Joint Doppler shift and time delay estimation by deconvolution of generalized matched filter

Resolution probability is the most important indicator for signal parameter estimator, including estimating time delay, and joint Doppler shift and time delay. In order to get high-resolution probability, some procedures have been suggested such as compressed sensing. Based on the signal’s sparsity, compressed sensing has been used to estimate signal parameters in recent research. After solving ℓ0 norm Optimization problem, the methods would achieve high resolution. These methods all require high SNR. In order to improve the performance in low SNR, a novel implementation is proposed in this paper. We give a sparsity representation for the generalized matched filter output, or ambiguity function, while the former methods utilized the sparsity representation for channel response in time domain. By deconvolving the generalized matched filter output, 2-dimension estimation for Doppler shift and time delay would be gotten by greedy method, optimization method based on relaxation, or Bayesian method. Simulation demonstrates our method has better performance in low SNR than the method by the channel sparsity representation.


Introduction
Compressed sensing, or compressive sampling, was proposed by David Donoho, Emmanuel Candès, Terence Tao, and Justin Romberg in the early twenty-first century. Compressed sensing started the revolution in sampling theorem and had got breakthrough applications in image compression, magnetic resonance imaging (MRI), super-broadband communication.
For signal parameter estimation, the source number is usually limited and the channel is sparse. Due to sparsity, compressed sensing (CS) can improve the performance for signal parameters estimation, including time delay, frequency, direction, and multiple parameters. In 2002, Cotter [1] proposed time delay estimation method for sparse channel by matching pursuit. Considering orthogonality, Karabulut [2] used orthogonal matching pursuit (OMP) to improved convergence speed and accuracy. Addressing the joint estimation issue, Doppler frequency and time delay were estimated by OMP and basis pursuit (BP) algorithms [3]. In [3], Beger also compared compressed sensing methods with subspace methods, such as MUSIC and ESPRIT, and the former outperformed the latter over realistic underwater acoustic channels. For direction estimation, Malioutov explored second-order cone programming to solve 1 norm problem and obtained signal's directions. Combined 1 norm and 2 norm by exploiting orthogonality between the noise-subspace and the overcomplete basis matrix, Zheng [4] proposed a weighted 1,2 -SVD (singular value decomposition) method to get more sparse solution for direction. Based on the likelihood ratio test with a sparsity promoting prior, ref [5] and [6] jointly detect the unknown number of noise-like jammers and angles of arrival. Analogously, the methods in [4][5][6][7] can also be used to estimate time delay and frequency after signal sparse reconstruction. Signal parameters estimation by compressed sensing can achieve more excellent resolution than conventional methods such as generalized cross correlation methods [8], WRELAX (weight Fourier transform and relaxation) [9] methods and subspace methods such as MUSIC [10,11]. But there are still some current problems: how to construct the overcomplete basis matrix when the true parameters are not in the finite set; the computation quantity is too large for high dimension scenario; moreover, the algorithms performance would be degraded severely in low SNR. Got inspired from image processing, Yang [12] suggested a deconvolved method to estimate direction, which also belonged to CS methods and obtained gain by beamforming. The method reconstructed sparse model in beam domain and could achieve better performance in low SNR. Convolution and deconvolution are common operations for image processing. Richardson and Lucy [13] proposed a classical deconvolution method, Richardson-Lucy deconvolution. These methods [14,15] restore a blurred image to a clear one by deconvolution.
As insights from the operation, time delay estimation may obtain gain from the matched filter. Matched filter is an indispensable step for active sonar, radar, and communication. Many conventional algorithms take advantage of the cross-relation between the transmitted signal and the received signal. Ideally, the peaks should appear in the points that are corresponding to the true time delays, and matched filter or correlation methods are usually used to estimate wideband signal's time delay. However, for narrowband signal, the matched filter output or correlation function is flat and difficult to search the peak especially for two close echoes. An ideal matched filter output is expected that nonzeros only being according to the time delays. According to the sparsity of the ideal matched filter output, or correlation function, a deconvolved method is suggested in this paper. Simulation results are provided to compare the methods based on the sparsity of channel impulse and matched filter output, and the new method has better performance in low SNR.

Signal model
Assume a single receiver, the received signal is where s(t) is the emitted source signal, T is the observation time and should be larger than s(t)'s time duration. n(t) is Gaussian white noise. The received signal x(t) is modeled by a sum of K echoes from multiple paths, with different time delay t i and amplitude varia- where ξ i is Doppler scale, ξ i = c+v i c−v i , and v i is the ith echo's radial velocity to the platform (to be positive when closer). Usually, the velocity is far less than acoustic speed c, and where B is bandwidth, Doppler frequency f i can take place of Doppler scale. Doppler frequency shift f i = (ξ i − 1)f c and f c is carrier frequency. Under the condition, Eq. (2) can be simplified as: Otherwise, the duration compression cannot be ignored.

Previous method by channel estimation
In order to estimate time delay, some researchers have suggested to solve the problem by CS methods. Most of the methods are based on sparse channel impulse response estimation. In [16], the observed signal is considered as a convolution of the transmitted signal and channel impulse response.
where the channel impulse response h(t) includes all of the paths: . With a sampling period T s and N samples, Eq. (1) can be written as discrete form: where The sampling error is ignored, and the true time delay must be contained in the set {0, T s , (N − 1)T s }. Due to Doppler effect, the received signal's pulse would be different from the transmitted signal's. And it cannot be ignored for wideband signal or larger Doppler scale. As a result, in order to cover the pulse variation, N should be larger than the maximum time delay plus duration. Then, the observed signal can be rewritten as cyclic convolution form.
The cyclic convolution matrix is constructed as 6.
In time domain, the number of paths is much smaller than that of time samples. As a result, a sparsity representation of signal is obtained as Eq. (5). The channel impulse should be sparse and estimated by solving the 0 -norm problem: In [16], we suggested to estimate time delays by relaxing 0 -norm problem, including greedy algorithm and 1 -norm problem by convex optimization. The compressed sensing methods achieved super resolution. However, some pseudo-peaks exist and the performance would degrade severely in low SNR scenario.

1D estimation for time delay
Matched filter(MF) is a necessary operation in radar/sonar area to improve SNR. Furthermore, it is also the most conventional method for time delay estimation. The targets' time delays can be estimated by searching the peaks of matched filter (MF) output or cross-correlation function. Definite y(τ ) to be "matched filter spectrum, " as the output for matched filter in time domain: where ( * ) is complex conjugate symbol. When τ = t i , the output y(τ ) will reach maximum. The discrete form is:  y (1,q) , . . . , y (N−1,q) ] T . In order to eliminate the impact of amplitude variation, normalized is suggested here, In For the signal as Eq. (4), the square vector of the MF output should be the sum of some weighted column vector.
whereŷ =[ŷ(0),ŷ(1) . . . ,ŷ(N − 1)] T , andŷ(m) = K i=1 a 2 i δ(m − τ i ). Therefore,ŷ is a sparse vector. Accordingly, another sparsity representation is obtained as Eq. (11). C is the dictionary matrix. The computation quantity can be cut down by pre-estimation. For instance, the echoes' time delays can be restricted in the duration [ 0, N t − 1] by priori knowledge. Hence, the dimension of C is reduced to N t × N t , while S ∈ C N×N t .

2D estimation for time delay and Doppler
Considering the Doppler scale, a 2-dimension estimation is needed. The finite set of 2-D parameter (τ , ξ) is defined as where ξ is Doppler scale, and ξ 0 is the possible minimum, ξ is the step.
In [16], the channel impulse response h(t, ξ) on the Doppler-time plane can be formulated as: Then, the 2D channel impulseĥ can be estimated by compressed sensing, and The dictionary matrixŜ is expanded to a N × The 2D channel estimation by CS has similar problem as 1-dimension (1D) estimation in low SNR. Similar to the deconvolution of matched filter output, the deconvolution on the Doppler-time plane could be expanded by a generalized matched filter or ambiguity function. The generalized matched filter output is: Ideally, we suppose the true time delays and Doppler scales are in the set of 2-D parameter as Eq. (12). Naturally, time delay and Doppler scale can be estimated jointly by deconvolution, which can be also achieved by compressed sensing. The dictionary matrix must be expanded to high dimension, After sparsity presentation is accomplished through channel impulse or generalized matched filter output, joint time delay and Doppler can be estimated by solving 0 norm optimization problem. In order to seeking solutions to NP (nondeterministic polynominal) hard problem, there are three categories of approaches, including optimization methods based on relaxation, greedy algorithms, or Bayesian methods. The methods by using convex optimization have stable calculation accuracy but large computation quantity. Furthermore, it is difficult to choose the relax factor. MFCUSS (multiple focal underdetermined system solver) in [17] solves an underdetermined system of equations and obtains similar precision as convex method. Greedy algorithms, such as basis pursuit, matching pursuit [1], and orthogonal matching pursuit [18], can get faster computation speed but lower resolving power. Based on the statistical properties of received signal, such as Laplace prior [19] or Gaussian prior [20], sparse Bayesian methods can complement 0 problem by linear programming or greedy algorithms. Without the need for sparsity in iterative process, Bayesian methods have better universality, but higher computation complexity.

Result and discussion
To demonstrate the algorithm, 1D and 2D estimation simulation are both designed. The CS methods based on channel impulse response and matched filter output(generalized matched filter output) are illustrated and compared.

1D estimation for time delay
Considering the target stable. The transmitted signal is CW signal and has duration T = 200 with normalized sampling frequency; the center frequency is 0.2. The received  In the numerical simulation, time delays are estimated by several CS tools that have been introduced in the last section, including orthogonal matching pursuit [21] (GOMP), optimization method based on relaxation [22] (SDP), and sparse Bayesian learning [23] (SBL). The methods with sparsity representation for matched filter output are short as MF-domain methods, and a subscript " mf " will be used to identify the methods. Meanwhile, the methods with sparsity representation for channel impulse response are short as time-domain methods (Fig. 3).   Table 1. SNR is set as 18 dB to ensure the two echoes can be distinguished, and average computation time is obtained through 200 times simulations. Optimization methods based on relaxation (SDP) are solved by quadratic programming, and get similar computation time. Other than that, the computation time of MF-domain methods are smaller than those of time-domain methods. The advantage is due to the smaller dimension of dictionary matrix in MF-domain methods.
Change SNR to observe different probability. τ 1 and τ 2 are the true time delays, whileτ 1 andτ 2 are the estimated ones. In a single trial, if |τ i − τ i | ≤ ζ , and |τ 1 − τ 1 | + |τ 2 − τ 2 | < |τ 1 −τ 2 |, we consider the two echoes are distinguished successfully; otherwise, they are  erly. Otherwise, some convergence problem may be occurred. GOMP is an iterative method with the minimal computation requirement, and the resolution is worst. SDP's performance is between the two. The result in Fig. 4a is in scenario of CW signal. More illustration will be analyzed for LFM signal. For matched filter, large bandwidth would improve the resolving ability deservedly. Simulated results demonstrate that the methods will also get gain from bandwidth both in time domain and MF domain. Illustrated in Fig. 4b, c, and d, the normalized bandwidth are 0.05, 0.1, and 0.2 respectively. The resolution probability is increased with the bandwidth.

2D estimation for time delay and Doppler
Considering the Doppler scale, the 2D estimation are shown in this subsection. The simulation conditions are listed in Table 2.
The super resolution estimations are obtained after sparsity representation in Figs. 5 and 6, when the transmitted pulses are CW and LFM respectively. SNR is set as 5 dB, and both of the methods can separate the two echoes in the two simulations. Moreover, MFdomain method gives more "clear" results than time-domain method as shown in the two figures.

Conclusion
In this paper, time delay estimation by compressed sensing has been studied. Besides the sparsity representation for channel impulse response, a novel sparsity representation for the matched filter output or correlation function is proposed. According to the matched filter output deconvolution, super resolution results would be obtained. For joint Doppler shift and time delay estimation, the method could be expanded by the generalized matched filter or ambiguity function. Compared to the channel sparsity representation, our method has better performance especially in low SNR scenario and smaller computation quantity for 1D estimation.