One-bit PAST subspace update method

Abstract

The present invention discloses a one-bit PAST subspace update method, comprising the following steps: S1, establishing a signal model to obtain the output signal of an array at time t; S2, performing one-bit quantization: quantizing the output signal of the array using a one-bit quantization method; S3, updating the signal subspace using a one-bit PAST algorithm. The present invention proposes a new subspace update method to track DOA in real time. Different from traditional methods, the present invention specifically targets one-bit quantized data, and designs a new cost function and a new iterative process to update the signal subspace in real time. Compared with existing methods, the one-bit method proposed in the present invention can significantly reduce hardware costs and system power consumption. In addition, the present method can be applied to both static and dynamic environments.

Description

Technical Field

[0001] The invention belongs to the technical field of array signal processing and provides a one-bit PAST subspace updating method suitable for DOA tracking, which is used to solve the problem of real-time tracking of the direction of arrival (DOA) of a signal source under one-bit sampling conditions. Background Art

[0002] Direction of arrival (DOA) estimation is widely used in the field of array signal processing, such as radar, sonar, wireless communication, and satellite navigation. At present, DOA estimation is mainly based on the MUSIC algorithm and the ESPRIT algorithm. However, in order to extract signals from array observations, the above methods require EVD decomposition of the covariance matrix or SVD decomposition of the measurement matrix. In a dynamic environment, since the signal changes over time, the EVD or SVD decomposition process needs to be repeated continuously, which significantly increases the amount of computation. To address this problem, many dynamic DOA estimation methods have emerged, such as the subspace averaging method (subspace smoothing algorithm), the Gauss-Newton iterative method, and the subspace tracking algorithm (subspace tracking algorithms). Taking the subspace tracking algorithm as an example, this method can update the signal subspace in real time with lower computational complexity. In addition, the PAST algorithm can also be used to improve the performance of DOA estimation in sparse linear array scenarios. The algorithm adopts a new cost function.

[0003] For array processing systems, analog-to-digital converters (ADCs) are widely used in the signal digitization process. In order to ensure accuracy, high-resolution quantization is mainly used. However, high-precision quantization will significantly increase hardware costs, system power consumption, data storage and transmission burdens, etc., especially for large-scale array systems. In order to overcome the above defects, low-precision quantization is used instead of high-precision quantization. A special low-precision quantization method is one-bit quantization, which can be achieved by a single comparator. When one-bit quantization is used, both hardware costs and system power consumption will be significantly reduced. Therefore, DOA estimation based on one-bit quantization has become a current research hotspot in the field of array signal processing. However, the existing DOA estimation method based on one-bit quantization mainly considers the situation where the signal source is static and cannot be used to track the DOA of time-varying signals. Summary of the invention

[0004] The purpose of the present invention is to overcome the shortcomings of the prior art and provide a method specifically for one-bit quantized data, designing a new cost function and a new iterative process to update the signal subspace in real time. Compared with the existing methods, the one-bit PAST subspace update method proposed in the present invention can significantly reduce the hardware cost and the power consumption of the system. In addition, the method can be applied to both static and dynamic environments.

[0005] The object of the present invention is achieved through the following technical scheme: a one-bit PAST subspace updating method, comprising the following steps:

[0006] S1. Establish a signal model: Assume that K uncorrelated narrowband signals located in the far field are in a time-varying direction {θ 1 (t),θ 2 (t),…,θ K (t)} is incident on an array containing M elements; at time t, the output signal of the array is expressed as:

[0007]

[0008] Where s(t)=[s 1 (t),s 2 (t),…,s K (t)] represents the signal waveform vector; n(t) is the noise vector, which is a uniform Gaussian white noise and has nothing to do with the incident signal; A(t) = [a(θ 1 (t)),a(θ 2 (t)),…,a(θ K (t))] is the array manifold matrix, where the steering vector a(θ k (t)) is expressed as:

[0009]

[0010] d represents the distance between adjacent sensors, λ represents the wavelength of the incident signal, and the superscript T represents the matrix transpose;

[0011] S2. One-bit quantization: The output signal of the array is quantized using a one-bit quantization method, and the array output signal is expressed as:

[0012]

[0013] Where Q(·) is a negative quantization function; sign(x) is a sign function: when x is a non-negative number, the function value is 1; otherwise, the function value is 0; represents the real part of x(t);

[0014] According to the inverse sine function law, the following relationship is obtained:

[0015]

[0016] in: and are the covariance matrices of the data x(t) and y(t), respectively. The superscript H represents the conjugate transpose, and E(·) represents the mathematical expectation. Σ is a diagonal matrix, and its diagonal elements are [Σ] q,q =[R xx ] q,q ;remember Matrix sine -1 The (p,q)th element of (A) is expressed as:

[0017]

[0018] Indicates [A] p,q The imaginary part of

[0019] The variance of uniform Gaussian white noise on different sensors is equal, so the power of the received signal on different sensors is equal, so all non-zero elements of the matrix Σ are the same, denoted as p; therefore, we get:

[0020]

[0021] Among them, σ 2 and denote signal power and noise power, respectively, δ(·) denotes the Dirac function;

[0022] Combining equation (4) with equation (6), we get:

[0023]

[0024] Matrix sine -1 The (p,q) element of (A) is expressed as:

[0025]

[0026] S3. Use the one-bit PAST algorithm to update the signal subspace, and continuously update the signal subspace through the cost function J(W(t)):

[0027]

[0028] In formula (22), v l (i) represents the equivalent signal vector of the signal received vector x(i) at time i, v l The projection approximation of (i) is u(i) = W H (i-1)v l (i).

[0029] The beneficial effects of the present invention are as follows: the present invention proposes a new subspace update method to track DOA in real time. Different from the traditional method, the present invention specifically targets one-bit quantized data, designs a new cost function and a new iterative process to update the signal subspace in real time. Compared with the existing methods, the one-bit PAST subspace update method suitable for DOA tracking proposed by the present invention can significantly reduce the hardware cost and power consumption of the system. In addition, the present method can be applied to both static and dynamic environments. DETAILED DESCRIPTION

[0030] The present invention is described by taking a uniform linear array (ULA) containing M elements as an example. For ease of description, the symbols and their meanings used in the present invention are as follows: [I] m represents the m-order identity matrix; [0] m×n represents an m×n-order zero matrix; (·) * 、(·) T and(·) H denote the unit complex conjugate, matrix transpose, and conjugate transpose, respectively. E(·) denotes the mathematical expectation; ||·|| denotes the Euler norm. and denote the real and imaginary parts of the complex number z, respectively. [A] p,q represents the (p,q)th element of the matrix A; δ(·) represents the Dirac function. Symbols Represents Hadamard convolution.

[0031] The one-bit PAST subspace updating method of the present invention comprises the following steps:

[0032] S1. Establish a signal model: Assume that K uncorrelated narrowband signals located in the far field are in a time-varying direction {θ 1 (t),θ 2 (t),…,θ K (t)} is incident on an array containing M elements; at time t, the output signal of the array is expressed as:

[0033]

[0034] Where s(t)=[s 1 (t),s 2 (t),…,s K (t)] represents the signal waveform vector; n(t) is the noise vector, which is a uniform Gaussian white noise and has nothing to do with the incident signal; A(t) = [a(θ 1 (t)),a(θ 2 (t)),…,a(θ K(t))] is the array manifold matrix, where the steering vector a(θ k (t)) is expressed as:

[0035]

[0036] d represents the distance between adjacent sensors, λ represents the wavelength of the incident signal, and the superscript T represents the matrix transpose;

[0037] S2. One-bit quantization: The output signal of the array is quantized using a one-bit quantization method, and the array output signal is expressed as:

[0038]

[0039] Where Q(·) is a negative quantization function; sign(x) is a sign function: when x is a non-negative number, the function value is 1; otherwise, the function value is 0; represents the real part of x(t);

[0040] According to the arcsine law, we get the following relationship:

[0041]

[0042] in: and are the covariance matrices of the data x(t) and y(t), respectively. The superscript H represents the conjugate transpose, and E(·) represents the mathematical expectation. Σ is a diagonal matrix, and its diagonal elements are [Σ] q,q =[R xx ] q,q ;remember Matrix sine -1 The (p,q)th element of (A) is expressed as:

[0043]

[0044] Indicates [A] p,q The imaginary part of

[0045] The variance of uniform Gaussian white noise on different sensors is equal, so the power of the received signal on different sensors is equal, so all non-zero elements of the matrix Σ are the same, denoted as p; therefore, we get:

[0046]

[0047] Among them, σ 2 and denote signal power and noise power, respectively, δ(·) denotes the Dirac function;

[0048] Combining equation (4) with equation (6), we get:

[0049]

[0050] Matrix sine -1 The (p,q) element of (A) is expressed as:

[0051]

[0052] S3, using one-bit PAST algorithm to update the signal subspace;

[0053] The classic PAST algorithm continuously updates the signal subspace through the following cost function:

[0054]

[0055] Among them, 0<β≤1 represents the forgetting factor, z(i)=W H (i-1)x(i) represents the projection approximation of the signal reception vector x(i); when the space spanned by the columns of the matrix W(t) is equal to the signal subspace, J(W(t)) can obtain the global minimum value; in the present invention, we do not directly use the above cost function to extract the signal subspace from the one-bit quantized observation value, but reconstruct the cost function through the following method.

[0056] When the DOA of the signal source is constant, the covariance matrix R at sampling time t is xx (t) can usually be estimated as the following sample covariance matrix:

[0057]

[0058] In the case of dynamic changes in the DOA of the signal, the signal subspace at time t is estimated using the following exponentially weighted sample covariance matrix:

[0059]

[0060] By comparing equation (10) with equation (11), we can see that in the covariance matrix R xx (t) and C xx (t) satisfy a certain relationship: when β = 1, C xx (t) = tR xx (t); Without loss of generality, these weighting coefficients [1,β,…,β t-1 ] constitutes a geometric sequence; the sum of the weighted coefficients is calculated as:

[0061]

[0062] get

[0063]

[0064] Similarly, we get

[0065]

[0066] Among them, C yy (t) represents the exponentially weighted sample covariance matrix of the one-bit quantized observation data y(t):

[0067]

[0068] According to equations (7), (13) and (14), the following relationship is derived:

[0069]

[0070] Right now:

[0071]

[0072] From equation (11), we get a new matrix Q(t):

[0073]

[0074] Substituting formula (17) into formula (18), we get:

[0075]

[0076] The lth column of the matrix Q(t) is expressed as:

[0077]

[0078] Among them, the matrix Indicated Phase;

[0079] v l The covariance matrix of (t) is defined as:

[0080]

[0081] σ is the square root of the signal power, because σ 2 / p 2 is a constant, from the covariance matrix and E{x(t)x H (t)}; therefore, v l (t) can be regarded as the equivalent signal vector of x(t); and then vl (t) replaces x(t) to track the signal subspace;

[0082] Consider the signal vector v l (t), construct a new cost function J(W(t)) to continuously update the signal subspace:

[0083]

[0084] In formula (22), v l (i) represents the equivalent signal vector of the signal received vector x(i) at time i, v l The projection approximation of (i) is u(i) = W H (i-1)v l (i) Mainly used to reduce the amount of calculation; v l (i) Determined using the incident signal at the previous sampling time.

[0085] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific variations and combinations that do not deviate from the essence of the present invention based on the technical revelations disclosed by the present invention, and these variations and combinations are still within the protection scope of the present invention.

Claims

1. One-bit PAST subspace update method, It is characterized in that The following steps are involved: S1. Establish signal model: Assume that K uncorrelated narrowband signals in the far field are in a time-varying direction. {θ 1 (t),θ 2 (t),…,θ K (t)} is incident on an array containing M elements; at time t, the output signal of the array is expressed as: Where s(t)=[s 1 (t),s 2 (t),…,s K (t)] represents the signal waveform vector; n(t) is the noise vector, which is a uniform Gaussian white noise and has nothing to do with the incident signal; A(t) = [a(θ 1 (t)),a(θ 2 (t)),…,a(θ K (t))] is the array manifold matrix, where the steering vector a(θ k (t)) is expressed as: d represents the distance between adjacent sensors, λ represents the wavelength of the incident signal, and the superscript T represents the matrix transpose; S2. One-bit quantization: The output signal of the array is quantized using a one-bit quantization method, and the array output signal is expressed as: Where Q(·) is a negative quantization function; sign(x) is a sign function: when x is a non-negative number, the function value is 1; otherwise, the function value is 0; represents the real part of x(t); According to the inverse sine function law, the following relationship is obtained: in: and are the covariance matrices of the data x(t) and y(t), respectively. The superscript H represents the conjugate transpose, and E(·) represents the mathematical expectation. Σ is a diagonal matrix, and its diagonal elements are [Σ] q,q =[R xx ] q,q ;remember Matrix sine -1 The (p,q)th element of (A) is expressed as: Indicates [A] p,q The imaginary part of The variance of uniform Gaussian white noise on different sensors is equal, so the power of the received signal on different sensors is equal, so all non-zero elements of the matrix Σ are the same, denoted as p; therefore, we get: where, σ 2 and represent the signal power and the noise power respectively, and δ(·) represents the Dirac function; Combining equation (4) with equation (6), we get: S3. Use the one-bit PAST algorithm to update the signal subspace, and continuously update the signal subspace through the cost function J(W(t)): In formula (8), 0<β≤1 represents the forgetting factor; when the space spanned by the columns of the matrix W(t) is equal to the signal subspace, J(W(t)) can obtain the global minimum value; v l (i) represents the equivalent signal vector of the signal received vector x(i) at time i, v l The projection approximation of (i) is u(i) = W H (i-1)v l (i).

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