This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2010 proceedings.
Orthogonal Pulse Compression Codes for MIMO Radar System Lei Xu
Qilian Liang
Department of Electrical Engineering University of Texas at Arlington Arlington, TX 76010 Email:
[email protected]
Department of Electrical Engineering University of Texas at Arlington Arlington, TX 76010 Email:
[email protected]
Abstract—Inspired by recent advances in MIMO radar, we introduce orthogonal pulse compression codes to MIMO radar system in order to gain better target direction finding performance. We propose the concept and the design methodology for the optimized ternary pulse compression codes called the optimized punctured Zero Correlation Zone sequence-Pair Set (ZCZPS). According to codes property analysis, our proposed codes are able to provide the optimized autocorrelation and cross correlation properties during ZCZ. We also present a generalized MIMO radar system model using our proposed codes as pulse compression codes and simulate the target direction finding performance of the fluctuating and nonfluctuating system. The simulation results show that the more antennas used, the better target direction finding performance could be provided.
I. I NTRODUCTION There has been considerable interest in MIMO radars which employ multiple antennas both at the transmitter and at the receiver. The present important research of MIMO radar includes all kinds of techniques. Direction finding [1] is such a technology that a well known waveform is transmitted by an omnidirectional antenna, and a target reflects some of the transmitted energy toward an array of sensors that is used to estimate some unknown parameters, e.g. bearing, range, or speed. Also, beamforming [2] is another important process generally used in direction finding process that an array of receivers can steer a beam toward any direction in space. The advantages of using an array of closely spaced sensors at the receiver are the lack of any mechanical elements in the system, the ability to use advanced signal processing techniques for improving performance, and the ability to steer multiple beams at once. However, MIMO radars, unlike phased array radars, could transmit different waveforms on the different antennas of the transmitter, which makes it necessary to do the waveform design for the system. In this paper, we design a set of orthogonal ternary codes which are used as pulse compression codes for the MIMO radar system. To the best of our knowledge, it is the first time to introduce pulse compression codes to MIMO radar system. Pulse compression, known as a technique to raise the signal to maximum sidelobe (signal-to-sidelobe) ratio to improve the target detection and range resolution abilities of the radar system, allows a radar to simultaneously achieve the energy of a long pulse and the resolution of a short pulse
without the high peak power [3]. The phase-coded waveform is a basic waveform design suitable for pulse compression. A common form of phase coding is binary phase coding, in which the phase of each subpulse is selected to be either 0 or π radians. Barker code [4] is one family of binary phase code widely used nowadays that can produce compressed waveforms with constant sidelobe levels equal to unity. It has special features with which its sidelobe structure contains the minimum energy which is theoretically possible for binary codes, and the energy is uniformly distributed among the sidelobes (the sidelobe level of the Barker codes is 1/N 2 that of the peak signal) [5]. Unfortunately, the length N of known binary and complex Barker codes is limited to 13 and 25, respectively [6], which may not be sufficient for the desired radar applications. In [7] [8] [9], polyphase codes, with better Doppler tolerance and lower range sidelobes such as the Frank and P1 codes, the Butler-matrix derived P2 code and the linearfrequency derived P3 and P4 codes were intensively analyzed. However, the low range sidelobe of the polyphase codes can not reach the level zero either, what is more, the structure of polyphase codes is more complicated and is not easy to generate comparing with binary codes. In this paper, we provide a generalized MIMO radar signal model using our orthogonal ternary codes as pulse compression codes which could achieve the high resolution and orthogonality of MIMO radar system simultaneously. We focuses on the direction finding performance of the system regarding the orthogonality of the codes and we will consider the resolution performance in the later work. The simulation results show that better direction finding performance could be obtained by combining MIMO radar and pulse compression codes together. The rest of the paper is organized as follows. Section 2 introduces the definition and properties of optimized punctured ZCZPS. A method using optimized punctured sequence-pair and Hadamard matrix to construct ZCZPS is also given. Section 3 presents and analyzes a generalized MIMO radar system for our proposed codes. In section 4, some simulation results are provided by using specific examples with different number of uniform linear antennas at the transmitter and receiver of MIMO radar system. In Section 5, conclusions are drawn on our newly provided orthogonal pulse compression
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codes and MIMO radar system. II. O RTHOGONAL P ULSE C OMPRESSION C ODES A set of orthogonal pulse compression codes could be used in the MIMO radar system to gain the diversity and improve the direction finding performance. In this section, we will propose and analyze the concept and design methodology for a new ternary codes which could be applied to MIMO radar system. A. Definition and Design for Optimized Punctured ZCZ Sequence-Pair Set Matsufuji and Torii have provided some methods of constructing ZCZ sequences in [10] [11]. In this section, a set of novel ternary codes, namely the optimized punctured ZCZ sequence-pair set, is constructed through applying the optimized punctured sequence-pair [12] to the zero correlation zone. In other words, optimized punctured ZCZPS is a specific kind of ZCZPS. Definition 2-1 [12] Sequence u = (u0 , u1 , ..., uN −1 ) is the punctured sequence for v = (v0 , v1 , ..., vN −1 ), 0, if uj is punctured (1) uj = vj , if uj is Non-punctured Where P is the number of punctured bits in sequence v, suppose vj ∈ (−1, 1), uj ∈ (−1, 0, 1), u is P -punctured binary sequence, (u, v) is called a punctured binary sequencepair. Definition 2-2 [12] The autocorrelation of punctured sequence-pair (u, v) is defined Ruv (τ ) =
N−1
ui v(i+τ )modN , 0 ≤ τ ≤ N − 1
(2)
i=0
If the punctured sequence-pair has the following autocorrelation property: E, if τ ≡ 0modN Ruv (τ ) = (3) 0, otherwise the punctured sequence-pair is called optimized punctured N−1 sequence-pair [12]. Where, E = i=0 ui v(i+τ )modN = N − P , is the energy of punctured sequence-pair. Definition 2-3 Assume (X, Y) to be sequence-pair set of length N and the number of sequence-pairs K, where i = 1, 2, 3, ..., N − 1, p = 0, 1, 2, ..., K − 1, if all the sequences in the set satisfy the following equation: Rx(p) y(q) (τ ) =
N−1
(p) (q)∗
xi y(i+τ )mod(N ) =
i=0
⎧ ⎨ λN, 0, = ⎩ 0,
N−1
(p) (q)∗
yi x(i+τ )mod(N )
i=0
for τ = 0, p = q for τ = 0, p = q for 0 < |τ | ≤ Z0
(4)
where 0 < λ ≤ 1, then (x(p) , y(p) ) is called a ZCZ sequencePair, ZCZP (N, K, Z0 ) is an abbreviation. (X, Y) is called
a ZCZ sequence-Pair Set, and ZCZP S(N, K, Z0 ) is an abbreviation. Definition 2-4 If (X, Y) in Definition 2-3 is constructed by optimized punctured sequence-pair and a certain matrix, such as Hadamard matrix or an orthogonal matrix, here (p)
xi
(q) yi
∈ (−1, 1),
i = 0, 1, 2, ..., N − 1
∈ (−1, 0, 1), i = 0, 1, 2, ..., N − 1.
Then (X, Y) can be called optimized punctured ZCZ sequencepair set. Based on odd length optimized punctured binary sequence pairs and a Hadamard matrix, an optimized punctured ZCZPS can be constructed on following steps: Step 1: Considering an optimized punctured binary sequence-pair (u, v) of odd length N1 . Step 2: A Hadamard matrix B (consisting of several Walsh sequences) of order N2 is considered. The length of the sequence of the matrix is also N2 . Step 3: Processing bit-multiplication on the optimized punctured binary sequence-pair and each row of Hadamard matrix B, then sequence-pair set (X, Y ) is obtained. Because of space limit, the steps would be described in detail in the later version, as well as the proof. Since the optimized punctured binary sequence-pairs here are of odd lengths and the lengths of Walsh sequence are 2n , n = 1, 2, ..., common divisor of N1 and N2 is 1, GCD(N1 , N2 ) = 1. The sequence-pair set (X, Y) is optimized punctured ZCZPS and N1 − 1 is the zero correlation zone Z0 . The length of each sequence in optimized punctured ZCZPS is N = N1 ∗ N2 . The number of sequence-pairs in optimized punctured ZCZPS rests on the order of the Hadamard matrix. The sequence x(p) in sequence set X and the corresponding sequence y(p) in sequence set Y construct a sequence-pair (x(p) , y(p) ) that can be used as a pulse compression code. B. Properties of Optimized Punctured ZCZ Sequence-pair Set In this section, the optimized punctured ZCZPS (X, Y) is constructed by 5-bit length optimized punctured binary sequence-pair (u, v), u = [+ + − + −], v = [+ + 000] (using + and − symbols for 1 and −1 ) and Hadamard matrix H of order 4. We follow the three steps presented in the previous part to construct an optimized punctured ZCZPS. The number of sequence-pairs here is 4, and the length of each sequence is 5 ∗ 4 = 20. Each row of each matrix X = [x(1) ; x(2) ; x(3) ; x(4) ] and Y = [y(1) ; y(2) ; y(3) ; y(4) ] constitute a certain optimized punctured ZCZP (x(p) , y(p) ). The autocorrelation property and cross correlation property of 20-bit length optimized punctured ZCZ sequence pair set (X, Y) are shown in Fig. 1. From the Fig. 1, the sidelobe of autocorrelation of the codes can be as low as 0 when the time delay is kept within Z0 = N1 = 5 (zero correlation zone) and the cross correlation value is kept as low as 0 during the whole time domain. It is known that a suitable criterion for evaluating pulse compression code of length N is the ratio of the peak signal divided by the peak signal sidelobe (PSR). The sidelobe of
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Fig. 1.
listed along the diagonal of the matrix S = diag(s1 , ..., sM ). The transmitted waveforms could be normalized such that |si |2 = 1/M . The normalizing method ensures that transmitted power is not dependent of the number of transmitting antennas. Suppose, all antennas transmit the same waveform, S = sIM , where the subscript M denotes the order of the unity matrix. Similar to the transmitter, the model for the array at the receiver could be developed, resulting in an N × P channel matrix K. Similarly, the first antenna on the receiving part will be taken as the reference with zero phase. The signal radiated by the receive antenna impinges at angle θ0 which is the angle of arrival (AOA) relative to the receiving array normal. The phase difference for n-th transmit antenna is ϕn = 2π((n − 1)dr /λ)sinθ0 . For phase-modulated pulse compression waveforms, the corresponding pulse compression P −1 (n) codes C (n) = p=0 Cp (t−pτc ) have to be applied to each receive antenna to implement the matched filter. The signal vector arrived at the n-th receive antenna could be given by
Correlation property of optimized punctured ZCZPS
Normalized amibiguity
Periodic Autocorrelation Property 1 0.8 0.6 0.4 0.2 0 −20
−15
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−5
0
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Normalized amibiguity
Delayτ/tb Periodic Cross Correlation Property 1
0.5
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−15
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Delayτ/t
b
the new code shown in Fig. 1 can be as low as 0, so the PSR can be as high as infinite which could effectively avoid masking mainlobes of the other targets. In addition, the zero cross correlation properties make those codes well cooperate in the MIMO radar system without introducing the interference. Hence, our proposed codes could be used as a set of orthogonal pulse compression codes in the MIMO radar system. III. S IGNAL M ODEL O F MIMO R ADAR In this section, we describe a signal model for the MIMO radar system using orthogonal pulse compression codes. Assume a radar system that utilizes an array with M antennas at the transmitter, and N antennas at the receiver. For simplicity, we assume that the arrays at the transmitter and receiver are parallel. A transmitting linear array made up of M elements equally spaced a distance d apart. The elements are assumed to be isotropic radiators in that they have uniform response for signals from all directions. The first antenna will be taken as the reference with zero phase. The signal radiated by the transmit antenna impinges at angle θ which is the angle of arrival (AOA) relative to the transmitting array normal. From simple geometry, the difference in path length between adjacent elements for signals transmitting at an angle θ with respect to the normal to the antenna, is dsinθ. This gives a phase difference between adjacent elements of φ = 2π(d/λ)sinθ, where λ is wavelength of the received signal. And the phase difference for m-th transmit antenna is φm = 2π((m − 1)d/λ)sinθ [3]. For convenience, we take the amplitude of the received signal at each element to be unity. N −1 (m) A pulse compression code C (m) = p=0 Cp (t − pτc ) is applied to m-th transmit antenna, and the signal vector induced by the m-th transmit antenna is given by (m)
g (m) = e−jφm [C0 1≤m≤M
(m)
, C1
(m)
, C2
(m)
, ..., CP −1 ]T , (5)
The signal vectors are organized in the M ×P transmit matrix G = [g (1) , g (2) , ..., g (M ) ]T . The transmitted waveforms are
(n)
k (n) = e−jϕn [C0
(n)
, C1
(n)
, C2
(n)
, ..., CP −1 ]T ,
1 ≤ n ≤ N, K = [k (1) , k (2) , ..., k (N ) ]T
(6)
Since we study the pulse compression technique in the MIMO radar system, the number of matched filters on the receiving side should be the same as the number of transmitting signals. For the further research in this paper, we assume the number of transmitting antennas and that of receiving antennas are the equivalent. Here, N = M . Assume there is a far field complex (multiple scatters) target and it is known that small changes in the aspect angle of complex targets can cause major changes in the radar cross section (RCS). Here, RCS for each receiver antenna is assumed to have isotropic reflectivity modeled by zero-mean, unit-variance per dimension, independent and identically distributed (i.i.d.) Gaussian complex random variables λi . The target is then modeled by the diagonal matrix ⎡ 1 Σ= √ 2M
⎢ ⎢ ⎢ ⎢ ⎣
λ0
0
0 .. .
λ1 .. .
0
...
··· .. . .. . 0
0 .. . 0 λM −1
⎤ ⎥ ⎥ ⎥ ⎥ ⎦
(7)
wherethe normalization factor makes the target average M −1 2 i=0 |λi | RCS= independent of the number of receiving 2M antennas in the model. The nonfluctuating target modeled using non-zero constants for λi is identified as ”Swerling0” or ”Swerling5” model. For the fluctuating target, if the target RCS is drawn from the Rayleigh pdf and vary independently from pulse to pulse, the target model represents a classical ”Swerling2” model. Processing the transmit matrix, the target matrix and the receive matrix together, the MIMO radar channel model is given by M × M matrix shown in (8). According to the (8), it is easy to notice that P −1 (m) (n) specified by the pulse compression codes p=0 Cp Cp
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⎡
⎤ ··· 0 ⎢ ⎥ . .. ⎢ ⎥ ⎢ ⎥ . .. λ1 ⎢ ⎥ ∗ ∗ H ∗ ⎢ 0 ⎥ H = K[G Σ] = ⎢ ⎥ [g1 g2 . . . gM ] ⎢ . ⎥ . . ⎣ ⎦ .. .. 0 ⎣ .. ⎦ kM 0 ... 0 λM −1 ⎡ (p) (p) (p) (p) P −1 P −1 λ1 ej(φ2 −ϕ1 ) p=0 C2 C1 λ0 ej(φ1 −ϕ1 ) p=0 C1 C1 P −1 (p) (p) P −1 (p) (p) ⎢ λ1 ej(φ2 −ϕ2 ) p=0 C2 C2 ⎢ λ0 ej(φ1 −ϕ2 ) p=0 C1 C2 = ⎢ .. ⎢ .. ⎣ . . P −1 (p) (p) P −1 (p) (p) j(φ1 −ϕM ) λ0 e λ1 ej(φ2 −ϕM ) p=0 C2 CM p=0 C1 CM ⎡
k1 k2 .. .
⎤
λ0
0
exists at each position of the matrix. If we select orthogonal pulse compression codes for transmit and receive antennas, it is satisfied that P −1 Es m = n (m) (n) Cp Cp = (9) 0 m = n p=0
The H matrix turns to be shown as (10). As a result, the signal vector received by the MIMO radar is given by r = HS + n
(11)
Where the additive white Gaussian noise vector n consists of i.i.d, zero-mean complex normal distributed random variables. In this case we assume that all antennas transmit the same waveform, S = sIM . If receiver antenna uses a beamformer to steer towards direction θ0 , ϕn = 2π((n − 1)dr /λ)sinθ0 . The beamformer is modeled by a diagonal matrix ⎡ ⎤ ··· 0 ejϕ1 0 ⎢ ⎥ ⎢ 0 ⎥ ejϕ2 · · · 0 ⎢ ⎥ β(θ0 ) = ⎢ . (12) . . ⎥ .. .. . .. ⎣ .. ⎦ 0
...
0
ejϕM
The output of the beamformer is shown in (13). Processing the output of y, we obtain the diagonal of the output matrix y and change it into a M × 1 vector y . The output of the beamformer at the receiver antenna is y
=
[λ0 ej(φ1 +ϕ1 −ϕ1 ) Es s, λ1 ej(φ2 +ϕ2 −ϕ2 ) Es s, · · · ,
λM −1 ej(φN +ϕN −ϕN ) Es s]T + n
(14)
where Es >> σ 2 (n ). In MIMO radar for direction finding (DF) purpose, the transmit antennas are sufficiently separated, so the phase shifts at the transmitter are set to zero. It is easy to see that when θ = 0, φm = 2π(d/λ)sinθ = 0 and g (m) = (0) (1) (2) (N −1) T [Cm , Cm , Cm , ..., Cm ] . If the beamformer can well estimate the direction θ0 at the receiver antenna, stating differently, θ0 ∼ = θ0 and ϕn = ϕn . The result at the MIMO receiver antennas is
y = [λ0 Es s, λ1 Es s, . . . , λM −1 Es s] + n T
(15)
(8)
... ... .. . ...
λM −1 ej(φM −ϕ1 )
P −1
p=0 P −1 λM −1 ej(φM −ϕ2 ) p=0
(p)
(p)
(p)
(p)
CM C1 CM C2
.. . P −1 (p) (p) λM −1 ej(φM −ϕM ) p=0 CM CM
⎤ ⎥ ⎥ ⎥. ⎥ ⎦
Here, we apply MSE (mean square error) to the output y of receiver antennas to estimate direction finding error. Similar to RAKE receiver, we can choose the path which could provide the minimum phase difference in y to find the direction of the target, which could be called Selective Combining. Considering target detection or recognition, we could also sum up all the paths in order to achieve the diversity gain, however, we only focus on the research of direction finding in this paper. IV. S IMULATIONS AND A NALYSIS In this section, we are running MATLAB simulations of the MIMO radar system using different number of antennas to see the direction finding performance. The numbers of transmitting and receiving antennas are both M , the transmit antennas are spaced sufficiently and the antenna array is used in the receiving part. The target fluctuating model in which the channel fluctuated according to a Rayleigh distribution is considered besides the nonfluctuating model. Estimation MSE is used as the common figure of merit for comparing the performance. We choose the path which provides the best performance before estimate MSE called Selective Combining method. Using nonfluctuating and fluctuating target model, the MIMO radar systems of different antennas are illustrated in Fig. 2. From the Fig. 2, it is easy to see that under the situation of both nonfluctuating and fluctuating models the system with more antennas could always achieve less MSE than the system with less antennas especially when the SNR value is not large. However, considering the nonfluctuating model, when the SNR increases the advantage becomes less distinct. Comparing the Fig.2(a) and 2(b), the performance of our system for fluctuating modeal is degraded because of the Rayleigh fading. According to the results, a general conclusion could be drawn that the more antennas MIMO radar system utilized the better direction finding performance could achieved in the both models. V. C ONCLUSIONS In this paper, we introduced the orthogonal pulse compression codes to the MIMO radar system which has the same number of transmit and receive antennas to improve the radar direction finding performance. We provided a set
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⎡ ⎢ ⎢ H = ⎢ ⎣
⎡ ⎢ ⎢ y = rβ(θ0 ) = HSβ(θ0 ) + n = ⎢ ⎣
0 λ1 ej(φ2 −ϕ2 ) Es .. .
··· ··· .. .
0 0 .. .
0
...
0
λM −1 ej(φM −ϕM ) Es
λ0 ej(φ1 −ϕ1 +ϕ1 ) Es s 0 .. . 0
Fig. 2.
⎤
λ0 ej(φ1 −ϕ1 ) Es 0 .. .
0 ... j(φ2 −ϕ2 +ϕ2 ) Es s . . . λ1 e .. .. . . 0 ...
MSE of beamforming at the receiver
⎥ ⎥ ⎥ ⎦
(10)
⎤
0 0 .. .
⎥ ⎥ ⎥ + n (13) ⎦
λM −1 ej(φM −ϕM +ϕM ) Es s
ACKNOWLEDGMENT
0
10
This work was supported in part by the National Science Foundation under Grants CNS-0721515, CNS- 0831902, CCF0956438, CNS-0964713, and Office of Naval Research (ONR) under Grant N00014-07-1-0395 and N00014-07-1-1024.
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[1] S. Pasupathy and A. N. Venetsanopoulos, “Optimum active array processing structure and space-time factorability,” IEEE Trans. Aerosp. Electron. syst., vol.10, pp. 770-778, 1974. [2] S. Haykin, J. Litva and T. J. Shepherd, Radar Array Processing, SpringerVerlag, New York, 1st edition, 1993. [3] S. Ariyavisitakul, N. Sollenberger, and L. Greenstein, Introduction to Radar System, Tata McGraw-Hill, 2001. [4] R. H. Barker, “Group Synchronizing of Binary Digital Sequences,” Communication Theory, pp. 273-287, 1953. [5] J. L. Eaves and E. K. Reedy, Principles of Modern Radar, Van Nostrand Reinhold, 1987. [6] L. Bomer and M. Antweiler, “Polyphase Barker sequences ,” Electronics Letters, 1577-1579, Dc. 1989. [7] R. L. Frank, “Polyphase codes with good nonperiodic correlation properties”, IEEE Transactions on Information Theory, IT. -9, pp. 43-45, Jan. 1963. [8] B. L. Lewis and F. F. Kretschuner, “A new class of polyphase pulse compression codes and techniques”, IEEE Transactions on Aerospace and Electronic Systems, AES-17, pp. 364-372, May. 1981. [9] B. L. Lewis and F. F. Kretschuner, “Linear frequency modulation derived polyphase pulse compression codes”, IEEE Transactions on Aerospace and Electronic Systems, AES-18, pp. 637-641, Sep. 1982. [10] S. Matsufuji, N Suehiro , N Kuroyanagi and P Z Fan, “Two types of polyphase sequence set for approximately synchronized CDMA systems,” IEICE Trans. Fundamentals, E862A(1): pp. 229-234, Jan. 2003. [11] H. Torii, M. Nakamura and N. Suehiro, “A new class of zero correlation zone sequences,” IEEE Tran. Inform.Theory, Vol.50: pp. 559-565, Mar. 2004. [12] T. Jiang, Research on Quasi-Optimized Binary Signal Pair and Perfect Punctured Binary Signal Pair Theory, Ph.D Dissertation: Yanshan University, 2003.
of new optimized triphase pulse compression codes, gave a specific example and analyzed the codes’ properties. We presented and analyzed a generalized MIMO radar system model for our provided framework. Simulation results showed that significant SNR gain could be obtained in MIMO radar system using orthogonal pulse compression codes. The MIMO radar system using more antennas outperforms the one having less antennas.
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