Alternative signal processing of complementary waveform returns for range sidelobe suppression

Signal Processing 159 (2019) 187–192

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Short communication

Alternative signal processing of complementary waveform returns for range sidelobe suppressionR Jiahua Zhu a, Ning Chu b,∗, Yongping Song c, Shuang Yi d, Xuezhi Wang d,e, Xiaotao Huang c, Bill Moran d,e a

College of Meteorology and Oceanology, National University of Defense Technology, Changsha, Hunan 410073, PR China College of Energy Engineering, Zhejiang University, Hangzhou, Zhejiang 310027, PR China College of Electronic Science, National University of Defense Technology, Changsha, Hunan 410073, PR China d School of Engineering, RMIT University, Melbourne, Victoria 3053, Australia e Department of Electrical and Electronic Engineering, University of Melbourne, Melbourne, Victoria 3010, Australia b c

a r t i c l e

i n f o

Article history: Received 7 November 2018 Revised 14 January 2019 Accepted 11 February 2019 Available online 11 February 2019 Keywords: Golay complementary waveforms Sidelobes suppression Pointwise addition processor Ambiguity function illumination

a b s t r a c t It has been an important technique to reduce sidelobes for processing nonzero Doppler target returns under Golay complementary waveforms. While the pointwise minimization processor in our early work shows an enhanced performance on the suppression of range sidelobes compared with existing approaches, it has a reduced detection probability to target with location uncertainties. In this paper, we present an alternative processing method which combines the radar returns in two separated pairs of Golay complementary waveforms through a pointwise addition processor. Our simulation result shows that this alternative method has a similar performance to that of the pointwise minimum processor for range sidelobes suppression and computational complexity but results in an improved target detection probability, with an acceptable decrease on the peak to peak-sidelobe ratio.

1. Introduction Techniques for range sidelobe suppression and resolution enhancement have long been used in advanced radar systems. As an instance, matched filtering between the received signal and the transmitted waveform is one of the simplest approach for these problems, which accumulates the target amplitude in the range bin and obtain an almost ideal pulse response. Based on this approach, waveforms that generates impulse-like autocorrelation functions are of great interest in the sidelobe and resolution improvement. Linear frequency modulation (LFM)/chirp waveform, one of the most common of such waveforms, is widely employed in practice due to its simplicity of generation and high time-bandwidth product. Nevertheless, traditional LFM waveform suffers from conflation on range and Doppler and associated range sidelobes, as reflected in the ambiguity function. Despite the previous works R This work was supported by the National Natural Science Foundation of China under grant 61701440 and 91634110, and the Asian Office of Aerospace Research & Development (AOARD)/AFRL under grant FA2386-15-1-4066. ∗ Corresponding author. E-mail addresses: [email protected] (J. Zhu), [email protected] (N. Chu), [email protected] (Y. Song), [email protected] (S. Yi), [email protected] (X. Wang), [email protected] (X. Huang), [email protected] (B. Moran).

https://doi.org/10.1016/j.sigpro.2019.02.012 0165-1684/© 2019 Elsevier B.V. All rights reserved.

© 2019 Elsevier B.V. All rights reserved.

several decades ago, numbers of representative literatures are still making efforts on the further improvement of LFM waveform in recent years. Joint design schemes on the LFM waveform and objective function are proposed by Cheng and Ciuonzo et al. [1,2] under both energy and similarity constraints, which make a global optimization on the signal-to-interference (plus noise) ratio. Rasool et al. [3], inspired by the echo-location method of bats, improve the LFM waveform to V-chirp waveform and enhanced the resolution of waveform with a slight loss of detection probability. The false targets problem in this scheme is solved by Zhu et al. [4], who extend the V-chirp waveform to a double V-chirp. In addition to the LFM waveform, Golay complementary waveforms are proved effective to produce high resolution pulse with essentially zero range sidelobes at the zero-Doppler in the ambiguity function [5], and could be widely employed in digital systems without large bandwidth requirement. However, enormous range sidelobes are observed in other Doppler bins due to the sensitivity to the mismatch of Doppler, which cannot be addressed through conventional sidelobe suppression methods like windowing. This problem is studied by Pezeshki and Calderbank et al. and they find that the carefully arranging of the transmission order of Golay complementary waveforms brings a modest reduction of range sidelobes near the zero-Doppler line in the ambiguity function [5,6]. Based on their idea, Dang et al. introduce an algo-

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rithm named Binominal Design (BD) algorithm [7], to expand the range sidelobe blanking area by weighting the receiver sequences for matched filtering, while decreases the Doppler resolution in the ambiguity function significantly. In a parallel direction, Suvorova et al. propose an algorithm to choose the suitable transmission order for Golay complementary waveforms according to the Reed– Müller codes [8], providing minimum range sidelobes at a given Doppler shift in the ambiguity function. On the other hand, cell-by-cell processing, or pointwise processing in other words, has been deeply discussed by researchers when handling radar images [9,10], which gives a further enhancement to the signal-to-noise ratio (SNR). To integrate the advantages of these works, we propose a signal processing procedure combining the outputs of a Weighted average Doppler (WD) algorithm and the existing BD algorithm via a pointwise minimization processor (PMP) in our early work [11], which is technical justified to have a large range sidelobe blanking area around the given Doppler value in the ambiguity function without significant loss of Doppler resolution, but the employment of nonlinear PMP may decrease the detection probability of target (or even lose the detection of target) due to the location uncertainties. In this continuous work, an alternative signal processing procedure by replacing the nonlinear processor as a linear pointwise addition processor (PAP) is applied, which is effective to obtain an enhanced detection probability of target with a similar performance on range sidelobe suppression, at the cost of the decrease of peak to peak-sidelobe ratio (PPSR) and Delay–Doppler resolution to some extent in the ambiguity function. The reminder of the paper is organized as follows. In Section 2, the concept of Golay complementary waveforms is briefly introduced, following by the illustration of PMP and the alternative PAP proposed in this work, with the comparison of target detection probability in the later. Simulations and discussions are put forward in Section 3 to verify the effectiveness of PAP. Conclusion and further avenues of research are provided in Section 4. Notation: We denote x(l) to be the lth element of sequence x, and Cx (k) is the autocorrelation output of x(l) at lag k. We write δ (k) as the Kronecker delta function with a pulse response at the kth index. The unit energy baseband signal is denoted by (t) in the time domain. A pulse train P = { p(0 ), p(1 ), . . . , p(N − 1 )} is written as P = { p(n )}N−1 . χ (t, FD ) means the t th row and the FD th n=0 column element of the Delay–Doppler map χ . The operation “min” and “norm” represent the pointwise minimization and pointwise addition, respectively. An independent and identically distributed (IID) Guassian distribution is expressed as N (τ , σ 2 ), whose mean value and variance are τ and σ 2 , separately.

2. Problem formation and the alternative approach 2.1. Golay complementary waveforms The Golay complementary waveforms consist of two unimodular ( ± 1) sequences x(l) and y(l) of length L (the x(l) and y(l) are called a Golay complementary pair, and details on the generation of Golay complementary pair can be found in [12]). The time extent of each chip in the pair is Tc and the total time duration of each pair is LTc . The autocorrelation of a Golay complementary pair satisfies Cx (k ) + Cy (k ) = 2Lδ (k ), k = −(L − 1 ), . . . , (L − 1 ),  T /2 A baseband pulse (t) with unit energy, i.e. −Tc c /2 |(t )|2 dt = 1, is modulated on each chip interval by each of the Golay complementary pair, so that the transmitted sequences are expressed  −1  −1 as x(t ) = Ll=0 x(l )(t − l Tc ), y(t ) = Ll=0 y(l )(t − l Tc ). A (P, Q) pulse train is used to determine whether x(t) or y(t) is transmitted in each pulse. Here P = { p(n )}N−1 is a binary sequence, so that n=0

the transmitted signal is as

N−1

zP (t ) =

n=0

p(n )x(t − nT ) + (1 − p(n ) )y(t − nT )

(1)

where T is the pulse repetition interval (PRI). The (n + 1 )th pulse in zP (t) is x(t) if p(n ) = 1 and is y(t) if p(n ) = 0. The alternating sequence P = {0, 1, 0, 1, . . .} is the standard transmission order for Golay complementary waveforms. The sequence Q = {q(n )}N−1 of n=0 positive real numbers is applied on the received signal to weight the returns. The pulse train for the matched filtering is

zQ (t ) =

N−1 n=0

q(n )[ p(n )x(t − nT ) + (1 − p(n ) )y(t − nT )]

(2)

The standard weighting sequence Q is an all 1 sequence. Then according to [13], the ambiguity function of Golay comlementary waveforms is

χPQ (t, FD ) =



+∞

−∞

zP (s ) exp( j2π FD s )zQ∗ (t − s )ds

(3)

where the superscript “∗ ” denotes complex conjugation. As aforementioned, the Golay complementary waveforms, transmitted in standard order and matched filtering with standard weighting sequence, causes enormous range sidelobes across the nonzero Doppler axes, which essentially increase the false alarm rate during the target detection. This problem is addressed by the PMP algorithm described below. 2.2. Pointwise minimization procedure The PMP algorithm proposed previously [11] for the range sidelobe suppression under Golay complementary waveforms is summarized in Fig. 1, where χWD (t, FD ) and χBD (t, FD ) are the Delay– Doppler maps for the WD and BD algorithms, respectively, χ (t, FD ) is the final output from the pointwise processor. Specifically in our early work [11], the “Pointwise Processor” in Fig. 1 represents the pointwise minimization processor (PMP), and we denote the final output of PMP as χPMP (t, FD ). For the BD algorithm [7], P is the n standard transmission order, and Q = α × {CN−1 }N−1 , where α = n=0 N−1 n n N/ n=0 CN−1 , CN−1 represents the number of n-combinations from a given set of N − 1 elements. For the WD algorithm [11], while Q is the standard weighting sequence, P is calculated through the method in [8] to select the optimal transmission order to minimize range sidelobes near the average targets’ Doppler f¯d weighted by the associated target amplitudes [14]

f¯d =

⎧ H f ⎨ h=1H dh ⎩

H

h=1

H

(1−Ah ) fdh

h=1 (1−Ah )

if all Ah are the same,

(4)

otherwise.

where H is the number of targets in the Delay–Doppler map, Ah and fdh are the normalized amplitude and Doppler of the hth target respectively. Target Doppler can often be estimated in a separate process, e.g., the detection of a Doppler radar, or the prior Doppler predicted by a tracker using past detections ([15] Chs. 2.3.4). A PMP is then employed in this procedure to combine the outputs of WD and BD algorithm:

χPMP (t, FD ) = min {χWD (t , FD ), χBD (t , FD )}

(5)

and the final result retains the large range sidelobe blanking area (the area where range sidelobes are less than −90dB) of BD algorithm as well as the Doppler resolution given by the WD algorithm theoretically. Nevertheless, a problem of PMP is that the radar cross section of target, or the “effective detectable area” of target reflected by the Delay–Doppler resolution may be reduced in the final output when the underlying target has location uncertainties. The PAP is motivated from this consideration and is demonstrated next.

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189

Fig. 1. Illustration of the signal processing procedure: Pointwise processor = minimum operator → PMP; Pointwise processor = addition operator → PAP.

2.4. Target detection probability

Fig. 2. Comparison of radar cross sections for a target of Swerling II model under PMP and PAP algorithms.

2.3. Pointwise addition procedure To solve the drawback induced by the PMP, we propose a pointwise addition processor (PAP) to replace the PMP. Denoted by χPAP (t, FD ), the final output of PAP is expressed as

χPAP (t, FD ) = norm{χWD (t, FD ) + χBD (t, FD )}

(6)

where “norm” is the normalization operation. The operation of PAP and PMP are based on the same theory, that is under the assumption that the target is stable during the whole radar illumination period (the location and magnitude of the target are stationary in two Delay–Doppler maps), but that of the range sidelobes are various. The use of PAP will prevent the drop of target detection probability caused by the nonlinear PMP, while maintaining a comparable range sidelobe blanking area as well as the Delay–Doppler resolution similar to the PMP in theory. In order to compare the performances of PAP and PMP, we use the peak to peak-sidelobe ratio (PPSR) in [15] as a measure. It is defined as

P P SR(FD ) =

|χ ( 0, 0 )|2 max |χ (t, FD )|2

(7)

t∈Sd

where Sd contains the delays where the range sidelobes are located. The PPSR measures the performance of range sidelobe suppression at a desired Doppler value, which in fact calculates the ratio of the energy of the peak in the ambiguity function (the target) to that of the largest range sidelobe at Doppler FD . Certainly, it can be expected that the PMP will induce a higher PPSR than the PAP. In the presence of a Swerling II target, the detection situation in the Delay-Doppler map by taking into account target location uncertainties may be described in Fig. 2 [13]. Intuitively, while the PMP can achieve a higher PPSR, it potentially results in a smaller radar cross section. On the other hand, the ambiguity function of PAP has a larger radar cross section at the cost of sacrificing the Delay-Doppler resolution and PPSR. In other words, the PAP actually performs worse than PMP in the view of range sidelobe suppression and Delay–Doppler resolution for a Swerling II target, but obtains larger radar cross section of the target (which invokes higher target detection probability). Nevertheless, in the simulation result shown in Fig. 5, the level reduced PPSR of PAP is remaining reasonably high for identifying a target from sidelobes.

As described before, miss detections may be caused when the target location fluctuates of a Swerling II target is significant, both for the PMP and PAP. The target location uncertainties comprise two independent fluctuations – the delay fluctuation and the Doppler fluctuation, respectively. They can be considered as IID Guassian distributions in the far field detection scene, i.e. N (τˆ , σT2 ) and N ( fˆd , σ 2 ), where tˆ and FˆD are the estimated (mean) values of D

target delay and Doppler, which could be obtained by a tracker, σT2 and σD2 are their variances. The probability density functions of these two fluctuations and their maximum boundaries τs and fds are shown in Fig. 3. Obviously, the delay fluctuation boundary should not exceed the interval [−LTc , LTc ], while the Doppler fluctuation boundary should be in the red rectangle shown in the top sub-figure of Fig. 4(column(b)), otherwise there will be definitely no target in the final output PMP, and two targets in that of PAP. Without loss of generality, we consider a probability of 99% that the fluctuations of a target are in the above maximum boundaries, which means that [16]



τs = 3 σ T f ds = 3 σ D

(8)

Next, a method is put forward to analyze the target detection probability of PMP and PAP by comparing their ambiguity functions. Under a given detection threshold DL, the corresponding boundaries that a target can be detected after the fluctuations are able to be obtained from the final outputs of ambiguity functions χBD (t, FD ) and χWD (t, FD ) after PMP and PAP, respectively (shown in Fig. 4), which are denoted as τ PMP (DL) and fdPMP (DL ), τ PAP (DL) and fdPAP (DL ), respectively. Therefore, according to the properties of Gaussian distribution, the four probabilities PaPMP , PbPMP , PaPAP , PbPAP for that the fluctuations are within the above four boundaries are calculated as



PaPMP =

a2

a1

 PbPMP =

b1

 PaPAP =

a4 a3

 PbPAP =

b2

b4 b3





1 1 exp − τ 2 dτ √ 2 2π





1 1 exp − fd2 d fd √ 2 2π



(10)



1 1 exp − τ 2 dτ √ 2 2π



(9)

(11)



1 1 exp − fd2 d fd √ 2 2π

(12)

where a1 = −τPMP (DL )/σT , a2 = τPMP (DL )/σT ; b1 = − fdPMP (DL )/σD , b2 = fdPMP (DL )/σD ; a3 = −τPAP (DL )/σT , a4 = τPAP (DL )/σT ; b3 = − fdPAP (DL )/σD , b4 = fdPAP (DL )/σD . Since the fluctuations are IID Gaussian distribution, the target detection probabilities after PMP and PAP are given as

PDPMP (DL ) = PaPMP × PbPMP

(13)

PDPAP (DL ) = PaPAP × PbPAP

(14)

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Fig. 3. The relationship between the probability density functions of two fluctuations in Swerling II target model and their boundaries: (a) delay fluctuation and (b) Doppler fluctuation.

Fig. 4. Plots of ambiguity functions χBD (t, FD ), χWD (t, FD ), χPMP (t, FD ) and χPAP (t, FD ) (column (a) from top to bottom) and the slices of them across the zero-delay (column (b) from top to bottom). (colorbar unit is dB)

J. Zhu, N. Chu and Y. Song et al. / Signal Processing 159 (2019) 187–192

191

Table 1 Comparison of the computational complexity.

χWD (t, FD ) =

Approaches

Total computational complexity

BD algorithm PMP output PAP output

(2L − 1 ) + NY 2[(2L − 1 ) + NY ] + Y 2[(2L − 1 ) + NY ] + Y

L−1 N−1  1  [Cx (k ) + Cy (k )] qWD (n )ξnk 2 k=−L+1

n=0

L−1 N−1  1  − [Cx (k ) − Cy (k )] (−1 ) pWD (n) qWD (n )ξnk 2 k=−L+1

 GWD (t, FD ) + SWD (t, FD )

Fig. 5. The comparison of the PPSR.

Fig. 6. The target detection probabilities of PMP and PAP.

Note that this proposed approach are just used to illustrate the improvement of target detection probability that PAP compared to PMP. In this method we do not consider the false alarm rate since we directly compare the ambiguity functions without the influence of clutter and noise. The following simulation results shown in Fig. 6 are persuadable under high SNR situation. However, the improvement of PAP may be less significant in practical comparison, due to the unavoidable false alarm in real illumination when clutter and noise are comparable to the magnitude of target. 3. Simulation and discussions The performance of PAP is confirmed by the simulation with the following parameters: radar carrier frequency fc = 1GHz, bandwidth B = 50MHz, sampling rate fs = 2B, PRI T = 50 μ s, pulse number N = 25 = 32. The Golay complementary waveforms have, for each element of the pair, L = 64 chips of values ± 1 in chip interval Tc = 0.1 μ s. Each chip has fs × Tc = 10 sampling points. As deduced in [11], the ambiguity functions of χBD (t, FD ) and χWD (t, FD ) are computed and divided into the following two parts

χBD (t, FD ) = −

L−1 N−1  1  [Cx (k ) + Cy (k )] qBD (n )ξnk 2

1 2

k=−L+1

n=0

L−1 

N−1 

[Cx (k ) − Cy (k )]

k=−L+1

 GBD (t, FD ) + SBD (t, FD )

(−1 ) pBD (n) qBD (n )ξnk

n=0

(15)

n=0

(16)

where ξnk = exp( j2π FD nT )C (t − kTc − nT ). Fig. 4 shows the ambiguity functions of the BD algorithm, WD algorithm, the PMP output and the PAP output, and slices across them at the zero-delay. Note that the first terms in (15) and (16) are contributed by the target return — corresponding to the peak of ambiguity function, which are highlighted in the red rectangles in Fig. 4. The second terms represent the sidelobes induced by signal processing. The last row of Fig. 4 illustrates the output of PAP. It indicates that the PAP has a similar performance with PMP in terms of the scale of range sidelobe blanking area, as well as the mainlobe width. According to (5), the computational complexity of BD algorithm, the results of PMP and PAP are compared in Table 1, where Y is the total number of the resolution cell in a [−T /2, T /2] × [−π , π ] Delay–Doppler scene. The computing complexity of matched filtering in delay and integration are 2L − 1 and NY, respectively; the computing complexity of pointwise minimization and pointwise addition are both Y. Therefore, the relative computational complexity ratio of these three approaches are about 1:2:2. Next, the PPSR plots for PAP and PMP are compared in Fig. 5. It is obvious that the PPSR curve of the PMP is always higher than that of the PAP within the Doppler period [0, π ]rad. This means the PAP generates larger range sidelobes than PMP shown in the Delay–Doppler maps. Nevertheless, the mean PPSR of the PAP output is greater than 95dB, which is high enough for differentiating the target returns from the range sidelobes. Furthermore, the target detection probabilities of PMP and PAP under different detection threshold are described in Fig. 6. 10 0 0 Monte Carlo runs are taken to calculate the target detection probability for each detection threshold. The results demonstrate a better target detection performance of PAP compared to the PMP, with a decrease of target detection probability under an enhanced detection threshold. 4. Conclusion In this paper, an alternative signal processing procedure using the PAP is proposed for the range sidelobe suppression in signal processing when radar transmit Golay complementary waveforms. Analytical and simulation results confirm that the PAP performs no worse than the existing PMP but with an enhanced target detection probability, though an affordable decrease of the PPSR values and Delay–Doppler resolution is observed through the ambiguity function. As an ongoing research, we intend to consider the case of complementary waveforms with more than 2 sequences and Golay complementary pair under multicarriers. In addition, we will implement the proposed processing methods in hardware for real experiment. The results will be reported elsewhere.

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References [1] X. Cheng, A. Aubry, D. Ciuonzo, et al., Robust waveform and filter bank design of polarimetric radar, IEEE Trans. Aerosp. Electron. Syst. 53 (1) (2017) 370–384. [2] D. Ciuonzo, A. De Maio, G. Foglia, et al., Intrapulse radar-embedded communications via multiobjective optimization, IEEE Trans. Aerosp. Electron. Syst. 51 (4) (2015) 2960–2974. [3] S.B. Rasool, M.R. Bell, Biologically inspired processing of radar waveforms for enhanced delay-Doppler resolution, IEEE Trans. Signal Process. 59 (6) (2011) 2698–2709. [4] J. Zhu, Y. Song, C. Fan, et al., Nonlinear processing for enhanced Delay-Doppler resolution of multiple targets based on an improved radar waveform, Signal Process. 130 (2017) 355–364. [5] R. Calderbank, S. Howard, B. Moran, Waveform diversity in radar signal processing, IEEE Signal Process. Mag. 26 (1) (2009) 32–41. [6] A. Pezeshki, R. Calderbank, W. Moran, et al., Doppler resilient Golay complementary waveforms, IEEE Trans. Inform. Theory 54 (9) (2008) 4254–4266. [7] W. Dang, A. Pezeshki, S. Howard, et al., Coordinating complementary waveforms for sidelobe suppression, in: 45th Asilomar Conf. Signals, Systems and Computers, 2011, pp. 2096–2100.

[8] S. Suvorova, S. Howard, B. Moran, et al., Doppler resilience, Reed-Müller codes and complementary waveforms, in: 41st Asilomar Conf. Signals, Systems and Computers, 2007, pp. 1839–1843. [9] M.A. Richards, J.A. Scheer, W.A. Holm, Principles of Modern Radar Volume I-Basic Principles, Scitech Publishing, USA, 2010. [10] D. Ciuonzo, On time-reversal imaging by statistical testing, IEEE Signal Process. Lett. 24 (7) (2017) 1024–1028. [11] J. Zhu, X. Wang, X. Huang, et al., Range sidelobe suppression for using Golay complementary waveforms in multiple moving target detection, Signal Process. 141 (2017) 28–31. [12] M. Golay, Complementary series, IRE Trans. Inform. Theory 7 (2) (1961) 82–87. [13] M.A. Richards, Fundamentals of Radar Signal Processing, McGraw-Hill Education, NY, 2005. [14] J. Zhu, X. Wang, X. Huang, et al., Golay complementary waveforms in Reed-Müller sequences for radar detection of nonzero Doppler targets, Sensors 18 (2018) 192. (1–20) [15] W. Dang, Signal Design for Active Sensing, Colorado State University, 2014 Ph. d dissertation. [16] S.M. Kay, Fundamentals of Statistical Signal Processing, Vol II - Detection Theory, Prentice Hall, 1998.