Quadrature Compressive Sampling for Radar Echo Signals

Feng Xi, Shengyao Chen and Zhong Liu Department of Electronic Engineering

Nanjing University of Science and Technology Nanjing, Jiangsu 210094, PRC

xf_njust@yahoo.com.cn, chen_shengyao@163.com, eezliu@mail.njust.edu.cn

Abstract—Quadrature sampling is an effective technique for ex-tracting digital in-phase and quadrature (I/Q) components in the modulated radio signals. Existing quadrature sampling tech-niques require the sampling rate to be at least twice of the band-width of the bandpass signal. The newly introduced compressive sampling theory makes sampling the analog signal at a low sam-pling rate possible if the signal has a sparse representation. This paper merges the two sampling techniques and develops a qua-drature compressive sampling (QuadCS) system to obtain the digital I/Q components with low-rate samplers. The operation principle of the QuadCS system is described and the formulation to recover the I/Q components from the low-rate samples is de-veloped. The simulation results demonstrate that the QuadCS is effective for acquiring and reconstructing the I/Q components sparse in waveform-matched dictionary. With the QuadCS sys-tem, the simulated signals can be sampled at about 10% Nyquist sampling rates. In addition, the output signal-to-noise ratio is improved about 15~20dB.

Keywords-Compressive sampling; Quadrature sampling; Sparse signal; Radar echo signal

I. INTRODUCTION

In modern radar systems, the wideband signals are widely applied as the transmitted waveform. A major challenge in radar signal processing is the requirement of high rate samplers to sample the radar echo signals. The demand of high sampling rate severely stresses current analog-to-digital converter (ADC) technology and imposes demanding challenges on the subse-quent storage and DSP processors.

The newly introduced compressive sampling (CS) [1~3], or compressive sensing, brings us new concepts on the low-rate data acquisition. The CS theory exploits the sparsity of signals and samples signals closer to their information rate instead of their bandwidth. With high probability, the CS can recover sparse signals from far fewer samples or measurements than the Nyquist samples. The fewer samples lead to a reduced sampling rate and, hence, to a reduced use of ADCs resources. Along with the CS theory, several schemes have been proposed to implement the CS of the analog signals. These include ran-dom sampling [4], random filtering [5], random demodulation [6, 7], segmented compressed sampling [8], Xampling [9], and

Fig.1 The structure of the QuadCS system.

so on. All of them exploit some type of the sparsity in wide-band analog signals and implement the sub-Nyquist sampling. For the bandpass signals, these techniques often first downcon-vert them to the baseband before implement the CS.

In coherent radars, there is often a need to produce the ba-seband in-phase and quadrature (I/Q) components from the received radar echo signal [10]. Then the information con-tained in the echo signal is extracted from the components. There are two fundamental demodulators to produce digital I/Q components, including the classical analog demodulator [11] and the digital quadrature sampling technique [12]. The digital quadrature sampling technique overcomes some implementa-tion difficulties in analog demodulator and has become a fun-damental module in coherent radar and communications sys-tems. It should be noted that these demodulators are based on the bandpass sampling theorem [12, 13] and hence the high-rate samples are delivered.

The purpose of this paper is to merge the CS theory and the digital quadrature sampling and develop a quadrature compres-sive sampling (QuadCS) system to directly sample the received bandpass signal with a low-rate ADC. The structure of the QuadCS system is schematically drawn in Fig. 1, which con-sists of two subsystems: the low-rate sampling subsystem and the quadrature demodulation subsystem. The former randomly projects the received signal to a compressive bandpass signal and samples the compressive signal with the low-rate ADC. Because the bandwidth of the compressive signal can be set to be much smaller than that of the received signal, the sampling

This work was partially supported by Natural Science Foundation of China (60971090, 61171166, 61101193).

978-1-4577-1010-0/11/$26.00 ©2011 IEEE

rate is greatly reduced. The latter implements the quadrature demodulation to extract the I/Q components from the output of the low-rate sampling subsystem. With the compressive mea-surements provided by the QuadCS system, the I/Q compo-nents can be recovered by solving a 1l -norm optimization prob-lem [14~16].

This paper is organized as follows. Section II defines the signal model and gives the problem description. Section III presents the structure of the QuadCS system. In Section IV, the formulation to recover the I/Q components from the compres-sive measurements is developed. Simulation results are demon-strated in Section V, and Section VI concludes this paper.

II. SIGNAL MODELS AND PROBLEM STATEMENT

A. Signal Model In radar receivers, the received waveform after downcon-

verting to intermediate frequency (IF) signal is usually de-scribed as

( ) ( ) ( ) ( ) ( )01

cos 2K

k k k k kk

r t a t t f t t t t n tσ π φ ϕ=

= − − + − + +⎡ ⎤⎣ ⎦∑ (1)

where 0f is the IF frequency, K is the number of the radar echoes with kt , kσ and kϕ representing the time delay, gain coefficient and phase offset of the k th (1 k K≤ ≤ ) echo, re-spectively; ( )a t and ( )tφ are a priori known envelope and phase of the transmitting waveform, respectively; ( )n t is the additive noise. Assume that the transmitting baseband width is B with 0B f . Then the waveform (1) has a spectrum cen-tered at 0f with bandwidth B .

The received waveform ( )r t without noise can be represented as the following form

( ) ( )0 0( ) ( )cos 2 ( )sin 2r t I t f t Q t f tπ π= − (2) where ( )I t and ( )Q t denote the I/Q components, respectively, of the bandpass signal ( )r t

( ) ( ) [ ]1

cos ( )K

k k k kk

I t a t t t tσ φ ϕ=

′= − − +∑ (3)

( ) ( ) [ ]1

sin ( )K

k k k kk

Q t a t t t tσ φ ϕ=

′= − − +∑ (4)

with 02k k kf tϕ ϕ π′ = − . The baseband complex envelope signal ( )s t of ( )r t is given by

( ) ( ) ( )

( )01

K

k kk

s t I t jQ t

s t tσ=

+

= −∑ (5)

where kjk ke

ϕσ σ ′= and ( ) ( ) ( )0 = j ts t a t e φ .

B. Waveform-Matched Dictionary

In active radar system, the receiver is aware of the exact model of transmitted signal. To achieve very sparse represen-tation of echo signals, we can construct a waveform-matched

dictionary [17] for the echo signal. The waveform-matched dictionary ψ

consists of all the time-delay versions of ( )0s t

at integral multiples of 0 1 Bτ = , i.e.,

( ){ }, 1,2, ,n t n Nψ= =ψ … (6) with

( ) ( )0 0 1,2, ,n t s t n n Nψ τ= − = … (7)

where 0N T τ= ⎡ ⎤⎢ ⎥ and T is the maximum time delay. The waveform-matched dictionary ψ discretizes the time-delay axis with the resolution 0 1 Bτ = .

With the assumption of { }0 0 0,2 , ,kt Nτ τ τ∈ … , the complex envelope signal ( )s t can be rewritten as

( ) ( )1

k

K

k nk

s t tσ ψ=

=∑ (8)

with 0k kn t τ= . In matrix form, we will have ( ) ( )s t t= Ψ σ (9)

where [ ]1 2, , , T

Nσ σ σσ

( ) ( ) ( ) ( )0 0 0 0 0 0, 2 , ,t s t s t s t Nτ τ τ− − −⎡ ⎤⎣ ⎦Ψ …

The ( )s t is K -sparse if K N . The sparsity level

0K = σ

exactly equals to the number of the target echoes. Based on the sparse representation of the radar echo signal,

this paper focuses on the design of a low-rate sampling system to extract the I/Q components from the low-rate samples of

( )r t .

III. STRUCTURE OF QUADCS SYSTEM

The QuadCS system consists of two subsystems: low-rate sampling and quadrature demodulation. Its operating prin-ciples are as follows.

A. Low-Rate Sampling Subsystem The low rate sampling subsystem is similar to the random

demodulation scheme in [6, 7] and implements the sub-Nyquist sampling of the input bandpass analog signals. Since the signal to be processed here is a bandpass signal with the known IF frequency rather than a lowpass signal as those in the random demodulation, the component operations are dif-ferent.

The IF signal ( )r t is firstly mixed by a random-binary signal ( )p t

( ) ), , 1 , 0,1,k p pp t t k B k B kε ⎡= ∈ + =⎣ … (10) where pB B≥ and 1kε = or 1− . The ( )p t is called as chip-ping sequence [7], which alternates between +1 and –1 ran-domly at or above the Nyquist rate of the baseband

signal. The

mixing operation will spread the frequency content of the ba-seband signal to full spectrum of ( )p t . Different from that in random demodulation, the chipping rate is determined by the

passband width of the received signal ( )r t instead of the up-

per frequency of ( )r t . The analog bandpass filter ( )bph t is centered at the fre-

quency 0f with the bandwidth csB B . The output of the filter ( )y t

is a compressive bandpass signal with center fre-

quency 0f and the bandwidth csB ( ) ( ) ( ) ( )( )

( ) ( ) ( ){ }02Re ( )

bp

bp

j f tcs

y t h t p t r t

h p t r t d

s t e π

τ τ τ τ∞

−∞

= ∗

= − −

=

∫ (11)

where ( )css t is the compressive envelope signal of ( )r t with

( ) ( ) ( )cs cs css t I t jQ t= + (12) and

( )csI t and ( )csQ t are the compressive I/Q components

( ) ( ) ( ) ( ) ( ) ( )I Qcs bp bpI t p t h I t h Q t dτ τ τ τ τ τ

+∞

−∞⎡ ⎤= − − + −⎣ ⎦∫ (13)

( ) ( ) ( ) ( ) ( ) ( )I Qcs bp bpQ t p t h Q t h I t dτ τ τ τ τ τ

+∞

−∞⎡ ⎤= − − − −⎣ ⎦∫ (14)

with ( ) ( ) ( )0cos 2Ibp bph t h t f tπ= and ( ) ( ) ( )0sin 2Q

bp bph t h t f tπ= .

Similar to (9), the compressive envelope signal ( )css t can also be sparsely described as

( ) ( )cs css t t= ψ σ (15) where

( ) ( ) ( ) ( )1 2, , ,cs cs cscs Nt t t tψ ψ ψ⎡ ⎤⎣ ⎦ψ …

and

( ) ( ) ( ) ( )020

j fcsn bpt h e p t s t n dπ τψ τ τ τ τ

+∞ −

−∞= − −∫

for 1,2, ,n N= … . Because the bandwidth csB of the signal ( )y t is smaller

than that B of the signal ( )r t , it can be sampled with a low-rate ADC. Let the lower and upper band edge of ( )y t be

0 2L csf f B= − and 0 2H csf f B= + . Then according to the bandpass sampling theorem [10, 11], the sampling frequency

csf can be chosen as 4 2

4 1L cs

csf Bf

l+=+

(16)

with l be a positive integer satisfying 2L csl f B≤ ⎢ ⎥⎣ ⎦ . The min-imum sampling frequency csf is 2 csB , provided that 2L csf B is an integer, or equivalently, ( )0 2 1 2 csf l B= + ( 1,2,l = … ).

The output of the low-rate sampling subsystem is a se-quence of the discrete-time samples [ ]{ }, 0,1,2,y k k = … with

( )( ) ( )( )( ) ( )

2

1 2

[ ]

1 is even

1 is odd

cs

kcs cs

kcs cs

y k y k f

I k f k

Q k f k+

=

⎧ −⎪= ⎨−⎪⎩

(17)

B. Quadrature Demodulation Subsystem

The quadrature demodulation subsystem is used to extract the two sampling sequences, digital I/Q components csI and

csQ , from the low-rate sampling sequence [ ]{ }, 0,1,2,y k k = … . The csI and csQ components will be subsequently applied to the recovery of the I/Q components of the ( )r t in the next sec-tion. As in any quadrature sampling techniques, the digital csI and csQ are directly available from [ ]y k , according to (17). However, there is a time shift between the two components. To make time alignment, we need to do some processing.

For the I channel, the sequence [ ]{ }, 0,1,2,y k k = … is firstly down-sampled by a factor of 2 and then multiplied by a se-quence ( )1 m− with 2k m= . Then the csI is given by

[ ] ( )cs cs mI m I t= (18)

with 2m cst m f= ( 0,1,2,m = … ). For the Q channel, [ ]y k is

first multiplied by ( )2sin 2kπ− to make a frequency shift of 1 4 such that

[ ] [ ] ( )( ) ( ) ( ) ( ) ( )

2 sin 2

sin coscs cs cs cs cs cs

y k y k k

Q k f I k f k Q k f k

ππ π

′ = −

= − +(19)

The Q component is obtained by lowpass filtering [ ]y k′ with a cutoff frequency of 1 4 and downsampling the filtered out-put by a factor of 2:

[ ] ( )cs cs mQ m Q t= (20) with 2m cst m f= ( 0,1,2,m = … ).

Then the samples of the compressive complex baseband signal ( )css t are given by

[ ] [ ] [ ]cs cs css m I m jQ m= + (21) with the sampling rate 2cs csf B≥ .

IV. I/Q COMPONENT RECOVERY METHODS

With the low-rate sampling sequence provided by the QuadCS system, the current problem is to recover the base-band I/Q components from the low-rate samples. As revealed by (3)~(5), the key to acquiring the I/Q components is to ob-tain the estimates of the sparse σ .

Now assume that the QuadCS system produces length- M low-rate sampling sequence, [ ] [ ] [ ]0 , 1 , , 1

Tcs cs cs css s s M⎡ ⎤= −⎣ ⎦s … .

Then following (15), we have matrix formulation

cs cs=s Ψ σ (22) where

( )( )

( )

0

1

1

cs

cscs

cs M

t

t

t −

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

ψ

ψΨ

ψ

with 2m cst m f= ( 0 1m M≤ ≤ − ). With the formulation (22), the sparse σ can be obtained as

usual 1l -norm recovery formulation,

0 0.04 0.08 0.12 0.16 0.20

10

20

30

40

50

60

70

80

90

100

Compression Factor (λ)

Pro

babi

lity

of S

ucce

ssfu

l Rec

over

y (%

)

K=5K=8K=10

Fig.2 The probability of successful recovery of the QuadCS system with

the different compression factor λ

( )( )

1

2

min

s.t. noise free

with noisecs cs

cs cs ε

⎧⎪⎪ =⎨⎪

− ≤⎪⎩

σ

s Ψ σ

s Ψ σ

(23)

The above 1l -norm optimization problem has be widely re-searched in the development of the CS theory. There now exist a wide variety of approaches to effectively solve the 1l -norm optimization [14~16]. Once the representation coefficients σ are obtained, we can recover the baseband complex envelope signal ( )s t (or equivalently, the I/Q components) according to (9). The recovered I/Q components can be further processed by utilizing the conventional DSP processors for specific ap-plications, for example, radar echo detection, target range es-timation and so on.

V. SIMULATIONS

In this section, we evaluate the performance of the QuadCS system through several simulation experiments. Both noise and noise-free cases are considered. In addition, the im-provement on the signal-to-noise ratio (SNR) is also simulated.

The input signal of the QuadCS system is a linear combi-nation of K time-delay versions of the LFM pulsed signal with the bandwidth 50B MHz= and the pulse width 10 sτ μ= . The IF is set as 0 300f MHz= . Different number of radar echoes are assumed, and their gain coefficients kσ (1 k K≤ ≤ )and time-delays kt (1 k K≤ ≤ ) are randomly over ( ]0,1 and the set { }1 ,2 , ,B B N B… , respectively.

We define the compression factor csB Bλ = to measure the reduction of the signal bandwidth over the input signal. The factor also indicates the reduction of the sampling rate over the Nyquist rate of the baseband signal in the QuadCS system.

0 5 10 15 200

20

40

60

80

100

Sparsity Level (K)

Pro

babi

lity

of S

ucce

ssfu

l Rec

over

y (%

)

λ=0.08λ=0.12λ=0.16

Fig.3 The probability of successful recovery of the QuadCS system with

the different sparsity level K .

A. Recovery of Noisy-free Signal

In our first simulation, the input signal is assumed to be noise-free. We compute the probability of successful recovery (PSR) with different sparsity level K and the compression factor λ in the QuadCS system. The simulation results are demonstrated in Fig.2 and Fig.3. As seen in these figures, the signal bandwidth can be compressed to about 10% of the orig-inal bandwidth for recovering the I and Q components with 90% PSR. As the sparsity level K increases, the PSR perfor-mance of the QuadCS system degrades more slowly when the compression factor is larger. However, as demonstrated in Fig.2, the PSR performance degrades rapidly once the com-pression factor λ is lower than some threshold value.

B. Recovery of Noisy Signal

Now we evaluate the performance of the QuadCS system with the noisy input signal. Fig.4 shows the recovery results under different noises and sparse positions. The additive noise introduces some small nonzero gain coefficients on the time-delay axis. However, the actual time delay and gain coeffi-cients are still recoverable from the noisy input. In Fig.5, we give the SNR performance of the QuadCS system. For the different values of the compression factor λ and the sparsity level K , the SNR is improved about 15dB~20dB by the QuadCS system. For the larger compression factor λ , the QuadCS system can achieve the higher SNR of the output signal. These results demonstrate that the QuadCS system can not only recover the I/Q components in the noisy situation, but also can improve the SNR of the recovered signal. With these results, it is believed that the performance of the signal processing after recovering can also be improved by using the QuadCS system

0 0.2 0.4 0.6 0.8 1

x 10-5

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Time Delay (s)

(a) SNR=10dB

Gai

n C

oeffi

cien

ts

Actual valueRecovered value

0 0.2 0.4 0.6 0.8 1

x 10-5

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Time Delay (s)

(b) SNR=0dB

Gai

n C

oeffi

cien

ts

Actual valueRecovered value

Fig. 4 The recovery performance of the QuadCS system with the noisy signals when 0.16λ = , 5K = . (a) The input SNR is 10dB; (b) The input SNR is 0dB.

-5 0 5 10 15 2010

15

20

25

30

35

40

Input SNR (dB)

Out

put S

NR

(dB

)

λ=0.08,K=2λ=0.16,K=2λ=0.24,K=2λ=0.16,K=5λ=0.24,K=5

Fig. 5 Output SNR vs. Input SNR of the QuadCS system with the noisy sig-

nals.

VI. CONCLUSION

In this paper, we have developed a new quadrature sam-pling system, QuadCS, for sampling the radar echo signals. The system combines the advantages of the CS theory and digital quadrature demodulation and can directly sample the bandpass signal with the sampling rate well below the Nyquist rate. The I/Q components is recovered from the low-rate sam-ples by solving a convex optimization problem. The structure of the QuadCS system is simple. All of its subsystems can be

conveniently constructed by using the off-the-shelf compo-nents. Because of space limitations, this paper focuses only on radar echo signals. Similar results can be obtained through setting different waveform-matched dictionaries in communi-cations and other modulated waveforms. It is expected that the QuadCS system can be widely applied in modern radar and communications system by replacing the current IF sampling system.

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