Robust Sparse Fourier Transform Based on The Fourier Projection-Slice Theorem
RRobust Sparse Fourier Transform Based on The FourierProjection-Slice Theorem
Shaogang Wang, Vishal M. Patel and Athina Petropulu
Department of Electrical and Computer EngineeringRutgers, the State University of New Jersey, Piscataway, NJ 08854, USA
Abstract —The state-of-the-art automotive radars employ mul-tidimensional discrete Fourier transforms (DFT) in order toestimate various target parameters. The DFT is implementedusing the fast Fourier transform (FFT), at sample and com-putational complexity of O ( N ) and O ( N log N ) , respectively,where N is the number of samples in the signal space. Wehave recently proposed a sparse Fourier transform based onthe Fourier projection-slice theorem (FPS-SFT), which appliesto multidimensional signals that are sparse in the frequencydomain. FPS-SFT achieves sample complexity of O ( K ) andcomputational complexity of O ( K log K ) for a multidimensional, K -sparse signal. While FPS-SFT considers the ideal scenario,i.e., exactly sparse data that contains on-grid frequencies, in thispaper, by extending FPS-SFT into a robust version (RFPS-SFT),we emphasize on addressing noisy signals that contain off-gridfrequencies; such signals arise from radar applications. This isachieved by employing a windowing technique and a voting-based frequency decoding procedure; the former reduces thefrequency leakage of the off-grid frequencies below the noise levelto preserve the sparsity of the signal, while the latter significantlylowers the frequency localization error stemming from the noise.The performance of the proposed method is demonstrated boththeoretically and numerically. Index Terms —Multidimensional signal processing, sparseFourier transform, automotive radar, Fourier projection-slicetheorem.
I. I
NTRODUCTION
With the rapid development of the advanced driver-assistance systems (ADAS) and self-driving cars, the auto-motive radar plays an increasingly important role in providingmultidimensional information on the dynamic environment tothe control unit of the car. Traditional automotive radars mea-sure range and range rate (Doppler) of the targets includingcars, pedestrians and obstacles using frequency modulationcontinuous waveform (FMCW). A digital beamforming (DBF)automotive radar [1] can provide angular information bothin azimuth and elevation [2] of the targets, which is moredesirable in the ADAS and self-driving applications.A typical DBF automotive radar uses uniform linear array(ULA) as the receive array. For such configuration and underthe narrow-band signal assumption, each radar target can berepresented by a D -dimensional ( D -D) complex sinusoid [3],whose frequency in each dimension relates to target param-eters, e.g., range, Doppler and direction of arrival (DOA).The simultaneous multidimensional parameter estimation of aDBF automotive radar requires intensive processing. The con-ventional implementation of such processing relies on a D -Ddiscrete Fourier transform (DFT), which can be implemented by the fast Fourier transform (FFT). The sample complexityof the FFT is O ( N ) , where N = (cid:81) D − i =0 N i is the numberof samples in the D -D data cube with N i the sample lengthfor the i th dimension. For N a power of , the computationalcomplexity of the FFT is O ( N log N ) . Since N is typicallylarge, the processing via FFT is still demanding for real-timeprocessing with low-cost hardware.The recently proposed sparse Fourier transform (SFT) [4]–[6] leverages the sparsity of signals in the frequency domainto reduce the sample and computational complexity of DFT.Different versions of the SFT have been investigated forseveral applications including medical imaging, radar signalprocessing, etc. [7], [8]. In radar signal processing, the numberof radar targets, K , is usually much smaller than N , whichmakes the radar signal sparse in the D -D frequency domain.Hence, it is tempting to replace the FFT with SFT in orderto reduce the complexity of radar signal processing. However,most of the SFT algorithms are designed for 1-dimensional( -D) signals and their extension to multidimensional signalsare usually not straightforward. This is because the SFT algo-rithms are not separable in each dimension since operationssuch as detection within an SFT algorithm must be consideredjointly for all the dimensions [8].Multidimensional SFT algorithms are investigated in [7],[9], [10]; those algorithms share a similar idea, i.e., reductionof a multidimensional DFT into a number of -D DFTs. TheSFT of [9] achieves the sample and computational complexitylower bounds of all known SFT algorithms by reducing a -dimensional ( -D) DFT into -D DFTs along rows andcolumns of a data matrix. However, such algorithm requires avery sparse signal (in the frequency domain) whose frequen-cies are uniformly distributed; such limitation stems from therestriction of applying DFT only along axes of the data matrix,which corresponds to projecting a -D DFT of the data matrixonto the two axes of such matrix. In [7], [10], the multidimen-sional DFT is implemented via the application of -D DFTson samples along a few lines of predefined and deterministicslopes. Although employing lines with various slopes leads tomore degrees of freedom in frequency projection of the DFTdomain, the limited choice of line slopes in [7], [10] is stillan obstacle in addressing less sparse signals. Moreover, thelocalization of frequencies in [7], [10] is not as efficient asthat of [9], which employs the phase information of the -DDFTs and recovers the significant frequencies in a progressivemanner. Thus, the SFT algorithms of [7], [10] suffer from a r X i v : . [ ee ss . SP ] D ec igher complexity as compared to the SFT of [9].We have recently proposed FPS-SFT [11], a multidimen-sional, Fourier projection-slice based SFT, which enjoys lowcomplexity while avoiding the limitations of the aforemen-tioned algorithms, i.e., it can handle less sparse data inthe frequency domain, with frequencies non-uniformly dis-tributed. FPS-SFT uses the low-complexity frequency local-ization framework of [9], and extends the multiple slopes ideaof [7], [10] by using lines of randomly runtime-generatedslopes. The abundance of randomness of line slopes enableslarge degrees of freedom in frequency projection in FPS-SFT.Thus, less sparse, non-uniformly distributed frequencies can beeffectively resolved (see Section III-A for details). Employingrandom lines is not trivial, since the line parameters, includingthe line length and slope set should be carefully designed toenable an orthogonal and uniform frequency projection (seeLemmas and in [11]). FPS-SFT can be viewed as a low-complexity, Fourier projection-slice approach for signals thatare sparse in the frequency domain. In FPS-SFT, the DFT ofa -D slice of the D -D data is the projection of the D -D DFTof the data to such line. While the classical Fourier projection-slice based method reconstructs the frequency domain of thesignal using interpolation based on frequency-domain slices,the FPS-SFT aims to reconstruct the signal directly based onfrequency domain projections; this is achieved by leveragingthe sparsity of the signal in the frequency domain.While the FPS-SFT of [11] considered the case with exactlysparse data containing frequencies on the grid, in this paperwe consider off-grid frequencies. FPS-SFT suffers from thefrequency leakage caused by the off-grid frequencies. Also,we address the noise that is contained in the signal. Thefrequency localization procedure of FPS-SFT is prone to error,since such low-complexity localization procedure is based onthe so-called OFDM-trick [4], which is sensitive to noise.Addressing these issues makes the FPS-SFT more applicableto realistic radar applications where the radar signal containsoff-grid frequencies and noise. We term this new extension ofFPS-SFT algorithm as RFPS-SFT.The off-grid frequencies are also addressed in [8], where weproposed a robust multidimensional SFT algorithm, i.e., RSFT.In RSFT, the computational savings is achieved by folding theinput D -D data cube into a much smaller data cube, on whicha reduced sized D -D FFT is applied. Although the RSFTis more computationally efficient as compared to the FFT-based methods, its sample complexity is the same as the FFT-based algorithms. Essentially, the high sample complexity ofRSFT is due to its two stages of windowing procedures, whichare applied to the entire data cube to suppress the frequencyleakage.Inspired by RSFT, the windowing technique is also appliedin RFPS-SFT to address the frequency leakage problem causedby the off-grid frequencies. Instead of applying the multidi-mensional window on the entire data as in RSFT, the windowin RFPS-SFT, while still designed for the full-sized data, isonly applied on samples along lines, which does not causeoverhead in sample complexity. To address the frequency localization problem in FPS-SFT stemming from noise, RFPS-SFT employs a voting-based frequency localization procedure,which significantly lowers the localization error. The perfor-mance of RFPS-SFT is demonstrated both theoretically andnumerically, and the feasibility of RFPS-SFT in automotiveradar signal processing is shown via simulations. Notation:
We use lower-case (upper-case) bold letters todenote vectors (matrix). [ · ] T denotes the transpose of a vector.The N -modulo operation is denoted by [ · ] N . [ S ] refers to theinteger set of { , ..., S − } . The cardinality of set S is denotedas | S | . The DFT of signal x is denoted by ˆ x . (cid:107) W (cid:107) , (cid:107) W (cid:107) are the l and l norm of matrix W , respectively.II. S IGNAL M ODEL AND P ROBLEM F ORMULATION
We consider the radar configuration that employs an ULAas the receive array. Assume that the ULA has N half-wavelength-spaced elements. The radar transmits FMCWwaveform with a repetition interval (RI) of T p . We also assumethat there exist K targets in the radar coverage. After de-chirping, sampling and analog-to-digital conversion for bothI and Q channels, the received signal within an RI can beexpressed as a superposition of K -D complex sinusoids andnoise [3], i.e., r ( n ) = y ( n ) + n ( n ) = (cid:88) ( a, ω ) ∈ S ae j n T ω + n ( n ) , (1)where n (cid:44) [ n , n ] T ∈ X (cid:44) [ N ] × [ N ] is the samplinggrid and N is the number of samples within an RI. y ( n ) (cid:44) (cid:80) ( a, ω ) ∈ S ae j n T ω is the signal part of the received signal; ( a, ω ) represents a -D sinusoid, whose complex amplitudeis a , and it holds that < a min ≤ | a | ≤ a max ; the -Dfrequency ω (cid:44) [ ω , ω ] T ∈ [0 , π ) represents the normalizedradian frequencies corresponding to targets’ range and DOA,respectively. The set S , with | S | = K contains all the -Dsinusoids. The noise, n ( n ) , is assumed to be i.i.d., circularlysymmetric Gaussian, i.e., CN (0 , σ n ) . The SNR of a sinusoidwith amplitude a is defined as SN R (cid:44) ( | a | /σ n ) .The target’s rang r , Doppler f d and DOA θ relate to ω as ω = 2 π (2 ρr/c + f d ) /f s , ω = π sin θ , where ρ, c, f s arethe chirp rate, the speed of wave propagation and samplingfrequency, respectively; the chirp rate is defined as the ratioof the signal bandwidth and the RI. Thus, the target parametersare embedded in frequencies ω , ω , which can detected in the -D N × N -point DFT of r ( n ) [3], i.e., ˆ r ( m ) (cid:44) N (cid:88) n ∈X w ( n ) r ( n ) e − j π (cid:16) m n N + m n N (cid:17) , = ˆ y ( m ) + ˆ n ( m ) , m (cid:44) [ m , m ] T ∈ X , (2)where w ( n ) is a 2-D window, introduced to suppress frequencyleakage generated by off-grid frequencies; N = N N ;and ˆ y ( m ) , ˆ n ( m ) are the DFTs of the windowed y ( n ) and n ( n ) , respectively. Assuming that the peak to side-lobe ratio(PSR) of the window is large enough, such that the side-lobe (leakage) of each frequency in S can be neglectedn the DFT domain, then ˆ y ( m ) is contributed by a set of -D sinusoids, whose frequencies are on-grid, i.e., S (cid:48) (cid:44) { ( a, ω ) : ω (cid:44) [2 πm /N , πm /N ] T , [ m , m ] T ∈ X } with K < | S (cid:48) | << N . Note that since the windowing degrades thefrequency resolution, each sinusoid in S is related to a clusterof sinusoids in S (cid:48) , which can be estimated from ˆ r ( u, v ) ; next,the estimation of S can be computed from the estimation of S (cid:48) via, for example, the quadratic interpolation method [12].The sample domain signal component associated with thewindow w ( n ) , n ∈ X and the set of sinusoids, S (cid:48) , can beexpressed as x ( n ) (cid:44) (cid:88) ( a, ω ) ∈ S (cid:48) ae j π (cid:16) m n N + m n N (cid:17) , [ n , n ] T ∈ X . (3)The state-of-the-art DBF automotive radars also measure thetarget Doppler f d by processing a -dimensional ( -D) datacube generated by N consecutive RIs [3]. The normalizedradian frequency ω in the Doppler dimension relates to theDoppler as ω = 2 πf d T p . The DBF automotive radars thatalso measure elevation DOA of targets introduce a -th dimen-sion of processing [2]; the DOA measurement in elevation issimilar to that of the azimuth DOA dimension. In those cases,the proposed RFPS-SFT algorithm can be naturally extendedto multidimensional cases, where the reductions of complexityof the signal processing algorithms are more significant.III. T HE RFPS-SFT A
LGORITHM
A. FPS-SFT
The FPS-SFT algorithm proposed in [11] applies to mul-tidimensional data of arbitrary size that is exactly sparse inthe frequency domain. In the -D case, FPS-SFT implementsa -D DFT as a series of -DFTs on samples extractedalong lines, with each line being parameterized by the randomslope parameters α (cid:44) [ α , α ] T ∈ X and delay parameters τ (cid:44) [ τ , τ ] T ∈ X . The signal along such line can beexpressed as s ( α , τ , l ) (cid:44) x ([ α l + τ ] N , [ α l + τ ] N )= (cid:88) ( a, ω ) ∈ S (cid:48) ae j π (cid:18) m α l + τ N N + m α l + τ N N (cid:19) , l ∈ [ L ] . (4)On taking an L -point DFT on (4) w.r.t. l , we get ˆ s ( α , τ , m ) (cid:44) L (cid:88) l ∈ [ L ] s ( α , τ , l ) e − j π lmL = 1 L (cid:88) ( a, ω ) ∈ S (cid:48) ae j π (cid:16) m τ N + m τ N (cid:17) (cid:88) l ∈ [ L ] e j πl (cid:16) m α N + m α N − mL (cid:17) ,m ∈ [ L ] . (5)The line length, L , which is the least common multiple(LCM) of N , N is designed such that the orthogonalitycondition for frequency projection is satisfied (see Lemma of [11] for details), i.e., for m ∈ [ L ] , [ m , m ] T ∈ X , ˆ f ( m ) (cid:44) L (cid:88) l ∈ [ L ] e j πl (cid:16) m α N + m α N − mL (cid:17) ∈ { , } , (6) then if (cid:20) m α N + m α N − mL (cid:21) = 0 , [ m , m ] T ∈ X , (7)the m th entry of (5) can be simplified as ˆ s ( α , τ , m ) = (cid:80) ( a, ω ) ∈ S (cid:48) ae j π (cid:16) m τ N + m τ N (cid:17) . The solutions of (7) with re-spect to m lie on a line with slope − α N / ( α N ) in the N × N -point DFT domain (see the proof of Lemma in[11]), i.e., for m , m satisfying (7), it holds that m = [ m (cid:48) + kα L/N ] N , m = [ m (cid:48) − kα L/N ] N , k ∈ Z , (8)where [ m (cid:48) , m (cid:48) ] T ∈ X is one of the solutions of (7).Hence each entry of the L -point DFT of the slice takenalong a time-domain line with slope α /α represents aprojection of the -D DFT along the line with slope − α N / ( α N ) , which is orthogonal to the time-domain line.This is closely related to the Fourier projection-slice theorem.In fact, FPS-SFT can be viewed as a low-complexity, Fourierprojection-slice based multidimensional DFT. This is achievedby exploring the sparsity nature of the signal in the frequencydomain, which is explained in the following.Assume that the signal is sparse in the frequency domain,i.e., | S (cid:48) | = O ( L ) . Then, if | ˆ s ( α , τ , m ) | (cid:54) = 0 , with highprobability, the m th bin is -sparse, i.e., contains the projectionof the DFT value from only one significant frequency, and itholds that ˆ s ( α , τ , m ) = ae j π (cid:16) m τ N + m τ N (cid:17) , ( a, ω ) ∈ S (cid:48) . Insuch case, the -D sinusoid, ( a, ω ) , can be ‘decoded’ by threelines of the same slope but different offsets. The offsets forthe three lines are designed as τ , τ (cid:44) [[ τ + 1] N , τ ] T , τ (cid:44) [ τ , [ τ + 1] N ] T , respectively; such design allows for thefrequencies to be decoded independently in each dimension.The sinusoid corresponding to the -sparse bin, m , can bedecoded as m = (cid:20) N π φ (cid:18) ˆ s ( α , τ , m )ˆ s ( α , τ , m ) (cid:19)(cid:21) N ,m = (cid:20) N π φ (cid:18) ˆ s ( α , τ , m )ˆ s ( α , τ , m ) (cid:19)(cid:21) N ,a = ˆ s ( α , τ , m ) e − j π ( m τ /N + m τ /N ) . (9)To recover all the sinusoids in S (cid:48) , each iteration of FPS-SFT adopts a random choice of line slope (see Lemma of [11]) and offset. Furthermore, the contribution of therecovered sinusoids in previous iterations is removed to cre-ate a sparser signal. Specifically, assuming that for currentiteration, the line slope and offset parameters are selectedas α , τ , respectively, the recovered sinusoids are projectedinto L frequency bins to construct the DFT along the line, ˆ s r ( α , τ , m ) (cid:44) (cid:80) ( a, ω ) ∈I m ae j π (cid:16) m τ N + m τ N (cid:17) , m ∈ [ L ] ,where I m , m ∈ [ L ] represent the subsets of the recoveredsinusoids that relate to the constructed DFT along line viaprojection, i.e., I m (cid:44) { ( a, ω ) : [ m α N + m α N − mL ] =0 , [ m , m ] T ∈ X } , m ∈ [ L ] . Next, the L -point inverseDFT (IDFT) is applied on ˆ s r ( α , τ , m ) , m ∈ [ L ] , from whichthe line, s r ( α , τ , l ) , l ∈ [ L ] due to the previously recoveredinusoids are constructed. Subsequently, those constructed linesamples are subtracted from the signal samples of the currentiteration. B. RFPS-SFT
FPS-SFT [11] was developed for data that is exactly sparsein the frequency domain. Also, the frequencies are assumed tobe on-grid of the N × N -point DFT. In the radar applicationhowever, the radar signal contains noise. Also, the discretizedfrequencies associated with target parameters are typicallyoff-grid. In the following, we propose RFPS-SFT, whichemploys the windowing technique to reduce the frequencyleakage produced by the off-grid frequencies and a voting-based frequency localization to reduce the frequency decodingerror due to noise.
1) Windowing:
To address the off-grid frequencies, weapply a window w ( n ) , n ∈ X on the signal of (1). The PSRof the window, ρ w , is designed such that the side-lobes ofthe strongest frequency are below the noise level, hence theleakage of the significant frequencies can be neglected and thesparsity of the signal in the frequency domain can be preservedto some extend. The following lemma reflects the relationshipbetween ρ w and the maximum SNR of the signal. Lemma 1. (Window Design):
Consider (2), which is the N × N -point DFT of signal of (1). Let W ∈ R N × N be the matrixgenerated by the window function w ( n ) , n ∈ X . The PSR ofthe window, ρ w , should be designed such that ρ w > (cid:107) W (cid:107) √ π (cid:107) W (cid:107) (cid:112) SN R max , (10) Where
SN R max (cid:44) a max /σ n . Note that unlike the RSFT that applies windows on theentire data cube, in RFPS-SFT, while the window is stilldesigned for the entire data cube, the windowing is appliedonly on the sampled locations. Thus, the windowing does notincrease the sample and computational complexity of RFPS-SFT.
2) Voting-based frequency decoding:
When the signal isapproximately sparse, the frequencies decoded by (9) are notintegers. Since we aim to recover the gridded frequencies,i.e., S (cid:48) of (3), the recovered frequencies are rounded to thenearest integers. When the SNR is low, the frequency decodingcould result into false frequencies; those false frequenciesenter the future iterations and generate more false frequencies.To suppress the false frequencies, motivated by the classical m -out-of- n radar signal detector [13], RFPS-SFT adopts an n d -out-of- n s voting procedure in each iteration. Specifically,within each iteration of RFPS-SFT, n s sub-iterations are ap-plied; each sub-iteration adopts randomly generated line slopeand offset parameters and recovers a subset of frequencies, S i , i ∈ [ n s ] . Within those frequency sets, a given recoveredfrequency could be recovered by n out of n s sub-iterations. Fora true significant frequency, n is typically larger than that of afalse frequency, thus only those frequencies with n ≥ n d areretained as the recovered frequencies of the current iteration. When ( n s , n d ) are properly chosen, the false frequencies canbe reduced significantly.
3) The lower bound of the probability of correct localiza-tion and the number of iterations of RFPS-SFT:
The proba-bility of decoding error relates to the SNR, signal sparsity andchoice of ( n s , n d ) in RFPS-SFT. In the following, we providethe lower bound for the probability of correct localization ofthe significant frequencies for each iteration of RFPS-SFT,from which one can derive the number of iterations of RFPS-SFT in order to recover all the significant frequencies ofsufficient SNR.According to Section II, a -D sinusoid ( a, ω ) ∈ S of (1)is associated with a cluster of -D sinusoids S ⊆ S (cid:48) of (3),whose frequencies are on the grid of the N × N -point DFT.Let’s assume that the sinusoid ( a d , π [ m /N , m /N ] T ) ∈ S with [ m , m ] T ∈ X has the largest absolute amplitudeamong the sinusoids in S . In addition we assume that the SNRof ( a, ω ) is sufficiently high, then the probability of correctlylocalizing [ m , m ] T in each iteration of RFPS-SFT is lowerbounded by P d (cid:44) n s (cid:88) n (cid:48) d = n d (cid:18) n s n (cid:48) d (cid:19) ( P P w ) n (cid:48) d (1 − P P w ) n s − n (cid:48) d , (11)where P (cid:44) (1 − | S (cid:48)(cid:48) | /N ) N/L − with L = LCM( N , N ) isthe probability of a sinusoid in S (cid:48)(cid:48) being projected to a -sparsebin, and S (cid:48)(cid:48) with S (cid:48)(cid:48) ⊆ S (cid:48) represents the remaining sinusoidsto be recovered in the future iterations of RFPS-SFT; P w (cid:44) (1 − P u )(1 − P v ) is the lower bound of the probability of correctlocalization for a -D sinusoid that is projected into a -sparsebin for one sub-iteration of RFPS-SFT; P u , P v are the upperbounds of the probability of localization error for the twofrequency components, m , m , respectively, which is definedas P u (cid:44) (cid:0) σ p (1 − f | a n | ( δ u )) (cid:1) , P v (cid:44) (cid:0) σ p (1 − f | a n | ( δ v )) (cid:1) , where δ u (cid:44) aπ (cid:107) W (cid:107) / (2 N N ) , δ v (cid:44) aπ (cid:107) W (cid:107) / (2 N N ) ,with W ∈ R N × N the window that is applied on the data; σ p with ≤ σ p ≤ π is the parameter determined bythe phases of the error vectors contained in the -sparsebin; f | a n | ( x ) is the cumulative distribution function (CDF)of the Rayleigh distribution, which is defined as f | a n | ( x ) (cid:44) − e − x / (2 σ a (cid:48) n ) , x > , where σ a (cid:48) n (cid:44) σ n (cid:107) W (cid:107) / (2 N L ) .Essentially, (11) represents the complementary cumulativebinomial probability resulted from the n d -out-of- n s votingprocedure, where the success probability of each experi-ment, i.e., localizing ( a d , π [ m /N , m /N ] T ) in each sub-iteration of RFPS-SFT is P P w . When | S (cid:48) | is known, (11)can be applied to estimate the largest number of iterations(the upper bound) of RFPS-SFT in order to recover all thesignificant sinusoids in S (cid:48) since the least number of recoveredsinusoids in each iteration can be estimated by | S (cid:48)(cid:48) | P d .
4) Complexity analysis:
The RFPS-SFT executes T it-erations; within each iteration, n s sub-iterations with ran-domized line parameters are invoked. The samples used ineach sub-iteration is L , since three L -length lines, with L = LCM( N , N ) are extracted to decode the two frequencyomponents in the -D case. Hence, the sample complexity ofRFPS-SFT is O (3 T n s L ) = O ( L ) .The core processing of RFPS-SFT is the L -point DFT,which can be implemented by the FFT with computationalcomplexity of O ( L log L ) . In addition to the FFT, each sub-iteration needs to evaluate O ( | S (cid:48) | ) frequencies. Hence thecomputational complexity of RFPS-SFT is O ( L log L + | S (cid:48) | ) .Assuming that | S (cid:48) | = O ( L ) , the sample and computationalcomplexity can be simplified as O ( | S (cid:48) | ) and O ( | S (cid:48) | log | S (cid:48) | ) ,respectively. Furthermore, since K = O ( | S (cid:48) | ) , the sampleand computational complexity of RFPS-SFT can be furthersimplified as O ( K ) and O ( K log K ) , respectively.
5) Multidimensional extension:
The multidimensional ex-tension of RFPS-SFT is straightforward and similar to that ofFPS-SFT (See Section . of [11] for details).IV. N UMERICAL R ESULTS
Effect of windowing on frequency localization:
For the datathat contains off-grid frequencies, the PSR of the requiredwindow is given in Lemma 1. However, the larger the PSR, thewider the main-lobe of the window, which results into largerfrequency clusters in the DFT domain and thus larger | S (cid:48) | (see(3)), i.e., a less sparse signal. Moreover, the larger the PSR,the smaller the SNR of the windowed signal, which leads tolarger frequency localization error. Hence, for a signal withknown maximum SNR, SN R max , there exists a window withthe optimal PSR in terms of frequency localization successrate, i.e., ratio of number of correctly localized frequenciesto the number of significant frequencies, which is | S (cid:48) | in oneiteration of RFPS-SFT. Fig. 1 shows the numerical evaluationof such optimal windows for signals for various values of SN R max and sparsity level, i.e., K = | S | . According to(10), for signals with SN R max equal to dB and dB ,the PSR of the window should be larger than dB and dB , respectively. The corresponding optimal PSR for theDolph-Chebyshev windows appear to be dB and dB ,respectively. Fig. 1 shows the success rate of the first iterationof RFPS-SFT, which is the lowest success rate of all theiterations.Fig. 2 demonstrates localization of off-grid -D frequenciesof RFPS-SFT using Dolph-Chebyshev window for variousvalues of PSR. A windows with insufficient PSR leads tomiss detections and false alarms (see Fig. 2 (a)), while awindow with sufficiently large PSR yields good performancein frequency localization, with a trade-off of causing largerfrequency cluster sizes (see Fig. 2 (b)). The ground truth inFig. 2 represents (3), which relates to the window. Effect of voting on frequency localization:
The n d -out-of- n s voting in frequency decoding procedure of RFPS-SFT cansignificantly reduce the false alarm rate. A low false alarmrate in each iteration of RFPS-SFT is required since the falsefrequencies would enter the next iteration of RFPS-SFT, whichcreates more false frequencies. For a fixed n s , the larger the n d /n s is, the smaller the false alarm rate is. However, thisinvolves a trade-off between false alarm rate and complexity;specifically, the smaller the false alarm rate, the larger the
40 50 60 70 80 9000.050.10.150.20.25 Lo c a li z a t i on s u cc e ss r a t e Fig. 1. Frequency localization success rate of the first iteration of RFPS-SFTversus window PSR. The Dolph-Chebyshev windows with various PSR isapplied. N = N = 256; ( n s , n d ) = (3 , . The results are averaged basedon iterations of Monte Carlo simulation. DFT gridGround truthEstimated
70 75 80 85 90175180185190195 (a)
DFT gridGround truthEstimated
70 75 80 85 90175180185190195 (b)Fig. 2. -D frequency recovery with different window. K = 10 , σ n =1 , a min = a max , SNR max = 30 dB, ( n s , n d ) = (3 , , T = 30 .Dolph-Chebyshev windows with various PSR are adopted. The ground truthrepresents (3), which relates to the window. A windows with insufficient PSRleads to miss detections and false alarms, while a window with sufficientlylarge PSR yields good performance in frequency localization, albeit resultinginto larger frequency cluster size. (a) ρ w = 45 dB . (b) ρ w = 70 dB . number of the iterations required to recover all the significantfrequencies.Figs. 3 and Fig. 2 (b) show the examples of -D fre-quency recovery using different ( n s , n d ) . In Fig. 3 (a), we set ( n s , n d ) = (1 , , which reduces to the frequency localizationin FPS-SFT, i.e., without voting. In this case, one can see thatmany false frequencies are generated. Figs. 3 (b) and Fig. 2(b) show the frequency localization result with ( n s , n d ) equalto (3 , and (3 , , respectively; while the former generateslarge amount of false frequencies, the latter exhibits idealperformance. DFT gridGround truthEstimated
70 75 80 85 90175180185190195 (a)
DFT gridGround truthEstimated
70 75 80 85 90175180185190195 (b)Fig. 3. Effect of voting on -D frequency recovery. K = 10 , σ n =1 , a min = a max , SNR max = 30 dB . T = 30 for (a)-(c). Dolph-Chebyshev windows with ρ w = 70 dB is applied. The n d -out-of- n s votingprocedure significantly improves frequency localization performance when ( n d , n s ) is properly designed. (a) ( n d , n s ) = (1 , . (b) ( n d , n s ) = (3 , .ABLE IR ADAR P ARAMETERS
Parameter Symbol Value
Center frequency f c GHz
Pulse bandwidth b w MHz
Pulse repetition time T p us Number of range bins N Number of PRI N Number of antenna elements N Maximum range R max m Effect of the SNR and the sparsity level on the number ofiterations of RFPS-SFT:
The number of iterations of RFPS-SFT to recover all the significant frequencies is affected by theSNR and the sparsity level of the signal. A low SNR and lesssparse signal requires large number of iterations. As discussedin Section III-B3, we are able to estimate the largest number ofiterations that recovers S (cid:48) . Figs. 4 (a) shows the predicted andmeasured number of recovered frequencies in each iterationof RFPS-SFT for | S (cid:48) | equal to . Fig. 4 (b) shows thepredicted and measured number of iterations of RFPS-SFT forsignal with various SNR and sparsity level. The figure showsthat the number of iterations upper bounds are consistent withthe measurements.
20 40 60 80 100 120 140
Iterations N u m be r o f r e c o v e r d f r equen c i e s PredictedMeasured (a)
200 300 400 500 600 700 800 900 100020406080100120140160180200220 N u m be r o f i t e r a t i on s (b)Fig. 4. Effect of SNR and sparsity level on number of iterations of RFPS-SFT.(a) | S | (cid:48) = 1000 , SNR = 30 dB, σ p = 1 / . (b) Comparison of predicted andmeasured number of iterations for various SNR and sparsity level, | S (cid:48) | . Radar target reconstruction:
We simulate the target re-construction for a DBF automotive radar via RFPS-SFT andcompare with the RSFT. The main radar parameters are listedin Table I; such radar configuration represents a typical long-range DBF radar [3] except that we set the number of antennaelements to be moderately large to provide a better angularmeasurement performance. Fig. 5 shows the target reconstruc-tion of radar targets via -D FFT, RFPS-SFT and RSFT. Allthe three algorithms are able to reconstruct all the targets.Compared to the FFT and RSFT, RFPS-SFT only requiresapproximately of data samples, which exhibits low samplecomplexity. However, we note that RFPS-SFT requires largerSNR than the FFT and the RSFT based methods. In nearrange radar applications, such as automotive radar, high SNRis relatively easy to obtain.V. C ONCLUSION
In this paper, we have proposed RFPS-SFT, a robust ex-tension of the SFT algorithm based on Fourier projection-slice theorem. We have shown that RFPS-SFT can address -20200 200 D O A ( deg r ee ) Velocity (m/s) Range (m)
FFTR-FPS-SFTRSFT (a)
Range (m)
Velocity (m/s) D O A ( deg r ee ) FFTR-FPS-SFTRSFT (b)Fig. 5. Radar target reconstruction via FFT, FPS-SFT and RSFT. (a)Reconstruction of three targets. (b) Details of the frequency locations thatare reconstructed for one of the three targets. multidimensional data that contains off-grid frequencies andnoise, while enjoys low complexity. Hence the proposedRFPS-SFT is suitable for the low-complexity implementationof multidimensional DFT based signal processing, such as thesignal processing in DBF automotive radar.R
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