, 8]. Although a denser grid reduces this effect, it tremendously increases the dimensionality of theproblem to solve. This recently motivated many contributions on continuous sparse signal reconstruction algorithms, notably used toperform off-the-grid estimations. For instance, in [9, 10, 11], the authors reformulate the reconstruction problem as a Semi-DefiniteProgram. In [12, 13, 14], an approximated formulation of the sparse reconstruction is solved by a gradient descent. Nonetheless, thehigh computational complexity of these methods makes them not suitable for real-time applications.In [15], a continuous version of the Basis Pursuit [16] is derived from the definition of an interpolated model that approximatesthe off-the-grid atoms. In [17, 18], the authors similarly designed a continuous adaptation of the Orthogonal Matching Pursuit(OMP) [19], namely the Continuous OMP (COMP), using the same interpolation concept. The low complexity of this algorithmmakes it of high interest for real-time radar applications. Still, COMP has only been formulated for the estimation of scalar parame-ters. In this paper, we propose a formulation of COMP for the estimation of both the ranges and speeds of targets from the receivedsignal of an FMCW radar. Conventionally, such radar signals are modelled by applying known simplifications that enable a factor-ized expression of the radar atoms [20, 21]. The resulting atoms factorize in a product of two sub-atoms, respectively dependingon the ranges and on the speeds of the targets. Among on-the-grid algorithms, the 2D factorized OMP (F-OMP) presented in [22]can take advantage of the factorization to reduce the dimensionality of problems. The simplification of the radar atoms, however,causes model mismatch which generates distortions in the reconstruction process [23, 24, 25]. Thereby, the conventional applicationof F-OMP to radar systems is affected by both the grid errors and the model mismatch .Our main contribution is to combine the concepts of interpolation and factorization to build the Factorized COMP (F-COMP)that enables a fast and accurate estimation of target parameters with FMCW radars. Our simulations show the superiority ofusing low-density grids with interpolated models — and hence, off-the-grid algorithms — instead of denser grids with on-the-grid algorithms. This superiority still holds when the reconstruction is affected by the model mismatch appearing from the factorizationof the radar model. From the simulations, we also evaluate the possible degradation of the performance of F-COMP due to themodel mismatch and its sensibility to radar system parameters.
Notations:
Matrices and vectors are denoted by bold symbols, j = √− , and c is the speed of light. The scalar product betweenthe vectors a and b reads (cid:104) a , b (cid:105) . The transpose and conjugate transpose of a matrix A are A (cid:62) and A H , respectively. Themodulo operator is mod , (cid:126) is the convolution operator, (cid:12) is the Hadamard (element-wise) product, (cid:107) · (cid:107) F is the Frobenius norm, [ N ] := { , · · · , N } , and C N (0 , σ ) is the centered complex normal distribution of variance σ . GM and LJ are funded by the Belgian FNRS. a r X i v : . [ ee ss . SP ] F e b . SIGNAL MODEL AND APPROXIMATIONS In this section, we formulate the radar signal as a linear combination of atoms parameterized by continuous parameters. Next, weapply simplifications to the resulting model to build a factorized expression of these atoms. We finally sample the parameter domainto obtain a grid from which we specify interpolation-based approximations for both the exact and factorized expressions of theatoms.
The transmitting antenna of an FMCW radar continuously emits a modulated wave which can be expressed by [26] s T ( t ) = exp (cid:0) j2 π (cid:82) t f c ( t (cid:48) )d t (cid:48) (cid:1) , (1)where the transmission power is arbitrarily set to 1, and f c ( t ) is the instantaneous carrier frequency at instant t . For a linear chirpmodulation, we have f c ( t ) = f + B ( tT c mod 1) , (2)where f is the lowest frequency, B is the bandwidth of the transmitted signal and T c is the chirp duration.We consider K point targets located at ranges { r k } K ⊂ R ⊂ R + and moving with radial speeds { v k } K ⊂ V ⊂ R . The speedare assumed to be constant during an acquisition frame. The target parameters are thus known to lie within the parameter domain P := X × V ⊂ R . The signal at the receiver side is expressed by s R ( t ) = s T ( t ) (cid:126) (cid:2) (cid:80) Kk =1 α k δ (cid:0) t − τ k ( t ) (cid:1)(cid:3) , (3)where α k are scattering coefficients modelling all effects occurring in the wave reflection process, including the radar cross sections.We consider an ideal noiseless and clutterless scenario. τ k ( t ) is a Delay-Doppler term defined by τ k ( t ) = c ( r k + v k t ) . (4)After coherent demodulation, the received baseband signal is e R ( t ) = (cid:80) Kk =1 α k exp (cid:0) − j2 π (cid:0) f c ( t ) τ k ( t ) − B T c ( τ k ( t )) (cid:1)(cid:1) . (5)The above signal is sampled at rate /T s with M s samples acquired per chirp and M c chirps in total. There are M := M s M c acquired samples in total. The resulting sampled measurement vector y ∈ C M c M s , such that y m c M s + m s = e R ( m c T c + m s T s ) ,reads y = (cid:80) Kk =1 α k a ( r k , v k ) , (6)where, for all ( r, v ) ∈ P , the vector a ( r, v ) is an atom of the continuous radar sensing dictionary, defined by D := { a ( r, v ) :( r, v ) ∈ P} . Thus, (6) formulates the complete radar signal as a linear combination of atoms taken in the parametric dictionary D . To reduce the dimensionality of the problem solved in Sec. 3, we aim to decouple the range r and the velocity v in the atom a ( r, v ) .From the sampling of (5), we can decompose a ( r, v ) as a m c M s + m s ( r, v ) = ψ m s ( r, v ) φ m c ( v ) θ m s ,m c ( r, v ) , (7)and ψ m s ( r, v ) = exp (cid:0) − j2 π BM s r + γv ) c m s (cid:1) , (8) φ m c ( v ) = exp (cid:0) − j2 πf T c v c m c (cid:1) , (9) θ m s ,m c ( r, v ) = exp (cid:0) − j π c BM s (cid:0) r c T s + m s (cid:1) v ( m c T c + m s T s ) (cid:1) · exp (cid:0) j π BM s T s v c ( m c T c + m s T s ) (cid:1) , (10)with γ = f M s T s B . Note that the above equations remain identical when time gaps are inserted between chirps ( i.e. , T c > M s T s ).Typically, factorizing the expression in (6) relies on a few assumptions [23, 24, 25]: (i) R is such that for all r ∈ R , r < c T s ,(ii) V is such that for all v ∈ V , | v | ≤ c f T c and (iii), the transmitted signal is narrowband and such that M c Bf < . Theconditions above enable the approximation θ m c M s + m s ( r, v ) (cid:39) . Therefore, given Y ∈ C M s × M c the matrix-shaped y with Y m s ,m c = y m c M s + m s , the factorized model reads Y (cid:39) (cid:80) Kk =1 α k A ( r k , v k ) , (11)Defining the sub-atoms ψ ( r + γv ) := (cid:0) ψ ( r, v ) , · · · , ψ M s ( r, v ) (cid:1) (cid:62) and φ ( v ) := (cid:0) φ ( v ) , · · · , φ M c ( v ) (cid:1) (cid:62) , we have A ( r, v ) := ψ ( r + γv ) φ ( v ) (cid:62) = ψ ( r (cid:48) ) φ ( v ) (cid:62) . (12)In (12), φ ( v ) links V to the rows of A ( r, v ) and ψ ( r + γv ) links R (cid:48) := { r (cid:48) = r + γv : r ∈ R , v ∈ V} , to the columns of A ( r, v ) .Hence (11) approximates the radar signal as a linear combination of atoms A ( r k , v k ) that are factorized by (12) and such that r (cid:48) and v are decoupled. Note that ( r, v ) is retrieved from ( r (cid:48) , v ) by substracting the deterministic offset γv from r (cid:48) . .3. Interpolated sparse representation: Conventional methods to recover parameter values from signals modelled as (6) require the assumption that these parameters aretaken from a grid which results from the sampling of P . The greedy algorithm “Continuous OMP” (COMP) [17] extends OMP andsucceeds to estimate off-the-grid parameters. COMP also operates with a parameter grid, but uses linear combinations of multipleatoms to approximate all atoms a ( r, v ) from the continuous dictionary. In other terms, it interpolates from the grid the atoms of D that are parameterized from off-the-grid parameters. Our algorithm F-COMP applies the same interpolation concept to the atoms A ( r, v ) , which are factorized by (12).More precisely, let us define the separable grid Ω P = { ω n } Nn =1 ⊂ P such that Ω P = Ω R × Ω V , with Ω R := { ¯ r n r } N r n r =1 ⊂ R and Ω V := { ¯ v n v } N v n v =1 ⊂ V , respectively the range and speed grids. We have N := N r N v and the indices n and n r , n v are linkedby ω n v N r + n r = (¯ r n r , ¯ v n v ) (cid:62) .In COMP, each atom a ( r k , v k ) in the model (6) is approximated from one of the grid bins with a generic interpolation model,inspired by [15, 17], that reads a ( r k , v k ) (cid:39) (cid:80) Ii =1 c ( i ) k d ( i ) [ n ( k )] , (13)where d ( i ) [ n ( k )] is the i -th interpolant atom associated to the n ( k ) -th bin from the grid Ω P , and n ( k ) is a grid index which dependson the interpolation scheme and on ( r k , v k ) . The coefficients c ( i ) k are obtained from a mapping function [17] denoted by C n ( r, v ) and defined from the interpolation scheme. More precisely, for all k ∈ [ K ] , ( c (1) k , · · · c ( I ) k ) = C n ( k ) ( r k , v k ) .In F-COMP, we aim to apply the same interpolation concept to the factorized model (12). To that end, we propose a “factoriza-tion over interpolation” strategy where each A ( r k , v k ) is interpolated by A ( r k , v k ) (cid:39) (cid:80) Ii =1 ˜ c ( i ) k D ( i ) [ n r ( k ) , n v ( k )] . (14)In this scheme, for all i ∈ [ I ] , we decompose the global interpolant atoms D ( i ) [ n r ( k ) , n v ( k )] using interpolant sub-atoms denotedby ξ ( i ) [ n r ( k )] and η ( i ) [ n v ( k )] , i.e. , D ( i ) [ n r ( k ) , n v ( k )] = ξ ( i ) [ n r ( k )] (cid:0) η ( i ) [ n v ( k )] (cid:1) (cid:62) . (15)As stated in the previous section, A ( r, v ) approximates the matrix-reshaped a ( r, v ) . Therefore, the right-hand side of (14) approx-imates the matrix-reshaped right-hand side of (13). We restrict our study to the simple case where the interpolation is obtained from a Taylor approximation of the atoms of D . Wepostpone to future work the application to radars of other interpolation schemes, such as polar interpolation [27, 28]. We approximatethe atoms of D by an order-1 Taylor expansion which is described for all n ∈ [ N ] from (13) by d (1) [ n ] = a ( ω n ) , d (2) [ n ] = ˜ R ∂ a ∂r ( ω n ) , d (3) [ n ] = ˜ V ∂ a ∂v ( ω n ) . (16) C n v N r + n r ( r, v ) = (cid:0) , ˜ R − ( r − ¯ r n r ) , ˜ V − ( v − ¯ v n v ) (cid:1) . (17)where ˜ R and ˜ V are constants for dimensionality normalization purpose. Similarly, from the factorized model (14) - (15), for all ( n r , n v ) ∈ [ N r ] × [ N v ] , we set ξ (1) [ n r ] = ξ (3) [ n r ] = ψ (¯ r n r ) , η (1) [ n v ] = η (2) [ n v ] = φ (¯ v n v ) , (18) ξ (2) [ n r ] = ˜ R − ψ (cid:48) (¯ r n r ) , η (3) [ n v ] = ˜ V − φ (cid:48) (¯ r n v ) . In this simple case, the interpolating coefficients in (14) are identical to the coefficients in (13), i.e. , ˜ c ( i ) k = c ( i ) k for all ( k, i ) ∈ [ K ] × [ I ] and are obtained from (17).The models we just defined from the grids Ω R and Ω V enable us, in the next section, to derive algorithms that recover parameters { ( r k , v k ) } Kk =1 lying off-the-grid. This is done either with higher complexity from (13) or faster but less accurately from (14).
3. OFF-THE-GRID ALGORITHMS3.1. Continuous OMP:
Alg. 1 formulates COMP for a generic interpolation scheme. This algorithm adapts the methodology of OMP to greedily minimize (cid:13)(cid:13) y − (cid:80) Kk =1 α k (cid:80) Ii =1 c ( i ) k d ( i ) [ n ( k )] (cid:13)(cid:13) . The greedy iterations provide, for each target, the index ˆ n ( k ) of a corresponding on-the-grid estimator and a set of I complex coefficients, instead of one in OMP. The recovered coefficients for the k -th target are gatheredin ˆ β k := ( ˆ β (1)1 , · · · ˆ β ( I ) k ) , where ˆ β ( i ) k estimates α k c ( i ) k . These coefficients contain information on both α k and the deviation of thetarget parameters from the grids. The correction step in (22) leverages this information from ˆ β k to compute off-the-grid estimators of { ( r k , v k ) } Kk =1 . Moreover, the joint computation of { β k (cid:48) } Kk (cid:48) =1 in each iteration enables to adaptively adjust the effective off-the-grid values of the previous estimators. This enables the distinction of closer targets with respect to OMP.The computational complexity of COMP is dominated by the index selection step. The inner minimization of (19) has a closeform involving the product ( d (1) [ n ] , · · · , d ( i ) ) H [ n ] ∈ C I × M with r ( k ) . Hence the complexity of (19) is O ( IMN ) . Using the lgorithm 1: Continuous OMP (COMP) for Radar
Input : K , y , (cid:8) d ( i ) [ n ] (cid:9) ( i,n ) ∈ [ I ] × [ N ] , Ω P . Output: { ˆ α k } Kk =1 , { (ˆ r k , ˆ v k ) } Kk =1 begin Initialization: r (1) = y , k = 1 ; While k ≤ K : ˆ n k = arg min n ∈ [ N ] (cid:0) min β ∈ C I (cid:13)(cid:13) I (cid:88) i =1 β i d ( i ) [ n ] − r ( k ) (cid:13)(cid:13) (cid:1) (19) (cid:8) ˆ β k (cid:48) (cid:9) kk (cid:48) =1 = arg min { ˆ β k (cid:48) ∈ C I } kk (cid:48) =1 (cid:13)(cid:13)(cid:13) k (cid:88) k (cid:48) =1 I (cid:88) i =1 β ( i ) k (cid:48) d ( i ) (cid:2) ˆ n k (cid:48) (cid:3) − y (cid:13)(cid:13)(cid:13) (20) r ( k +1) = y − k (cid:88) k (cid:48) =1 I (cid:88) i =1 ˆ β ( i ) k (cid:48) d ( i ) (cid:2) ˆ n k (cid:48) (cid:3) (21) k ← k + 1 for all k ∈ [ K ] , (ˆ α k , ˆ r k , ˆ v k ) = arg min α ∈ C , ( r,v ) ∈P (cid:13)(cid:13) α C ˆ n ( k ) ( r, v ) − ˆ β k (cid:13)(cid:13) . (22)Taylor interpolation scheme we detailed in Sec. 2 d), the index n ( k ) is expected to correspond to the closest bin of Ω P to theparameter ( r k , v k ) . In that case, we can approximate the index selection of (19) by its corresponding formulation in OMP, whosecomplexity is O ( MN ) and which is formulated by ˆ n k = arg max n ∈ [ N ] |(cid:104) d (1) [ n ] , r ( k ) (cid:105)| . (23)Advanced analysis of the implication of this simplification, as well as comparisons with the non-simplified implementation, aregoing to be presented in an extension of this contribution. Our algorithm, the Factorized Continuous OMP (F-COMP), leverages the factorized interpolated model (14) to reduce the complex-ity of the estimation of the parameters. It follows the same steps as COMP and greedily minimizes (cid:13)(cid:13) Y − (cid:80) Kk =1 α k (cid:80) Ii =1 c ( i ) k D ( i ) [ n r ,k ] (cid:13)(cid:13) F .We apply the same simplification of the index selection step (23) that we used for COMP. From (15), the indices ˆ n r ( k ) and ˆ n v ( k ) are selected by (ˆ n r ( k ) , ˆ n v ( k )) = arg max ( n r ,n v ) ∈ [ N r ] × [ N v ] (cid:12)(cid:12) ( ξ (1) [ n r ]) H R ( k ) ( η (1) [ n v ]) ∗ (cid:12)(cid:12) , where R ( k ) is the residual of the k -th iteration. This maximization is computed with a complexity O ( N min( M c , M s )) .To leverage the factorization in the computation of { ˆ β k (cid:48) } Kk (cid:48) =1 , we propose the following procedure, inspired by the implemen-tation of the Factorized 2D-OMP (F-OMP) in [22], (ˆ β (cid:62) , · · · , ˆ β (cid:62) k ) = H − f , (24)where H = H ξ (cid:12) H η ∈ C Ik × Ik where H ξ = Ξ H Ξ with Ξ = (cid:16) ξ (1) [ˆ n r , ] , · · · , ξ ( I ) [ˆ n r , ] , · · · ξ ( I ) [ˆ n r ,k ] (cid:17) (25)and f = ( f (1)1 , · · · f ( I )1 , f (1)2 , · · · , f ( I ) k ) ∈ C Ik where f ( i ) k (cid:48) = ( ξ ( i ) [ˆ n r ,k (cid:48) ]) H Y ( η ( i ) [ˆ n v ,k (cid:48) ]) ∗ . (26)The implementations above enable F-COMP to estimate the target parameters faster than COMP. Still, this estimation is expectedto be less accurate than COMP because the factorized model in (11) — from which F-COMP derives — is an approximation of theexact radar signal (6). ig. 1: Comparison of (a) Computation Time (seconds), (b) Miss Rate and (c) AverageError Within Successes of OMP, COMP, F-OMP and F-COMP in function of the numberof bins the location and in the velocity grids ( N ∗ = N r = N v ). The simulated systemhas M ∗ = 16 . Fig. 2:
Comparison of (a) Computation Time, (b) Miss Rate and (c) AHE of F-OMP andF-COMP in function of the number of bins in the location and velocity grids and with M ∗ = 64 . Fig. 3: (a)
Comp. Time of F-COMPComp. Time of F-OMP , (b)
MR of F-COMPMR of F-OMP and (c)
AHE of F-COMPAHE of F-OMP infunction of M s and M c . The final step of both F-COMP and COMP is the correction step formulated by (22). For a given k ∈ [ K ] , we estimate the scatteringcoefficient α k and the off-the-grid deviations that we denote by δ r := ˜ R − ( r k − ¯ r ˆ n r ( k ) ) and δ v := ˜ V − ( v k − ¯ v ˆ n v ( k ) ) .We implementthis step for the order-1 Taylor interpolation scheme we described in Sec. 2. The substitution of (17) in (22) leads to the followingcoupled relations, ˆ α k = ˆ β (1) k +ˆ β (2) k ˆ δ r +ˆ β (3) k ˆ δ v δ +ˆ δ . (27) ˆ δ r = (cid:60){ ˆ β (2) k / ˆ α k } , ˆ δ v = (cid:60){ ˆ β (3) k / ˆ α k } . (28)The coupled equations above are easy to solve by iteratively computing (27) and (28) with the intialization ˆ δ r = ˆ δ v = 0 . Then, foreach k ∈ [ K ] , we directly compute ˆ r k and ˆ v k from (ˆ r k , ˆ v k ) (cid:62) = ω ˆ n ( k ) + (cid:0) ˜ R ˆ δ r , ˜ V ˆ δ v (cid:1) (cid:62) , (29)which ends the algorithms COMP and F-COMP.
4. NUMERICAL RESULTS4.1. Simulated system and performance metrics:
The effectiveness of F-COMP with respect to both COMP and F-OMP is validated with simulated signals from an FMCW radar asdefined in Sec. 2. The simulated system is characterized by B = 200 MHz, f = 24 GHz, T s = 5 µ s and T c = M s T s .We study the evolution of the computation time and estimation performance with respect to the number of grid bins N ∗ := N r = N v . The performance is evaluated with (i) the Miss Rate (MR) and (ii) the Average Hit Error (AHE). They are computedfrom the estimation error which, for each k ∈ [ K ] in a given realisation of K targets , is defined by E k = (cid:113)(cid:0) ˆ r k − r k ρ r (cid:1) + (cid:0) ˆ v k − v k ρ v (cid:1) . (30)In (30), ρ r := c B and ρ v := c f M c T c characterize the resolutions of, respectively, the range and the speed estimations. Moreprecisely, they correspond to the width of the main lobe of the cardinal sine-shaped approximated ambiguity functions. The k -thestimator is a miss if E k > . Each dot of all curves is obtained by averaging the values of the metrics resulting from the applicationof the different algorithms on 10,000 realisations of random sets of K = 5 independent targets. Those realisations are characterizedby α k ∼ C N (0 , and r k (resp. v k ) uniformly chosen in R (resp. V ) for all k ∈ [ K ] . We set R =]0 , M s c B ] (meters) and V =] − c f T c , c f T c ] (meters per second). For a given realisation of K targets, the estimators { ˆ r k , ˆ v k } Kk =1 are sorted and associated to the exact values { r k , v k } Kk =1 without repetitionand such that the miss rate is the smallest. .2. Comparative results: In real radar applications, the “non-factorized algorithms” (OMP and COMP) which exploit the exact signal model are often notpracticable because of their excessively high memory and time requirements. Before comparing the two “factorized algorithms”(F-OMP and F-COMP) with realistic radar parameter’s values, we first compared the results of all four algorithms (F-)(C)OMPwith low numbers of acquired samples, i.e. , M ∗ := M s = M c = 16 . This enables separable observations of the effect of thefactorization and the interpolation.From Fig. 1(a), it is clear that the computation times of non-factorized algorithms grow faster with N ∗ than the factorizedones. The gaps in computation time between the Continuous algorithms (F-COMP and COMP) and their respective on-the-grid counterpart (F-OMP and OMP) correspond the computation time of the off-the-grid corrections. Fig. 1(b) and (c) show that thecontinuous algorithms provide better estimations than on-the-grid algorithms for all values of N ∗ .The performance loss caused by the approximation used by the factorized algorithms only appears with a dense grid (highvalue of N ∗ /M ∗ ). Indeed, Fig. 1(b) and (c) shows that when the grid density is low, F-COMP and COMP almost have identicalperformance because the effect of this approximation is dominated by the grid error. Therefore, non-factorized algorithms are onlyadvantageous when using a dense grid, in which case they are not practicable in term of computation time and memory requirement.Regarding the factorized algorithms, F-COMP is only slower than F-OMP by a constant factor (which depends of the interpolationscheme) while providing significant improvement by enabling off-the-grid target estimation. For this reason F-COMP appears asthe best trade-off between performance and computation time for most values of MR and AHE it can reach.Fig. 2 presents the result from a simulated radar with the more realistic value M ∗ = 64 . Fig. 2 (b) and (c) reveal that theF-COMP curves saturate to greater ( i.e. , worse) values of MR and AHE than the results in Fig. 1. This deterioration is caused by anincrease of the model mismatch because M c Bf is larger, which diminishes the validity of neglecting the distorsion in (11). We analysed the effect of M s and M c on the performance gain between F-COMP and F-OMP. This effect is due to their impact onthe model mismatch. We varied these numbers from 8 to 256 while maintaining N r = 2 M s and N v = 2 M c . Fig. 3 (a), (b) and (c)show the ratios respectively between, the computation times, the MR and the AHE of F-COMP and F-OMP. Increasing M s (resp. M c ) improves the resolution of the range (resp. speed) estimation and leads to a reduction of the miss rate of F-COMP with respectto F-OMP (Fig. 3 (b)). Yet, this improvement is dominated by the increasing model mismatch as M c gets larger. This causes thedeterioration of the MR and AHE ratios in Fig 3 (b) and (c) for large values of M c . To sum up, Fig. 3 reveals that for all values M s and M c , F-COMP always exhibits better performance than F-OMP. Yet, the improvement tends to fade as M c Bf increases.
5. CONCLUSION AND FUTURE WORK
In this contribution, we studied the detection of multiple off-the-grid moving targets using FMCW radars. This motivated usto extend the Continuous OMP to factorized models, and hence to design the F-COMP. We showed that F-COMP gathers theadvantages of both COMP and F-OMP and provides the best trade-off between complexity and accuracy of estimation. In radarapplications, we showed how some values of the radar system parameters affect, but do not remove, the performance gain of F-COMP over F-OMP. This paper proposed a simple implementation of the concepts of F-COMP. Several enhancements can increaseits effectiveness over other greedy algorithms. More precisely, in future work we may investigate the extension of more sophisticatedinterpolation schemes to factorizable dictionaries, and study the compromise between the interpolation order and the required griddensity for optimal correction. Moreover, the presented algorithms can be extended to include mismatch compensation [25] oradaptive grid [29] strategies.
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