CN103269223A - A Method of Compressing Sampling of Analog Signals - Google Patents

Abstract

The present invention provides an analog signal compression sampling method. On the basis of the framework, a compressed measurement matrix is proposed. Through the synchronous transformation of the compressed sampling value sequence and the compressed measurement matrix, the measured analog signal duplicates of the matrix-vector relational correlation pair are removed. Finally, the measured analog signal is reconstructed according to the relationship, and the sampling sequence of the measured analog signal is obtained. In this way, when the integrator integrates the demodulated signal within the sampling period, reset processing is not performed, which solves the problem of incomplete signal sampling caused by the discharge time of the integrator, and improves the performance of the analog information conversion compression sampling system.

Description

A Method of Compressing Sampling of Analog Signals

technical field

The invention belongs to the technical field of high-speed periodic signal sampling, and more specifically relates to an analog signal compression sampling method capable of reducing the difficulty of system design.

Background technique

Compressive sampling technology is an under-sampling method based on compressed sensing theory. This technology utilizes the sparse characteristic that only a small number of frequency components have significant amplitude after the Fourier transform of the periodic measured analog signal, and the amplitude of most of the frequency components is zero. The signal is randomly demodulated, the demodulated output signal is compressed by an integrator, and finally the compressed signal is sampled by a low-speed ADC, and the original signal, that is, the measured analog signal, can be accurately reconstructed through an optimization algorithm.

The implementation principle of the existing compressed sampling technology is shown in Figure 1. The expression of the mth sampling value y[m] is:

$\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; \&Integral;</span>\begin{array}{c}\mathrm{<span\; class="notranslate">\; m</span>}{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}\\ <span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}\end{array}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; d\&tau;</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

Where x(t) is the measured analog signal, p _c (t) is a pseudo-random sequence, and T _s is the sampling period.

In the existing compressed sampling method, the integrator needs to be reset through the auxiliary circuit after each sampling, so as to avoid the information coupling between two adjacent samples. However, the reset of the integrator by the auxiliary circuit is complicated. And the time required for each reset of the integrator is unknown, therefore, a uniform longer reset time must be adopted when it is reset. At the same time, during the reset process of the integrator, the energy of the measured analog signal cannot be collected, which will cause information leakage. This effect is reflected in the sampling value as an incomplete sampling of the measured analog signal, and when the sampled value is used to reconstruct the measured analog signal, the reconstructed signal will be distorted.

Contents of the invention

The purpose of the present invention is to overcome the deficiencies of the prior art, and provide a method for compressed sampling of analog signals to solve the incomplete sampling effect caused by the reset time of the integrator and improve the performance of compressed sampling signal reconstruction.

To achieve the above object, the analog signal compression sampling method of the present invention is characterized in that, comprising the following steps:

(1) The measured analog signal and the pseudo-random sequence with the Nyquist frequency of the signal are randomly demodulated by the mixer, and the demodulated output signal will carry the spectrum information of the signal in the entire frequency band, and the integrator realizes Compress the demodulated signal, and finally sample the integrated output signal at a sampling rate much lower than the Nyquist frequency of the signal to obtain a sequence of compressed sampling values;

During the sampling period, when the integrator integrates the demodulated signal, no reset processing is performed;

(2), according to the mathematical behavior model of the compressed sampling system, construct the compressed measurement matrix after synchronous transformation;

(3) Perform synchronous transformation on the obtained compressed sampling value sequence, and obtain a matrix-vector relational expression for removing correlation according to the constructed synchronously transformed compressed measurement matrix, and finally use this relational expression to reconstruct the measured analog signal, The sampling sequence of the analog signal to be tested is obtained, and the compressed sampling of the analog signal to be tested is completed.

The purpose of the present invention is achieved like this:

The analog signal compressed sampling method of the present invention proposes a compressed measurement matrix on the basis of the framework, and removes the impact of the matrix-vector relational correlation on the reconstruction performance of the measured analog signal by synchronously transforming the compressed sampling value sequence and the compressed measurement matrix , and finally reconstruct the measured analog signal according to the relationship, and obtain the sampling sequence of the measured analog signal. In this way, when the integrator integrates the demodulated signal within the sampling period, reset processing is not performed, which solves the problem of incomplete signal sampling caused by the discharge time of the integrator, and improves the performance of the analog information conversion compression sampling system.

Description of drawings

Fig. 1 is a schematic block diagram of traditional compressed sampling;

Fig. 2 is a functional block diagram of a specific embodiment of the analog signal compression sampling method of the present invention.

Detailed ways

Specific embodiments of the present invention will be described below in conjunction with the accompanying drawings, so that those skilled in the art can better understand the present invention. It should be noted that in the following description, when detailed descriptions of known functions and designs may dilute the main content of the present invention, these descriptions will be omitted here.

Fig. 2 is a functional block diagram of a specific embodiment of the analog signal compression sampling method of the present invention.

As shown in Figure 2, in this embodiment, the analog signal compression sampling method of the present invention includes the following steps:

Step ST1: Obtain a sequence of compressed sampling values.

The measured analog signal x(t) and the pseudo-random sequence P _c (t) with the Nyquist frequency of the signal are randomly demodulated by using a mixer. In this embodiment, in order to meet the requirements of circuit realizability and compressed sensing theory on the compressed measurement matrix, the pseudo-random sequence P _c (t) is composed of a Ledmarch pseudo-random sequence.

The integrator realizes the compression of the demodulated signal, and the integrator can be realized by a voltage integrator. In the present invention, when the integrator integrates the demodulated signal within the sampling period 0~m·T _s , no reset processing is performed. This is different from the reset process in the prior art, where integration is performed every sampling period T _s .

Finally, the integral output signal is sampled at a sampling rate much lower than the Nyquist frequency of the signal to obtain Compared with the prior art, the present invention does not need to reset the integrator.

The expression of compressed sampling value sequence y[m] is:

$\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; \&Integral;</span>\begin{array}{c}\mathrm{<span\; class="notranslate">\; m</span>}{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; d\&tau;</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$

Among them, m=1,2,...,M, M is the sequence length of the compressed sampling value.

Step ST2: Construct a compressed measurement matrix Φ after synchronous transformation.

In the present invention, the function of the integrator is equivalent to summing the compressed sampling values of each sampling period, therefore, the matrix form of the integrator can be expressed as:

In the formula, the matrix C is an M×N dimensional matrix, q=N/M, N is the length of the signal to be reconstructed, and the elements in the positions other than 1 in the matrix are 0. When N is not divisible by M, two adjacent rows share the information of a sampling value, and in matrix C, it is represented by fractions according to the proportional relationship of time length. For example:

When M=3, N=15, the expression of matrix C is:

$\mathrm{<span\; class="notranslate">\; C</span>}<span\; class="notranslate">\; =</span>\left[\begin{array}{ccccccccccccccc}\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

When M=3, N=14, the expression of matrix C is:

$\mathrm{<span\; class="notranslate">\; C</span>}<span\; class="notranslate">\; =</span>\left[\begin{array}{cccccccccccccc}\mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 3</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 3</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

The constructed compressed measurement matrix Φ after synchronous transformation is:

$\mathrm{<span\; class="notranslate">\; \&Phi;</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; C</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; =</span>\left[\begin{array}{c}{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\\ {\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="c"\; filenames="HDA00003135205000012.png"\; state="\{\{state\}\}">c</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\\ <span\; class="notranslate">\; \&CenterDot;</span>\\ <span\; class="notranslate">\; \&CenterDot;</span>\\ <span\; class="notranslate">\; \&CenterDot;</span>\\ {\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\end{array}\right]<span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

P的矩阵表示为：In the formula, c ₁ ...c _M is the row vector of the integrator matrix C, and C' is the integrator matrix after synchronous transformation. P is an N×N-dimensional diagonal matrix composed of pseudo-random sequence P _c (t) as diagonal elements. If the pseudo-random sequence is [ε ₁ ε ₂ ... ε _N ], the value of vector element ε _i is 1 Or -1, and the probability of the value distribution satisfies:

The matrix of P is expressed as:

The values of the off-diagonal elements in the formula are all 0.

Step ST3: Perform synchronous transformation on the compressed sampling value sequence to obtain a matrix-vector relational expression for which correlation is removed, and then reconstruct the measured analog signal.

In the analog signal compressed sampling method of the present invention, because the integrator is not reset, the current compressed sampling value contains the information of the compressed sampling value in each sampling period, and there is a strong correlation between the sampling values. In order to remove this The impact of correlation on the performance of a compressed sampling system, using the current sampled value minus the synchronous transformation of the previous adjacent sampled value. The sampled value sequence y after synchronous transformation and the compressed measurement matrix Φ after synchronous transformation have the following matrix-vector relationship for de-correlation:

y=Φx=Φ·Ψα, (9)

Among them, y=[y[1] y[2]-y[1] ... y[M]-y[M-1]], where Ψ is the frequency domain sparse representation base, which is composed of discrete Fourier transform vectors, Ψ is an N×N-dimensional matrix, α is the transformation coefficient of the base Ψ sparsely represented in the frequency domain by the sampling sequence x of the analog signal to be reconstructed, and the sequence length of α and x is N.

Through the compression sensing signal reconstruction algorithm, the transformation coefficient α is obtained, and finally the sampling sequence x of the measured analog signal x(t) is obtained through the inverse Fourier transform.

Reconstructing the measured analog signal belongs to the prior art, and will not be repeated here.

The compressed sampling value sequence proposed by the invention is transformed synchronously with the compressed measurement matrix, which can reduce the correlation.

The correlation coefficient μ(Φ,Ψ) of the compressed measurement matrix is defined as:

$\mathrm{<span\; class="notranslate">\; \&mu;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&Phi;</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; \&Psi;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="max"\; filenames="HDA00003135205000012.png"\; state="\{\{state\}\}">max</figure-callout></span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; \&lang;</span>{\mathrm{<span\; class="notranslate">\; \&phi;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&psi;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; \&rang;</span><span\; class="notranslate">\; |</span>$

In the formula, φ _i and ψ _j are respectively the i-th row vector and the j-th column vector of the matrices Φ and Ψ.

Assume that the probability of the correlation coefficient before the synchronous transformation between the compressed sampling value sequence and the compressed measurement matrix (Φ=C P) is less than the constant u (u>0) satisfies the condition:

The probability that the correlation coefficient is smaller than the constant u after the synchronous transformation of the compressed sampling value sequence and the compressed measurement matrix (Φ=C′·P) satisfies the condition:

p ₁ and p ₂ are normal numbers greater than 0, then p ₂ >p ₁ , that is, the probability that the correlation coefficient is smaller than the constant u after transformation is greater.

prove:

为矩阵C的第k行第i列元素的共轭转置，ψ_kj为矩阵Ψ的第k行第j列元素。由霍夫丁不等式有如下关系：Correlation coefficient μ(Φ,Ψ) before synchronous transformation, since Φ=C P, so the correlation coefficient can be rewritten as μ(C P,Ψ)=μ(C,P Ψ), because $\&lang;c_i,P{\mathrm{\&psi;}}_j\&rang;={\mathrm{\&Sigma;}}_{k=1}^N{\mathrm{\&epsiv;}}_k{c}_{\mathrm{the\; ki}}^{*}{\mathrm{\&psi;}}_{\mathrm{kj}}={\mathrm{\&Sigma;}}_{k=1}^N{\mathrm{\&epsiv;}}_k{a}_k^{\mathrm{ij}},$ here $a_k^{\mathrm{ij}}=c_{\mathrm{the\; ki}}^{*}{\mathrm{\&psi;}}_{\mathrm{kj}},$ ε _k is the kth diagonal element of matrix P, c _i is the ith row vector of matrix C,

is the conjugate transpose of the element in row k and column i of matrix C, and ψ _kj is the element in row k and column j of matrix Ψ. According to Hoffding's inequality, there is the following relationship:

概率的联合界为：For all constants u>0, 1≤i, j≤N, with

The joint bound for the probability is:

so:

Similarly, it can be proved that the correlation coefficient μ(Φ,Ψ)=μ(C'·P,Ψ) after synchronous transformation satisfies the following conditions:

显然，根据矩阵C与C′的定义可得 ${\left|\right|b^{\mathrm{ij}}\left|\right|}_2^2<{\left|\right|a^{\mathrm{ij}}\left|\right|}_2^2.$ 假定In the formula

Obviously, according to the definition of matrices C and C', we can get ${\left|\right|b^{\mathrm{ij}}\left|\right|}_2^2<{\left|\right|a^{\mathrm{ij}}\left|\right|}_2^2.$ assumed

${\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="p"\; filenames="HDA00003135205000012.png"\; state="\{\{state\}\}">p</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\underset{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span>$

and

${\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\underset{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; b</span>}}^{\mathrm{<span\; class="notranslate">\; ij</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; )</span>$

所以p₂>p₁，即：变换以后相关系数小于常数u的概率更大，能够降低相关性。because

So p ₂ >p ₁ , that is, the probability that the correlation coefficient is smaller than the constant u after transformation is greater, which can reduce the correlation.

Although the illustrative specific embodiments of the present invention have been described above, so that those skilled in the art can understand the present invention, it should be clear that the present invention is not limited to the scope of the specific embodiments. For those of ordinary skill in the art, As long as various changes are within the spirit and scope of the present invention defined and determined by the appended claims, these changes are obvious, and all inventions and creations using the concept of the present invention are included in the protection list.

Claims (4)

1. an analog signal compression sampling method is characterized in that, may further comprise the steps:

(1), tested analog signal is carried out demodulation at random with the pseudo random sequence with signal nyquist frequency through frequency mixer, signal after the demodulation output all will carry the spectrum information of signal on whole frequency band, integrator is realized the compression to signal after the demodulation, last far below the sample rate of signal nyquist frequency integral output signal is sampled, obtain the compression sampling value sequence;

In the sampling time section, when the signal of integrator after to demodulation carries out integration, do not carry out reset processing;

(2), according to the mathematics behavior model of compression sampling system, matrix is measured in the compression of constructing after the synchronous conversion;

(3), the compression sampling value sequence that obtains is carried out synchronous conversion, measure matrix according to the compression after the synchronous conversion of structure, obtain removing the matrix-vector relational expression of correlation, with this relational expression tested analog signal is reconstructed at last, obtain the sample sequence of tested analog signal, finish the compression sampling to tested analog signal.

2. analog signal compression sampling method according to claim 1 is characterized in that, the pseudo random sequence described in the step (1) is Rider horse contract pseudo random sequence.

3. analog signal compression sampling method according to claim 2 is characterized in that, the compression described in the step (2) is measured matrix and is:

$\mathrm{\&Phi;}=C^{\&prime;}\&CenterDot;P=\left[\begin{array}{c}{c}_1\\ c_2-c_1\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ c_M-c_{M-1}\end{array}\right]\&CenterDot;P$

C in the formula ₁C _MBe the row vector of integrator Matrix C, C ' is the integrator matrix after the synchronous conversion; P is pseudo random sequence P _c(t) N * N that constitutes as diagonal element ties up diagonal matrix, if pseudo random sequence is [ε ₁ε ₂ε _N], vector element ε then _iValue be 1 or-1, and the probability that value distributes satisfies:

The matrix notation of P is:

The value of off-diagonal element is 0 in the formula;

The integrator Matrix C can be expressed as:

In the formula, Matrix C is that M * N ties up matrix, q=N/M, and N is for treating reconstruction signal length, the element of non-1 position is 0 in the matrix; When N can not be divided exactly by M, adjacent two row were shared the information of a sampled value, in Matrix C, adopted fraction representation by time length ratio relation.

4. analog signal compression sampling method according to claim 3 is characterized in that, the matrix-vector relational expression of the removal correlation described in the step (3):

y=Φx=Φ·Ψα,

Wherein, sampled value sequence y=[y[1 after the conversion synchronously] y[2]-y[1] ... y[M]-y[M-1]], Ψ is that frequency-domain sparse is represented base in the formula, constituted by the discrete Fourier transform (DFT) vector, Ψ is that N * N ties up matrix, to be the sample sequence x that treats the tested analog signal of reconstruct represent the conversion coefficient of basic Ψ at frequency-domain sparse to α, and the sequence length of α and x is N.

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