A Proposed Compressive Sensing Based LFMCW Radar Signal Processor

Sameh G.   Salem; Fathy M.    Ahmed; Mamdouh H.   Ibrahim; Abdel Rahman H.    Elbardawiny

doi:10.17577/IJERTV4IS040893

Volume 04, Issue 04 (April 2015)

A Proposed Compressive Sensing Based LFMCW Radar Signal Processor

DOI : 10.17577/IJERTV4IS040893

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Authors : Sameh G. Salem, Fathy M. Ahmed, Mamdouh H. Ibrahim, Abdel Rahman H. Elbardawiny
Paper ID : IJERTV4IS040893
Volume & Issue : Volume 04, Issue 04 (April 2015)
DOI : http://dx.doi.org/10.17577/IJERTV4IS040893
Published (First Online): 21-04-2015
ISSN (Online) : 2278-0181
Publisher Name : IJERT
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A Proposed Compressive Sensing Based LFMCW Radar Signal Processor

Sameh G. Salem, Fathy M. Ahmed, Mamdouh H. Ibrahim, Abdel Rahman H. Elbardawiny

Military Technical College Cairo, Egypt

Abstract:- Application of Compressive Sensing (CS) in Linear Frequency Modulation Continuous Wave (LFMCW) radar had been investigated and proved by the authors in [8]. In this paper, a new approach for applying CS in LFMCW radar signal processing is introduced. The proposed approach depends on apply CS in range processing direction by acquiring the base band radar signal with a sampling rate, according to a Pseudo Random (PN) sequence, less than that of the well known Nyquist rate. The information of the received radar signal (target range and speed) are reconstructed by the use of Complex Approximate Message Passing (CAMP) reconstruction algorithm. The superiority of the proposed LFMCW radar signal processor (Based on CS) compared to that of the traditional one (based on Fast Fourier Transform (FFT)) from points of view of detection performance through Receiver Operating Characteristics (ROC) curves, resolution performance and hardware complexity is validated.

Keywords: LFMCW Radar, Compressive Sensing, CAMP, FFT.

INTRODUCTION

The small size, simplicity and economy of LFMCW radar systems were the basic reasons of wide applications in many areas such as aviation, military, security, navigation, and automotive. Signal processing plays an important role in such radars. The early work of Nyquist, Shannon [1] and Moores law [2] on sampling and representation of continuous signals, signal processing are moved from the analog to the digital domain. The general block diagram of digital LFMCW radar receiver is shown in the Fig.(1) [3].

challenges are met by the radar system to acquire and process wideband signals;
1. Acquiring signals at very high sampling rates,
2. Storing the resulting large amounts of data, and
3. Processing/analyzing large amounts of data.
Sampling of very wide bandwidth signals may require ADC hardware that is unavailable or very expensive. So, an efficient way to deal with these wide bandwidths and high data rates as well as the large amount of data is the emerging field of Compressive Sensing (CS) [4]. CS theory states that, given certain circumstances, it is possible to reconstruct a signal which is sampled at a rate below the Nyquist rate [5,6]. Applying CS theory in LFMCW radar signal processing may lead to a reduction in sampling rate, complexity, power consumption, and cost. On the other hand, performance is a critical point to be considered.

In this paper, a new approach is introduced for LFMCW radar signal processing based on the concept of compressive sensing (CS). However, while the traditional LFMCW radars use well established processing algorithms and detection schemes, such as Fast Fourier Transform (FFT) and Constant False Alarm Rate (CFAR) detectors to extract target range and velocity, the reconstruction of the target scene from the CS measurements involves the use of highly nonlinear algorithms such as L1-minimization and Complex Approximate Message passing (CAMP). These algorithms have several free parameters that must be tuned properly in order to achieve good performance [6-8].

Threshold

Signal processor

ADC

Baseband signal (beat frequency)

Figure 1 Block diagram of digital LFMCW radar receiver

Hit report

The rest of this paper is organized as follows; section 2 gives a survey on the principle of CS theory. Section 3 illustrates the application of LFMCW radar based on CS. Section 4 describes the proposed CS-based LFMCW radar signal processor. Performance evaluation of LFMCW radar signal processing using CS compared to the traditional one is provided in section 5. Finally, conclusion comes in section 6.

There are many requirements of advanced LFMCW radar such as: small size, low cost, weight and complexity. But the main major factors are accuracy and resolution. For most modern high-resolution multi-channel radar systems, one of the major problems to deal with is the huge amount of data to be acquired, processed and/or stored. But why all these data are needed?. According to the well known Nyquist-Shannon sampling theorem, natural signals have to be sampled at least twice the signal bandwidth to prevent aliasing. Many
PRINCIPLES OF COMPRESSIVE SENSING

Considering a discrete time (or space) signal having N samples. This signal can be represented as a complex N- length vector x CN. Known by the Matrix theory, any signal in the space CN can be represented by orthonormal basis {xi} where i from 1 to N with linear form. Orthonormal basis {xi}

can be rewritten as x=[x1, x2 xN] in the form of Matrix. Any signal S CN can be represented in terms of x as [4-6]:

N

in Fig.(2). Sparse discrete signal of LFMCW radar is generated after multiplying the received base band radar signal (after ADC) with a transformation matrix of

dimension N x N. The transformation matrix ( ) should be

S i xi

i1

(1)

Fourier matrix to satisfy the sparsity condition. The generated sparse signal is N samples. This sparse signal is used to

Where, i is the coordinate of x. The specific form of i is i S, xi .

generate a compressed or measurement vector, y, with length M where M < N after multiplication with measurement matrix, (N x M).

Eq. (1) can be rewritten as [9]:

T

S = x (2)

X(t)

ADC

X(n)

Sparse signal S

Compressed sparse signal

Y(n)

Reconstruction algorithm

Decision

Where,

[1,2

,……,N ] .

Base band

signal

1 xN 1 xN

1 xM

It is clear that

and S are equivalent representations of the

Sensing matrix (A)

Transformation Matrix ()

( NxN)

Measurement Matrix ()

( NxM )

same signal, X. In other words, S is the time domain signal, and is the representation in the transformation domain. Suppose K be a non-zero number of elements of . If K is smaller than N, it can be shown that the signal S is sparse or compressible. Meanwhile, basis vector {xi} is the sparse basis of the signal S [8,9].

The information contained in many natural signals can be represented more concisely if the signals are looked at in a proper transform domain. Then the complex signal of interest x CN can be recovered from an under sampled set of linear projections which is called measurement vector y CM, where M < N, using one of a reconstruction algorithm. The sensing model can be mathematically represented as [5]:

= (3)

where the M x N matrix A is called the sensing matrix. The sensing mechanism of acquiring M measurements via linear projection through the operator is mathematically represented as in Eq.(3). Using the sparse representation, then the compressed vector, y can be written as [5]:

= = S (4)

Where is N x N measurement CS matrix. For the special class of matrices with Gaussian independent and identically distributed (i.i.d.) random entries, it has been demonstrated that the Restricted Isometry Property (RIP) condition is satisfied with high probability to the following equation [3]

Figure 2 Application of CS in LFMCW radar [8].

According to [8,9], The target information (range and speed) contained in the original signal can be reconstructed from the compressed measurement using different reconstruction algorithms. According to [8], CAMP algorithm 10-12] gives the best result. The Nyquist rate based CS (CAMP) approach has been found to give the highest detection performance and an extra amount of calculations. In the next section, a proposed approach is introduced to reduce the sampling rate below the Nyquist and reduce the extra calculations (complexity).

THE PROPOSED CS-BASED LFMCW RADAR

SIGNAL PROCESSOR

The compressed samples (M) are then transformed into sparse (measurement vector, y) by applying FFT on azimuth direction. To get the required target parameters (range and speed), a reconstruction process based on CAMP algorithm is performed on the obtained measurement, y. To achieve certain false alarm probability using the internal adaptive thresholding function in the CAMP algorithm, a gain factor

(G) is used. The idea behind the proposed approach, shown in Fig.(3), is how to get the compressed sparse signal ,y(n), directly from the base band signal without performing multiplication with a sensing matrix and without acquiring the base band signal with Nyquist rate.

M CKlog(N K) (5)

Compressed signal in range

Compressed sparse signal in range

Where N is the total number of the original signal samples, M

X(t)

W(n)

FFT

Y(n)

Reconstruction

Decision

is the number of measurement vector such that M < N and C

Base band signal

ADC

is a constant dependent on the total number of samples and the number of useful signals samples (K) [3].

( ZXN)

( ZXM)

(azimuth direction) ( ZXM)

algorithm

( ZXN)
1. APPLICATION OF CS IN LFMCW RADAR
  
  Application of CS in LFMCW radar signal processing had
  
  PN- code
  
  Sensing matrix(A) ( MXN)
  
  Gain(G)
  
  been investigated and proved by the authors in [8] as shown
  
  Figure 3 General block diagram of the proposed CS-based LFMCW radar signal processor
  
  This is achieved by sampling the base band signal with a PN sequence with length M < N achieving the required reduction ratio () in range samples where =M/N according to Eq.(5).
  
  A simple description for the block diagram of the proposed approach is shown in Fig.(4) where the reconstruction algorithm is initialized by the sensing matrix (A), the compressed sparse signal (y), and the threshold gain (G).
  
  with the receiver thermal noise and known target returns. The thermal noise simulated is Additive White Gaussian with zero mean and unity variance. The setup procedure for evaluating the performance of the proposed LFMCW radar signal processor based on CS compared to the traditional LFMCW radar signal processor based on FFT is shown in Fig.(5). The comparison aspects are the detection performance, resolution performance and hardware complexity.
  
  Received radar signal x(t)
  
  Obtain measurement W(n)
  
  Obtain sparse signal Y= FFT(W)
  
  PN- code ( range)
  
  Random sequence
  
  Generate sensing matrix(A)
  
  ADC
  
  Traditional LFMCW radar (FFT based)
  
  Fs (Nyquist)
  
  Base band signal (beat frequnecy)
  
  Decision
  
  CFAR
  
  Decision
  
  Reconstruction algorithm
  
  The proposed approach (CS based)
  
  ADC
  
  Random sequence
  
  Obtain the estimated signal
  
  Y * A
  
  1 Target present
  
  PN code
  
  Sensing matrix Gain (A)
  
  Gain
  
  Compute the threshold
  
  Comparator
  
  0 Target absent
  
  Figure 5 Setup procedure for evaluating the proposed approach compared to the traditional one
PERFORMANCE EVALUATION OF THE PROPOSED LFMCW RADAR SIGNAL PROCESSOR

The performance of the proposed LFMCW radar signal processor based on CS is evaluated using Matlab. The base band radar signal is generated using the simulation model outlined in [13]. The number of total range-Doppler samples to be processed is assumed to be 128 range samples and 32 Doppler samples. It simulates a composite radar environment

Detection performance is evaluated for both the proposed algorithm and the traditional one through ROC by calculating both probability of false alarm (Pfa) and probability of detection (Pd). The probability of false alarm (Pfa) is calculated using the CFAR processing in the traditional LFMCW radar. But in the proposed approach, it is computed after the reconstruction algorithm where the internal threshold is used to control the output false alarm probability. The reconstruction algorithm output can directly indicate the target existence or not. Fig.(6) shows the comparison between the probabilities of false alarm (Pfa) and the threshold gains for both the proposed approach and the traditional LFMCW radar. This figure is used to set the correct threshold gains in both the proposed approach and the traditional one which achieve the same probability of false alarm insuring fair detection comparison.

The detection performance of the proposed approach is evaluated using different reduction percentages in range samples; 25%, 50% and 75%. This detection performance is compared with that of the traditional one at different SNRs and at Pfa of 10-5.

The detection performance through ROC of both the proposed approach and the traditional one is evaluated for three cases representing the existence of different number of targets and at different reduction ratios.

– Case I: assumes that a single target is located in one Coherent Processing Interval (CPI) such that its expected range cell is 64 and its normalized Doppler frequency is 0.5 and at different reduction ratios () in range samples. The detection performance of this case is shown in Fig.(7) and its range-Doppler representation is shown in Fig.(8).

– Case II: assumes that nine targets are located such that their expected range cells are 16,32, 64 ,48,52,70,80,96 and 105 with the same normalized Doppler frequency as in case I. The detection performance in this case is shown in Fig.(9) and its range-Doppler representation is shown in Fig.(10).

Normalized amplitude

1

0.8

0.6

1

Normalized amplitude

0.8

0.6

– Case III: assumes that ten targets are located such that their expected range cells are 16, 32, 64, 48, 52, 70, 80, 96, 105 and 114 with the same normalized target Doppler frequency of 0.5. The detection performance of this case is shown in Fig.(11) and its range-Doppler representation is shown in Fig.(12).

0.4

0.2

0

40

30

20

10

Doppler

0 0

(a)

100

50

Range

150

0.4

0.2

0

40

30

20

10

Doppler

0 0

(b)

100

50

Range

150

100

Traditional LFMCW radar

Figure 8 Range-Doppler processing reresentation for case I, SNR= -5dB and Pfa = 10-5.

CONCLUSION

In this paper, LFMCW radar signal processing based on CS theory is performed to extract target information such as range and speed. The proposed CS-based LFMCW radar signal processor is based on range samples reduction. The reconstructed signal has been obtained using CAMP algorithm. Performance evaluation of the proposed approach is compared with that of the traditional one through ROC

Figure 15 Resolution in Doppler domain of both the proposed approach and

traditional LFMCW radar for two targets (Doppler cells numbers 10 and 11 at SNR=-5dB and Pfa = 10-5.

Traditional LFMCW radar.
The proposed approach (=50%).

Hardware Complexity

The expected complexity is evaluated regarding the required memory, number of complex operation, and consequently the processing time. Initial calculations for the proposed approach of Fig.(3) compared to that of the traditional one of Fig.(1) are summarized in Table (2). This comparison is performed considering a reduction ratio of 50% in range samples (=0.5) for the proposed approach.

From this table, it is found that, the complexity and the processing time of the proposed approach are approximately half that of the traditional one.

Table 2 Initial complexity comparison between the proposed approach at 50% reduction ratio in range samples and the traditional one, (128 range cells and 32 Doppler cells).

Comparison aspects Approach	Samp- ling frequ- ency	Memo- ry size (cells)	Time of proce- ssing (clock)	Complexity
Comparison aspects Approach	Samp- ling frequ- ency	Memo- ry size (cells)	Time of proce- ssing (clock)	operations	Calcul- ations (comp- lex addit- ion and multipl- ication)
Traditional (N-pionts)	fs	32×128 x2 (8192)	Two CPI (8192)	128 point FFT + 32 point FFT + CFAR threshold	1072
The			One CPI (4096)	32 point FFT + y *A + Thresholding
proposed		32×64+
approach (Range	fs/2	64×128			479
reduction)		(10240)
(M=N/2)

curves. The detection performance of the proposed approach is approximately similar to that of the traditional one in case of 50% reduction ratio in range samples. As the reduction ratio increases, the detection performance increases. Time of processing and the expected hardware complexity of the proposed approach is half that of the traditional one in case of 50% reduction ratio in range samples. The resolution performance of the proposed approach is similar to that of the traditional one in both range and Doppler domains.

The proposed approach may be considered as a good alternative to the traditional LFMCW radar signal processor with less sampling rate, less hardware complexity, and less processing time. Real time implementation of the proposed approach using FPGA may be considered as a future work.

REFERENCES

T. Strohmer, J. Tanner, "Implementations of Shannon's Sampling Theorem, a Time-Frequency Approach ", Sampling Theory in Signal and Image Processing, Sampling Publishing, vol.4, pp.1-17, Jan. 2005.
David C. Brock, Gordon E. Moore, " Understanding Moore's Law: Four Decades of Innovation ", Chemical Heritage Foundation, 2006.
Merrill I. Skolnik, "Introduction to Radar Systems", 3rd edition, by The McGraw-Hill Companies, 2008.
E.Candes and M. Wakin. "An Introduction to Compressive Sampling". IEEE Signal Process. Mag., vol.(25) : 21-30, Mar. 2008.
D.L.Donoho, Compressed Sensing, IEEE Trans.on Information Theory, vol. 52, pp. 1289-1306, 2006.
E.Candes, Compressive Sampling, in Proc. Int. Congress of Mathematics, pp. 1433 1452, Aug. 2006.
E.Candes, J. Romberg, and T. Tao, Robust Uncertainty Principles: Exact Signal Reconstruction From Highly Incomplete Frequency Information, IEEE Trans. on Information Theory, vol. 52, no. 2, pp. 489509, 2006.
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Zhu Lei, Qiu Chunting, Zhu Lei," Application of Compressed Sensing Theory to Radar Signal Processing", IEEE Trans. Signal processing, vol.(1): 978-981,2010.
D. L. Donoho, A. Maleki, and A. Montanari. "Message Passing Algorithms For Compressed Sensing". Proc. Natl. Acad. Sci., 106(45):1891418919, 2009.
A. Maleki, L. Anitori, Y. Zai, and R. G. Baraniuk. "Asymptotic Analysis of Complex LASSO Via Complex Approximate Message Passing (CAMP)". IEEE Trans. Inf. Theory, 2011.
L. Anitori, M. Otten, P. Hoogeboom, Arian Maleki ,Richard G. Baraniuk," Compressive CFAR Radar Detectors ", IEEE Journal of selected Topics in signal proc.,pp.0320-0325,2012.
Nadav Levanon and Eli Mozeson, "Radar Signals", Published by John Wiley & Sons, Inc., Hoboken, New Jersey,2004.

Number of targets (K)	Minimum number of measurements (M)	Minimum reduction ratio (=M/N)
1	21	0.16
9	58	0.46
10	67	0.51

A Proposed Compressive Sensing Based LFMCW Radar Signal Processor

Reconstruction algorithm

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