Joint Estimation Method of Frequency and DOA Parameters Based on Undersampling Structure

Abstract

The present invention proposes a joint frequency and DOA parameter estimation method based on an undersampling structure, which can use a ULA to complete the task of joint frequency and DOA parameter estimation at an under-Nyquist rate, overcoming the problems of high hardware resource consumption and high total sampling rate. First, an improved receiving structure is designed, adding a simple delay network to the first sensor of a uniform linear array, directly connecting the remaining array elements to the ADC, and all channels complete sampling at an under-Nyquist rate. Then, the covariance matrix is established using the output signals of multi-coset channels, and the pseudo-spectrum is obtained based on the MUSIC algorithm. The frequency estimation value is obtained by searching for the spectral peak. Finally, the output signals of all channels are combined, a new steering vector is constructed according to the sampling structure, and the MUSIC algorithm is used to estimate the DOA corresponding to each carrier frequency respectively, thus overcoming the ambiguity problem and having a good estimation effect in a low signal-to-noise ratio environment.

Description

Technical Field

[0001] The present invention belongs to the field of array signal processing and relates to a method for jointly estimating frequency and DOA parameters under undersampling. Background Art

[0002] In modern signal processing, signals have relatively high carrier frequencies and are distributed over a relatively wide frequency spectrum range. Therefore, the problem of estimating the frequencies of narrowband sources and DOA (direction of arrival) under an under-Nyquist sampling rate has attracted extensive attention in recent years. Achieving these tasks under an under-Nyquist sampling rate is crucial because it not only relaxes the sampling requirements for analog-to-digital converters (ADCs), but also eliminates subsequent high-speed signal processing operations.

[0003] Liu et al. proposed adding a multi-coset structure after each array element to jointly estimate frequency and DOA, and using under-Nyquist sampling for each channel. However, this method also brings high hardware pressure. Even though the sampling rate of each channel is lower than the Nyquist sampling rate, the total sampling rate is still quite high. To overcome the problems of high hardware resource consumption and high total sampling rate, Kumar et al. proposed a new algorithm. A simple under-Nyquist sampling structure was proposed and the corresponding algorithm was given. However, this algorithm can only be applied to non-uniform linear arrays, otherwise there will be ambiguity in estimating DOA, and this method is sensitive to noise. Summary of the Invention

[0004] The present invention proposes a method for jointly estimating frequency and DOA parameters based on an undersampling structure, designs an improved receiving structure, and adds a simple delay network to the first sensor. The carrier frequency and DOA of the incoming signal can be estimated using a uniform linear array at an under-Nyquist rate, and a new estimation method is proposed based on the idea of the MUSIC algorithm. This algorithm can pair the carrier frequency and DOA, thus overcoming the ambiguity problem faced.

[0005] In the array structure, the joint estimation of frequency and the corresponding DOA is effectively achieved with a very low total sampling rate and fewer delay channels. It can meet the design requirements in engineering.

[0006] The technical solution of the present invention:

[0007] Define the sources as K far-field narrowband signals with uncorrelated baseband information, distributed in the two-dimensional space of angular frequency, and the angular range is The signals are distributed over a wide frequency spectrum range, where the Nyquist rate corresponds to the entire frequency spectrum. Let x(t) represent the combination of K narrowband signals in the time domain, which can be expressed as

[0008]

[0009] Among them, the signal s k (t) is a baseband signal, and f k is the carrier frequency of the signal.

[0010] The present invention uses an improved array receiving structure to receive signals. The array receiving structure consists of a linear array with M array elements. Among them, the first array element is connected to a multi-coset sampling structure with N independent channels and performs synchronous sampling at the same sampling rate. By constructing a new array time-delay matrix, the carrier frequency is automatically paired with its corresponding DOA, realizing the joint estimation of frequency and DOA.

[0011] The joint estimation method includes the following steps:

[0012] S1 In the sub-Nyquist sampling structure, time-domain data is usually converted into frequency-domain data for analysis. Therefore, the discrete-time Fourier transform of the nth sampling signal of the multi-coset sampling structure under the first array element is expressed as:

[0013]

[0014] where n = 1,..., N, represents the aliased spectrum of the kth signal, is a set of integers, is the modulation signal of the Fourier transform, f sub is the sampling frequency, f sub = f nyq / L, f nyq is the Nyquist sampling frequency, L is the undersampling multiple; c n is the delay factor of different channels, which can be designed as a sparse value similar to a sparse array. The unit delay τ = 1 / f nyq , is the discrete-time Fourier transform representation of the noise of the nth sampling signal;

[0015] The multi-coset output signals of N channels are combined into the following matrix form:

[0016]

[0017] where, S(f) = [S1(f), S2(f), …, S K (f)] T , N = [N1,..., N M T .

[0018] is the first row of the A a matrix, A​a = [a(f1,θ1),..., a(f K ,θ K )] is the array manifold matrix including frequency information and direction-of-arrival information, d m is the array element position coefficient, A d = [a d (f1), …, a d (f K )] is the time-delay manifold matrix only including frequency information,

[0019] S2. Calculate the covariance matrix of the received data of the multi-coset sampling structure:

[0020]

[0021] where R s = E[S(f)S H (f)] = diag(σ1,..., σ K ). Due to the uncorrelated assumption of the signal source baseband, this covariance matrix is a diagonal matrix, and σ K represents the power of the k-th signal, σ 2 is the noise power, I N is the N-dimensional identity matrix, L is the undersampling multiple, and the noise power increases to Lσ n because the noise power spectrum is folded under sub-Nyquist sampling.

[0022] Perform eigenvalue decomposition on to obtain the corresponding matrix U composed of the eigenvectors corresponding to the noise subspace. n Using the obtained noise subspace, the scanning function of the MUSIC pseudo-spectrum can be written as

[0023]

[0024] The MUSIC algorithm obtains the pseudo-spectrum, and the frequency corresponding to the spectral peak is the estimated value of the frequency of the incoming wave signal.

[0025] S3. The output signal of the m-th (m ∈ [2, M]) antenna element can be expressed as

[0026]

[0027] where, d is the element spacing, θ k is the direction of arrival of the k-th signal, f k is the carrier frequency of the k-th signal, d m is the array element position coefficient, N m(f) is the discrete-time Fourier transform representation of the signal noise of the m-th array element;

[0028] Combine the array element output signals of M - 1 channels into the following matrix form:

[0029]

[0030] X a = [X2, X3,..., X M T , is the 2nd to M-th rows of the A a matrix, N a = [N2, N3,..., N M T ;

[0031] Combine the multi-coset sampling signal X d with the sampling signals X a of the remaining array elements to obtain X:

[0032]

[0033] Let The joint steering matrix G(θ, f) contains both the direction information and frequency information corresponding to the signal sources one by one.

[0034] Calculate the covariance R X :

[0035] R X = E[XX H = GR s G H +Lσ 2 I (M+N-1)

[0036] where, R s = E[S(f)S H (f)] = diag(σ1,..., σ K ), I (M+N-1) is the (M + N - 1)×(M + N - 1) identity matrix, σ 2 is the noise power, and L is the undersampling multiple;

[0037] Perform eigenvalue decomposition on the covariance matrix R X to obtain the eigenvector U w .

[0038] S4. Construct the steering vectors corresponding to each carrier frequency

[0039] ​​

[0040] Among them

[0041] The estimated value of the frequency obtained by combining S2 Calculate the MUSIC spectrum, and search for the paired angle estimation value through spectrum peak search

[0042]

[0043] where k = [1, 2,..., K];

[0044] Spectrum peak search The abscissa corresponding to the peak value is the DOA estimation value Finally, the combined value paired with the k-th signal is obtained and

[0045] Repeated estimation using the above estimation method yields K sets of one-to-one corresponding parameter estimation values

[0046] The beneficial effects of the present invention are as follows: The present invention proposes a joint frequency and DOA estimation method based on an undersampling structure, which can complete the estimation of the carrier frequency and DOA of incoming signals using a uniform linear array at an under-Nyquist rate, overcoming the problems of high hardware resource consumption and high total sampling rate. And it can automatically pair the carrier frequency and DOA, overcoming the ambiguity problem, improving the performance of DOA estimation, and having good estimation effects in a low signal-to-noise ratio environment. Description of the drawings

[0047] Figure 1 It is a structural diagram of an undersampling array.

[0048] Figure 2 It is the frequency estimation pseudo-spectrum when the carrier frequencies of signals are the same.

[0049] Figure 3 It is the DOA estimation pseudo-spectrum when the carrier frequencies of signals are the same.

[0050] Figure 4 It is the frequency estimation pseudo-spectrum when the incoming directions of signals are the same.

[0051] Figure 5 It is the DOA estimation pseudo-spectrum when the incoming directions of signals are the same.

[0052] Figure 6 It is a graph of the variation of carrier frequency estimation RMSE with SNR.

[0053] Figure 7 It is a graph of the variation of DOA estimation RMSE with SNR. Specific implementation manners

[0054] The present invention will be described in detail below in conjunction with embodiments:

[0055] Embodiment 1

[0056] This example verifies the performance of the above algorithm when the incoming wave signals have the same carrier frequency

[0057] Simulation conditions and parameters:

[0058] Assume that the carrier frequencies of the narrowband signals are distributed within the frequency band of , then f nyq = 5 GHz. The number of array elements M of the ULA is 10, the element spacing d = c / (2f nyq ), the number of multi-coset delay channels N is 10 units, the delay factor τ = 1 / f nyq , and the delay coefficient C = [0, 4, 7, 8, 12, 14, 16, 20, 21, 24]. The number of incoming wave signals is 5, the incoming wave directions are θ = [-60°, -40°, 30°, -15°, 60°], the carrier frequencies are f = {1.1, 2.1, 3.1, 4.1, 4.1} GHz, and signals 4 and 5 have the same carrier frequency. The undersampling frequency f s is 500 MHz, the signal-to-noise ratio SNR is 10 dB, the number of snapshots is 200, and the carrier frequency estimation pseudo-spectrum is as shown in Figure 2 , and the DOA estimation pseudo-spectrum is as shown in Figure 3 .

[0059] Embodiment 2

[0060] This example verifies the performance of the above algorithm when the incoming wave signals have the same DOA

[0061] Simulation conditions and parameters:

[0062] Assume that the carrier frequencies of the narrowband signals are distributed within the frequency band of , then f nyq = 5 GHz. The number of array elements M of the ULA is 10, the element spacing d = c / (2f nyq ), the number of multi-coset delay channels N is 10 units, the delay factor τ = 1 / f nyq , and the delay coefficient C = [0, 4, 7, 8, 12, 14, 16, 20, 21, 24]. The number of incoming wave signals is 4, the incoming wave directions are θ = [30°, -40°, 30°, 60°], the carrier frequencies are f = {1.1, 2.1, 3.1, 4.1} GHz, and signals 1 and 3 have the same incoming wave direction. The undersampling frequency f s is 500 MHz, the signal-to-noise ratio SNR is 10 dB, the number of snapshots is 200, and the carrier frequency estimation pseudo-spectrum is as shown in Figure 4 , and the DOA estimation pseudo-spectrum is as shown in Figure 5 .

[0063] Embodiment 3

[0064] This example verifies the influence of SNR on the estimation performance of this algorithm

[0065] Assume that the carrier frequencies of narrowband signals are distributed within the frequency band of , then f nyq = 5 GHz. The number of array elements M of the ULA is 10, the element spacing d = c / (2f nyq ), the number of multi-coset delay channels N = 10 units, the delay factor τ = 1 / f nyq , and the delay coefficient C = [0, 4, 7, 8, 12, 14, 16, 20, 21, 24]. The number of incoming signals is 5, the directions of arrival are θ = [-60°, -40°, 30°, -15°, 60°], the carrier frequencies are f = {2.1, 2.6, 3.1, 3.6, 4.1} GHz, the undersampling frequency f s is 500 MHz, the signal-to-noise ratio SNR varies from 10 dB to 30 dB, the number of snapshots is 200, and 500 Monte Carlo experiments are carried out. The variation of the RMSE of carrier frequency estimation with SNR is as shown in Figure 6 , and the variation of the RMSE of DOA estimation with SNR is as shown in Figure 7 .

Claims

1. A joint estimation method for frequency and DOA parameters based on an undersampling structure, characterized in that, A linear array with M array elements is used to receive K far-field narrowband signals with uncorrelated baseband information. Among them, the first array element is connected to a multi-coset sampling structure with N independent channels and performs synchronous sampling at the same sampling rate. Let x(t) represent the combination of K narrowband signals in the time domain: where the signal s k (t) is a baseband signal, and f k is the carrier frequency of the signal, the wide spectral range is f nyq is the Nyquist sampling frequency; The estimation method includes: S1. The discrete-time Fourier transform of the n-th sampling signal of the multi-coset sampling structure under the first array element is expressed as: where n = 1, ..., N, k ∈ {1, 2, … K}, f ∈ [0, f sub ) represents the aliased spectrum of the k-th signal, is a set of integers, is the modulation signal is the Fourier transform of, f sub is the sampling frequency, f sub = f nyq / L, where L is the undersampling multiple, c n is the delay factor for different channels, and the unit delay τ = 1 / f nyq , is the discrete-time Fourier transform representation of the noise of the n-th sampled signal; The multi-coset output signals of N channels are combined into the following matrix form: Among them, S(f) = [S1(f), S2(f), …, S K (f)] T , is A a the first row of the matrix, A a = [a(f1,θ1),..., a(f K ,θ K )] is an array manifold matrix including frequency information and direction-of-arrival information, d m is the element position coefficient, A d = [a d (f1), …, a d (f K )] is a time-delay manifold matrix including only frequency information, S2. Calculate the covariance matrix of the received data of the multi-coset sampling structure: where R s = E[S(f)S H (f)] = diag(σ1,...,σ K ), defines the covariance matrix as a diagonal matrix, σ K represents the k-th signal power, σ 2 is the noise power, I N is the N-dimensional identity matrix, and L is the undersampling multiple; For perform eigenvalue decomposition to obtain the corresponding matrix U composed of the eigenvectors corresponding to the noise subspace n , and use the obtained noise subspace to establish the scanning function of the MUSIC pseudospectrum as: The pseudo-spectrum is obtained by the MUSIC algorithm, and the abscissa corresponding to the spectral peak is the estimated value of the carrier frequency of the incoming wave signal S3. The output signal of the m-th antenna array element is: where m = 2,..., M, d is the element spacing, θ k is the arrival direction of the k-th signal, f k is the carrier frequency of the k-th signal, d m is the element position coefficient, N m (f) is the discrete-time Fourier transform representation of the noise of the m-th element signal; The array element output signals of M - 1 channels are combined into the following matrix form: X a = [X2, X3,..., X M T , is the 2nd to Mth rows of matrix A, N a = [N2, N3,..., N a M T ;​​​ Merge the multi-coset sampling signal X d with the sampling signals X of the remaining array elements a to obtain X: Let The joint flow pattern matrix G(θ,f) simultaneously contains the direction information and frequency information corresponding to the signal source one by one; Calculate covariance R X : R X = E[XX H = GR s G H + Lσ 2 I (M+N-1) where R s = E[S(f)S H (f)] = diag(σ1,...,σ K ), I (M+N-1) is the (M + N - 1)×(M + N - 1) identity matrix, σ 2 is the noise power, and L is the undersampling multiple; For the covariance matrix R X perform eigenvalue decomposition to obtain the eigenvector U corresponding to the noise subspace w ; S4. Construct the steering vectors corresponding to each carrier frequency where θ ∈ [-90°, 90°], Estimated value of the frequency obtained in combination S2 Calculate the MUSIC spectrum where k = [1, 2,..., K]; Spectral peak search The abscissa corresponding to the peak value is the DOA estimation value Finally, the combined value of the k-th signal pairing is obtained and

Images