WO2019242058A1

WO2019242058A1 - Phase retrieval method based on array antenna

Info

Publication number: WO2019242058A1
Application number: PCT/CN2018/096328
Authority: WO
Inventors: 李强; 黄磊; 裴灿; 黄敏; 赵博; 张亮; 周汉飞
Original assignee: 深圳大学
Priority date: 2018-06-22
Filing date: 2018-07-19
Publication date: 2019-12-26
Also published as: CN108845975B; CN108845975A

Abstract

Provided in the present invention is a phase retrieval method based on an array antenna: a received signal to be phase retrieved is captured via an antenna array, the received signal is transmitted to a receiver used for phase retrieval; and, the receiver employs an iterative interpolation algorithm to perform a phase retrieval with respect to the received signal and outputs the phase-retrieved received signal. The disclosed method introduces the principle of phase retrieval of the field of optical and image processing to the field of array signal processing, with an array antenna serving as the background of research, hypothesizes that the array antenna detects only the strength of a signal, establishes a phase retrieval model based on the amplitude of a measured value, and utilizes the iterative interpolation algorithm for phase retrieval, thus implementing the complete retrieval of an incoming signal.

Description

Phase recovery method based on array antenna

Technical field

The present invention relates to the technical field of array antenna signal processing, and in particular, to a phase recovery method based on an array antenna.

Background technique

The technique of restoring the complete signal based on the linear measurement of the signal's strength / amplitude information, such as the Fourier transform, is often called phase recovery. Phase recovery technology is widely used in astronomy, crystallography, optical imaging, microscopy, and audio signal processing.

The phase recovery problem has been widely studied in the past few decades, and a variety of different algorithms have emerged. There are mainly iterative algorithms based on the Fourier transform, and based on the Transport of Intensity Equation (TIE). Algorithms, as well as phase recovery algorithms based on convex optimization that have been proposed in recent years.

From a mathematical point of view, phase recovery is to recover the original complex vector signal containing phase information from the strength of several measured signals. Suppose that in a noisy environment, the array antenna has measured N amplitudes, which is expressed as

Then the signal model measured by the antenna array is expressed as:

In the above formula, the vector x is a M-dimensional original incident signal, the vector n is an N-dimensional noise, and A is a steering vector matrix of the incident signal.

The phase recovery problem is a non-convex non-linear problem. Generally, the number of measurements N is much larger than the dimension M of the signal in order to accurately recover the original signal. In theory, the number of measurements N needs to meet at least O (M log) to recover the original signal with high probability. Phase recovery algorithms are roughly divided into two categories. One is optical measurement. As far as the measurement method is concerned, an optical system can be established to achieve it, such as a Hilbert transform system. You can also avoid the idea of digital algorithms and directly design an experimental system to measure the phase factor, such as using high-order optical correlation, or using a four-wave mixing technology to generate a mirrored light field, etc .; the other is a digital algorithm that uses the light intensity Iteratively iteratively obtains the phase distribution Gerchberg-Saxton (GS) algorithm, and the method to obtain the phase distribution by solving the light intensity distribution transmission equation (Fourier transform method, Green function method, Zernike polynomial method, wavefront transmission equation), etc. Among them, the GS algorithm, its iterative control is controlled according to the trend of the mean square error. The GS algorithm has the disadvantage of easily falling into the local minima in the calculation of the phase recovery problem.

Therefore, the prior art needs to be further improved.

Summary of the Invention

In view of the above-mentioned shortcomings in the prior art, an object of the present invention is to provide a phase recovery method, which overcomes the defect that the carrier phase of an incident desired signal is lost and the original signal cannot be completely recovered in the prior art.

An embodiment of the present invention discloses a phase recovery method for an array antenna, which includes:

Step A: Use an antenna array to collect a received signal for phase recovery;

Step B: transmitting the received signal to a receiver for phase recovery;

In step C, the receiver uses an iterative interpolation algorithm to perform phase recovery on the received signal, and outputs the received signal after the phase recovery.

Optionally, the step C further includes:

Step C1: Establish a phase recovery model based on the array antenna according to the steering vector of the original signal corresponding to the received signal;

Step C2: An alternating iterative method and a least square method are used to establish an objective function corresponding to the phase recovery model;

Step C3: Iteratively solve the objective function by using an iterative interpolation algorithm to obtain a restored original signal.

Optionally, after step C1, the method further includes:

Step C11: Introduce a substitute function, and transform the non-convex form of the phase recovery model into an easy-to-solve convex function.

Optionally, the steering vector matrix of the incident signal in step B1 is:

A _i is a column vector in the above expression, and

, Where i = 1,2, ..., M, d is the distance between the antenna elements, λ is the wavelength of the incident signal, M is the dimension of the signal, N is the number of antenna elements, and ψ _i is the ith signal The angle of incidence.

Optionally, the step of iteratively solving the objective function by using an iterative interpolation algorithm includes:

Calculate a new interpolation variable z based on the value of x ^(k) and use the least square method to obtain it;

These include:

Step C31, at the k-th iteration, let

According to the objective function in step C11 and combined with the least squares method, a parameter x ₁ is obtained as:

In the above formula, (·) ^H represents a conjugate transpose;

Step C32, let

Similarly, we get:

According to the intermediate variables x ₁ and x ₂ , two new vectors f and g are defined, which are expressed as

f = x ₁ -x ^(k)

g = (x ₂ -x ₁ ) -f

Step C33, calculate an intermediate parameter based on the vectors f and g:

Define the intermediate variable z, expressed as:

z = x ^(k) -2αf + α ² g

In step C34, according to the least squares method, let c ₃ = e ^{j∠ (Az)} . Then, at the k + ^1th time, the recovered vector is expressed as:

Optionally, the array antennas in step A are evenly spaced and uniformly distributed in a linear array, and the received signals collected include N signal amplitudes.

Beneficial effects, the present invention provides a phase recovery method based on an array antenna. A received signal to be phase-recovered is collected through the antenna array, and the received signal is transmitted to a receiver for phase recovery. The receiver uses iterative interpolation The algorithm performs phase recovery on the received signal and outputs the received signal after the phase recovery. The method disclosed in the present invention introduces the phase recovery theory in the fields of optics and image processing into the field of array signal processing. With the array antenna as the research background, it is assumed that the array antenna only measures the signal strength and establishes a phase based on the amplitude of the measured value The recovery model uses the iterative interpolation algorithm to recover the phase, so as to realize the complete recovery of the incident signal.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flowchart of steps of the array antenna-based phase recovery method provided by the present invention; FIG.

FIG. 2 is a comparison diagram of the mean square error curve between the recovered signal and the original signal under different iterations of the phase recovery method and the GS method according to the present invention; FIG.

FIG. 3a is a simulation diagram of a recovered signal when an iterative number is 1 in an iterative interpolation phase recovery method based on an array antenna;

FIG. 3b is a simulation diagram of the recovered signal when the number of iterations is 10 in an iterative interpolation phase recovery method based on an array antenna.

detailed description

In order to make the objectives, technical solutions, and advantages of the present invention clearer and more specific, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

In the baseband signal processing operation in the prior art, the traditional baseband signal processing processes are: receiving signals through an antenna array at the radio frequency front end, and then using a conventional navigation receiver to capture and track the received signals, and then The tracked signal is transmitted to the information solving device for information solving. The phase recovery method disclosed in the present invention is a new signal processing method applied in a navigation receiver, which realizes the capture and tracking of the received signals in the antenna array through the iterative interpolation technology, and realizes the complete restoration of the original signals.

An embodiment of the present invention discloses a phase recovery method for an array antenna. As shown in FIG. 1, the method includes:

Step S1: Use an antenna array to collect a received signal to be phase-recovered.

Step S2: transmitting the received signal to a receiver for phase recovery.

Step S3: The receiver uses an iterative interpolation algorithm to perform phase recovery on the received signal, and outputs the received signal after the phase recovery.

An antenna array provided at the radio frequency front end receives signals, and transmits the received signals to a receiver for phase recovery.

Specifically, the array antennas set in this method are uniformly spaced linear arrays, and the received signals collected include N signal amplitudes. The receiver uses an iterative interpolation algorithm to perform phase recovery on the received signal and outputs it.

Specifically, the receiver in step S3 uses an iterative interpolation algorithm to perform phase recovery on the received signal, and the content of the received signal output after the phase recovery further includes:

Step S31: Establish a phase recovery model based on the array antenna according to the steering vector of the original signal corresponding to the received signal;

Step S32: Establish an objective function corresponding to the phase recovery model by using an alternate iterative method and a least square method;

Step S33: Iteratively solve the objective function by using an iterative interpolation algorithm to obtain a restored original signal.

Preferably, in order to achieve the establishment of the objective function, after step S31, the method further includes:

Step S311: Introduce a substitute function to transform the non-convex form of the phase recovery model into a convex function that is easy to solve.

The step of iteratively solving the objective function by using an iterative interpolation algorithm includes:

These include:

Step S331, at the k-th iteration, let

According to the objective function in step S311 and combined with the least squares method, a parameter x ₁ is obtained as:

In the above formula, (·) ^H represents a conjugate transpose;

Step S332, make

Similarly, we get:

f = x ₁ -x ^(k)

g = (x ₂ -x ₁ ) -f

Step S333, calculate an intermediate parameter based on the vectors f and g:

Define the intermediate variable z, expressed as:

z = x ^(k) -2αf + α ² g

In step S334, according to the least squares method, let c ₃ = e ^{jA (Az)} . Then, at the k + ^1th time, the recovered vector is expressed as:

The method provided by the present invention is described in detail below according to specific application embodiments.

Step S1: Consider the background of the array antenna. When the phase of the incident signal is lost, a phase recovery model based on the array antenna is established according to the incident signal steering vector.

From a mathematical point of view, the phase recovery is to recover the original complex vector signal x from the intensity of the N measurement signals, and the vector x is an M-dimensional incident signal. Now suppose that in a noisy environment, the array antenna has measured N signal amplitudes, which is specifically expressed as

Then the signal measured by the antenna array

The model is represented as

The vector x in equation (1) is a M-dimensional original incident signal, and the vector n is an N-dimensional noise.

Is a known measurement signal, A is the vector matrix of incident signal steering, expressed as

A _i is a column vector in the above expression, and

Where i = 1,2, ..., M, d is the distance between the antenna elements, λ is the wavelength of the incident signal, M is the dimension of the signal, N is the number of antenna elements, and ψ _i is the ith signal incident The corresponding angle.

For the designed iterative interpolation phase recovery method based on the array antenna, the model we choose is:

In the above formula, ‖‖ ₂ means L2 or Frobenius norm,

Is the estimated value of the incident signal x.

Step S2: Introduce a substitute function, and transform the non-convex form of the phase recovery model into an easy-to-solve convex function.

Equation (3) is a non-convex nonlinear problem, which will be solved using the idea of substitution function below.

First, a new vector c = e ^{j∠ (Ax) is introduced} , where ∠ is an angle operation, and the equation (3) is equivalent to:

among them

Is the vector

Let it be a diagonal matrix formed by its main diagonal.

Step S3: Solve the iterative closed-form solution of the phase recovery model by using the alternating iterative method and the least square method to establish an objective function corresponding to the phase recovery model.

In this step, an alternating iterative method and a least square method are used to establish an objective function corresponding to the phase recovery model in step 2. This objective function contains part of the phase information of the original signal, and then iteratively solves the objective function. Iteratively updates the global phase variable c and the recovery variable x to recover the original signal.

When solving the problem iteratively, the phase variable c and the recovery variable x are alternately updated. First assume the value of the vector x at time k, that is, x ^(k) , and then calculate the value c ^{(k + 1) of the} vector c at time ^{k + 1} , and then calculate the updated value x ^{(k +1)} . And so on until the loop is terminated until the set iteration termination condition is met. The specific expression is as follows:

The iterative update expression of the vector c is

The iterative update of the vector x is expressed as

From the least squares method, we get

In the above formula, (·) ^H represents a conjugate transpose. When the iteration is terminated, x ^{(k + 1)} at this time is regarded as the original signal that is finally recovered.

Step S4: Using an iterative interpolation technique, an iterative interpolation phase recovery method based on an array antenna is proposed, which can quickly recover the original signal.

The iterative interpolation model achieves superlinear convergence speed under the condition that only parameter updates are required. It does not directly update x ^{(k + 1)} from the k-th iteration. The iterative interpolation model first finds the intermediate point z based on x ^(k) . Then update the next point x ^{(k + 1)} from this intermediate point. The specific operations on the iterative interpolation model are as follows:

First, at the k-th iteration, let

According to equations (6) and (7),

Then, make

Still according to equations (6) and (7), we get

f = x ₁ -x ^(k) (10)

g = (x ₂ -x ₁ ) -f (11)

According to equations (10) and (11), calculate an intermediate parameter

Define the intermediate variable z, expressed as

z = x ^(k) -2αf + α ² g (13)

Finally, according to equations (5)-(7), let c ₃ = e ^{j∠ (Az)} , then at the k + ^1th time, the recovered vector is expressed as

And so on until the loop is terminated until the set iteration termination condition is met. Finally, when the number of iterations satisfies the appropriate iteration conditions, the iteration is terminated, and x ^{(k + 1)} at this time is the last recovered signal.

To prove the effectiveness of the present invention, simulation verification was performed.

Assume that our array antenna is a uniform linear array, the corresponding number of array elements is 128, the array element spacing is half the wavelength of the incident signal, the number of signals is 16, and the corresponding angle is randomly distributed between 0 ° and 90 °. The incident signal is assumed to be a random Gaussian distribution. The noise power is set to 1, the signal-to-noise ratio is 25dB, and the maximum number of iterations is 50 times.

FIG. 3a and FIG. 3b show the MSE curve diagram between the recovered signal and the original signal in the present invention under different iteration times. It can be clearly seen from the figure that as the number of iterations increases, the MSE value of the method of the present invention decreases rapidly. When the number of iterations is 10, the MSE reaches a steady state value, which proves that the algorithm can effectively and quickly recover the original signal in the absence of phase information. The MSE curve of the GS method converges linearly, and the convergence speed is slow.

In order to show the process of recovering the signal by the iterative interpolation method in the present invention, FIG. 3 shows the distribution signal of the recovered signal when the number of iterations is 1 and 10, respectively. For convenience, the original signal distribution is also shown in the figure. Since the assumed initial value of the recovered signal is a random Gaussian distribution, it can be seen from the figure that when the first iteration is completed, the recovered signal is very different from the original signal. When the 10th iteration is completed, the recovered signal is basically close to the original signal, which proves the effectiveness of the algorithm. The recovery process in FIG. 3 corresponds to the MSE shown in FIG. 2.

Claims

A phase recovery method for an array antenna is characterized in that it includes:

Step A: Use an antenna array to collect a received signal for phase recovery;

Step B: transmitting the received signal to a receiver for phase recovery;

In step C, the receiver uses an iterative interpolation algorithm to perform phase recovery on the received signal, and outputs the received signal after the phase recovery.
The phase recovery method for an array antenna according to claim 1, wherein said step C further comprises:

Step C1: Establish a phase recovery model based on the array antenna according to the steering vector of the original signal corresponding to the received signal;

Step C2: An alternating iterative method and a least square method are used to establish an objective function corresponding to the phase recovery model;

Step C3: Iteratively solve the objective function by using an iterative interpolation algorithm to obtain a restored original signal.
The phase recovery method for an array antenna according to claim 2, wherein after step C1, the method further comprises:

Step C11: Introduce a substitute function to transform the non-convex form of the phase recovery model into an easy-to-solve convex objective function, expressed as

Where matrix A is the steering vector matrix of the incident signal x,
The signal measured for the array antenna,
Is the vector
Let it be a diagonal matrix formed by its main diagonal, the vector c = e j∠ (Ax) , ∠ is the angle operation, || || 2 represents L2 or Frobenius norm.
The phase recovery method for an array antenna according to claim 3, wherein the steering vector matrix of the incident signal x in the step C11 is:

A i is a column vector in the above expression, and
Where i = 1, 2, ..., M, d is the distance between antenna elements, λ is the wavelength of the incident signal, M is the dimension of the signal, N is the number of antenna elements, and ψ i is the ith signal incident The corresponding angle.
The phase recovery method for an array antenna according to claim 4, wherein the step of iteratively solving the objective function by using an iterative interpolation algorithm comprises:

Calculate a new interpolation variable z according to the value of x (k) , and obtain the solution of the objective function by the method of least squares;

These include:

Step C31, at the k-th iteration, let
According to the objective function in step C11, combined with the least squares method, a parameter x1 is obtained as:

In the above formula, (·) H represents a conjugate transpose;

Step C32, let
Similarly, we get:

According to the intermediate variables x 1 and x 2 , two new vectors f and g are defined, which are expressed as

f = x 1 -x (k)

g = (x 2 -x 1 ) -f

Step C33, calculate an intermediate parameter based on the vectors f and g:

Define the intermediate variable z, expressed as:

z = x (k) -2αf + α 2 g

In step C34, according to the least squares method, let c 3 = e j∠ (Az) . Then, at the k + 1th time, the recovered vector is expressed as:
The phase recovery method for an array antenna according to claim 4, wherein the array antennas in the step A are distributed in an evenly spaced linear array, and the received signals include N signal amplitudes .