CHEN Shengyaoand XI Feng

School of Electronic and Optical Engineering,Nanjing University of Science and Technology,Nanjing 210094,China

1.Introduction

Pulsed radar systems usually emit a train of modulated pulses and receive the echo signals reflected from illuminated targets which are delayed and Doppler shifted replicas of the emitted pulses.For high resolution applications,the transmitted pulses commonly have large bandwidth and thus the reception of the echo signals requires the use of high-rate analog-to-digital converters,which results in large volume of sampled data and leads to high loads in subsequent storage and processing.

With advance of the compressive sampling(CS)theory[1–3],analog-to-information conversion(AIC)[4]has been developed to low-rate sample sparse analog signals,where the sampling rate is related to the information rate rather than the bandwidth. By sparse reconstruction techniques[5],the Nyquist samples of analog signals can be recovered from the compressive samples. In CS radar[6],the echoes are sampled by an AIC after some analog down-converting.As the information rate of echoes attributes to the number of targets which is rather sparse in real radar scenes,the sampling rate is much less than the bandwidth of transmitted pulses.Recently,there are several structures to implement the AIC,such as random demodulation(RD)[7],random modulator pre-integrator(RMPI)[8],Xampling [9],and quadrature compressed sensing(QuadCS)[10,11].Theoretical and experimental results show that these AICs are efficient for low-rate acquisition of analog sparse signals with a large bandwidth.

In practical radar applications,it is not necessary to recover the Nyquist samples of echo signals.More attention is paid to the time-delay and Doppler shift which stem from the illuminated targets.Several kinds of techniques have been developed to estimate delay-Doppler from compressive samples directly[12–17].However,there is no unified method to measure delay-Doppler estimation performance for compressive sampling pulse-Doppler(CSPD)radar with different AIC structures.In this paper,we consider the deterministic Cramer-Rao bound(CRB)of CSPD radar as a unified metric to quantitatively analyze the delay-Doppler estimation performance.From the theo-retical derived CRBs of a single target and multiple on-grid targets radar scenes,we know that the estimate accuracy of both time-delay and Doppler shift is inversely proportional to the received target signal-to-noise ratio(SNR),the number of transmitted pulses and the sampling rate of AIC systems.Moreover,the CRB of Doppler shifts also lies on the coherent processing interval of radar echoes,while the CRB of time-delays is independent on it.These results are also validated by numerical experiments.Moreover,the simulations reveal that typical AIC structures have weak influence on the CRBs of time-delays and Doppler shifts.

Therefore,in view of delay-Doppler estimation,the AIC structures can be flexibly selected for the implementation of CSPD radar.

It is notable that the result stemming from the deterministic CRB is different from that based on the typical metrics in the CS theory,such as coherence and restricted isometry property[18],which focus on signal reconstruction performance and require sparse echo signals.With no sparsity assumption,our result is applicable to non-sparse target scenes.

The rest of this paper is organized as follows.The problem of interest is formulated in Section 2.The deterministic CRB of delay-Doppler estimation is derived in Section 3.The performance of CSPD radar with different parameters and AIC structures is analyzed and compared by simulations in Section 4.Section 5 concludes the paper.

Before proceeding,we briefly introduce some mathematical notation that we will use throughout the paper.To avoid confusion,a real signal and a complex signal are represented as s(t)andrespectively.Bold letters denote the matrices or vectors.Re{·}and Im{·}take the real part and imaginary part of{·}.(·)?,(·)Tand(·)Hrepresent complex conjugation,transposition,and conjugate transposition of(·),respectively.? denotes the Kronecker product,vec(S)means column-wise vectorization of S,and?·?2is the Euclidean norm.

2.Signal model and problem formulation

Coherent radar systems usually transmit multiple periodic pulses and perform coherent processes to obtain the estimation of target information.In this paper,we consider a classical pulse Doppler radar with a co-located receiver and transmitter.Let the radar transmitted signal be

where αkis the complex reflectivity of the k th target,and the approximation follows from the stop-and-hop assumption[19],where the phase variation due to the Doppler shift is negligibly small during the duration of one pulse.For unambiguous delay estimation,τkis con fined into[0,T-Tp).The time-delay resolution and the Doppler shift resolution are τ0=1/B and v0=1/LT,respectively.

For parametric representation of(2),the waveform matched dictionary is defined as the time-delay versions ofwith the unknown time-delays τk,i.e.,.The complex baseband echo signal of the lth pulse(0≤l≤L-1)can be expressed as

In real radar scenarios,the received signal inevitably contains noise besides the target echoes.With the presence of additive white Gaussian noise(AWGN),the signal(t)is translated into

In CSPD radar,the echo signal is sampled by an AIC system to attain compressive measurements[6].The sampling process can be described as a matrix form.Denoteandas the Nyquist-rate sampling vectors ofandin the time window t∈[lT,(l+1)T),respectively.The typical AIC systems[7–11]can be modeled as a measurement matrix M ∈ CM×N,where M＜N and the ratio M/N is called the compressive rate.The sampling process is equivalent to multiplication of rl+nlby the matrix M.Lettingdenote the compressive measurement vector of the lth pulse,we can express it as

One main goal of CSPD radar signal processing is to accurately estimate the unknown time-delays τkand Doppler shifts νkin terms of the compressive measurementsof all L pulses.In the rest of this paper,we will discuss the deterministic CRB of the joint delay-Doppler estimation.

Before proceeding,we declare some SNR definitions of the targets,which will be utilized in the derivation of the CRB.define the received SNR for the kth target as

For the lth pulse,the compressive measurements of the kth target is.And the corresponding compressive SNR is

Note that the output SNR of the AIC systems is the same as the input SNR[14],namely,Then we have

With proper parameter setting of AIC systems,we can ensure thatwhich means the AIC systems preserve the energy of the received signal after compressed sampling,and the variance ofis NN0B/MIM[20].

3.CRB

The CRB provides a theoretical limit on the variance of any unbiased parameter estimator.This section derives the deterministic CRB for the time-delay and Doppler shift estimation of CSPD radar.

By stacking the measurement vectorsandasand n?=respectively,we have

For the case where ηiis a time-delay or Doppler shift,the partial derivatives required to evaluate(10)can be written as

The partial derivatives with respect to Re{αi}and Im{αi}can be easily represented as

The FIM in(10)can be rewritten as the following matrix partition:

where

with 1≤i≤K and 1≤j≤K.Denote as

Then the CRB of time-delays and Doppler shifts is

Equation(16)is the general mathematicalre presentation of the CRB.This CRB is applicable to arbitrary CSPD radar scenes,no matter what the targets are on-grid or off-grid.However,we cannot explicitly find the relationship between the CRB and the important parameters in the CSPD radar,such as SNR,the compressed rate of AIC systems,the number of pulses and the coherent processing interval.In the following,we will present the specific expression for the CRBs of time-delays and Doppler shifts separately in two typical cases,respectively.

(i)Single off-grid target scene

We firstly consider the simplest case that a single target exists.Asand a(vi)Ha(vi)=L,it easily follows thatand FIMαIαR=0,which implies that

we have

and

Note that the target’s SNR satisfies

Then the CRB of time-delay and Doppler shift is

From(21),we find that both the CRBs of time-delay and Doppler shift are inversely proportional to the target’s SNR, the compressive rateM/N, and the number of pulses L.Moreover,the CRB of Doppler shift also lies on the coherent processing interval(CPI)LT and has no relation to the AIC structure,while the CRB of time-delay rests with the AIC structure and also associates with the transmitted waveform and its derivation.The reason is that AIC systems only compressively sample the radar echoes in time domain but not in Doppler domain.

(ii)Multiple on-grid targets scene

We secondly discuss the case of multiple onthe-grid tar-gets,which is utilized in several delay-Doppler estimation methods[12,14,15].For simplicity,assume that both the grid spaces of the time-delay and Doppler shift equal to the corresponding resolutions,and all the targets have the same SNR.Obviously,for all the K targets,we haveandwith typical radar waveforms[10].Similar to the case of single target,

we have

where

and

Therefore,we have

where

and the(i,j)th element of the matrix F is

Note that the matrices D and E are both invertible.Then the CRBs of the time-delays and Doppler shifts equal to

and

respectively.

From(29)and(30),we find that the target’s SNR,the compressive rate M/N and the number of pulses L are also inversely proportional to the CRBτand CRBv,which is the same as the single target scene. However, both CRBτand CRBvare affected by the time-delays and Doppler shifts of unknown targets. In(29)and(30),the effect of target parameters on the CRBs is represented by the matrix F.However,we cannot explicitly separate it from the CRBτand CRBvby some matrix manipulations.Note that(21)can be described by(29)and(30)with F=0.Therefore,we can quantify the effect of F through comparing the CRBs of the single target scene and the multiple on-grid targets scene.In the next section,the numerical results will show that target parameters almost have no effect on the CRBs in the multiple on-grid targets scene.

Remark 1For the multiple off-grid targets scene,no specific expression of CRBτand CRBvcan be derived effectively.Through the numerical simulations in the next section,we will validate that the dominant variables,which are determined in the single off-grid target and multiple ongrid targets scenes,still control the CRBs in the multiple off-grid targets scene.

Remark 2It is noticed that the effect of AIC systems on the CRBs consists of two aspects:the compressive rate M/N and the structure of the measurement matrix M.From(21),(29)and(30),we find that the compressive rate plays an important role in the CRBs.On the other hand,the effect of the AIC structure is not explicitly exposed.According to simulations in Section 4,the results of experiment(ii)reveal that the structure of M has slight influence on the CRBs with a fixed compressive rate.These results are compatible with the conventional CS theory,in which a higher compressive rate means more recoverable targets and the structures of AIC systems have weak effect on the number of recoverable targets.

4.Numerical results

In this section,several experiments are simulated to analyze the CRB of the joint delay-Doppler estimation in compressive domain.Three kinds of radar echo scenes are included,a single off-grid target, multiple on-grid targets andmultiple off-grid targets. In simulation experiments, the received signalis a linear combination of delayed and Doppler-shifted versions of the linear frequency modulation(LFM)pulsed signal with bandwidth B=100 MHz and pulse width Tp=10 μ s.The PRI is T=100 μ s and the number of pulses is L=100.With the assumed parameters,the unambiguous time-delays and Doppler shifts are 0–90 μ s and 0–10 kHz.The delay resolution is τ0=0.01 μ s and the Doppler resolution is v0=0.1 kHz,respectively.Without special statements,an RD AIC system is used to sample the echo signalwith SNR=20 dB and compressive rate M/N=0.2.Five targets exist in the multiple targets scenes.For the off-grid target,the time delay τkand the Doppler shift νkof the kth target are randomly chosen from the interval(0,10)μ s and the interval(0,10)kHz.For on-grid targets,the time-delay τkand the Doppler shift νkare randomly chosen on the grids{0,τ0,...,(Nτ-1)τ0}and{0,v0,...,(L-1)v0},respectively,where.The amplitude of reflectivity αkis set to 1 for all targets and the phase is uniformly distributed in(0,2π].

(i)Effect of universal parameters

We firstly evaluate the relationship between the CRBs and the universal parameters of AIC systems and transmitted pulses.To eliminate the effect of the transmitted waveforms,we normalize the CRBs of time-delay and Doppler shift by the corresponding resolutions τ0and ν0,respectively,namely,the normalized CRB is equal to the CRB divided by the resolution.Fig.1 shows the variation of the normalized CRBs as a function of the SNR.The sub- figures on the lower-left corner are the local enlarged region of the CRBs between SNR=4 dB and SNR=11 dB.It shows that both the CRBτand CRBvare inversely proportional to the SNR.The CRBs are almost identical in different radar target scenes,and only the CRBτof the multiple off-grid targets scene slightly increases.Comparing the CRBs of the single target scene and the multiple on-grid targets scene,we know that the unknown target parameters almost have no effect.Fig.2 displays the curves of the normalized CRBs versus the compressive rate M/N.Similar to Fig.1,both CRBτand CRBvare inversely proportional to the compressive rate,too.Since the abscissa is not logarithmic in Fig.2,the curves have different shapes from that in Fig.1.Fig.3 illustrates the dependence of the normalized CRBs on the number of pulses L,where the CPI LT is set as 10 ms with variable T.Obviously,the effect of L on the CRBs is similar to that of the compressive rate.As a comparison,Fig.4 displays the relationship between the normalized CRBs and the CPI LT,where the number of pulses L=100is fixed but the PRI T is tunable.Different from the dependence shown in Fig.3,CPI has no effect on the CRBτ,but only influences the CRBv,which is identical to the analysis in Section 3(i).The above results validate the relationship between the CRBs and the universal parameters of AIC systems and transmitted pulses established in the single target scene and the multiple on-grid targets scene,which still holds in the multiple off-grid targets scenes.

Fig.1 CRBs vs.SNR

Fig.2 CRBs vs.the compressive rate M/N

Fig.3 CRBs vs.the number of pulses L

Fig.4 CRBs vs.the CPI LT

(ii)Effect of AIC system structures

We secondly judge the influence of different AIC structures on the CRB of time-delay and Doppler shift estimation with five off-grid targets.Four typical AIC structures,RD[7],RMPI[8],Xampling[9],and QuadCS[10]are utilized in the simulation.As a benchmark,an idea random AIC system is also exploited to sample echo signals,where the measurement matrix is populated with i.i.d.Bernoulli random variables.There into,the RMPI consists of four RD channels,where each channel has one-fourth sampling rate.Fig.5 shows the variation of the normalized CRBs as a function of the SNR with different AIC structures.The sub- figures on the lower-left corner are the local enlarged region of the CRBs between SNR=4 dB and SNR=6 dB.From Fig.5,we know that all but the RD structure have similar time-delays and Doppler shifts estimation performance.This may be because that the RD structure stems from the random projection in the time domain and is more suitable for sampling frequency-domains parse signals[7].The RMPI is developed from the RD to sample the pulse radar echo through increasing the randomness of the mea-surement matrix.On the other hand,the Quad CS and Xam-pling execute the random projection in frequency domain and thus they are applicable to pulse radar.In fact,the performance gap between the RD and the other AIC structures is rather small. Stated differently, the structures of AIC systems have weak influenceon the CRBτand CRBv.Therefore,from the point of view of delay-Doppler estimation,the AIC systems can be flexibly selected for the implementation of CSPD radar.

Fig.5 CRBs vs.AIC structures

5.Conclusions

In this paper, we derive the deterministic CRB for the joint delay-Doppler estimation of CSPD radar, which can be exploited as a unified metric to evaluate the performance of CSPD radar with different AIC structures. Different from the analysis in general CS theory with sparsity assumption,the result stemming from the deterministic CRB is applicable to the non-sparse target scene. Theoretical and numerical results show that the CRBs of both time-delays and Doppler shifts are inversely proportional to the received target SNR, the number of transmitted pulses and the sampling rate of AIC system, but they are weakly influenced by the AIC structures. The result implies that the AIC structures can be flexibly selected in CSPD radar.

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