Complementary Waveforms for Range-Doppler Sidelobe Suppression Based on a Null Space Approach
aa r X i v : . [ c s . I T ] J a n Complementary Waveforms for Range-DopplerSidelobe Suppression Based on a Null SpaceApproach
Jiahuan Wang, Pingzhi Fan, Des McLernon and Zhiguo Ding
Abstract —While Doppler resilient complementary waveformshave previously been considered to suppress range sidelobeswithin a Doppler interval of interest in radar systems, theircapability of Doppler resilience has not been fully utilized. In thispaper, a new construction of Doppler resilient complementarywaveforms based on a null space is proposed. With this newconstruction, one can flexibly include a specified Doppler intervalof interest or even an overall Doppler interval into a termwhich results in range sidelobes. We can force this term tozero, which can be solved to obtain a null space. From the nullspace, the characteristic vector to control the transmission ofbasic Golay waveforms, and the coefficients of the receiver filterfor Golay complementary waveform can be extracted. Besides,based on the derived null space, two challenging non-convexoptimization problems are formulated and solved for maximizingthe signal-to-noise ratio (SNR). Moreover, the coefficients of thereceiver filter and the characteristic vector can be applied to fullypolarimetric radar systems to achieve nearly perfect Dopplerresilient performance, and hence fully suppress the inter-antennainterferences.
Index Terms —Doppler resilience, coefficients of the receiverfilter, window function, ambiguity function, coordinate descent.
I. I
NTRODUCTION
In pulsed radar systems, pulse compression technology [1][2] has been commonly used to obtain high pulse energy,large bandwidth, and improved range resolution. Through theuse of a matched filter receiver, the returned signal reflectedby a target goes through a filter matched to the reverse andconjugate version of the transmitted pulse. Then the echosignal is compressed into a short pulse which is shown in thematched filter output along with the well-known maximumSNR. However, the matched filter output also has undesiredrange sidelobes if the pulses are not carefully chosen. Therange sidelobes of a strong target may mask main the peakof a weak target near the strong target. Therefore, low rangesidelobes are desirable when dealing with multi-targets.In order to obtain these low range sidelobes, phase codingis usually used in radar for digital pulse compression. For aphase coded waveform, it is phase coded by a unimodular
This work was supported by NSFC Project No.62020106001/No.61731017,National Key R&D Project No.2018YFB1801104, and 111 project No.111-2-14.J. Wang and P. Fan are with the School of Information Science andTechnology, Southwest Jiaotong University, Chengdu, China (e-mail: [email protected], [email protected])D. McLernon is with the School of Electronic and Electrical Engineering,University of Leeds (e-mail: [email protected])Z. Ding is with the School of Electrical and Electronic Engineering, TheUniversity of Manchester (e-mail: [email protected]) code or sequence. The matched filter output of a phase codedwaveform is controlled by an aperiodic auto-correlation func-tion of a code (sequence). For bi-phase codes, the Barker codeis a famous code whose aperiodic auto-correlation functionhas low sidelobes with only one element amplitude value.In addition, polyphase codes proposed by Heimiller [3] andChu [4] also have low sidelobes of aperiodic auto-correlationfunctions. However, it is impossible to achieve zero sidelobesof an aperiodic auto-correlation with one unimodular sequence[5]. This has resulted in the use of Golay complementary pairsin phase coding.In radar, Golay waveforms phase coded by Golay pairs arecoherently transmitted, and the returned signals are also co-herently processed through the matched filter. Then the sum ofthe matched filter outputs has no range sidelobes since Golaypairs have an impulse-like aperiodic autocorrelation function.But the Golay pairs are sensitive to Doppler effects whichresult from the moving targets. In other words, as the inter-pulse Doppler shift changes the phase of the complementarywaveforms, the matched filter outputs’ sum of complementarywaveforms have fairly high range sidelobes. In order to solvethis problem, some methods for constructing Doppler resilientwaveforms have been proposed. These existing constructionmethods fall into two categories.The first category only concerns the transmission of thebasic Golay waveforms. The transmission is determinedby space-time codes. Examples of these codes that playa key role in constructing Doppler resilient Golay wave-forms (pulse train) are first-order Reed-M¨uller codes [6],Prouhet–Thue–Morse (PTM) sequences [5], oversampled PTMsequences [7], generalized PTM sequences [8], and equal sumsof powers (ESP) sequences [9]. The first-order Reed-M¨ullercodes decrease the range sidelobes (less than -60dB) at aspecific Doppler value (e.g., 0.25 rad). PTM sequences canalmost clear range sidelobes (which are approximately equalto -80dB) near zero Doppler (i.e., [ − . , . ] in rad). The over-sampled PTM sequences can also almost clear range sidelobesnot only near zero Doppler but also in all rational Dopplershifts (in rad). The generalized PTM sequence is generallyused for a complementary set, but it is also compatible withthe traditional PTM sequence when only a complementary pairis considered. Therefore, the generalized PTM sequence for acomplementary waveform set can almost clear range sidelobesnear zero Doppler. The ESP sequence has almost the sameDoppler resilient performance compared to a PTM sequence.In fact, ESP and PTM are closely related to the solutions of the Prouhet-Tarry-Escott (PTE) problem. However, ESPsequences need two antennas to transmit the Golay waveformsin some pulse repetition intervals (PRIs) that will result ininter-waveform interferences.The second category focuses on not only the transmissionof the basic Golay waveforms but also the coefficients ofthe receiver filter. The pulse weighing technology is similarto traditional time-domain window design. In the windowingmethod, the coefficients of the matched filters are determinedfrom known time-domain windows such as the rectangularwindow, B-spline windows, the triangular window, Hann andHamming windows, which can achieve range sidelobe sup-pression. However, when Doppler effects are present, thesewell-known traditional windows cannot be directly used tosuppress the sidelobes down to a very low level. Wu et al . [10]jointly considered the coefficients of the receiver and a givenwindow function (e.g., Hamming and rectangular windows)with the higher-order Doppler null and max-SNR constraints,so that the traditional window function can be indirectlyused to suppress the range sidelobes in a given Dopplerinterval. Generally, these known windows are not suitable forDoppler resilience. Dang et al . [11] [12] [13] proposed thebinomial design (BD) that puts binomial coefficients as thecoefficients of the receiver filter, and alternatively transmitsGolay waveforms at the transmitter. The BD method hasa relatively large Doppler resilient interval, in which rangesidelobes are suppressed down to almost zero.So in this paper, the construction of Doppler resilientcomplementary waveforms based on a null space approachis proposed. The main contributions of this paper are listed asfollows: • For the defined ambiguity function [13], it is provedtheoretically that there exists a totally Doppler resilienttype ambiguity function, as well as the delay resilient typeambiguity function, under certain conditions by solvinga linear system and finding the null space. • A Doppler resilient transmission waveform based onGolay pairs which can suppress the range sidelobes ina specified Doppler interval of interest, or even overallDoppler interval, is designed. Here, the design problemis formulated as a linear system with one key term whichresults in range sidelobes. • By forcing the above key term in the formulated linearsystem to zero, the linear system can be solved to obtain anull space. From the null space, the characteristic vectorto control the transmission of basic Golay waveforms,and the coefficients of the receiver filter, in the intervalsof interest or the entire Doppler interval, are extracted. • Based on the derived null space, a complex optimizationproblem, which is non-convex and non-concave in nature,is formulated for maximizing the signal-to-noise ratio(SNR). A heuristic coordinated descent (HCD) algorithmis proposed to obtain a sub-optimal solution of theformulated optimization problem, where the null space istaken as a means to transform the problem into a simplerone, i.e, finding the coefficients of a linear combinationof null space basis vectors. • The above characteristic vector in the transmission wave- form, and the coefficients of the receiver filter, whenapplied to fully polarimetric radar systems, can achievenearly perfect Doppler resilient performance and fullysuppress the inter-antenna interference. • The delay resilient problems are also investigated andsolved by using the frequency-domain phase coding,also based on the null space algorithm. In fact, thenull space algorithm is used to obtain the frequency-domain characteristic vector, and also the coefficients ofthe frequency-domain filter at the receiver.
A. Notation
The superscripts ( · ) T , ( · ) ∗ and ( · ) H denote transpose, com-plex conjugate, and conjugate transpose, respectively. In addi-tion, ‘ ◦ ’ denotes the Hadamard product and Null ( E ) denotesnull space of matrix E .II. S IGNAL M ODEL
A. Time-Domain Signal Model
A pair of biphase sequences x and y is called Golay pair orcomplementary pair if C x [ k ] + C y [ k ] = L δ k , k = − L + , · · · , , · · · , L − C x [ k ] is the auto-correlation function of x at lag k , δ k isthe Kronecker delta function, and x = [ x [ ] , x [ ] , · · · , x [ L − ]] T , y = [ y [ ] , y [ ] , · · · , y [ L − ]] T .In the signal model, the basic Golay complementary wave-forms s x ( t ) and s y ( t ) are phase coded by the Golay comple-mentary pair ( x , y ) [7], [11], [12], i.e., s x ( t ) = L − ∑ l = x [ l ] u ( t − lT c ) , s y ( t ) = L − ∑ l = y [ l ] u ( t − lT c ) , (2)where u ( t ) is a unit-energy baseband pulse shape, and T c isthe chip length.Let the vector p = [ p , p , · · · , p N − ] T be the characteristicvector to control the transmission of basic Golay waveforms s x ( t ) and s y ( t ) , where N is the number of pulses and p n = −
1. If p n = s x ( t ) is transmitted, otherwise s y ( t ) is trans-mitted. Thus the characteristic vector p and the basic Golaywaveforms, s x ( t ) and s y ( t ) constitute the Golay transmissionwaveform or complementary waveform, Z p ( t ) , i.e., Z P ( t ) = N − ∑ n = [( + p n ) s x ( t − nT ) + ( − p n ) s y ( t − nT )] , (3)where T denotes the pulse repetition interval (PRI).Let the input to the matched filter be Z P ( t ) e j ν t , where ν = π f d , and f d is the Doppler shift in Hz. Also, Z P ( t ) e j ν t passesthrough the linear filter with impulse response Z ∗ W ( − t ) , where Z W ( t ) = N − ∑ n = w ∗ n [( + p n ) s x ( t − nT )+ ( − p n ) s y ( t − nT )] , (4) w n ∈ C is the coefficient of receiver filter and w =[ w , w , · · · , w N − ] T . Then the output, i.e., the cross-ambiguity function is givenby χ P , W ( τ , ν ) = ˆ + ∞ − ∞ Z P ( t ) Z ∗ W ( t − τ ) e j ν t dt . (5)The radar parameters (such as chip length T c and PRI T ) arechosen to ensure that LT c is much less than T and ν T is almostequal to zero. After carefully choosing the radar parameters,the center lobe of χ P , W ( τ , ν ) depends on the discrete crossambiguity function [10] [13] A P , W ( k , θ ) = [ C x [ k ] + C y [ k ]] N − ∑ n = w n e jn θ + [ C x [ k ] − C y [ k ]] N − ∑ n = p n w n e jn θ , (6)where θ = ν T = π f d T is the Doppler shift in radians.The first part of (6) only determines the shape of A P , W ( , θ ) since the Golay complementary pair ( x , y ) makes the firstformula of (6) vanish at nonzero k . However, the second partof (6) determines the range sidelobes around the Doppler shift θ , in which the choices of p n and w n are important. In [13][10], two performance metrics (i.e., Doppler resilience andSNR) are chosen to judge the Doppler resilient complementarywaveform specified by { p , w } . B. Frequency-Domain Pulse Amplitude Modulation (PAM)Signal Model
The frequency-domain PAM waveforms for x and y aregiven by [14] [15]ˆ x ( ω ) = L − ∑ l = x [ l ] Ω ( ω − lW c ) , ˆ y ( ω ) = L − ∑ l = y [ l ] Ω ( ω − lW c ) , (7)where W c denotes the subcarrier spacing and Ω ( ω ) denotesthe subcarrier complex amplitude.Then the transmitted frequency PAM pulse train is ex-pressed asˆ z P ( ω ) = N − ∑ n = p n ˆ x ( ω − nW ) + ( − p n ) ˆ y ( ω − nW ) , (8)and the impluse response of the filter in the frequency domainis ˆ z Q ( − ω ) , whereˆ z Q ( ω ) = N − ∑ n = w ∗ n [ p n ˆ x ( ω − nW ) + ( − p n ) ˆ y ( ω − nW )] , (9)where W is the frequency-domain PRI, where W ≫ W c .Then the cross ambiguity function of ˆ z P ( ω ) and ˆ z Q ( ω ) isgiven by χ fP , W ( ν , τ ) = ˆ + ∞ − ∞ ˆ z P ( ω ) ˆ z ∗ Q ( ω − ν ) e − j τω d ω . (10)After applying the inverse continuous-time Fourier transform,the transmitted frequency domain PAM waveform ˆ z P ( ω ) is anOFDM waveform in the time-domain: Z P ( t ) = π N − ∑ n = L − ∑ l = ( p n x [ l ] + ( − p n ) y [ l ]) · e j ( nW + lW c ) t ˆ Ω ( t ) , (11)and similarly the inverse Fourier transform of ˆ z Q ( ω ) is givenby Z Q ( t ) = π N − ∑ n = L − ∑ l = w n ( p n x [ l ]+ ( − p n ) y [ l ]) e j ( nW + lW c ) t ˆ Ω ( t ) , (12)where ˆ Ω ( t ) = F − { πΩ ( ω ) } and F − denotes the inverseFourier transform.The discrete ambiguity function based on χ fP , W ( ν , τ ) is givenby B P , W ( i , α ) = [ C ˆ x [ i ] + C ˆ y [ i ]] N − ∑ n = w n e jn α + [ C ˆ x [ i ] − C ˆ y [ i ]] N − ∑ n = ( − ) p n w n e jn α , (13)where α = τ W is the time shift.III. D OPPLER / DELAY RESILIENCE BASED WINDOWING
A. Doppler ResilienceDefinition 1:
The ambiguity function A P , W ( k , θ ) is aDoppler resilient type if the following conditions hold: | A P , W ( , θ ) | > , θ ∈ Θ , (14)and | A P , W ( k , θ ) / A P , W ( , θ ) | ≤ η , k = , θ ∈ Θ , (15)where Θ = [ , D I ] is the specified Doppler interval of interest, D I is a positive real number, and η is a very small positivereal number.We will show that the range sidelobes may almost vanisharound the specified Doppler interval of interest Θ = [ , D I ] ,and maintain | A ( , θ ) | 6 =
0. The best case is that η in (15) istotally equal to zero such that the discrete ambiguity functionsatisfies the following equation A P , W ( k , θ ) = A P , W ( , θ ) δ k , θ ∈ Θ . (16) Definition 2:
The ambiguity function is a totally Dopplerresilient type for θ ∈ Θ if A P , W ( k , θ ) = A P , W ( , θ ) δ k , θ ∈ Θ , (17)where | A P , W ( , θ ) | > E is proposed as follows: E = e j θ e j θ e j θ · · · e j ( N − ) θ e j θ e j θ e j θ · · · e j ( N − ) θ ... ... ... ... ... e j θ M − e j θ M − e j θ M − · · · e j ( N − ) θ M − . (18) where Θ ∆ = { θ , θ , · · · , θ M − } and Θ ∆ ⊂ Θ . Proposition 1: If ( x , y ) is a Golay pair, then the discreteambiguity function defined in (6) can be rewritten as A P , W ( k , θ ) = ( C x ( k ) − C y ( k )) N − ∑ n = p n w n e jn θ , k = , L N − ∑ n = w n e jn θ , k = . (19) Proof.
When k =
0, since ( x , y ) is a Golay pair, then C x [ k ] + C y [ k ] =
0, so that A P , W ( k , θ ) = ( C x ( k ) − C y ( k )) N − ∑ n = p n w n e jn θ . (20)When k = C x ( ) = C y ( ) = L , and then it holds that A P , W ( k , θ ) = L N − ∑ n = w n e jn θ . (21) Remark 1: ( C x ( k ) − C y ( k )) ∑ N − n = p n w n e jn θ denotes therange sidelobes, which we should clear. Lemma 1:
Let z n = p n w n , n = , , , · · · , N −
1. Then | A P , W ( k , θ ) | is bounded by | f z ( θ ) | , i.e., | A P , W ( k , θ ) | ≤ L | f z ( θ ) | , k = , (22)where the key term f z ( θ ) is given by f z ( θ ) = N − ∑ n = z n e jn θ . (23) Proof.
Since | C x ( k ) | ≤ L and | C y ( k ) | ≤ L , then12 | C x ( k ) − C y ( k ) | ≤ L , for all k ∈ [ − ( L − ) , L − ] . (24)When k =
0, we have | A P , W ( k , θ ) | = | [ C x ( k ) − C y ( k )] N − ∑ n = z n e jn θ |≤ | [ C x ( k ) − C y ( k )] || N − ∑ n = z n e jn θ | = L | N − ∑ n = z n e jn θ | = L | f z ( θ ) | . (25)From lemma 1, we konw that range sidelobes can becontrolled by f z ( θ ) . Therefore, clearing the range sidelobesmeans f z ( θ ) = Lemma 2: f z ( θ ) = θ ∈ Θ ∆ if and only if z = [ z , z , z , · · · , z N − ] T ∈ Null ( E ) , where z n = p n w n . Proof. f z ( θ ) = θ ∈ Θ ∆ iff N − ∑ n = z n e jn θ = , for all θ ∈ Θ . (26) i.e., ∑ N − n = z n e jn θ ∑ N − n = z n e jn θ ... ∑ N − n = z n e jn θ M − = , (27) = ⇒ Ez = . (28)In other words, z ∈ Null ( E ) . Theorem 1: If p and w satisfy w = , w / ∈ Null ( E ) and p ◦ w ∈ Null ( E ) , the ambiguity function A P , W ( k , θ ) is a totallyDoppler resilient type ambiguity function, i.e., A P , W ( k , θ ) = , k = , θ ∈ Θ ∆ . (29) Proof.
This is easily obtained from Lemma 1 and Lemma2. Based on Theorem 1, the range sidelobes in Θ ∆ will vanish.However, the Doppler interval of interest Θ = [ , D I ] or theoverall Doppler interval [ , π ] is truely focused, within whichthe range sidelobes are suppressed. In other words, we hopethe range sidelobes of the ambiguity function A P , W ( k , θ ) canbe no more than -90dB in the Doppler interval of interest [ , D I ] . Proposition 2:
In the Doppler interval of interest [ , D I ] , thediscrete Doppler shifts can be chosen as θ m = mD I / ( M − ) , m = , , · · · , M −
1. If f z ( θ m ) = θ m = , , · · · , M −
1, thenthe range sidelobes of A P , W ( k , θ ) can be suppressed for all θ ∈ [ , D I ] , i.e., A P , W ( k , θ ) → Ez = , we find the nullspace of E . Then p and w can be found based on Null ( E ) .Supposed that ˆ z ∈ Null ( E ) , then p and w are solved as follows: p n = ( + , if Re { ˆ z } ≥ , − , if Re { ˆ z } < , (30) w n = ( + ˆ z n , if Re { ˆ z } ≥ , − ˆ z n , if Re { ˆ z } < . (31)It is easy to verify that w = , w / ∈ Null ( E ) . Then we designan algorithm for finding p and w in algorithm 1. Algorithm 1:
Null space (NS) algorithm for obtaining p and w : Input N , D I , and θ m = mD I / ( M − ) , m = , , · · · , M −
1. Here, M = N − Generate matrix E shown in (18). Compute the null space of E , i.e., Null ( E ) . Select a solution from Null ( E ) , called ˆ z . Obtain p and w as (30) and (31) Remark 2:
In the practical operations of Algorithm 1, we usethe MATLAB instruction “null(E)” to obtain the basis vectors (which can span the null space). Then we select a solutionfrom Null ( E ) . Usually, we can select the basis vector as oursolution. If SNR is considered, we should carefully choosethe vector by some algorithms which is also introduced inthis paper. Remark 3:
The number M of the discrete Doppler shifts θ m ( m = , , · · · , M − ) is limited by the number of pulses N .This fact results from the solutions of the linear equations, i.e,if M < N , Ez = must have nontrivial solutions. Therefore, M can be chosen as M = N −
1. Of course, the smaller theDoppler interval and the more the number of Doppler shifts,the better suppression of range sidelobes of A P , W ( k , θ ) . B. Delay ResilienceDefinition 3:
The ambiguity function B P , W ( i , α ) is a delayresilient type if the following formulas hold: | B P , W ( , α ) | > , α ∈ Γ , (32)and | B P , W ( i , α ) / B P , W ( i , ) | ≤ η , i = , α ∈ Γ , (33)where Γ = [ , T I ] is the delay interval of interest, T I is a positivereal number, and η is a very small positive real number. Definition 4:
The ambiguity function B P , W ( i , α ) is a totallydelay resilient type for θ ∈ Θ if B P , W ( i , α ) = B P , W ( i , ) δ i , (34)where | B P , W ( i , ) | > T is proposed as follows: T = e j α e j α e j α · · · e j ( N − ) α e j α e j α e j α · · · e j ( N − ) α ... ... ... ... ... e j α M − e j α M − e j α M − · · · e j ( N − ) α M − . (35)where Γ ∆ = { α , α , · · · , α M − } , and Γ ∆ ⊂ Γ = [ , T I ] . Lemma 3:
The first term of B P , W ( i , α ) in (13) is a deltafunction for any given θ ∈ Θ , i.e.,12 [ C x [ i ] + C y [ i ]] N − ∑ n = w n e jn α = L ( N − ∑ n = w n e jn α ) δ i , (36)if and only if w = and w / ∈ Null ( T ) . Lemma 4:
The second term of B P , W ( i , α ) in (13) is zero forall given α ∈ Γ ∆ , i.e.,12 [ C x ( i ) − C y ( i )] N − ∑ n = p n w n e jn α = , (37)if and only if p ◦ w ∈ Null ( T ) . Proof.
Since C x ( i ) − C y ( i ) =
0, when k = N − ∑ n = p n w n e jn α = , for all α ∈ Γ ∆ , (38) i.e., Tz = , (39)where z = [ z , z , · · · , z N − ] T , z n = p n w n , n = , , · · · , N − θ m ∈ Θ , m = , , · · · , M −
1. In other words, p ◦ w ∈ Null ( E ) . Theorem 2: If p and w satisfy w = , w / ∈ Null ( T ) and p ◦ w ∈ Null ( T ) , then the ambiguity function A P , W ( k , θ ) is atotally delay resilient type ambiguity function, i.e., B P , W ( i , α ) = , k = , θ ∈ Γ ∆ . (40) C. Signal-to-Noise Ratio (SNR)
The SNR [11]–[13] is described asSNR = L σ b N k w k k w k , (41)where σ b is the power of the target and N is the power spec-tral density (PSD) of the white noise [13]. We can maximizethe SNR by maximizing k w k / k w k under some constraints.For the constraints, the Doppler resilience constraint shouldbe still satisfied, i.e., Ez = , (42)where z = p ◦ w , p n ∈ { , − } . Then the optimization isproposed as follows:max w , p k w k k w k s . t . Ez = = p ◦ w p n ∈ { , − } . (43)This optimization problem (43) is a challenging optimizationproblem because it contains binary variables p n and complex-valued variables z n . Besides, the objective function in (43) isnot a concave function and the constraint set is not a convexset either.Because of the difficulty of the proposed optimizationproblem, we have to transform the complex style into amuch simpler form. We will analyze the constraint set first.Suppose z , z , · · · , z U ∈ Null ( E ) , and λ = [ λ , λ , · · · , λ U ] T isan arbitrary vector with real number elements, then Z λ ∈ Null ( E ) . (44)i.e., E ( Z λ ) = . (45)where Z = [ z z · · · z U ] . Moreover, for the second and thirdconstraint, it is easy to verify that k Z λ k = k p ◦ w k = k w k , (46)where k · k means either k · k or k · k .Therefore, the procedure for solving the optimization prob-lem (43) can be transformed into two steps: • Step 1 . Solve the followingmax λ k Z λ k k Z λ k . (47) • Step 2 . Implement (30) and (31).
D. First Algorithm for SNR
The optimization problem (47) is still a difficult problem,since its objective function is not concave. In order to solve it,two algorithms are proposed. The first algorithm is a simplerone, where the constraint set related to the first algorithm islimited to a much smaller set than that to the second algorithm.The second algorithm is slightly more difficult than the firstalgorithm, but the second one can find a better solution. Sothe first algorithm will be introduced first. Before introducingthe first algorithm, a theorem will be proposed.
Theorem 3:
Let λ u be nonnegative real number with ∑ Uu = λ u =
1, thenmax λ u k U ∑ u = λ u z u k = max u {k z u k } , u = , , · · · , U . (48) Proof.
According to the triangle inequality of norm, i.e., k U ∑ u = λ u z u k ≤ U ∑ u = λ u k z u k , (49)then max λ u k U ∑ u = λ u z u k ≤ max λ u U ∑ u = λ u k z u k = max u {k z u k } . (50)Since k ∑ Uu = λ u z u k = k z u k , when λ u = u = , , · · · , U ,then k z v k ≤ max λ u k U ∑ u = λ u z u k , (51) = ⇒ max u {k z u k } ≤ max λ u k U ∑ u = λ u z u k . (52)Based on (50) and (52), we can getmax λ u k U ∑ u = λ u z u k = max u {k z u k } , u = , , · · · , U . (53)As regards the first method, λ u should be limited by λ u ≥ ∑ Uu = λ u =
1. Based on this fact, Theorem 3 impliesthat λ has only one nonzero element, i.e.,1, and the otherelements are zeros, which means that the elements of λ should be limited to ∑ Uu = λ u = λ u ∈ { , } . Withoutloss of generality, assume z , z , · · · , z U are normalized, i.e., k z k , k z k , · · · , k z U k are equal to 1, then we get k Z λ k = . (54)Therefore, the optimization problem (47) is equivalent tomax λ k Z λ k (55) Without loss of generality, we assume k z k ≥ k z k ≥· · · , k z U k . Again, based on Theorem 3, the optimal solutionto (55) is λ u = ( , if u = , otherwise . (56)Let ˆ z = z , then p and w can be obtained from (30) and(31). Algorithm 2:
Basis selection (BS) method in nullspace: z , z , · · · , z U ∈ Null ( E ) are the basis vectors. Compute k z k , k z k , · · · , k z U k . if k z u k is the largest one, choose z u . E. Second Algorithm for SNR
Although the first proposed algorithm has a low computa-tional complexity, it has a very limited performance becausethe elements of λ are restricted to the set { , } . In order toimprove the performance, here we proposed a second method –termed a Heuristic Coordinated Descent (HCD) method wherethe binary constraint on λ is removed and λ ∈ C U . HCD canbe viewed as a relatively effective method to deal with the non-convex optimization problem with a much larger constraint set.Generally speaking, the Coordinate Descent (CD) algo-rithms [16] are iterative methods. The most common CDalgorithm is by fixing other elements of the variable vector andobtaining the new iteration point by minimizing (maximizing)the objective function with respect to a single element ofvariable vector. In other words, when an optimization problemwas considered, i.e., min x ∈ C N f ( x ) , (57)then the CD algorithm starts with some initial vector x ( ) =( x ( ) , x ( ) , · · · , x ( ) N − ) and repeats the following iteration x ( k ) = argmin x f (cid:16) x , x ( k − ) , x ( k − ) , · · · , x ( k − ) N − (cid:17) , x ( k ) = argmin x f (cid:16) x ( k ) , x , x ( k − ) , · · · , x ( k − ) N − (cid:17) , x ( k ) = argmin x f (cid:16) x ( k ) , x ( k ) , x , · · · , x ( k − ) N − (cid:17) , ... x ( k ) N − = argmin x N − f (cid:16) x ( k ) , x ( k ) , x ( k ) , · · · , x N − (cid:17) , (58)where k = , , , · · · .Since the objective value of (47) is always no less than 0,maximizing it is equivalent to minimizing the reciprocal, i.e.,min λ k Z λ k k Z λ k . (59)In the optimization problem (59), the objective function isstill a non-convex function, so that it is difficult for us to obtain the global optimal solution. Based on these difficulties,a heuristic Coordinated Descent (HCD) is proposed.The algorithm is based on the CD algorithm. It also startswith some initial vector λ ( ) = ( λ ( ) , λ ( ) , · · · , λ ( ) n ) and thenrepeats the procedure as (60). However, the difference betweenthe HCD and the general CD is that the HCD will revert to the ( k − ) -th state if the objective value of k -th state is higher thanthe one from the previous iteration. For details, the followingformula shows the k -th itertation of u -th element: λ ( k ) u = argmin λ u f (cid:16) λ ( k ) , λ ( k ) , · · · , λ ( k ) u − , λ u , λ ( k ) u + , · · · , λ ( k ) U (cid:17) , (60)and then λ ( k ) u will revert to λ ( k − ) u if f (cid:16) λ ( k ) , λ ( k ) , · · · , λ ( k ) u − , λ ( k ) u , λ ( k ) u + , · · · , λ ( k ) U (cid:17) ≥ f (cid:16) λ ( k ) , λ ( k ) , · · · , λ ( k − ) u − , λ ( k ) u , λ ( k ) u + , · · · , λ ( k ) U (cid:17) . (61)The second difference is that HCD generates many initial vec-tors. For every initial vector, we repeat the iteration procedureand obtain a solution when it satisfies the stop criteria. Forall these initial vectors, we have many solutions from whichwe can choose the best solution that has the smallest objectivevalue. The algorithm is summarized in algorithm 3. Algorithm 3:
Heuristic Coordinated Descent (HCD)in null space:for i = I randomized initial vectors λ ( ) for k = K for u = U Compute λ ( k ) u as (60)if (61) satisfies, then λ k − u = λ k − u .if k λ ( k ) − λ ( k − ) k ≤ ε ; ˆ λ i = λ ( k ) ; endend u end k end i choose ˆ λ i as the best λ such that (59) minimized.IV. W INDOWING FOR F ULLY P OLARIMETRIC R ADAR S YSTEMS
The fully polarimetric radar systems can transmit and re-ceive on two orthogonal polarizations at the same time. Theuse of two orthogonal polarizations increases the degrees offreedom and can result in significant improvement in detectionperformance.Two pulse trains Z VP ( t ) and Z HP ( t ) , transmitted simulta-neously from vertical polarization and horizontal polarization, are written by Z VP ( t ) = N − ∑ n = [( + p n ) s x ( t − nT ) − ( − p n ) ˜ s y ( t − nT )] , (62)and Z HP ( t ) = N − ∑ n = [( − p n ) ˜ s x ( t − nT ) + ( + p n ) s y ( t − nT )] , (63)where ˜ · means reversal, i.e.,˜ s x ( t ) = L − ∑ l = x [ L − − l ] u ( t − lT c ) , (64)˜ s y ( t ) = L − ∑ l = y [ L − − l ] u ( t − lT c ) . (65)In the proposed transmission mode, two orthogonal polar-izations have different waveforms in a PRI. For example,if s x ( t ) (or − ˜ s y ( t ) ) is transmitted from vertical polarization,then s y ( t ) (or ˜ s x ( t ) ) is transmitted from horizontal polarization.Besides, this mode also contains the famous Alamouti time-space coding when two different waveforms are transmitted inthe adjoint two PRIs . For example, in the n -th PRI, horizontalpolarization and horizontal polarization transmit s x ( t ) and s y ( t ) respectively; in the ( n + ) -th PRI, vertical polarization andhorizontal polarization transmit s y ( t ) and ˜ s x ( t ) respectively,which constitudes the famous Alamouti matrix " s x ( t ) − ˜ s y ( t ) s y ( t ) ˜ s x ( t ) , (66)which can eliminate polarization interference when the targetis static. For a Doppler shift resulting in polarization interfer-ence, a moving target is considered with a Doppler shift ω inHz.For the vertical polarization antenna, the returned signal isgiven by R V ( t ) = ( h VV Z VP ( t ) + h VH Z HP ( t )) e j ω t . (67)Also, similarly, for the horizental polarization antenna, thereturned signal is given by R H ( t ) = ( h HV Z V P ( t ) + h HH Z HP ( t )) e j ω t , (68)where h VH denotes the scattering coefficient into the verticalpolarization channel from a horizontally polarized incidentfield [5]. Note that h VV , h VH , h HV , h HH constitute a scatteringmatrix H = " h VV h VH h HV h HH . (69)At the receiver with two polarization antennas, each an-tenna has two responses of matched filters, i.e., Z ∗ VW ( − t ) and Z ∗ HW ( − t ) , where Z VW ( t ) = N − ∑ n = w n [( + p n ) s x ( t − nT ) − ( − p n ) ˜ s y ( t − nT )] , (70) and Z HW ( t ) = N − ∑ n = w n [( − p n ) ˜ s x ( t − nT )+ ( + p n ) s y ( t − nT )] . (71)The returned signals go through the matched filters then theoutputs of the matched filters are given by U ( τ ) = " RZ ( τ ) RZ ( τ ) RZ ( τ ) RZ ( τ ) (72)where RZ ( τ ) = ˆ + ∞ − ∞ R V ( t ) Z ∗ HW ( t − τ ) dt , (73) RZ ( τ ) = ˆ + ∞ − ∞ R V ( t ) Z ∗ VW ( t − τ ) dt , (74) RZ ( τ ) = ˆ + ∞ − ∞ R H ( t ) Z ∗ HW ( t − τ ) dt , (75) RZ ( τ ) = ˆ + ∞ − ∞ R H ( t ) Z ∗ VW ( t − τ ) dt . (76)After substituting R V ( t ) of (67) into (73) and (74), andbringing R H ( t ) of (68) into (75) and (76), then we get thefollowing proposition. Proposition 3:
The output matrix U ( τ ) in (72) can betransformed into U ( τ ) = " h VV h VH h HV h HH χ VP , VW ( τ , ω ) χ VP , HW ( τ , ω ) χ HP , VW ( τ , ω ) χ HP , HW ( τ , ω ) . (77)In (77), the cross ambiguity function χ a , b ( τ , ω ) is definedas χ a , b ( τ , ω ) = ˆ + ∞ − ∞ Z a ( t ) Z ∗ b ( t − τ ) e j ω t dt , (78)where a = V P or HP and b = VW or HW .Based on the transform from (5) to (6), the discrete crossambiguity function χ VP , VW ( k , θ ) is given by A VP , VW ( k , θ ) = [ C x ( k ) + C y ( k )] N − ∑ n = w n e jn θ + [ C x ( k ) − C y ( k )] N − ∑ n = p n w n e jn θ . (79)Similarily, the discrete cross ambiguity function χ VP , VW ( k , θ ) is given by A HP , HW ( k , θ ) = [ C x ( k ) + C y ( k )] N − ∑ n = w n e jn θ − [ C x ( k ) − C y ( k )] N − ∑ n = p n w n e jn θ . (80)Also, the discrete cross ambiguity function χ VP , HW ( k , θ ) is given by A VP , HW ( k , θ ) = [ C xy ( k ) − C ˜ y ˜ x ( k )] N − ∑ n = w n e jn θ + [ C xy ( k ) + C ˜ y ˜ x ( k )] N − ∑ n = p n w n e jn θ (81) = C xy ( k ) N − ∑ n = p n w n e jn θ , (82)and the discrete cross ambiguity function χ HP , VW ( k , θ ) is givenby A HP , VW ( k , θ ) = [ − C ˜ x ˜ y ( k ) + C yx ( k )] N − ∑ n = w n e jn θ + [ C ˜ x ˜ y ( k ) + C yx ( k )] N − ∑ n = p n w n e jn θ (83) = C yx ( k ) N − ∑ n = p n w n e jn θ . (84) Proposition 4:
After discretization, the output matrix U ( τ ) in (77) can be transformed into U ( τ ) = " h VV h VH h HV h HH A V P , VW ( k , θ ) A VP , HW ( k , θ ) A HP , VW ( k , θ ) A HP , HW ( k , θ ) . (85)From Proposition 4, to obtain the scattering coefficients, twoconditions must be satisfied: • the range sidelobes of A V P , VW ( k , θ ) and A HP , HW ( k , θ ) should be reduced to zero. • A V P , HW ( k , θ ) and A HP , VW ( k , θ ) should be equal to zero.The range sidelobes of A V P , VW ( k , θ ) and A HP , HW ( k , θ ) arisefrom the second terms of (79) and (80) , i.e.,12 [ C x ( k ) − C y ( k )] N − ∑ n = p n w n e jn θ . (86)Besides, A VP , HW ( k , θ ) and A HP , VW ( k , θ ) depend on C xy ( k ) N − ∑ n = p n w n e jn θ or C yx ( k ) N − ∑ n = p n w n e jn θ . (87)In summary, A VP , VW ( k , θ ) , A HP , HW ( k , θ ) , A VP , HW ( k , θ ) and A HP , VW ( k , θ ) are determined by N − ∑ n = p n w n e jn θ . (88)Therefore, to obtain the scattering coefficients, the followingequation must hold: N − ∑ n = p n w n e jn θ = , for all θ ∈ Θ . (89) Theorem 4:
The range sidelobes of A VP , VW ( k , θ ) are vanishedand the value of A V P , HW ( k , θ ) is zero if and only if N − ∑ n = p n w n e jn θ = , for all θ ∈ Θ . (90)Since (90) is the same as (26), then to solve (26) or (90) toobtain p and w , we can use Algorithm 1. V. N
UMERICAL R ESULTS AND D ISCUSSIONS
In this section, numerical examples are given to verify theresults in Sections III and IV. Also, the proposed null space(NS) Doppler resilient scheme and the binomial design (BD)scheme are verified and discussed.
Remark 4:
In some figures, ”Amb fcn” is the abbreviationof the ambiguity function.
A. Doppler Resilience in an Interested Doppler Interval for aSingle Antenna System
At first, to show the performance which flexibly suppressesthe range sidelobes in the Doppler interval of interest basedon the proposed null space algorithm, the specified Dopplerinterval of interest is given by θ ∈ [ , ] . In each continuousDoppler interval, the sampling resolution is D I / ( M − ) , where D I = M = N −
1. Besides, the number of pulses is N =
48. Based on the null space algorithm (algorithm 1),matrix E is generated as (18), thus the null space of E canbe calculated, and p , w are also easily obtained based onAlgorithm 1. Besides, the Golay pair is length-64 and is givenby x = [ , , , − , , , − , , , , , − , , , − , , , , , − , − , − , , − , , , , − , − , − , , − , , , , − , , , − , , − , − , − , , − , − , , − , − , − , − , , , , − , , , , , − , − , − , , − ] , (91) y = [ , − , , , , − , − , − , , − , , , , − , − , − , , − , , , − , , , , , − , , , − , , , , , − , , , , − , − , − , − , , − , − , − , , , , − , , − , − , , − , − , − , , − , , , − , , , ] . (92)Fig. 1 shows the value of p n along the PRI n and the mod-ulus of coefficients along the PRI n . Then the complementarytransmission waveform Z P ( t ) in (3) is determined by p n , and Z W ( t ) in (4) is determined by p n and w n .Fig. 2 shows two ambiguity functions (6) under θ ∈ [ , ] with N =
48 based on algorithm 1 and [7], respectively. InFig. 2(a) which is based on Algorithm 1, the sidelobes withinthe Doppler interval of interest are obviously lower than thesidelobes outside the Doppler interval of interest. Moreover,in Fig. 2(a), although [ , ] is considerd, the range sidelobesare still very low within [ , . ] . In Fig. 2(b), based on theoversampled-PTM sequence in [7], the ambiguity function hashiger range sidelobes than those shown in Fig. 2(a). B. Doppler Resilience in the Overall Doppler Interval for aSingle Antenna System
We will now discuss the Doppler resilience in the overallDoppler interval [ , π ) .Fig. 3 shows an ambiguity function (6) in the overallDoppler area [ , π ] with N =
48 based on Algorithm 1. For thewhole Doppler area, the range sidelobes are lower than -90dBwhich is an ultra low level.Fig. 4 is an ambiguity function (6) in the overall Dopplerarea [ , π ] with N =
48 based on Binomial design, which is -1-0.8-0.6-0.4-0.200.20.40.60.810 5 10 15 20 25 30 35 40 45 50
PRI n (a) The value of p n PRI n xy (b) The modulus of the coefficients of the receiver filter Fig. 1: The value of p n along the PRI n and the modulus of w n along the PRI n . a baseline of Fig. 3. In Fig. 4, the range sidelobes graduallyincrease when the Doppler shift is larger than about 2.2 rad.From Fig. 3 and Fig. 4, it is obvious that the Dopplerresilience based on the proposed method is significantly betterthan that of the BD method in the overall Doppler interval [ , π ] .Fig. 5 and Fig. 6 show the Doppler profile and peak rangesidelobe level (PRSL), respectively. From Fig. 5, it is observedthat the null space (NS) method keeps the mainlobes at a highlevel but the BD method gradually loses the mainlobe whenthe Doppler shift increases. From Fig. 6, BD also has lowsidelobes as well as NS when the Doppler shift is not verybig, but when the Doppler increases to a high value, the BDhas a high sidelobe level. Besides, from Fig. 6, the NS has anoverall low sidelobe level compared with BD. Moreover, thetwo figures indicate that the NS method has better Dopplerresilience in the overall Doppler interval [ , π ] .Fig. 7 shows that the value of k w k / k w k increases whenthe pulse number N increases. It is obvious that the proposedtwo methods have a significantly higher SNR than BD. Be-sides, for the two proposed methods, the heuristic coordinateddescent (HCD) slightly outperforms basis selection (BS). C. Doppler Resilience for Fully Polarimetric Radar Systems
Based on the analysis in section IV, the transmission schemeis proposed in (70) and (71). For the vertical polarization an-tenna (V-antenna), the candidate Golay waveform is s x ( t ) and (a) AF for the Doppler interval of interest based on Algorithm 1.(b) AF for the Doppler interval of interest based on [7] and [17]. Fig. 2: Comparison between (a) AF for the Doppler interval of interest based on algorithm1, and (b)AF for the Doppler interval of interest based on [7] and [17].Fig. 3: AF for overall Doppler interval [ , π ) based on Algorithm 1. Fig. 4: AF for all Doppler interval [ , π ) based on the binomial design. Doppler shift [rad] -90-80-70-60-50-40-30-20-100
NSBD
Fig. 5: Doppler profile. − ˜ s y ( t ) . For the horizontal polarization antenna (H-antenna),the candidate Golay waveforms are s y ( t ) and ˜ s x ( t ) . If s x ( t ) is transmitted at the V-antenna, then at the same time, s y ( t ) should be transmitted. Similarly, if − ˜ s y ( t ) is transmitted at theV-antenna, then at the same time, ˜ s x ( t ) should be transmitted.It is noted that here the pulse number N is still 48, and theGolay pair is shown in (91) and (92).To verify the flexible Doppler resilience in fully polari-metric radar systems, the Doppler area is chosen as before,i.e, θ ∈ [ , ] . p and w are generated via the null spacemethod. After plotting the ambiguity functions A VP , VW ( k , θ ) Doppler shift [rad] -90-80-70-60-50-40-30-20-100 P R S L [ d B ] NSBD
Fig. 6: Peak range sidelobe level.1
40 50 60 70 80 90 100
Pulse numbers N
HCDBSBD
Fig. 7: k w k / k w k versus the number of pulses N .Fig. 8: AF for the intersted Doppler interval about polarimetry interference based on NS. and A HP , HW ( k , θ ) , it is shown that they have the same shapeas Fig. 2. Besides, the ambiguity functions A V P , HW ( k , θ ) and A HP , VW ( k , θ ) also have the same shape (shown in Fig. 8).From Fig. 8, it is observed the values of A V P , HW ( k , θ ) or A HP , VW ( k , θ ) in dB are of a very low level (no more than-90dB) in the interested Doppler area, i.e., θ ∈ [ , ] . Insummary, the range sidelobes and polarimetry interferencesare flexibly controlled in the interested Doppler area. Fig. 9: AF for the overall Doppler interval about polarimetry interference based on NS.
Now, the Doppler resilience in the overall Doppler area [ , π ] is also investigated. After plotting the ambiguity func-tions A V P , VW ( k , θ ) and A HP , HW ( k , θ ) , it is shown that they Fig. 10: AF for the overall Doppler interval about polarimetry interference based on BD. have the same shape as Fig. 3. Besides, the ambiguity func-tions A VP , HW ( k , θ ) and A HP , VW ( k , θ ) also have the same shape(shown in Fig. 9). From Fig. 9, it is observed the valuesof A VP , HW ( k , θ ) or A HP , VW ( k , θ ) in dB are no more than -90dB, which means that polarimetry interferences are clearlyvanished in overall Doppler area [ , π ] .Thirdly, Fig. 10 is a cross ambiguity function generated byBD. It also has excellent low function values, but the valuesincrease when the Doppler shift is bigger than 1 rad, so that ithas worse polarimetry interferences as the Doppler increases. D. Discussion
Based on the above numerical examples, in which theDoppler resilience is good no matter what the Doppler intervalof interest or the overall Doppler interval [ , π ) , one may askwhy do we consider the Doppler interval of interest. In fact,a tradeoff exists between the number of pulses N and theDoppler interval.If the number of pulses N is too small or the Dopplerinterval is too big, then E is a tall matrix which may have fullcolumn rank for which the linear system is inconsistent , sothat Algorithm 1 cannot find any solution of the linear system,i.e., Null ( E ) = ∅ . Therefore, It makes sense to consider theDoppler interval of interest.Moreover. if the Doppler interval of interest [ , D I ] is given,then the range sidelobes can be suppressed better if thenumber M of the discrete Doppler shifts θ m ∈ [ , D I ] increases.However, the number M of the discrete Doppler shifts islimited by the number of pulses N . This constraint is to ensurethe existence of a solution for the addressed linear equations.According to the theory of solutons of linear equations, Ez = has infinite nontrivial solutions if M < N . Therefore, M canbe chosen as M = N − ONCLUSIONS
We have proposed a method based on a null space approachto obtain the Doppler resilient transmission scheme withbasic Golay waveforms, which can ensure that the discreteambiguity function is free of range sidelobes in the Doppler It is easy to verify that E is full rank if M > N and θ m θ m ( mod2 π ) ,where m , m ∈ { , , ··· , M − } and m = m . interval of interest or even in overall Doppler interval [ , π ) .Besides, the null space method can be also applied to OFDMsignals and obtain the delay resilient OFDM waveform. More-over, max-SNR is also considered and optimized by a basisselection method and heuristic coordinate descent methodswhich are based on the null space. Finally, we have extendedthe proposed methods to fully polarimetric radar so that rangesidelobes and inter-antenna interferences vanish in the overallDoppler interval. R EFERENCES[1] E. C. Farnett and G. H. Stevens, “Pulse compression radar,”
Radarhandbook , vol. 2, pp. 10–1, 1990.[2] B. R. Mahafza,
Radar systems analysis and design using MATLAB .CRC press, 2002.[3] R. Heimiller, “Phase shift pulse codes with good periodic correlationproperties,”
IRE Trans. Inf. Theory , vol. 7, no. 4, pp. 254–257, Oct.1961.[4] D. Chu, “Polyphase codes with good periodic correlation properties(corresp.),”
IEEE Trans. Inf. Theory , vol. 18, no. 4, pp. 531–532, Jul.1972.[5] A. Pezeshki, A. R. Calderbank, W. Moran, and S. D. Howard, “Dopplerresilient Golay complementary waveforms,”
IEEE Trans. Inf. Theory ,vol. 54, no. 9, pp. 4254–4266, Sep. 2008.[6] S. Suvorova, S. Howard, B. Moran, R. Calderbank, and A. Pezeshki,“Doppler resilience, reed-m¨uller codes and complementary waveforms,”in
Conf. Rec. Forty-first Asilomar Conf. Signals, Syst., Comput. , Nov.2007, pp. 1839–1843.[7] Y. Chi, A. Pezeshki, R. Calderbank, and S. Howard, “Range sidelobesuppression in a desired Doppler interval,” in
Proc. 2009 Int. WaveformDiversity Des. Conf., Kissimmee, FL, USA , Feb. 2009, pp. 8–13.[8] J. Tang, N. Zhang, Z. Ma, and B. Tang, “Construction of Dopplerresilient complete complementary code in mimo radar,”
IEEE Trans.Signal Process. , vol. 62, no. 18, pp. 4704–4712, Sep. 2014.[9] H. D. Nguyen and G. E. Coxson, “Doppler tolerance, complementarycode sets, and generalised thue–morse sequences,”
IET Radar, Sonarand Navigation , vol. 10, no. 9, pp. 1603–1610, Dec. 2016.[10] Z.-J. Wu, C.-X. Wang, P.-H. Jiang, and Z.-Q. Zhou, “Range-dopplersidelobe suppression for pulsed radar based on golay complementarycodes,” arXiv preprint arXiv:2003.11726 , 2020.[11] W. Dang, A. Pezeshki, S. Howard, W. Moran, and R. Calderbank,“Coordinating complementary waveforms for sidelobe suppression,” in
Forty-fifth Asilomar Conf. Signals, Syst., Comput., Pacific Grove, CA ,Nov. 2011, pp. 2096–2100.[12] W. Dang, “Signal design for active sensing,”
Dissertations and Theses- Gradworks , 2014.[13] W. Dang, A. Pezeshki, S. D. Howard, W. Moran, and R. Calderbank,“Coordinating complementary waveforms for suppressing range side-lobes in a Doppler band,” arXiv preprint arXiv:2001.09397 , 2020.[14] A. Pezeshki, R. Calderbank, and L. L. Scharf, “Sidelobe suppressionin a desired range/Doppler interval,” in
Proc. IEEE Radar Conf. , May.2009, pp. 1–5.[15] J. Wang, P. Fan, Y. Yang, and Y. L. Guan, “Range/doppler sidelobesuppression in moving target detection based on time-frequency bino-mial design,” in
Proc. IEEE 30th Annu. Int. Symp. Pers., Indoor MobileRadio Commun. (PIMRC) , Sep. 2019, pp. 1–5.[16] S. J. Wright, “Coordinate descent algorithms,”
Math. Program. , vol. 151,no. 1, pp. 3–34, Mar. 2015.[17] Y. Chi, A. Pezeshki, R. Calderbank, and S. Howard, “Complementarywaveforms for sidelobe suppression and radar polarimetry,” in