- Email: [email protected]
IRCI-free OQAM-OFDM radar pulse compression
IRCI-Free OQAM-OFDM Radar Pulse Compression
Journal Pre-proof
IRCI-Free OQAM-OFDM Radar Pulse Compression Qiao Shi , Xueting Li , Tianxian Zhang, Xinyu Liu , Lingjiang Kong PII: DOI: Reference:
S0165-1684(20)30413-8 https://doi.org/10.1016/j.sigpro.2020.107869 SIGPRO 107869
To appear in:
Signal Processing
Received date: Revised date: Accepted date:
9 June 2020 12 October 2020 2 November 2020
Please cite this article as: Qiao Shi , Xueting Li , Tianxian Zhang, Xinyu Liu , Lingjiang Kong , IRCI-Free OQAM-OFDM Radar Pulse Compression, Signal Processing (2020), doi: https://doi.org/10.1016/j.sigpro.2020.107869
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
Highlights • A pulse compression method is proposed with the offset quadrature ampli- tude modulation based orthogonal frequency division multiplexing (OQAM- OFDM) signal, without inserting any cyclic prefix (CP). • The proposed method is concise and easily implemented, which realizes the inter-range-cell interference (IRCI) free pulse compression perfectly, and improves the power efficiency. • The proposed method achieves that the scope of observation will not be restrained by the number of subcarriers. • The signal-to-noise ratio (SNR) after the pulse compression is derived. It is proved that the SNR gain for the proposed method can reach to the same as that of the method using the matched filter.
1
IRCI-Free OQAM-OFDM Radar Pulse Compression Qiao Shi, Xueting Li, Tianxian Zhang*, Xinyu Liu and Lingjiang Kong School of Information and Communication Engineering University of Electronic Science and Technology of China Chengdu, Sichuan, P.R. China, 611731 Corresponding author: Tianxian Zhang Tel: +86-028-61830768, Email: [email protected]
Abstract In this paper, a pulse compression method is proposed with the offset quadrature amplitude modulation based orthogonal frequency division multiplexing (OQAM-OFDM) signal. By establishing the received OQAM-OFDM signal as a linear model form and without inserting any cyclic prefix (CP), we propose a concise and easily implemented pulse compression method. The proposed method realizes the inter-range-cell interference (IRCI) free pulse compression perfectly, which improves the power efficiency, and the scope of observation will not be restrained by the number of subcarriers. Besides, the signal-to-noise ratio (SNR) after the pulse compression is derived. It is proved that the SNR gain for the proposed pulse compression method can reach to the same as that of the method using the matched filter. Finally, numerical simulation results are provided and discussed. Keywords: OQAM-OFDM radar, pulse compression, SNR gain. 1. Introduction Orthogonal frequency division multiplexing (OFDM) signal has been researched in radar systems for many years. To resist the inter-symbol interference (ISI) caused by the multipath, a cyclic prefix (CP) is added at the beginning of the OFDM signal, resulting in a reduction in power efficiency for the radar system [1, 2]. In addition, the rectangular filter, exploited by the OFDM signal, has relatively large sidelobes in the frequency response, which results in the inter-subcarrier interference (ICI) and makes the OFDM radar quite sensitive to the Doppler shift [3]. As an alternative to improve the power efficiency, the offset quadrature amplitude modulation based OFDM (OQAM-OFDM) signal has attracted a Preprint submitted to Signal Processing
November 6, 2020
lot of attention recently [4, 5], which does not require the CP to mitigate the ISI. Furthermore, a well-localized pulse shaping filter in time and frequency domain is applied in the OQAM-OFDM signal, which makes the OQAMOFDM signal less sensitive to the Doppler shift, compared with the OFDM signal [3]. Currently, some researchers have considered applying the OQAM-OFDM signal into radar systems. By performing the correlation processing on the transmitted OQAM-OFDM signal and the received signal, they provide the ambiguity function of the OQAM-OFDM signal, and verify that the OQAMOFDM signal can be used as a reasonable radar signal [6, 7]. However, the sidelobes level is still high because of the correlation processing. In [8], without using the correlation processing, we propose a pulse compression method for the OQAM-OFDM radar. Though good pulse compression performance can be achieved, the method is highly constrained (i.e., the number of subcarriers needs to be larger than ten times of the number of range cells.), which makes it difficult to be applied in practical applications. Besides, the realization process is quite complex, which directly follows the flow of communication processing (i.e., the demodulation process). However, for the radar system, only the target information is concerned. The demodulation process is unnecessary, and can be replaced by an estimation method, which has not been investigated in the open literatures. In this paper, we propose a concise and easily implemented pulse compression method for the OQAM-OFDM radar, without the insertion of the CP. The received OQAM-OFDM signal is established as a linear model form. The pulse compression is implemented by proposing an linear model-based estimation method, without using the demodulation process, and inter-range-cell interference (IRCI) free pulse compression is perfectly realized. Meanwile, the scope of observation will not be restrained by the number of subcarriers. Furthermore, we comprehensively analyze the signal-to-noise ratio (SNR) after the pulse compression, and find that the SNR gain can reach to the same as that of the method using the matched filtering. Finally, numerical simulations are provided to verify the effectiveness of the proposed method. 2. Signal Model We first give a brief introduction of the OQAM-OFDM radar signal model. With K subcarriers, the transmitted OQAM-OFDM signal can be
3
expressed as s (t) =
K−1 X
dk h (t)ej2πk∆f t ejπk/2 ,
(1)
k=0
where, ∆f = 1/T = B/K represents the subcarrier spacing, B is the signal bandwidth, h (t) [11] denotes a symmetrical real-valued pulse shaping filter, with unit energy, overlapping factor N and duration N T , dk denotes the real weight that is transmitted over the subcarrier k. For simplicity, we rewrite dk in vector notation as d = [d0 , d1 , · · · , dK−1 ]T ∈ RK , where [·]T denotes the transpose operator, R denotes the real field. At the receiver, let gr be the response of the target within the rth range cell, which includes the attenuation caused by the propagation process, and the radar cross section (RCS) coefficient from the scatterers in the rth range cell. Thus, the received signal for the radar system can be given by K−1 X X j2πk 2Rr 2Rr −j4πfc Rcr (2) dk h t − e T (t− c ) ejπk/2 + ω (t) , gr e ν (t) = c r k=0 where, c represents the speed of light, Rr is the range between the radar and the rth range cell, fc is the carrier frequency, ω (t) denotes the white Gaussian noise with zero mean and variance σ 2 . 3. Pulse Compression We suppose that the width of radar detection area is G, with R range cells, as shown in Fig. 1, where, G is smaller than the maximum unambiguous range of the radar. R0 represents the range from the radar to the nearest range cell. Thus, for the rth range cell, one can obtain Rr = R0 + rγ, where γ = c/2B is the radar range resolution. Let the received signal be sampled with a sample interval Ts = 1/B. One may find that Ts = 2γ/c and T = KTs . Therefore, for the delay with the rth range cell, 2Rr /c = 2R0 /c + rTs . The sampling begins when the first version of the transmitted signal is arrived (i.e., after 2R0 /c seconds). When there is a line-of-sight path between the radar and each of the range cell, ν (t) in (2) can be converted to the discrete time linear convolution of the weighting coefficient vector b = [b0 , b1 , . . . , bR−1 ]T ∈ CR and the transmitted signal sequence, and expressed as R−1 X νi = br si−r + ωi , i = 0, 1, . . . , Lν − 1, (3) r=0
4
where, the transmitted signal sequence is given by si =
K−1 X k=0
dk h (i)ej2πki/K ejπk/2 , i = 0, 1, . . . , Ls − 1,
(4)
Rr
the weighting coefficient br = gr e−j4πfc c , C denotes the complex field, Lν = N K + R − 1 is the length of received signal, Ls = N K is the length of transmitted signal, ωi denotes the sampling sequence of ω(t). In view of (3), the received signal can be rewritten in vector notation as v = Sb + w,
(5)
where, v = [v0 , v1 , . . . , vLν −1 ]T ∈ CLν denotes the received signal sequence, w = [ω0 , ω1 , . . . , ωLν −1 ]T ∈ CLν is the noise sequence with covariance matrix σ 2 I, S is a Lν × R matrix representing the transmitted signal matrix and can be written as s0 s−1 · · · s−(R−1) s1 s0 · · · s−(R−2) S = .. (6) .. .. , . . ··· . sLv −1 sLv −2 · · · sLv −R
where, si = 0 if i ∈ / [0, Ls − 1]. Thus, to achieve the pulse compression, the problem becomes how to estimate the weighting coefficient vector b with the received signal sequence v and the transmitted signal matrix S. Since w ∼ N (0, σ 2 I) and Lν = N K +R−1 > R, (5) constitutes a typical linear model in [9]. Besides, SH S is invertible, because the columns of S are linearly independent. Therefore, according to [9], b can be estimated by ˆ = SH S −1 SH v = b + SH S −1 SH w. (7) b
It is straightforward to show that the estimate of b is only related to itself. There is no interference from other range cells, and the real values can be estimated when the noise is not considered, which means the IRCI-free pulse compression is perfectly achieved. 4. SNR Analysis After Pulse Compression
In order to theoretically analyze the performance of the proposed method, the SNR after the pulse compression is derived. In view of (7), the noise 5
covariance matrix after the pulse compression can be expressed as −1 H H −1 H H H C =E S S S w S S S w h i h iH −1 −1 − E SH S SH w E SH S SH w −1 H −1 H H H H =E S S S ww S S S −1
=σ 2 SH S
(8)
,
where, E [·] denotes the statistical expectation. Thus, the SNR for the rth range cell after the pulse compression can be written as H bb rr |br |2 , SN Rr = (9) = [C]rr var ˆbr
ˆbr denotes the estimate of weighting coefficient for the rth range cell where, br , var ˆbr represents the variance of ˆbr . Obviously, SN Rr can reach to the maximum value only when the variance var ˆbr takes the minimum value. It is worth noting that var ˆbr can be written as var ˆbr = eTr Cer ,
(10)
T R where h er = [0, 0, .i. . , 0, 1, 0, . . . , 0] ∈ R and “1” is in the rth element. Since −1 E SH S SH w = 0 and C is the covariance matrix, C and C−1 are posiH tive definite matrixs. Therefore, C−1 can be decomposed as Φ Φ, where, 2 Φ −1 is a R×R reversible matrix. Additionally, notice that eTr ΦH ΦH er = −1 1. Let ξ1 = Φer and ξ2 = ΦH er , resulting in ξ1H ξ2 = 1. Then, according to the Cauchy-Schwarz inequality, we can obtain 2 (11) ξ1H ξ2 ≤ ξ1H ξ1 ξ2H ξ2 .
Therefore, we have
1 ≤ eTr ΦH Φer
eTr Φ−1 ΦH
−1
er = eTr C−1 er eTr Cer .
6
(12)
From (10) and (12), we can obtain 1 σ2 var ˆbr ≥ T −1 = H . er C er [S S]rr
(13)
−1 er . Thus, to The equality holds if and only if ξ1 = cξ2 or Φer = cr ΦH acquire the maximum SN Rr , the condition becomes ΦH Φer = cr er . In light of C−1 = ΦH Φ and (8), the condition can be given by c0 0 . . . 0 0 c1 . . . 0 SH S = σ 2 .. .. . . (14) .. . . . . . 0 0 . . . cR−1
Based on (4) and (6), we know that S is related to the transmitted signal. Thus, to obtain the maximum SN Rr , the problem converts to make S satisfy the condition in (14) by designing the transmitted real weights d. Then, in light of (6), it is notable that [S]lr = sl−r . Thus, Lν X H S S lr = si−r s∗i−l ,
(15)
i=1
where, {l, r} ∈ {1, 2, . . . , R}, ∗ represents the complex conjugate transpose. For large Lν , (15) can be approximated as H S S lr ≈
Ls −1−|l−r|
X
si s∗i+|l−r| .
(16)
i=0
P roof : For large Lν , (15) can be written as
∞ X S S lr ≈ si−r s∗i−l . H
(17)
i=−∞
According to (6), we know that si = 0 if i < 0 or i > Ls −1. Thus, only about 2R additional terms will be added in (17) compared with (15). If Lν R, these terms will be negligible. While Lν = N K +R−1, the condition Lν R is easy to be satisfied. Let r ≥ l and j = i − r, and (17) can be rewritten as
Ls −1−(r−l) ∞ X X ∗ S S lr = sj sj+r−l = sj s∗j+r−l . H
j=−∞
j=0
7
(18)
Similarly, for r < l , H S S lr =
Ls −1−(l−r)
X
sj s∗j+l−r .
(19)
j=0
Thus, (16) is certificated. It can be seen that (16) is the correlation function of the transmitted signal sequence. SH S is approximated as the autocorrelation matrix, and can be given as χ0 χ∗1 . . . χ∗R−1 χ1 χ0 . . . χ∗R−2 H S S = Ls .. (20) .. .. , . . . . . . χR−1 χR−2 . . . χ0 where,
L −1−m 1 sX sj s∗j+m , m = 0, 1, . . . , R − 1, χm = L s j=0
(21)
which can be regarded as the autocorrelation function of the transmitted signal sequence. To make (20) satisfy the condition in (14), χm = 0 if m 6= 0, which means SH S = Ls χ0 I. It is the approximate implementation, if the pseudo-random noise (PRN) sequence is utilized as the transmitted signal [9]. Thus, if the transmitted signal in (4) is equivalent to the PRN sequence, by designing the transmitted real weights d, the maximum SN Rr can be acquired. The variance of ˆbr can be given as var ˆbr = σ 2 /Ls χ0 . Typically, the transmitted signal is normalized, meaning that Ls χ0 = 1. Thus, according to (9), the maximum SN Rr can be given by SN Rmax,r
|br |2 Ls χ0 |br |2 = 2 . = σ2 σ
(22)
In addition, in view of (5), the mean transmitted power for the transmitted sequence is 1/Ls . Therefore, the SNR for the rth range cell before pulse compression can be expressed as SN Rr0 = 8
|br |2 . Ls σ 2
(23)
Based on (22) and (23), we can obtain SN Rmax,r = Ls SN Rr0 .
(24)
We can find that the SNR gain for the pulse compression is the length of the transmitted signal, which is exactly equal to that of the method using the matched filter. That is to say, the proposed method can realize the IRCI-free pulse compression and achieve the maximum SNR output without inserting the CP. 5. Simulation Results In this section, numerical simulation results are presented to evaluate the proposed pulse compression method. The pulse compression method proposed in [10] is used as a benchmark for comparison. In the simulations, a point target is located at 5256 m away from the radar. The Bandwidth is 150 MHz. The number of subcarriers is selected as K = 1024. The number of range cells is R = 512. The length of CP used in the OFDM signal is R − 1 = 511. The pulse shaping filter for the OQAM-OFDM signal is the raised cosine filter, with the overlapping factor N = 4. Without considering the noise, the results for the two pulse compression methods are shown in Fig. 2. It is obvious that the sidelobes for the two methods are really low and almost the same. Then, in order to verify the SNR gain obtained by our proposed method, we provide the pulse compression results and the output SNR curves with different received signal SNR, which are shown in Fig. 3 and Fig. 4. Theoretically, the SNR gain for the CP-OFDM signal is the number of subcarriers and equal to 10log10 K = 30.10 dB, while it is the length of the transmitted signal and equal to 10log10 (N K) = 36.12 dB for the OQAM-OFDM signal. From Fig. 3 and Fig. 4, we can find that the SNR gain is actually equal to the expected value. It means that our proposed method can completely exploit the transmitted energy, as the well-known matched filter does. Meanwhile, the energy of the CP is wasted by the CP-OFDM signal, resulting in a loss of energy of 10log10 ((K + R − 1)/K) = 1.76 dB. Thus, compared with the method in [10], our proposed method can improve the power efficiency. 6. conclusion In this paper, a concise and easily implemented pulse compression method has been proposed for the OQAM-OFDM radar. Firstly, we established 9
the OQAM-OFDM radar signal model. Then, without the insertion of the CP, the pulse compression method has been derived, achieving the IRCI-free pulse compression, and the scope of observation will not be restrained by the number of subcarriers. By analyzing the SNR after the pulse compression, we have found that the SNR gain can reach to that of the method using the matched filter. In addition, compared with the method using the CP-OFDM signals, our proposed method realizes the increase of the power efficiency. Finally, numerical simulations have verified the effectiveness of the proposed pulse compression method. Some future researches may be needed for the proposed OQAM-OFDM radar system. Since the pulse compression is completed, how can we apply the OQAM-OFDM radar into the target detection [12, 13] and the target tracking [14] is now under our current investigations. 7. Acknowledgment This work was supported by the National Natural Science Foundation of China under Grants 61971109, U19B2017, the Chang Jiang Scholars Program, the GF Science and Technology Special Innovation Zone Project, and the Fundamental Research Funds of Central Universities under Grant 2672018ZYGX2018J009. References [1] A. Gusmao, P. Torres, R. Dinis, and N. Esteves, “A reduced-CP approach to SC/FDE block transmission for broadband wireless communications,” IEEE Transactions on Communications, vol. 55, no. 4, pp. 801-809, Apr. 2007. [2] T. Zhang, X. Xia, and L. Kong, “IRCI free range reconstruction for SAR imaging with arbitrary length OFDM pulse,” IEEE Transactions on Signal Processing, vol. 62, no. 18, pp. 4748-4759, Sept. 2014. [3] B. Farhang-Boroujeny, “OFDM versus filter bank multicarrier,” IEEE Signal Processing Magazine, vol. 28, no. 3, pp. 92-112, May 2011. [4] D. Mattera, M. Tanda, and M. Bellanger, “Performance analysis of some timing offset equalizers for FBMC/OQAM systems,” Signal Processing, vol. 108, pp. 167-182, Mar. 2015. 10
[5] E. Kofidis, D. Katselis, A. Rontogiannis, and S. Theodoridis, “Preamblebased channel estimation in OFDM/OQAM systems: A review,” Signal Processing, vol. 93, no. 7, pp. 2038-2054, Jul. 2013. [6] W. Cao, J. Zhu, X. Li, W. Hu and J. Lei, “Feasibility of multicarrier modulation signals as new illuminators of opportunity for passive radar: orthogonal frequency division multiplexing versus filter-bank multi-carrier,” IET Radar, Sonar and Navigation, vol. 10, no. 6, pp. 1080-1087, Jun. 2016. [7] W. Xu, C. Wang, G. Cui, W. Wang and Y. Zhang, “The M-sequence encoding method for radar-communication system based on filter bank multi-carrier,” 2017 9th International Conference on Advanced Infocomm Technology (ICAIT), Chengdu, pp. 255-259, 2017. [8] Q. Shi, X. Li, T. Zhang, G. Cui, and L. Kong, “OQAM-OFDM radar approximated IRCI-Free pulse compression,” IEEE Transactions on Vehicular Technology, 2020. [9] S. M. Kay, “Fundamentals of statistical signal processing,” Technometrics, 1993. [10] T. Zhang, and X. Xia, “OFDM synthetic aperture radar imaging with sufficient cyclic prefix,” IEEE Transactions on Geoscience and Remote Sensing, vol. 53, no. 1, pp. 394-404, 2015. [11] M. Bellanger, “FBMC physical layer: A primer,” PHYDYAS, 2010. [12] X. Li, Z. Sun, T. S. Yeo, T. Zhang, W. Yi, G. Cui, and Lingjiang Kong, “STGRFT for detection of maneuvering weak target with multiple motion models,” IEEE Transactions on Signal Processing, vol. 67, no. 7, pp. 1902-1917, Apr. 2019. [13] W. Liu, J. Liu, Y. Gao, G. Wang, and Y. Wang, “Multichannel signal detection in interference and noise when signal mismatch happens,” Signal Processing, vol. 166, 2020. [14] J. Yan, W. Pu, S. Zhou, H. Liu, and M. S. Greco, “Optimal resource allocation for asynchronous multiple targets tracking in heterogeneous radar networks,” IEEE Transactions on Signal Processing, vol. 68, pp. 4055-4068, 2020. 11
ls cel
R
ge ran
g0
... g
...
gR1
r
R0
D
n ctio ete
aG Are
Figure 1: Illustration diagram.
12
0 OQAM-OFDM CP-OFDM
Amplitude(dB)
-50 -100 -150 -200 -250 -300 -350 5050
5150
5250
5350
5450
Range (m)
Figure 2: Range profiles of the point spread function.
13
SNR=-20dB CP-OFDM
X 256 Y 11.1
20
SNR=-20dB OQAM-OFDM
40
Amplitude(dB)
Amplitude(dB)
40
0
-20
X 256 Y 16.01
20
0
-20
5050
5150
5250
5350
5450
5050
5150
Range (m)
(a) Amplitude(dB)
Amplitude(dB)
5450
SNR=0dB OQAM-OFDM
40
X 256 Y 30.1
20
5350
(b)
SNR=0dB CP-OFDM
40
5250
Range (m)
0
X 256 Y 35.92
20
0
-20
-20 5050
5150
5250
5350
5050
5450
5150
5250
5350
Range (m)
Range (m)
(c)
(d)
Figure 3: Pulse compression results with different received signal SNR.
14
5450
50 45
CP-OFDM OQAM-OFDM CP-OFDM without energy loss
Output SNR (dB)
40 35 30 25 20 15 10 5 -20
-15
-10
-5
0
5
10
Received signal SNR (dB)
Figure 4: The output SNR after the pulse compression with different received signal SNR.
15
We declare that we have no known competing nancial interests or personal relationships that could have appeared to influence the work reported in this paper.
16
Credit Author Statement Qiao Shi: Conceptualization, Methodology, Software, Validation, WritingOriginal Draft, Writing-Review & Editing. Xueting Li: Investigation, Writing-Review & Editing. Tianxian Zhang*: Conceptualization, Methodology, Writing-Review & Editing, Funding acquisition. Xinyu Liu: Investigation, Writing-Review & Editing. Lingjiang Kong: Investigation, Writing-Review & Editing.
17
Journal Pre-proof
IRCI-Free OQAM-OFDM Radar Pulse Compression Qiao Shi , Xueting Li , Tianxian Zhang, Xinyu Liu , Lingjiang Kong PII: DOI: Reference:
S0165-1684(20)30413-8 https://doi.org/10.1016/j.sigpro.2020.107869 SIGPRO 107869
To appear in:
Signal Processing
Received date: Revised date: Accepted date:
9 June 2020 12 October 2020 2 November 2020
Please cite this article as: Qiao Shi , Xueting Li , Tianxian Zhang, Xinyu Liu , Lingjiang Kong , IRCI-Free OQAM-OFDM Radar Pulse Compression, Signal Processing (2020), doi: https://doi.org/10.1016/j.sigpro.2020.107869
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
Highlights • A pulse compression method is proposed with the offset quadrature ampli- tude modulation based orthogonal frequency division multiplexing (OQAM- OFDM) signal, without inserting any cyclic prefix (CP). • The proposed method is concise and easily implemented, which realizes the inter-range-cell interference (IRCI) free pulse compression perfectly, and improves the power efficiency. • The proposed method achieves that the scope of observation will not be restrained by the number of subcarriers. • The signal-to-noise ratio (SNR) after the pulse compression is derived. It is proved that the SNR gain for the proposed method can reach to the same as that of the method using the matched filter.
1
IRCI-Free OQAM-OFDM Radar Pulse Compression Qiao Shi, Xueting Li, Tianxian Zhang*, Xinyu Liu and Lingjiang Kong School of Information and Communication Engineering University of Electronic Science and Technology of China Chengdu, Sichuan, P.R. China, 611731 Corresponding author: Tianxian Zhang Tel: +86-028-61830768, Email: [email protected]
Abstract In this paper, a pulse compression method is proposed with the offset quadrature amplitude modulation based orthogonal frequency division multiplexing (OQAM-OFDM) signal. By establishing the received OQAM-OFDM signal as a linear model form and without inserting any cyclic prefix (CP), we propose a concise and easily implemented pulse compression method. The proposed method realizes the inter-range-cell interference (IRCI) free pulse compression perfectly, which improves the power efficiency, and the scope of observation will not be restrained by the number of subcarriers. Besides, the signal-to-noise ratio (SNR) after the pulse compression is derived. It is proved that the SNR gain for the proposed pulse compression method can reach to the same as that of the method using the matched filter. Finally, numerical simulation results are provided and discussed. Keywords: OQAM-OFDM radar, pulse compression, SNR gain. 1. Introduction Orthogonal frequency division multiplexing (OFDM) signal has been researched in radar systems for many years. To resist the inter-symbol interference (ISI) caused by the multipath, a cyclic prefix (CP) is added at the beginning of the OFDM signal, resulting in a reduction in power efficiency for the radar system [1, 2]. In addition, the rectangular filter, exploited by the OFDM signal, has relatively large sidelobes in the frequency response, which results in the inter-subcarrier interference (ICI) and makes the OFDM radar quite sensitive to the Doppler shift [3]. As an alternative to improve the power efficiency, the offset quadrature amplitude modulation based OFDM (OQAM-OFDM) signal has attracted a Preprint submitted to Signal Processing
November 6, 2020
lot of attention recently [4, 5], which does not require the CP to mitigate the ISI. Furthermore, a well-localized pulse shaping filter in time and frequency domain is applied in the OQAM-OFDM signal, which makes the OQAMOFDM signal less sensitive to the Doppler shift, compared with the OFDM signal [3]. Currently, some researchers have considered applying the OQAM-OFDM signal into radar systems. By performing the correlation processing on the transmitted OQAM-OFDM signal and the received signal, they provide the ambiguity function of the OQAM-OFDM signal, and verify that the OQAMOFDM signal can be used as a reasonable radar signal [6, 7]. However, the sidelobes level is still high because of the correlation processing. In [8], without using the correlation processing, we propose a pulse compression method for the OQAM-OFDM radar. Though good pulse compression performance can be achieved, the method is highly constrained (i.e., the number of subcarriers needs to be larger than ten times of the number of range cells.), which makes it difficult to be applied in practical applications. Besides, the realization process is quite complex, which directly follows the flow of communication processing (i.e., the demodulation process). However, for the radar system, only the target information is concerned. The demodulation process is unnecessary, and can be replaced by an estimation method, which has not been investigated in the open literatures. In this paper, we propose a concise and easily implemented pulse compression method for the OQAM-OFDM radar, without the insertion of the CP. The received OQAM-OFDM signal is established as a linear model form. The pulse compression is implemented by proposing an linear model-based estimation method, without using the demodulation process, and inter-range-cell interference (IRCI) free pulse compression is perfectly realized. Meanwile, the scope of observation will not be restrained by the number of subcarriers. Furthermore, we comprehensively analyze the signal-to-noise ratio (SNR) after the pulse compression, and find that the SNR gain can reach to the same as that of the method using the matched filtering. Finally, numerical simulations are provided to verify the effectiveness of the proposed method. 2. Signal Model We first give a brief introduction of the OQAM-OFDM radar signal model. With K subcarriers, the transmitted OQAM-OFDM signal can be
3
expressed as s (t) =
K−1 X
dk h (t)ej2πk∆f t ejπk/2 ,
(1)
k=0
where, ∆f = 1/T = B/K represents the subcarrier spacing, B is the signal bandwidth, h (t) [11] denotes a symmetrical real-valued pulse shaping filter, with unit energy, overlapping factor N and duration N T , dk denotes the real weight that is transmitted over the subcarrier k. For simplicity, we rewrite dk in vector notation as d = [d0 , d1 , · · · , dK−1 ]T ∈ RK , where [·]T denotes the transpose operator, R denotes the real field. At the receiver, let gr be the response of the target within the rth range cell, which includes the attenuation caused by the propagation process, and the radar cross section (RCS) coefficient from the scatterers in the rth range cell. Thus, the received signal for the radar system can be given by K−1 X X j2πk 2Rr 2Rr −j4πfc Rcr (2) dk h t − e T (t− c ) ejπk/2 + ω (t) , gr e ν (t) = c r k=0 where, c represents the speed of light, Rr is the range between the radar and the rth range cell, fc is the carrier frequency, ω (t) denotes the white Gaussian noise with zero mean and variance σ 2 . 3. Pulse Compression We suppose that the width of radar detection area is G, with R range cells, as shown in Fig. 1, where, G is smaller than the maximum unambiguous range of the radar. R0 represents the range from the radar to the nearest range cell. Thus, for the rth range cell, one can obtain Rr = R0 + rγ, where γ = c/2B is the radar range resolution. Let the received signal be sampled with a sample interval Ts = 1/B. One may find that Ts = 2γ/c and T = KTs . Therefore, for the delay with the rth range cell, 2Rr /c = 2R0 /c + rTs . The sampling begins when the first version of the transmitted signal is arrived (i.e., after 2R0 /c seconds). When there is a line-of-sight path between the radar and each of the range cell, ν (t) in (2) can be converted to the discrete time linear convolution of the weighting coefficient vector b = [b0 , b1 , . . . , bR−1 ]T ∈ CR and the transmitted signal sequence, and expressed as R−1 X νi = br si−r + ωi , i = 0, 1, . . . , Lν − 1, (3) r=0
4
where, the transmitted signal sequence is given by si =
K−1 X k=0
dk h (i)ej2πki/K ejπk/2 , i = 0, 1, . . . , Ls − 1,
(4)
Rr
the weighting coefficient br = gr e−j4πfc c , C denotes the complex field, Lν = N K + R − 1 is the length of received signal, Ls = N K is the length of transmitted signal, ωi denotes the sampling sequence of ω(t). In view of (3), the received signal can be rewritten in vector notation as v = Sb + w,
(5)
where, v = [v0 , v1 , . . . , vLν −1 ]T ∈ CLν denotes the received signal sequence, w = [ω0 , ω1 , . . . , ωLν −1 ]T ∈ CLν is the noise sequence with covariance matrix σ 2 I, S is a Lν × R matrix representing the transmitted signal matrix and can be written as s0 s−1 · · · s−(R−1) s1 s0 · · · s−(R−2) S = .. (6) .. .. , . . ··· . sLv −1 sLv −2 · · · sLv −R
where, si = 0 if i ∈ / [0, Ls − 1]. Thus, to achieve the pulse compression, the problem becomes how to estimate the weighting coefficient vector b with the received signal sequence v and the transmitted signal matrix S. Since w ∼ N (0, σ 2 I) and Lν = N K +R−1 > R, (5) constitutes a typical linear model in [9]. Besides, SH S is invertible, because the columns of S are linearly independent. Therefore, according to [9], b can be estimated by ˆ = SH S −1 SH v = b + SH S −1 SH w. (7) b
It is straightforward to show that the estimate of b is only related to itself. There is no interference from other range cells, and the real values can be estimated when the noise is not considered, which means the IRCI-free pulse compression is perfectly achieved. 4. SNR Analysis After Pulse Compression
In order to theoretically analyze the performance of the proposed method, the SNR after the pulse compression is derived. In view of (7), the noise 5
covariance matrix after the pulse compression can be expressed as −1 H H −1 H H H C =E S S S w S S S w h i h iH −1 −1 − E SH S SH w E SH S SH w −1 H −1 H H H H =E S S S ww S S S −1
=σ 2 SH S
(8)
,
where, E [·] denotes the statistical expectation. Thus, the SNR for the rth range cell after the pulse compression can be written as H bb rr |br |2 , SN Rr = (9) = [C]rr var ˆbr
ˆbr denotes the estimate of weighting coefficient for the rth range cell where, br , var ˆbr represents the variance of ˆbr . Obviously, SN Rr can reach to the maximum value only when the variance var ˆbr takes the minimum value. It is worth noting that var ˆbr can be written as var ˆbr = eTr Cer ,
(10)
T R where h er = [0, 0, .i. . , 0, 1, 0, . . . , 0] ∈ R and “1” is in the rth element. Since −1 E SH S SH w = 0 and C is the covariance matrix, C and C−1 are posiH tive definite matrixs. Therefore, C−1 can be decomposed as Φ Φ, where, 2 Φ −1 is a R×R reversible matrix. Additionally, notice that eTr ΦH ΦH er = −1 1. Let ξ1 = Φer and ξ2 = ΦH er , resulting in ξ1H ξ2 = 1. Then, according to the Cauchy-Schwarz inequality, we can obtain 2 (11) ξ1H ξ2 ≤ ξ1H ξ1 ξ2H ξ2 .
Therefore, we have
1 ≤ eTr ΦH Φer
eTr Φ−1 ΦH
−1
er = eTr C−1 er eTr Cer .
6
(12)
From (10) and (12), we can obtain 1 σ2 var ˆbr ≥ T −1 = H . er C er [S S]rr
(13)
−1 er . Thus, to The equality holds if and only if ξ1 = cξ2 or Φer = cr ΦH acquire the maximum SN Rr , the condition becomes ΦH Φer = cr er . In light of C−1 = ΦH Φ and (8), the condition can be given by c0 0 . . . 0 0 c1 . . . 0 SH S = σ 2 .. .. . . (14) .. . . . . . 0 0 . . . cR−1
Based on (4) and (6), we know that S is related to the transmitted signal. Thus, to obtain the maximum SN Rr , the problem converts to make S satisfy the condition in (14) by designing the transmitted real weights d. Then, in light of (6), it is notable that [S]lr = sl−r . Thus, Lν X H S S lr = si−r s∗i−l ,
(15)
i=1
where, {l, r} ∈ {1, 2, . . . , R}, ∗ represents the complex conjugate transpose. For large Lν , (15) can be approximated as H S S lr ≈
Ls −1−|l−r|
X
si s∗i+|l−r| .
(16)
i=0
P roof : For large Lν , (15) can be written as
∞ X S S lr ≈ si−r s∗i−l . H
(17)
i=−∞
According to (6), we know that si = 0 if i < 0 or i > Ls −1. Thus, only about 2R additional terms will be added in (17) compared with (15). If Lν R, these terms will be negligible. While Lν = N K +R−1, the condition Lν R is easy to be satisfied. Let r ≥ l and j = i − r, and (17) can be rewritten as
Ls −1−(r−l) ∞ X X ∗ S S lr = sj sj+r−l = sj s∗j+r−l . H
j=−∞
j=0
7
(18)
Similarly, for r < l , H S S lr =
Ls −1−(l−r)
X
sj s∗j+l−r .
(19)
j=0
Thus, (16) is certificated. It can be seen that (16) is the correlation function of the transmitted signal sequence. SH S is approximated as the autocorrelation matrix, and can be given as χ0 χ∗1 . . . χ∗R−1 χ1 χ0 . . . χ∗R−2 H S S = Ls .. (20) .. .. , . . . . . . χR−1 χR−2 . . . χ0 where,
L −1−m 1 sX sj s∗j+m , m = 0, 1, . . . , R − 1, χm = L s j=0
(21)
which can be regarded as the autocorrelation function of the transmitted signal sequence. To make (20) satisfy the condition in (14), χm = 0 if m 6= 0, which means SH S = Ls χ0 I. It is the approximate implementation, if the pseudo-random noise (PRN) sequence is utilized as the transmitted signal [9]. Thus, if the transmitted signal in (4) is equivalent to the PRN sequence, by designing the transmitted real weights d, the maximum SN Rr can be acquired. The variance of ˆbr can be given as var ˆbr = σ 2 /Ls χ0 . Typically, the transmitted signal is normalized, meaning that Ls χ0 = 1. Thus, according to (9), the maximum SN Rr can be given by SN Rmax,r
|br |2 Ls χ0 |br |2 = 2 . = σ2 σ
(22)
In addition, in view of (5), the mean transmitted power for the transmitted sequence is 1/Ls . Therefore, the SNR for the rth range cell before pulse compression can be expressed as SN Rr0 = 8
|br |2 . Ls σ 2
(23)
Based on (22) and (23), we can obtain SN Rmax,r = Ls SN Rr0 .
(24)
We can find that the SNR gain for the pulse compression is the length of the transmitted signal, which is exactly equal to that of the method using the matched filter. That is to say, the proposed method can realize the IRCI-free pulse compression and achieve the maximum SNR output without inserting the CP. 5. Simulation Results In this section, numerical simulation results are presented to evaluate the proposed pulse compression method. The pulse compression method proposed in [10] is used as a benchmark for comparison. In the simulations, a point target is located at 5256 m away from the radar. The Bandwidth is 150 MHz. The number of subcarriers is selected as K = 1024. The number of range cells is R = 512. The length of CP used in the OFDM signal is R − 1 = 511. The pulse shaping filter for the OQAM-OFDM signal is the raised cosine filter, with the overlapping factor N = 4. Without considering the noise, the results for the two pulse compression methods are shown in Fig. 2. It is obvious that the sidelobes for the two methods are really low and almost the same. Then, in order to verify the SNR gain obtained by our proposed method, we provide the pulse compression results and the output SNR curves with different received signal SNR, which are shown in Fig. 3 and Fig. 4. Theoretically, the SNR gain for the CP-OFDM signal is the number of subcarriers and equal to 10log10 K = 30.10 dB, while it is the length of the transmitted signal and equal to 10log10 (N K) = 36.12 dB for the OQAM-OFDM signal. From Fig. 3 and Fig. 4, we can find that the SNR gain is actually equal to the expected value. It means that our proposed method can completely exploit the transmitted energy, as the well-known matched filter does. Meanwhile, the energy of the CP is wasted by the CP-OFDM signal, resulting in a loss of energy of 10log10 ((K + R − 1)/K) = 1.76 dB. Thus, compared with the method in [10], our proposed method can improve the power efficiency. 6. conclusion In this paper, a concise and easily implemented pulse compression method has been proposed for the OQAM-OFDM radar. Firstly, we established 9
the OQAM-OFDM radar signal model. Then, without the insertion of the CP, the pulse compression method has been derived, achieving the IRCI-free pulse compression, and the scope of observation will not be restrained by the number of subcarriers. By analyzing the SNR after the pulse compression, we have found that the SNR gain can reach to that of the method using the matched filter. In addition, compared with the method using the CP-OFDM signals, our proposed method realizes the increase of the power efficiency. Finally, numerical simulations have verified the effectiveness of the proposed pulse compression method. Some future researches may be needed for the proposed OQAM-OFDM radar system. Since the pulse compression is completed, how can we apply the OQAM-OFDM radar into the target detection [12, 13] and the target tracking [14] is now under our current investigations. 7. Acknowledgment This work was supported by the National Natural Science Foundation of China under Grants 61971109, U19B2017, the Chang Jiang Scholars Program, the GF Science and Technology Special Innovation Zone Project, and the Fundamental Research Funds of Central Universities under Grant 2672018ZYGX2018J009. References [1] A. Gusmao, P. Torres, R. Dinis, and N. Esteves, “A reduced-CP approach to SC/FDE block transmission for broadband wireless communications,” IEEE Transactions on Communications, vol. 55, no. 4, pp. 801-809, Apr. 2007. [2] T. Zhang, X. Xia, and L. Kong, “IRCI free range reconstruction for SAR imaging with arbitrary length OFDM pulse,” IEEE Transactions on Signal Processing, vol. 62, no. 18, pp. 4748-4759, Sept. 2014. [3] B. Farhang-Boroujeny, “OFDM versus filter bank multicarrier,” IEEE Signal Processing Magazine, vol. 28, no. 3, pp. 92-112, May 2011. [4] D. Mattera, M. Tanda, and M. Bellanger, “Performance analysis of some timing offset equalizers for FBMC/OQAM systems,” Signal Processing, vol. 108, pp. 167-182, Mar. 2015. 10
[5] E. Kofidis, D. Katselis, A. Rontogiannis, and S. Theodoridis, “Preamblebased channel estimation in OFDM/OQAM systems: A review,” Signal Processing, vol. 93, no. 7, pp. 2038-2054, Jul. 2013. [6] W. Cao, J. Zhu, X. Li, W. Hu and J. Lei, “Feasibility of multicarrier modulation signals as new illuminators of opportunity for passive radar: orthogonal frequency division multiplexing versus filter-bank multi-carrier,” IET Radar, Sonar and Navigation, vol. 10, no. 6, pp. 1080-1087, Jun. 2016. [7] W. Xu, C. Wang, G. Cui, W. Wang and Y. Zhang, “The M-sequence encoding method for radar-communication system based on filter bank multi-carrier,” 2017 9th International Conference on Advanced Infocomm Technology (ICAIT), Chengdu, pp. 255-259, 2017. [8] Q. Shi, X. Li, T. Zhang, G. Cui, and L. Kong, “OQAM-OFDM radar approximated IRCI-Free pulse compression,” IEEE Transactions on Vehicular Technology, 2020. [9] S. M. Kay, “Fundamentals of statistical signal processing,” Technometrics, 1993. [10] T. Zhang, and X. Xia, “OFDM synthetic aperture radar imaging with sufficient cyclic prefix,” IEEE Transactions on Geoscience and Remote Sensing, vol. 53, no. 1, pp. 394-404, 2015. [11] M. Bellanger, “FBMC physical layer: A primer,” PHYDYAS, 2010. [12] X. Li, Z. Sun, T. S. Yeo, T. Zhang, W. Yi, G. Cui, and Lingjiang Kong, “STGRFT for detection of maneuvering weak target with multiple motion models,” IEEE Transactions on Signal Processing, vol. 67, no. 7, pp. 1902-1917, Apr. 2019. [13] W. Liu, J. Liu, Y. Gao, G. Wang, and Y. Wang, “Multichannel signal detection in interference and noise when signal mismatch happens,” Signal Processing, vol. 166, 2020. [14] J. Yan, W. Pu, S. Zhou, H. Liu, and M. S. Greco, “Optimal resource allocation for asynchronous multiple targets tracking in heterogeneous radar networks,” IEEE Transactions on Signal Processing, vol. 68, pp. 4055-4068, 2020. 11
ls cel
R
ge ran
g0
... g
...
gR1
r
R0
D
n ctio ete
aG Are
Figure 1: Illustration diagram.
12
0 OQAM-OFDM CP-OFDM
Amplitude(dB)
-50 -100 -150 -200 -250 -300 -350 5050
5150
5250
5350
5450
Range (m)
Figure 2: Range profiles of the point spread function.
13
SNR=-20dB CP-OFDM
X 256 Y 11.1
20
SNR=-20dB OQAM-OFDM
40
Amplitude(dB)
Amplitude(dB)
40
0
-20
X 256 Y 16.01
20
0
-20
5050
5150
5250
5350
5450
5050
5150
Range (m)
(a) Amplitude(dB)
Amplitude(dB)
5450
SNR=0dB OQAM-OFDM
40
X 256 Y 30.1
20
5350
(b)
SNR=0dB CP-OFDM
40
5250
Range (m)
0
X 256 Y 35.92
20
0
-20
-20 5050
5150
5250
5350
5050
5450
5150
5250
5350
Range (m)
Range (m)
(c)
(d)
Figure 3: Pulse compression results with different received signal SNR.
14
5450
50 45
CP-OFDM OQAM-OFDM CP-OFDM without energy loss
Output SNR (dB)
40 35 30 25 20 15 10 5 -20
-15
-10
-5
0
5
10
Received signal SNR (dB)
Figure 4: The output SNR after the pulse compression with different received signal SNR.
15
We declare that we have no known competing nancial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Credit Author Statement Qiao Shi: Conceptualization, Methodology, Software, Validation, WritingOriginal Draft, Writing-Review & Editing. Xueting Li: Investigation, Writing-Review & Editing. Tianxian Zhang*: Conceptualization, Methodology, Writing-Review & Editing, Funding acquisition. Xinyu Liu: Investigation, Writing-Review & Editing. Lingjiang Kong: Investigation, Writing-Review & Editing.
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