Joint design transmission waveform and sensing matrix ... - IEEE Xplore

2014 9th International Symposium on Communication Systems, Networks & Digital Sign (CSNDSP)

Joint Design Transmission Waveform and Sensing Matrix for the Compressive Sensing Radar Peng Chen1, Lenan Wu2 School of Information Science and Engineering, Southeast University, Nanjing, China 210096 Email: [email protected], [email protected] problems. The reference [13] considers optimal sensing matrix design with the optimality criterion depending on the coherence of the sensing matrix.

Abstract—The problems of optimal design transmission waveform and sensing matrix in the compressive sensing radar (CSR) system were considered. A new iteration algorithm was proposed for joint design transmission waveform and sensing matrix. This algorithm had better rate of convergence and less mutual coherence than traditional design methods. We gave simulation results of the target detection performance of the CSR system using proposed algorithm in the delay-Doppler plane. The simulation results showed that the optimal design CSR system had better target detection performance than random choosing radar system under conditions of different SNR, number of measurements, and number of targets.

However, optimizing the transmission waveform [2226] and sensing matrix [13] separately limits the detection performance of the CSR system. The reference [15] proposes a method for optimizing the transmission waveform and sensing matrix separately and simultaneously to decrease the cross correlations between different target responses. This method give the CSR system better reconstruction performance than the traditional separately design CSR system, but the realization of this method is very simple and we propose many improvements in this paper.

Keywords—compressive sensing radar (CSR); orthogonal matching pursuit (OMP) algorithm; joint optimization; mutual coherence; iteration algorithm

I.

INTRODUCTION

The sparsity of targets in the range-Doppler plane is exploited by the compressive sensing radar (CSR) system [1-7]. With sufficiently incoherent transmission waveform and sensing matrix, the target scene can be reconstructed by the technique of compressive sensing (CS) with much less sampling rate [8-10]. It has been demonstrated that CSR system can provide high-resolution [11, 12] than conventional radars [13, 14]. Furthermore, the CSR system eliminates the need for the pulse compression matched filter at the radar receiver [15]. These properties make it possible to use a low sampling rate, simple receiver designing, and inexpensive processing to realize a radar system [16-18]. Similar to traditional radar systems [19], the performance of the CSR is affected by the transmission waveform [20]. Many literatures have given some methods to design the transmitted signal. For example, when a direct optimization of the sensing matrix with respect to the waveform is prohibitive, [21] proposes a method to minimize a weighted coherence of the sensing matrix to design waveform. And [15] gives a simplified optimization objective function to design the transmission waveform. These proposed algorithms generate the waveform with smaller coherence and inner products compared to alltop sequence [12], so it can achieve better performance of CS reconstruction. In the CS theory, the sensing matrix also affects the recovery performance. A random Gaussian measurement matrix is typically used in CS recovery

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558

In this paper, a new iteration algorithm is proposed to jointly optimize the transmission waveform and sensing matrix. The algorithm includes two separately parts, which are the design of optimization transmission waveform and that of optimization sensing matrix. In both optimization parts Frobenius norm is used as the distance to measure the difference between the design and optimization ones, and random rotations of the signal and sensing matrix are used to get rid of the local optimization in iteration steps. After separately design these parts we jointly design them iteratively. The paper is organized as follows. In Section 2, we give a model of the CSR system. A new iteration algorithm is given in Section 3 to calculate the optimization transmission waveform, followed by a new algorithm to get the optimization sensing matrix in Section 4. And a joint optimization algorithm is given in Section 5. We give simulation results in Section 6. Finally, Section 7 concludes the paper. II.

RADAR SYSTEM MODEL

Targets can be represented by some points in the discrete delay-Doppler plane, and the sparsity of targets makes it possible to use the CS theory to detect the range and velocity of targets. As proposed by the reference [15], the received signal r in the CSR system can be written as (1) r Ȍx n where x is a sparse vector and nonzero elements indicate targets with corresponding range and velocity. The received signal is disturbed by additive Gaussian white


noise n . And Ȍ is the dictionary matrix, which represents all possible delay and Doppler frequencies of the transmission waveform s and can be written as

Ȍ

F T s, F T s, } , F 0

0

0

1

T L1s

M 1

arg min g G s,ĭ

s.t. s n

(2)

Tl LuN

III.

m

§ ZM0 0 " 0 · ¨ ¸ # ¸ 0 ZM1 % (4) FLmuL ¨ ¨ # % % 0 ¸ ¨¨ ¸ " 0 ZML1 ¸¹ © 0 describes the l delay and m Doppler frequency, respectively. N is the length of the transmission waveform s , and L N 1 is the number of discrete delay, and M is the number of discrete Doppler frequencies. In the CS theory, the sparse vector can be reconstructed by much less samples than the number of samples proposed by Nyquist theory. Consider the sensing matrix, which is the sample matrix in CS theory, can be represent as ĭ , which is often chosen random matrices, such as Gaussians or r1 randomly distributed matrices. In this paper, a new iteration algorithm is proposed to design this sensing matrix which has better performance than the random matrices and the algorithm proposed by [15]. After the compressed sample, the sample signal can be represent as ĭr

ĭȌx ĭn

°

½° (6) ¾ °¯ R i 2 · R j 2 ¿° The mutual coherence describes the off-diagonal elements of the matrix G R H R , so we can use the method proposed by [15] to represent the mutual coherence indirectly, which is g G

R iH R j

G G0

2 F

(8)

2 F

OPTIMIZATION METHOD FOR DESIGN THE TRANSMISSION WAVEFORM

1) Initialize the transmission waveform as a random N complex vector s 0 ^sn `n 1 , and set the iteration counter

k 0; 2) According to [15], the problem of (8) can be write as R Uk G0

2 F

and use the formula (23) in the reference

[15] to calculate the matrix U k ; 3) Use the formula (31) in the reference [15] to calculate the optimal signal s k under the condition of U k ; 4) Calculate the value of the optimization objective 2 function g s k , U k G k G k 0 F . If g s k , U k g s k 1 , U k 1 , set s = s k and go to step 2, otherwise go to step 5; 5) Rotate the transmission signal with M angles randomly, i.e., sˆ m s k 1 G exp jT m , where m 0,}, M 1 , ˆ under the condition and calculate the optimal matrix U m

of sˆ m , then we can get the optimization objective function ˆ . The optimal rotation angle is T T , where g sˆ , U

m

m

0

m0 arg min g sˆ m , Uˆ m

(5)

Defining the matrix R = ĭȌ , the performance of reconstruction the sparse vector x using some greedy algorithms, such as basis pursuit (BP) [27] or orthogonal matching pursuit (OMP) [28], can be represent by the mutual coherence of R , which is

P R max ® iz j

F

In this section, a new method is proposed to design the transmission waveform s when the sensing matrix is set as ĭ . The algorithm flowchart is shown in Fig.1 and described as follows:

(3)

and

y

2

1, n 1, 2,}, N

ĭ ĭ/ ĭ

where

§ 0l u N · ¨ ¸ ¨ I N uN ¸ ¨0 ¸ © L N l uN ¹

G G0

(7)

where G 0 is a diagonal matrix with same diagonal elements of G . So we should design the transmission waveform and sensing matrix to minimize the function g G . The problem of optimal design transmission waveform and sensing matrix can be represented as an optimization problem:

559

m

m0

. If g sˆ m0 , Uˆ m0 g s k 1 , U k 1 , let

s = sˆ m0 and go to step 2, otherwise stop the iteration algorithm and return the optimization transmission waveform s k 1 . Fig.2 shows the comparison of different optimization objective functions for design transmission waveform, which is the formula (8) used in our paper and the formula (22) used in the reference [15]. The simplified formula shown in [15] can describe the coherence of the matrix R approximately, but this optimization objective function has a great effect on the convergence of the signal design iteration algorithm proposed in this paper and even makes this algorithm cannot coverage to a transmission waveform with low coherence. So the iteration algorithm uses (7) as the optimization objective function to design transmission signal in this paper.

Fig.3 depicts the convergence of different iterative algorithms for design transmission signal, where the compared algorithm is proposed by [15] and (7) is used as the optimization objective function. We can see from this figure that the algorithm proposed in this paper can converges to a smaller value of mutual coherence than the method in [15], which even does not follow the direction to minimize the objective function. According to the CS recovery algorithm, such as OMP or BP, lower coherence transmission waveform gives a better reconstruction


performance, so the algorithm proposed in this paper to design the radar transmission waveform has a better convergence performance.

IV.

In this section, a new iteration algorithm is proposed to optimize the sensing matrix given the transmission waveform s , and this algorithm almost has the same steps with the algorithm proposed in this paper to design the optimization transmission signal, but vectorization is used to convert the sensing matrix to a vector, which can rotate with some random angles. The algorithm flowchart is shown in Fig.4 and the proposed algorithm is described as follows:

S

The Objective Function

Step 2, 3, 4 S S

Step 2, 3, 4 S

Step 5 S

1) Initialize the sensing matrix as a Gaussian random complex matrix ĭ 0 ĭ 0 / ĭ 0 F , and set the iteration

Step 2, 3, 4

counter k 0 ; 2) According to [15], the problem of (8) can be write as

S

S

Step 2, 3, 4

S

R Uk G0

Step 5

The Number of Iterations

Fig. 1. the Flowchart to Calculate the Optimization Transmission Waveform

2.30

2.24

4.8

2.22 4.6

The Optimal Objective Function The Objective Function in [15] 0

10

20

30

40

50

60

70

80

90

The Objective Function in [15]

The Optimal Objective Function

m

2.28

2.26

matrix · converts the vector to the corresponding matrix, then we can get the optimization target function ˆ ,U ˆ . The optimal rotation angle is T T , where g ĭ 0 m0 m m

2.20 100

m0


4.8

4.4

4.0

Our Algorithm Algorithm in [15] 20

30

40

50

60

70

80

90

ˆ ,U ˆ . If g ĭˆ , Uˆ g ĭ , U , let arg min g ĭ m m m0 m0 k 1 k 1 m

ˆ and go to step 2, otherwise stop sensing matrix ĭ = ĭ m0 the iteration algorithm and return the optimization sensing matrix ĭ k 1 . Fig.5 shows the comparison of different optimization objective functions for design sensing matrix. As described in the reference [15] that the approximation optimization objective function can be simplified, but from Fig.5 we can see the difference between the formula (8) and approximate one. Although the approximate objective function can describe the original one, the difference between the approximate one and the formula (8) may let the sensing matrix design algorithm proposed in this paper converge to the wrong sensing matrix. So in this paper we use (7) as the optimization objective function.

5.2

10

m

Fig. 2. the Comparison of Different Optimization Target Functions for Design Transmission Waveform

0

k 1

ˆ under the condition and calculate the optimal matrix U m ˆ , where function ˆ of sensing matrix ĭ matrix I m m


3.6

and use the formula (23) in the reference

F

ĭ = ĭ k and go to step 2, otherwise go to step 5; 5) Firstly, vectorize the sensing matrix to a vector I vec ĭ , then rotate this vector with M ' angles randomly, i.e., Iˆ I G exp jT , where m 0,}, M ' 1 ,

5.2

5.0

2

[15] to calculate the matrix U k ; 3) Use the formula (31) in the reference [15] to calculate the optimal signal s k under the condition of U k ; 4) Calculate the value of the optimization objective 2 function g s k , U k G k G k 0 F . if g s k , U k g s k 1 , U k 1 , set

S

0

OPTIMIZATION METHOD FOR DESIGN THE SENSING MATRIX

100


Fig.6 gives the convergence of different iterative algorithms for design sensing matrix, where the compared one is proposed by [15]. It can conclude from this figure that the sensing matrix design algorithm proposed by this paper has faster rate of convergence and do not fluctuate with the iteration. Furthermore, the sensing matrix calculated by this algorithm has better coherence performance.

Fig. 3. Convergence of Different Iterative Algorithms for Design Transmission Signal

560


Gaussian random complex matrix ĭ 0 The Objective Function

Step 2, 3, 4 ĭ ĭ

Step 2, 3, 4 ĭ

Step 5 ĭ

Step 2, 3, 4 ĭ

ĭ

Step 2, 3, 4

ĭ


Fig. 4. the Flowchart to Calculate the Optimization Sensing Matrix

2.30

7.0

2.25

5.5

2.20

5.0 2.15

4.5 4.0

2.10

3.5 3.0

2.05

The Optimal Objective Function The Optimal Objective Function

2.0

8


6.0

The Objective Function in [15]


6.5

2.5

2.00

0

10


Fig. 5. the Comparison of Different Optimization Target Functions for Design Sensing Matrix 7


, and

Algorithm in [15] Our Algorithm

7

6

5

4

3 0

4

8

12

16

20


6

Fig. 7. the Convergence Performance of Different Algorithms to Joint Design Transmission Waveform and Sensing Matrix

5

VI. 4

0

2

4

6

8

10

12

14

SIMULATION RESULTS

In this section, we give some simulation results. The CSR system transmits a waveform of length N 50 and measures a target scene with L 20 range and M 20 Doppler bins. And the number of targets which distribute in the discrete delay-Doppler plane uniformly is 3 . OMP algorithm is used as the CS recovery algorithm in this paper.

Our Algorithm Algorithm in [15]

3

16


Fig. 6. Convergence of Different Iterative Algorithms for Design Sensing Matrix

V.

F

and let k k 1 . If k ! K , go to step 5, otherwise go to step 2; 5) The optimal transmission signal and sensing matrix is s K and ĭ K , respectively. In this algorithm, K is the maximum number of iterations. And Fig.7 describes the convergence performance of different algorithms to joint design transmission waveform and sensing matrix, and the compared algorithm is proposed by reference [15]. We can see from this figure that the joint algorithm proposed by this paper has a faster rate of convergence and lower coherence. Although this joint design algorithm is more complex, the complexity is only caused by the random rotation.

ĭ

Step 5

0

ĭ0 / ĭ0

set the iteration counter k 0 ; 2) In the kth iteration, the sensing matrix is ĭ k 1 and use the transmission waveform design algorithm proposed in this paper to calculate the optimal signal s k ; 3) Use the sensing matrix design algorithm proposed in this paper to calculate the optimal sensing matrix ĭ k , when the transmission waveform is s k ; 4) Calculate the optimization objective function 2 H g G k G k G k 0 F , where G k ĭ k Ȍ k ĭ k Ȍ k ,

ĭ

Firstly, we compare the delay and Doppler frequency detection performance of the target using the joint optimization algorithm with the one using random transmission waveform and sensing matrix in the CSR system, where the SNR is from 0dB to 30dB. As shown in the Fig.8, the CSR system using joint optimization algorithm proposed in this paper has better target detection performance in the delay-Doppler plane with the whole simulation SNR range than the randomly choosing transmission signal and sensing matrix CSR system.

JOINT OPTIMIZATION TRANSMISSION WAVEFORM AND SENSING MATRIX

In this section, a joint optimization transmission waveform and sensing matrix algorithm is proposed, which integrates the optimal transmission waveform and sensing matrix calculated by algorithms proposed in this paper, and this algorithm is described as follows: 1) Initialize the transmission waveform as a random N complex vector s 0 ^sn `n 1 and the sensing matrix as a

561

100

The Error Probability of Targets Detection in the Delay-Doppler Plane



10-1

10-2

10-3

-4

10

The CSR System with Optimal Transmission Waveform and Sensing Matrix The CSR System with Random Transmission Waveform and Sensing Matrix 20

30

40

50

60

70

100

10-1

10-2

10-3

The CSR System with Optimal Transmission Waveform and Sensing Matrix The CSR System with Random Transmission Waveform and Sensing Matrix

10-4 0

The Number of Measurements

5

10

15

20

25

30

SNR (dB)

Fig. 8. Delay and Doppler Frequency Detection Performance with Different Number of Measurements

Secondly, we give the delay and Doppler frequency detection performance with different numbers of measurements. As we all known, more measurements will improve the reconstruction performance in the CS theory, but this requires high speed ADC and increases the cost of CSR system, so there is a tradeoff between the number of measurements and reconstruction performance. In this paper, the Nyquist sample number should be N ny N L 1 69 , and we simulate the CSR system using numbers of samples from 20 to 65 . SNR is set as the ratio between compressed signal and noise. From the simulate results shown in Fig.9 we can see that the CSR system with joint optimization design of transmission waveform and sensing matrix has better target detection performance than the system using random signal and sensing matrix with different numbers of measurements. Finally, Fig. 10 represents the target detection performance in the delay-Doppler plane with different numbers of targets. As the figure shown, increasing the number of targets will lead to the decrease of sparisity of the CSR system, and make the target detection performance worse. However, the CSR system using the joint optimization algorithm proposed in this paper gives better target detection performance with different numbers of targets than the CSR system without optimization. VII. CONCLUSION Because of the sparsity of targets in the range-Doppler plane, CS theory can be used in the radar system. And the sufficiently incoherent transmission waveform and sensing matrix makes it possible to detect the range and velocity of targets using CS reconstruction methods. In this paper, a new iteration algorithm to joint design transmission waveform and sensing matrix has been proposed based on the joint design method proposed by Z. Jindong [15]. This algorithm includes three steps. In the first step, we design the radar transmission waveform given the sensing matrix, then we design the sensing matrix under the condition of the optimal transmission waveform in the first step. Finally,

562


Fig. 9. Delay and Doppler Frequency Detection Performance with Different SNR 100

10-1

10-2

The CSR System with Optimal Transmission Waveform and Sensing Matrix The CSR System with Random Transmission Waveform and Sensing Matrix

-3

10

0

4

8

12

16

20

The Number of Targets Fig. 10. Delay and Doppler Frequency Detection Performance with Different Number of Targets

we update the transmission waveform based on the sensing matrix designed in the second step and go to the second step again. In this iteration algorithm we bring in the random rotation in design parts of transmission signal and sensing matrix, which can help to get rid of the local optimization in the iteration process, and we use better objective functions in both design parts to solve these optimal design problems. The theoretical analysis and simulate results show that our iteration algorithm has better performance in the rate of convergence and converges to lower mutual coherence between the transmission waveform and sensing matrix than the algorithm proposed by [15]. Finally, we compare targets detection performance in the delayDoppler plane using the optimal signal and sensing matrix with random ones under three conditions, which are different SNR, different numbers of measurements, and different numbers of targets. Simulation results have illustrated that the proposed algorithm is consistently stable


and demonstrated it superior performance under a wide range of condition. ACKNOWLEDGMENT This work was supported in part by National Natural Science Foundation of China (Grant No.61271204).

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