CN107026650B - Analog low-pass filter, analog information converter and filtering method - Google Patents
Analog low-pass filter, analog information converter and filtering method Download PDFInfo
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Abstract
The invention discloses an analog low-pass filter, an analog information converter and a filtering method applied to AIC. The analog low-pass filter comprises an addition unit, a state conversion unit, a first delay unit and at least one second delay unit; the adding unit is used for superposing the received mixing signal and the signal from the state conversion unit to obtain a synthesized signal; the first delay unit is used for delaying the synthesized signal by unit time; the second delay unit is used for delaying the signal from the state conversion unit by unit time; the state conversion unit is used for weighting the signals from the first delay unit and the second delay unit by using a cyclic period matrix and correspondingly inputting the signals to the adder and the second delay unit respectively; the parameters of the periodic cycle matrix meet the conditions that any two columns of the measurement matrix are irrelevant; one of the first delay cell or the at least one second delay cell is further adapted to input the delayed signal to a sampler coupled to the analog low pass filter.
Description
Technical Field
The present invention relates to the field of compressed sensing, and in particular, to an analog low-pass filter, an analog information converter, and a filtering method.
Background
According to the nyquist sampling theorem, for a wideband signal, the lowest sampling rate needs to be greater than twice the highest frequency in the signal in order to ensure distortion-free recovery of the original signal. That is, the higher the highest frequency of the signal, the higher the requirement for the lowest sampling rate, and the difficulty in developing Analog-to-Digital converters (ADCs) with higher sampling rates is increasing.
In order to solve this problem, researchers find that, in many practical applications, such as radar, sonar, medical detection systems, etc., the information rate carried by a signal is limited, that is, the signal has sparsity, and if the signal with sparsity can be compressed without basically affecting the information carried by the signal, the highest frequency of the compressed signal is far lower than that of the original signal, so that the requirement on the sampling rate of the ADC can be reduced. After the sampling signal is obtained, the original signal can be reconstructed by using the information carried by the sampling signal so as to meet the requirement of using the original signal. The theory of Compressive Sensing (CS) is based on such assumptions. Compressed sensing theory states that if a signal is compressible or sparse in a transform domain, a high-dimensional signal can be projected into a low-dimensional space through an observation matrix that is not related to the transform domain, and then the high-dimensional signal can be reconstructed with high probability from a small amount of low-dimensional data by solving an optimization problem. Such a device that converts an Analog signal into a digital signal using the compressive sensing principle and extracts sparse Information therefrom is called an Analog-to-Information converter (AIC). The analog information converter generally includes a mixer, an analog low-pass filter and a sampling unit, wherein the mixer is used for mixing an initial signal with a Pseudo-random sequence (Pseudo-random sequence) signal, and a low-frequency part of the mixed signal contains important information capable of expressing the initial signal; the analog low-pass filter is used for filtering the mixed signal and extracting the low-frequency part of the mixed signal; the sampler is used to sample the filtered signal, and the sampling unit is typically an ADC.
In order to realize the compressed sensing of the signal, the condition that any two columns of the observation matrix are irrelevant must be met, the condition is closely related to the design of the analog low-pass filter, and the condition cannot be met if the design of the analog low-pass filter is unreasonable, so that the compressed sensing of the signal cannot be realized. In view of the foregoing, there is a need in the art for an analog low pass filter that enables compressed sensing.
Disclosure of Invention
In order to solve the above technical problems, embodiments of the present invention provide an analog low pass filter that can be applied to AIC, an analog information converter that can implement compressed sensing, and a filtering method.
In a first aspect, an embodiment of the present invention provides an analog low-pass filter, where the analog low-pass filter includes:
the circuit comprises an adding unit, a state conversion unit, a first delay unit and at least one second delay unit;
the output end of the state conversion unit is connected with the addition unit, the input end of the state conversion unit is connected with the first delay unit, and the addition unit is connected with the first delay unit;
the output end and the input end of the state conversion unit are respectively connected with the at least one second delay unit;
the adding unit is configured to superimpose the mixing signal received by the analog low-pass filter and the signal from the state converting unit to obtain a synthesized signal, and input the synthesized signal to the first delay unit, where the mixing signal is a signal obtained by mixing an initial signal and a random sequence signal;
the first delay unit is configured to delay the synthesized signal from the addition unit by a unit time, and input a signal delayed by the first delay unit to the state transition unit;
the at least one second delay unit is used for delaying the signal from the state conversion unit by unit time and inputting the signal after the delay of the at least one second delay unit to the state conversion unit;
the state conversion unit is configured to perform weighting processing on the signals from the first delay unit and the at least one second delay unit by using a cyclic periodic matrix, and input the weighted signals to the adder and the at least one second delay unit respectively; the parameters of the periodic cyclic matrix are obtained according to the coefficients of the frequency domain impulse response function of the analog low-pass filter, and the coefficients of the frequency domain impulse response function meet the following conditions: any two columns of the measurement matrix obtained according to the frequency domain impulse response function and the sampling function are irrelevant;
the first delay unit, or one of the at least one second delay unit, is further configured to input the delayed signal to a sampler connected to the analog low-pass filter.
In a first possible implementation manner of the first aspect, the frequency-domain impulse response function H at time jj(z) is:
wherein i is a time number, T is a period of the periodic cyclic matrix, N is the number of row vectors or column vectors of A (T) 'lambda'k,i,jThe k characteristic value of Ψ (T,0), λ ″k,i,jIs the k-th characteristic value, λ ″, of Ψ (i-1,0) Ψ (T, j)'k,i,jThe kth eigenvalue of Ψ (i-1, j);
where A (t) is a periodic circulant matrix.
With reference to the first possible implementation manner of the first aspect, in a second possible implementation manner, any two columns of the uncorrelated values of the measurement matrix are specifically represented by the following formula:
wherein F phii＝D(z)Hi(z)，Hi(z) is the frequency domain impulse response function at time i, and D (z) is the sampling function; phiiIs D (z) Hi(z) a function after Fourier transform, namely a measurement matrix corresponding to the time i, wherein F is Fourier transform; f phij＝D(z)Hj(z)，ΦjIs D (z) Hj(z) a function after Fourier transform, namely a measurement matrix corresponding to the moment j, wherein F is Fourier transform; (F.phi.)i)HTo F phiiFind the conjugate transpose, | | F Φi||2To find F phii2 norm of the modulus, | F Φj||2To find F phijIs the allowable fluctuation value, said V is the 2 norm of the modulus oflFor measuring the matrix F phijAny two columns of.
With reference to the first possible implementation manner of the first aspect, in a third possible implementation manner, the condition further includes: the poles of the system are within the unit circle.
With reference to the third possible implementation manner of the first aspect, in a fourth possible implementation manner, the poles of the system are represented in a unit circle by the following formula:
wherein, | λ'k,i,j||2Is to obtain lambda'k,i,j2 norm of the norm.
With reference to the first possible implementation manner of the first aspect, in a fourth possible implementation manner, the condition further includes: the system pole is greater than or equal to a preset value.
With reference to the fourth possible implementation manner of the first aspect, in a fifth possible implementation manner, the system pole being greater than or equal to the preset value is represented by the following formula:
min{||λ'k,i,j||2}≥γ
wherein gamma is a preset value and represents the lower bound of the distance between the pole and the origin.
With reference to the first aspect, in a sixth possible implementation manner, the condition further includes: and the parameters of the periodic cyclic matrix are in a preset range.
In a second aspect, an embodiment of the present invention provides an analog information converter, where the analog information converter includes a mixer, a sampler, and an analog low-pass filter of any one of the first to sixth possible implementations of the first aspect and the first aspect, where the mixer is connected to the analog low-pass filter, and the analog low-pass filter is connected to the sampler;
the mixer is used for mixing the initial signal with the random sequence signal to obtain a mixing signal;
the analog low-pass filter is used for filtering the mixing signal to obtain a filtering signal;
the sampler is used for sampling the filtering signal.
In a third aspect, an embodiment of the present invention provides an analog information converter, including:
a processor;
a memory; and a communication port;
wherein the processor is configured to execute the instructions stored by the memory to perform the following steps when executing the instructions:
receiving a mixing signal from a mixer through the communication port, wherein the mixing signal is a signal obtained by mixing an initial signal and a random sequence signal, and the mixer is connected with an analog low-pass filter;
superposing the mixing signal and the weighted signal to obtain a synthesized signal;
delaying the synthesized signal for a unit time, and delaying the weighted signal for a unit time;
weighting the delayed signals by using a periodic cyclic matrix to obtain weighted signals; the parameters of the periodic cyclic matrix are obtained according to the coefficients of the frequency domain impulse response function of the analog low-pass filter, and the coefficients of the frequency domain impulse response function meet the following conditions: any two columns of the measurement matrix obtained according to the frequency domain impulse response function and the sampling function are irrelevant;
the delayed signal is input to a sampler connected to the analog low pass filter through the communication port.
In a first possible implementation manner of the third aspect, the frequency-domain impulse response function H at time jj(z) is:
wherein i is a time number, T is a period of the periodic cyclic matrix, N is the number of row vectors or column vectors of A (T) 'lambda'k,i,jThe k characteristic value of Ψ (T,0), λ ″k,i,jIs the k-th characteristic value, λ ″, of Ψ (i-1,0) Ψ (T, j)'k,i,jThe kth eigenvalue of Ψ (i-1, j);
where A (t) is a periodic circulant matrix.
With reference to the first possible implementation manner of the third aspect, in a second possible implementation manner, any two columns of uncorrelation of the measurement matrix are specifically represented by the following formula:
wherein F phii＝D(z)Hi(z)，Hi(z) is the frequency domain impulse response function at time i, and D (z) is the sampling function; phiiIs D (z) Hi(z) a function after Fourier transform, namely a measurement matrix corresponding to the time i, wherein F is Fourier transform; f phij＝D(z)Hj(z)，ΦjIs D (z) Hj(z) a function after Fourier transform, namely a measurement matrix corresponding to the moment j, wherein F is Fourier transform; (F.phi.)i)HTo F phiiFind the conjugate transpose, | | F Φi||2To find F phii2 norm of the modulus, | F Φj||2To find F phijIs the allowable fluctuation value, said V is the 2 norm of the modulus oflFor measuring the matrix F phijAny two columns of.
In a third possible implementation manner of the third aspect, the condition further includes: the poles of the system are within the unit circle.
In a fourth possible implementation manner of the third aspect, the condition further includes: the system pole is greater than or equal to a preset value.
In a fifth possible implementation manner of the third aspect, the condition further includes: and the parameters of the periodic cyclic matrix are in a preset range. In a fourth aspect, an embodiment of the present invention provides a filtering method, where the method includes:
the analog low-pass filter receives a mixing signal from the mixer, wherein the mixing signal is a signal obtained by mixing an initial signal and a random sequence signal, and the mixer is connected with the analog low-pass filter;
the analog low-pass filter superposes the mixing signal and the weighted signal to obtain a synthesized signal;
the analog low-pass filter delays the synthesized signal by unit time and delays the weighted signal by unit time;
the analog low-pass filter performs weighting processing on the delayed signals by utilizing a periodic cyclic matrix to obtain weighted signals; the parameters of the periodic cyclic matrix are obtained according to the coefficients of the frequency domain impulse response function of the analog low-pass filter, and the coefficients of the frequency domain impulse response function meet the following conditions: any two columns of the measurement matrix obtained according to the frequency domain impulse response function and the sampling function are irrelevant;
the analog low-pass filter inputs the delayed signal to a sampler connected to the analog low-pass filter.
In a first possible implementation manner of the fourth aspect, the frequency-domain impulse response function H at time jj(z) is:
wherein i is a time number, T is a period of the periodic cyclic matrix, N is the number of row vectors or column vectors of A (T) 'lambda'k,i,jThe k characteristic value of Ψ (T,0), λ ″k,i,jIs the k-th characteristic value, λ ″, of Ψ (i-1,0) Ψ (T, j)'k,i,jThe kth eigenvalue of Ψ (i-1, j);
where A (t) is a periodic circulant matrix.
With reference to the first possible implementation manner of the fourth aspect, in a second possible implementation manner, any two columns of uncorrelation of the measurement matrix are specifically represented by the following formula:
wherein F phii＝D(z)Hi(z)，Hi(z) is the frequency domain impulse response function at time i, and D (z) is the sampling function; phiiIs D (z) Hi(z) a function after Fourier transform, namely a measurement matrix corresponding to the time i, wherein F is Fourier transform; f phij＝D(z)Hj(z)，ΦjIs D (z) Hj(z) a function after Fourier transform, namely a measurement matrix corresponding to the moment j, wherein F is Fourier transform; (F.phi.)i)HTo F phiiFind the conjugate transpose, | | F Φi||2To find F phii2 norm of the modulus, | F Φj||2To find F phijIs the allowable fluctuation value, said V is the 2 norm of the modulus oflFor measuring the matrix F phijAny two columns of.
In a third possible implementation manner of the fourth aspect, the condition further includes: the poles of the system are within the unit circle.
In a fourth possible implementation manner of the fourth aspect, the condition further includes: the system pole is greater than or equal to a preset value.
In a fifth possible implementation manner of the fourth aspect, the condition further includes: and the parameters of the periodic cyclic matrix are in a preset range.
In a fifth aspect, an embodiment of the present invention provides a filtering method, where the method is applied to the first aspect and any one of the first to sixth possible implementation manners of the first aspect, and the filtering method includes:
receiving a mixing signal from a mixer, wherein the mixing signal is a signal obtained by mixing an initial signal and a random sequence signal, and the mixer is connected with the analog low-pass filter;
the adding unit superposes the mixing signal and the signal from the state conversion unit to obtain a synthesized signal;
the first delay unit delays the synthesized signal from the addition unit by a unit time;
the at least one second delay unit delays the signal from the state transition unit by a unit time;
the state conversion unit performs weighting processing on signals from the first delay unit and the at least one second delay unit by using a cyclic period matrix, and correspondingly inputs the weighted signals to the adder and the at least one second delay unit respectively; the parameters of the periodic cyclic matrix are obtained according to the coefficients of the frequency domain impulse response function of the analog low-pass filter, and the coefficients of the frequency domain impulse response function meet the following conditions: any two columns of the measurement matrix obtained according to the frequency domain impulse response function and the sampling function are irrelevant;
the delayed signal is input to a sampler connected to the analog low pass filter using the first delay cell or one of the at least one second delay cell.
In a first possible implementation form of the fifth aspect, the frequency-domain impulse response function H at time jj(z) is:
wherein i is a time number, T is a period of the periodic cyclic matrix, N is the number of row vectors or column vectors of A (T) 'lambda'k,i,jThe k characteristic value of Ψ (T,0), λ ″k,i,jIs the k-th characteristic value, λ ″, of Ψ (i-1,0) Ψ (T, j)'k,i,jThe kth eigenvalue of Ψ (i-1, j);
where A (t) is a periodic circulant matrix.
With reference to the first possible implementation manner of the fifth aspect, in a second possible implementation manner, any two columns of uncorrelation of the measurement matrix are specifically represented by the following formula:
wherein F phii＝D(z)Hi(z)，Hi(z) is the frequency domain impulse response function at time i, and D (z) is the sampling function; phiiIs D (z) Hi(z) a function after Fourier transform, namely a measurement matrix corresponding to the time i, wherein F is Fourier transform; f phij＝D(z)Hj(z)，ΦjIs D (z) Hj(z) a function after Fourier transform, namely a measurement matrix corresponding to the moment j, wherein F is Fourier transform; (F.phi.)i)HTo F phiiFind the conjugate transpose, | | F Φi||2To find F phii2 norm of the modulus, | F Φj||2To find F phijIs the allowable fluctuation value, said V is the 2 norm of the modulus oflFor measuring the matrix F phijAny two columns of.
In a third possible implementation manner of the fifth aspect, the condition further includes: the poles of the system are within the unit circle.
In a fourth possible implementation manner of the fifth aspect, the condition further includes: the system pole is greater than or equal to a preset value.
In a fifth possible implementation manner of the fifth aspect, the condition further includes: and the parameters of the periodic cyclic matrix are in a preset range.
According to the technical scheme, after the mixing signal is input into the adding unit, the mixing signal is superposed with the signal from the state conversion unit through the adding unit, and the first delay unit, the at least one second delay unit and the state conversion unit are subjected to delay and weighting cyclic processing, so that the filtering of the mixing signal is realized.
The parameters of the periodic cyclic matrix in the state conversion unit are obtained according to the frequency domain impulse response function obtained by the analog low-pass filter designed by the invention, the parameters of the frequency domain impulse response function obtained by the analog low-pass filter need to meet the conditions that any two columns of the analog low-pass filter are irrelevant, so that the analog low-pass filter can be suitable for an analog information converter, and the measurement matrix is obtained according to the frequency domain impulse response function and the sampling function obtained by the analog low-pass filter.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to these drawings without creative efforts.
Fig. 1 is a schematic structural diagram of an analog low-pass filter according to an embodiment of the present invention;
fig. 2 is a schematic structural diagram of a second-order analog low-pass filter according to an embodiment of the present invention;
fig. 3 is a schematic structural diagram of an analog-to-digital converter according to a second embodiment of the present invention;
fig. 4 is a schematic structural diagram of an analog-to-digital converter according to a third embodiment of the present invention.
Detailed Description
Before the technical solution of the present invention is introduced, the basic theory of compressed sensing is first introduced. It is assumed that the signal can be represented by a signal x (signal vector) of length n, when the signal x is represented by a set of orthogonal baseline linear combinations, i.e., where Ψ is a sparse transform basis matrix. If the projection coefficient θ has only K significant coefficients (i.e. coefficients with relatively large absolute values), and K is much smaller than n, we say that the signal x has sparsity, and the signal x is also called K-sparse signal. Since the signal x is sparse and thus can be compressed, assuming that the compressed signal is y (signal vector), the length is m, and m is much smaller than n, the signal y can be called an observation vector. The observation vector is a linear projection of the signal vector, which can be represented by a measurement matrix Φ, i.e., y ═ Φ x, where the measurement matrix Φ is a rank-deficient matrix. That is, assume that the projection coefficient is (θ)1,θ2,…,θn) The sparse transform basis matrix Ψ is
Measurement matrix phi of
Then
Since m is much smaller than n, the purpose of compressing the signal is achieved. In addition, we refer to the product A of φ and Ψ as the observation matrix.
When the compressed signal y is obtained, it needs to be restored to the signal x before it can be used. In order to achieve signal recovery, it is necessary to ensure that the correlation between any two columns of the observation matrix a (a ═ Φ Ψ) is as small as possible. Since the sparse transform basis matrix Ψ is relatively fixed, it is only necessary to ensure that the correlation between any two columns of the measurement matrix Φ is as small as possible.
In the embodiment of the present invention, an analog information converter AIC is used to implement compressed sensing on an original signal, that is, the signal is subjected to edge-compression and edge-sampling according to the principle described above. Based on the above analysis, in order to fulfill the requirement of compressed sensing, the design of AIC needs to ensure that the correlation of any two columns of the measurement matrix Φ is as small as possible. Whereas for AIC it typically comprises three parts, a mixer, a low-pass analog filter and a sampler, among which the measurement matrix Φ can be derived based on the low-pass analog filter function of the low-pass analog filter and the sampling function of the sampler. The design of the analog low-pass filter is particularly critical in the design of AIC, since the sampling function is generally only sample rate dependent and is relatively easy to obtain. The requirement of making any two columns of the measurement matrix phi irrelevant is met by designing the analog low-pass filter, so that the requirement of compressed sensing is met.
In the invention, an inventor firstly designs a simulation low-pass filter, determines a relational expression of a frequency domain impulse response function of the simulation low-pass filter according to the connection relation of each unit in the simulation low-pass filter, determines and obtains a value of a variable in the relational expression based on the requirement that the correlation of any two columns of the measurement matrix phi is as small as possible, and determines and obtains matrix parameters of corresponding units in the simulation low-pass filter by combining the value of the variable, thereby completing the design of the simulation low-pass filter in the AIC. The above design is described in detail below.
Design of composition and connection relation of analog low-pass filter
In the invention, the analog low-pass filter comprises an adding unit, a state conversion unit, a first delay unit and at least one second delay unit. The output end of the state conversion unit is connected with the addition unit, the input end of the state conversion unit is connected with the first delay unit end, and the addition unit is connected with the first delay unit; and the output end and the input end of the state conversion unit are respectively connected with the at least one second delay unit. The adding unit is configured to superimpose the mixing signal received by the analog low-pass filter and the signal from the state converting unit to obtain a synthesized signal, and input the synthesized signal to the first delay unit. The first delay unit is configured to delay the combined signal from the adding unit by a unit time, and input the signal delayed by the first delay unit to the state transition unit. The at least one second delay unit is configured to delay the signal from the state transition unit by a unit time, and input the signal delayed by the at least one second delay unit to the state transition unit. The state conversion unit is configured to perform weighting processing on the signals from the first delay unit and the at least one second delay unit by using a cyclic periodic matrix, and correspondingly input the weighted signals to the at least one second delay unit and the adder, respectively, so as to perform a cycle. The first delay unit, or one of the at least one second delay unit, is further configured to input the delayed signal to a sampler connected to the analog low-pass filter.
According to the above-mentioned composition and connection relation of analog low-pass filter, the frequency domain impulse response function H can be obtainedj(z)。
The key for determining whether the analog low-pass filter can be applied to the AIC system by the parameters of the periodic circular matrix in the state conversion unit is the determination of the parameters of the periodic circular matrix, and the primary condition met by the parameters of the periodic circular matrix is that any two columns of the measurement matrix are uncorrelated. Because of the filter function HjThe parameters of (z) are determined by the parameters of a periodic circulant matrix, and the measurement matrix is determined by the frequency domain impulse response function Hj(z) and a sampling function D (z) where the known, frequency-domain impulse response function H is determinedj(z) known, but frequency-domain impulse response function HjUnder the condition that the (z) parameter is unknown, according to the frequency domain impulse response function HjAny two columns of matrix after multiplication of (z) and sampling function D (z) and Fourier transform are uncorrelated to obtain frequency domain impulse response function Hj(z) and then according to said frequency domain impulse response function HjThe coefficients of (z) are inversely derived from the parameters of the periodic circulant matrix.
The first embodiment is as follows:
based on the above-described operating principle of the analog low-pass filter, the present invention provides an embodiment of an analog low-pass filter applied to AIC, and referring to fig. 1, the analog low-pass filter includes:
an adding unit 101, a state transition unit 102, one first delay unit 103 and at least one second delay unit 104.
Wherein, the output end of the state conversion unit 102 is connected with the addition unit 101, the input end of the state conversion unit 102 is connected with the first delay unit 103, and the addition unit 101 is connected with the first delay unit 103;
the output and input of the state transition unit 102 are respectively connected to the at least one second delay unit 104.
The adding unit 101 is configured to superimpose the mixing signal received by the analog low-pass filter and the signal from the state converting unit 102 to obtain a synthesized signal. The mixing signal is from a mixer, and specifically, the mixer mixes an initial signal with a random sequence signal to obtain the mixing signal.
The first delay unit 103 is configured to delay the synthesized signal from the adding unit 101 by a unit time, and input the signal delayed by the first delay unit 103 to the state transition unit 102.
The at least one second delay unit 104 is configured to delay the signal from the state transition unit 102 by a unit time, and input the signal delayed by the at least one second delay unit 104 to the state transition unit 102;
the state transition unit 102 is configured to perform weighting processing on the signals from the first delay unit 103 and the at least one second delay unit 104 by using a cyclic periodic matrix, and correspondingly input the weighted signals to the adder 101 and the at least one second delay unit 103 for circulation. The periodic cyclic matrix refers to a cyclic matrix with periodically changed parameters, and is characterized in that each parameter of a row vector is a result of sequentially moving each parameter of a previous row vector to the right by one position. In this embodiment, the periodic cyclic matrix may be represented as:
the periodic cyclic matrix has a characteristic of periodic variation, that is, a (T) ═ a (T + T), and T is a period of the periodic cyclic matrix. When the T is 0, the periodic cycle matrix is called a time-invariant matrix and is suitable for a stable random initial signal, and the stable random meaning is that any n-dimensional distribution function or probability density function of the signal is irrelevant to a time starting point; and when the T is larger than 0, the periodic cycle matrix is a time-varying matrix and is suitable for initial signals of non-stationary random, wherein the non-stationary random refers to correlation between any n-dimensional distribution function or probability density function of the signals and a time starting point.
The parameters of the periodic cyclic matrix are obtained according to the coefficients of the frequency domain impulse response function of the analog low-pass filter, and the coefficients of the frequency domain impulse response function meet the following conditions: and any two columns of the measurement matrix obtained according to the frequency domain impulse response function and the sampling function are irrelevant.
The first delay unit 103 is further configured to input the signal delayed by the first delay unit 103 to a sampler connected to the analog low-pass filter; or, one of the second delay units 104 in the at least one second delay unit 104 is further configured to input the signal delayed by the second delay unit 104 to the sampler connected to the analog low-pass filter.
The working principle of the analog low-pass filter is described below by taking a first delay unit 103 and a second delay unit 104 as examples. Referring to fig. 2, assuming that the mixing signal output by the mixer at different time t is x (t), the periodic cyclic matrix is
The mixing signal x (t) is input to the adding unit 101, and the adding unit 101 adds the mixing signal x (t) and the signal y from the state converting unit 1021(t) the signal after synthesis is z (t). The resultant signal z (t) is delayed by the first delay unit 103 to become a signal x1(t), the signal x1(t) is divided into two paths, one path is output, and the other path is input to the state conversion unit 102. At the same time, the signal x from the second delay unit 1042(t) is also input to the state transition unit 102. The signal x1(t) sum signal x2(t) constitutes a signal vector, which is multiplied by the periodic circulant matrix A (t) of the state transition unit 102 to pair the signal x1(t) sum signal x2(t) weighting and outputting the weighted signal y1(t) and y2(t) wherein the signal y1(t) input to the addition unit 101, signal y2(t) is input to the second delay unit 104 for the next cycle.
Specifically, x (0), x (1), and x (2) are mixing signals output from the mixer at times t equal to 0, 1, and 2, respectively, where t equal to 0 represents an initial time. Let a (0), a (1), a (2) be the corresponding periodic cyclic matrices at times t equal to 0, 1, 2, respectively, where,
z (0), z (1), and z (2) are signals output from adding section 101 when time t is 0, 1, and 2, respectively; x is the number of1(0)、x1(1)、x1(2) Signals output by first delay section 103 when time t is 0, 1, and 2, respectively; x is the number of2(0)、x2(1)、x2(2) Signals output by the second delay unit 103 when time t is 0, 1, and 2, respectively; y is1(0)、y1(1)、y1(2) Signals output from state converting section 102 to adding section 101 when time t is 0, 1, and 2, respectively; y is2(0)、y2(1)、y2(2) The signals output from the state transition unit 102 to the second delay unit 104 when the time t is 0, 1, and 2, respectively.
When t is 0, one can obtain:
z(0)＝x(0)
x1(0)＝0,x2(0)＝0
y1(0)＝0,y2(0)＝0
it should be noted that at the time when t is 0, the mixing signal x (0) passes through the adding unit 101, since y is the initial time1(0) Therefore, the synthesized signal z (0) of the output adding unit 101 is still the mixed signal x (0).
When t is 1, one can obtain:
x1(1)＝z(0)＝x(0),x2(1)＝0
z(1)＝y1(1)+x(1)
it should be noted that at the time when t is 1, the signal x (0) changes to x after passing through the first delay unit 1031(1) But signal x1(1) The same value as the signal x (0). The time delayed by the first delay unit 103, i.e. the unit time, is the time difference between t 1 and t 0. At the time t is 1, no signal flows out from the second delay unit 104, and thus the signal x is output2(1) The value of (d) is 0.
When t is 2, one can obtain:
x1(2)＝z(1)＝y1(1)+x(1)＝a1(1)x(0)+x(1)
x2(2)＝y2(1)＝a2(1)x(0)
z(2)＝y1(2)+x(2)＝(a1(2)a1(1)+a2(2)a2(1))x(0)+a1(2)x(1)+x(2)
the structure and the working principle of the analog low-pass filter are mainly described above, and in order to be applied to an AIC system to realize compression of a signal, parameters in a periodic cyclic matrix need to be designed. The calculation of the parameters of the periodic circular matrix is described in detail below.
First, according to the low-pass analog filter designed in this embodiment, the frequency domain impulse response function H at time j can be obtainedj(z)：
Wherein i is a time number, T is a period of a periodic cyclic matrix, N is the number of row vectors or column vectors of A (T) ', λ'k,i,jThe k characteristic value of Ψ (T,0), λ ″k,i,jIs the k-th characteristic value, λ ″, of Ψ (i-1,0) Ψ (T, j)'k,i,jIs the k-th eigenvalue of Ψ (i-1, j).
Function Ψ (n)2,n1) Is defined as:
wherein I is an identity matrix, i.e. elements on a diagonal (called a main diagonal) from the upper left corner to the lower right corner of the matrix are all 1, and other elements are all 0.
It is thus possible to obtain:
where A (t) is a periodic circulant matrix.
The physical meaning of the characteristic value is: assume that the matrix a (e.g., Ψ (T,0), Ψ (i-1,0), Ψ (T, j), Ψ (i-1, j)) can be represented as: ax is Ax, where a is not 0, then a represents the eigenvalue of matrix a and the corresponding eigenvector is x. The matrix a may be considered to be present in the direction of the eigenvector x, and the presence value in the eigenvector x is a. If a is larger, the larger the power the representation matrix a contributes in the x-direction, the more information is, here for solving the coefficients of the analog low-pass filter.
Obtaining the frequency domain impulse response function H at the moment jjAfter (z), assuming that the sampling function corresponding to the sampler 31 is D (z), the frequency domain impulse response function H is taken asj(z) multiplied by a sampling function d (z) to yield:
FΦj＝D(z)Hj(z) (2)
wherein phijIs D (z) Hj(z) a function after fourier transformation, i.e. a measurement matrix. F refers to fourier transform.
As described above, in order to realize that the analog low-pass filter can be applied to an AIC system to realize compressed sensing of signals, it is necessary to ensure that the correlations between any two columns of the measurement matrix are uncorrelated, but in practical applications, for various reasons, this condition is hardly fulfilled, so that it is sufficient to ensure that the correlations between any two columns of the measurement matrix are as small as possible, and therefore, the following conditions are provided:
wherein F phiiFor the measurement matrix corresponding to time i, F phijIs the measurement matrix corresponding to time j, (F phi)i)HTo F phiiFind the conjugate transpose, | | F Φi||2To find F phii2 norm of the modulus, | F Φj||2To find F phijIs the allowable fluctuation value, said V is the 2 norm of the modulus oflFor measuring the matrix F phijAny two columns of. That is, the measurement matrix F ΦjThe correlation between any two columns is less than or equal to the preset valueIs a value of allowable fluctuation of, i.e. can be regarded as the measurement matrix F phijAny two columns are uncorrelated.
In summary, since the inventor designs the low-pass analog filter with the above structure, the frequency-domain impulse response function H can be obtainedj(z), the frequency domain impulse response function HjThe parameters of (z) are obtained according to the parameters of the periodic cyclic matrix A (t). According to the frequency domain impulse response function HjThe product of (z) and the sampling function D (z) can obtain a measurement matrix, and the frequency domain impulse response function H can be obtained by using the condition that any two columns of the measurement matrix are uncorrelated (namely formula (3))j(z) and may then utilize the frequency domain impulse response function HjParameters of (z) the parameters of the periodic circulant matrix a (t) are deduced back from equation (1). After the parameters of the periodic cyclic matrix a (t) are obtained, the low-pass analog filter with the above structure can be used to realize the compressed sensing of the signal.
In addition, in practical application, the frequency domain impulse response function H of the low-pass analog filterjThe parameter (z) should satisfy the following condition besides the condition that any two columns of the measurement matrix are irrelevant:
1. as the analog low-pass filter, it is necessary to ensure that the system itself is stable, that is, all poles of the system are within a unit circle, the poles represent the positions of the pass band of the analog low-pass filter, and the unit circle is a circle composed of points with an origin of 0 and an euclidean distance of 1 from the origin. Namely, the following conditions need to be satisfied:
wherein, | λ'k,i,j||2Is to obtain lambda'k,i,j2 norm, λ 'of the pattern of'k,i,jSee formula (1).
In this embodiment, when the periodic cyclic matrix satisfies the conditions of the formula (3) and the formula (4), the analog low-pass filter designed in this embodiment can be applied to AIC to realize compressed sensing of signals on the basis of ensuring the stability of the analog low-pass filter in its own system.
2. In order to ensure a higher recovery probability, the observed value needs to carry more information of the input value, so that the analog low-pass filter needs to be designed to have a longer system response. To achieve this object, the conditions further include: the system pole is greater than or equal to the preset value, and is specifically represented by formula (5).
min{||λ'k,i,j||2}≥γ (5)
Wherein gamma is a preset value and represents the lower bound of the distance between the pole and the origin.
3. In order to ensure the convergence speed of the algorithm, the parameters of the periodic cyclic matrix can be set within a preset range, such as [ -5,5], when being calculated.
Example two:
based on the analog low-pass filter designed by the inventor, the invention also provides an analog information converter, see fig. 3, comprising a mixer 11, an analog low-pass filter 21 and a sampler 31. The mixer 11 is connected to the analog low-pass filter 21, and the analog low-pass filter 21 is connected to the sampler 31.
The mixer 11 mixes the initial signal with the random sequence signal to obtain a mixed signal.
The mixer 11 may be any mixer suitable for AIC in the prior art, and the present invention is not limited in particular. The mixer 11 is configured to mix the initial signal with a random sequence signal, wherein the random sequence signal is generally composed of a random sequence taking only { ± 1} two values, and the sampling rate of the random sequence signal is higher than the sampling rate of the initial signal in order to mix all the initial signals within the bandwidth. The part of the mixed signal that can express the characteristics of the original signal is concentrated in the low frequency part of the mixed signal, so that the low frequency part of the mixed signal can be extracted through the analog low pass filter 21, and the compression of the original signal is realized.
The composition and connection relationship of the analog low-pass filter 21 refer to the first embodiment, which is not described again in this embodiment.
The sampler 31 is configured to sample the filtered signal.
Since the filtered signal output from the analog low-pass filter 21 has a highest frequency lower than that of the original signal, the sampling rate of the sampler 31 is lower than that of the original signal directly according to the nyquist theorem, i.e. the requirement for the sampling rate of the sampler 31 is reduced.
In practical applications, the sampler 31 may be any sampler suitable for use in an AIC system, and the present invention is not limited in particular. In the present embodiment, the sampler 31 is an ADC (Analog to Digital Converter).
EXAMPLE III
In order to further reduce the requirement for the sampling rate of the sampler, the AIC system designed by the inventor comprises at least two sets of mixers, analog low-pass filters and samplers mentioned in the first embodiment in parallel, referring to fig. 4, the analog information converter provided in this embodiment comprises at least one mixer 51, at least one analog low-pass filter 52 and at least one sampler 53, wherein the three devices are the same in number, and the functions of the three devices are the same as those of the mixer 11, the analog low-pass filter 21 and the sampler 31 in the first embodiment, which is not described herein again. It should be noted that, in order to ensure that the correlation between any two columns of the AIC system measurement matrix is as small as possible, the random sequence signals in each mixer 51 should be different, and the corresponding periodic cyclic matrix in each analog low-pass filter 52 should be different, and the parameters of each periodic cyclic matrix need to be solved in conjunction with the conditions that each analog low-pass filter 52 should satisfy.
Since the random sequence signals in the mixer 51 are different from each other and the corresponding cyclic period matrix in each analog low-pass filter 52 is also different from each other, although the signals input to the AIC are the same, the signals output from each analog low-pass filter 52 are different from each other, and by sampling each output signal, different sampling data can be obtained, and the requirement for reducing the sampling rate of the sampler 53 is met. For example, assuming that the sampling rate of the sampler 53 is required to be fs, if there are two sets of mixers, analog low pass filters and samplers, then the sampling rate of each sampler may be fs/2. The larger the number of groups, the lower the sampling rate requirements for each sampler.
It should be noted that each unit in the analog low-pass filters mentioned in the first to third embodiments may be implemented by software, or may be implemented by hardware, for example, the adding unit may be implemented by an adder, the delay unit may be implemented by a delay, the state converting unit may be implemented by a memory and an adder, and specifically, the adder and the delay may be implemented by a resistor, an operational amplifier, a capacitor, a diode, and the like. The present invention is not limited to the above embodiments, and those skilled in the art can select the above embodiments according to actual needs. If the implementation is realized by software, for example, if a matlab tool (matlab is software) is adopted, a mixer, an adder and a delayer model in the simulink tool of matlab can be used for building, so as to realize the function of a corresponding device; if matlab's code is used, the mixer directly simulates the function of the corresponding device with "+" command, z ═ x × y, the adder with "+" command, z ═ x + y, and the delay with z (n) ═ x (n-1).
Example four
Based on the analog low-pass filter provided by the first embodiment, the present invention further provides an embodiment of a filtering method, where the filtering method is applied to the analog low-pass filters mentioned in the first to third embodiments, and the method includes:
receiving a mixing signal from a mixer, wherein the mixing signal is a signal obtained by mixing an initial signal and a random sequence signal, and the mixer is connected with the analog low-pass filter;
the adding unit superposes the mixing signal and the signal from the state conversion unit to obtain a synthesized signal;
the first delay unit delays the synthesized signal from the addition unit by a unit time;
the at least one second delay unit delays the signal from the state transition unit by a unit time;
the state conversion unit performs weighting processing on signals from the first delay unit and the at least one second delay unit by using a cyclic period matrix, and correspondingly inputs the weighted signals to the adder and the at least one second delay unit respectively; the parameters of the periodic cyclic matrix are obtained according to the coefficients of the frequency domain impulse response function of the analog low-pass filter, and the coefficients of the frequency domain impulse response function meet the following conditions: any two columns of the measurement matrix obtained according to the frequency domain impulse response function and the sampling function are irrelevant;
the delayed signal is input to a sampler connected to the analog low pass filter using the first delay cell or one of the at least one second delay cell.
For the working principle of the filtering method provided in this embodiment, please refer to embodiment one, which is not described herein again.
EXAMPLE five
Based on the analog low-pass filter provided by the first embodiment, the invention also provides another embodiment of a filtering method, which includes:
the analog low-pass filter receives a mixing signal from the mixer, wherein the mixing signal is a signal obtained by mixing an initial signal and a random sequence signal, and the mixer is connected with the analog low-pass filter;
the analog low-pass filter superposes the mixing signal and the weighted signal to obtain a synthesized signal;
the analog low-pass filter delays the synthesized signal by unit time and delays the weighted signal by unit time;
the analog low-pass filter performs weighting processing on the delayed signals by utilizing a periodic cyclic matrix to obtain weighted signals; the parameters of the periodic cyclic matrix are obtained according to the coefficients of the frequency domain impulse response function of the analog low-pass filter, and the coefficients of the frequency domain impulse response function meet the following conditions: any two columns of the measurement matrix obtained according to the frequency domain impulse response function and the sampling function are irrelevant;
the analog low-pass filter inputs the delayed signal to a sampler connected to the analog low-pass filter.
For the working principle of the filtering method provided in this embodiment, please refer to embodiment one, which is not described herein again.
EXAMPLE six
The present invention also provides another embodiment of an analog low-pass filter, comprising:
a processor;
a memory; and a communication port;
wherein the processor is configured to execute the instructions stored by the memory to perform the following steps when executing the instructions:
receiving a mixing signal from a mixer through the communication port, wherein the mixing signal is a signal obtained by mixing an initial signal and a random sequence signal, and the mixer is connected with an analog low-pass filter;
superposing the mixing signal and the weighted signal to obtain a synthesized signal;
delaying the synthesized signal for a unit time, and delaying the weighted signal for a unit time;
weighting the delayed signals by using a periodic cyclic matrix to obtain weighted signals; the parameters of the periodic cyclic matrix are obtained according to the coefficients of the frequency domain impulse response function of the analog low-pass filter, and the coefficients of the frequency domain impulse response function meet the following conditions: any two columns of the measurement matrix obtained according to the frequency domain impulse response function and the sampling function are irrelevant;
the delayed signal is input to a sampler connected to the analog low pass filter through the communication port.
For details of the working principle of the analog low-pass filter provided in this embodiment, please refer to embodiment one, which is not described herein again.
The "first" of the first router mentioned in the embodiments of the present invention is used for name identification only, and does not represent the first in sequence. The rule applies equally to "second".
It should be noted that, in the present specification, all the embodiments are described in a progressive manner, and the same and similar parts among the embodiments may be referred to each other, and each embodiment focuses on the differences from the other embodiments. Especially, for the second to sixth embodiments, since they are substantially similar to the first embodiment, the description is simple, and the relevant points can be referred to the partial description of the first embodiment. The above-described embodiments of the apparatus are merely illustrative, and the units described as separate parts may or may not be physically separate, and the parts displayed as units may or may not be physical units, may be located in one place, or may be distributed on a plurality of network units. Some or all of the modules may be selected according to actual needs to achieve the purpose of the solution of the present embodiment. One of ordinary skill in the art can understand and implement it without inventive effort.
The above description is only for the preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope of the present invention are included in the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.
Claims (21)
1. An analog low-pass filter, characterized in that the analog low-pass filter comprises:
the circuit comprises an adding unit, a state conversion unit, a first delay unit and at least one second delay unit;
the output end of the state conversion unit is connected with the addition unit, the input end of the state conversion unit is connected with the first delay unit, and the addition unit is connected with the first delay unit;
the output end and the input end of the state conversion unit are respectively connected with the at least one second delay unit;
the adding unit is configured to superimpose the mixing signal received by the analog low-pass filter and the signal from the state converting unit to obtain a synthesized signal, and input the synthesized signal to the first delay unit, where the mixing signal is a signal obtained by mixing an initial signal and a random sequence signal;
the first delay unit is configured to delay the synthesized signal from the addition unit by a unit time, and input a signal delayed by the first delay unit to the state transition unit;
the at least one second delay unit is used for delaying the signal from the state conversion unit by unit time and inputting the signal after the delay of the at least one second delay unit to the state conversion unit;
the state conversion unit is configured to perform weighting processing on the signals from the first delay unit and the at least one second delay unit by using a cyclic periodic matrix, and input the weighted signals to the adder and the at least one second delay unit respectively; the parameters of the periodic cyclic matrix are obtained according to the coefficients of the frequency domain impulse response function of the analog low-pass filter, and the coefficients of the frequency domain impulse response function meet the following conditions: any two columns of the measurement matrix obtained according to the frequency domain impulse response function and the sampling function are irrelevant;
the first delay unit, or one of the at least one second delay unit, is further configured to input the combined signal delayed by the unit time or the weighted signal delayed by the unit time to a sampler connected to the analog low pass filter.
2. An analog low-pass filter according to claim 1, characterized by a frequency-domain impulse response function H at time jj(z) is:
wherein i is a time number, T is a period of the periodic cyclic matrix, N is the number of row vectors or column vectors of A (T) 'lambda'k,i,jK characteristic value of Ψ (T,0), λ "k,i,jIs the k-th characteristic value, λ'k,i,jThe kth eigenvalue of Ψ (i-1, j);
where A (t) is a periodic circulant matrix.
3. The analog low-pass filter according to claim 2, wherein any two columns of uncorrelation of the measurement matrix are specifically expressed by the following formula:
wherein F phii＝D(z)Hi(z)，Hi(z) is the frequency domain impulse response function at time i, and D (z) is the sampling function; phiiIs D (z) Hi(z) a function after Fourier transform, namely a measurement matrix corresponding to the time i, wherein F is Fourier transform; f phij＝D(z)Hj(z)，ΦjIs D (z) Hj(z) a function after Fourier transform, namely a measurement matrix corresponding to the moment j, wherein F is Fourier transform; (F.phi.)i)HTo F phiiFind the conjugate transpose, | | F Φi||2To find F phii2 norm of the modulus, | F Φj||2To find F phijIs the allowable fluctuation value, said V is the 2 norm of the modulus oflFor measuring the matrix F phijAny two columns of.
4. The analog low pass filter of claim 2, wherein the conditions further comprise: the poles of the system are within the unit circle.
5. An analog low-pass filter according to claim 4, characterized in that the poles of the system are represented in the unit circle by the formula:
wherein, | λ'k,i,j||2Is to obtain lambda'k,i,j2 norm of the norm.
6. The analog low pass filter of claim 2, wherein the conditions further comprise: the system pole is greater than or equal to a preset value.
7. The analog low-pass filter according to claim 6, wherein the system pole is greater than or equal to a predetermined value by the following equation:
min{||λ'k,i,j||2}≥γ
wherein gamma is a preset value and represents the lower bound of the distance between the pole and the origin.
8. The analog low pass filter of claim 1, wherein the conditions further comprise: and the parameters of the periodic cyclic matrix are in a preset range.
9. An analog information converter, characterized in that the analog information converter comprises a mixer, a sampler, and an analog low-pass filter according to any one of claims 1 to 8, the mixer being connected to the analog low-pass filter, the analog low-pass filter being connected to the sampler;
the mixer is used for mixing the initial signal with the random sequence signal to obtain a mixing signal;
the analog low-pass filter is used for filtering the mixing signal to obtain a filtering signal;
the sampler is used for sampling the filtering signal.
10. An analog low pass filter, comprising:
a processor;
a memory; and a communication port;
wherein the processor is configured to execute the instructions stored by the memory to perform the following steps when executing the instructions:
receiving a mixing signal from a mixer through the communication port, wherein the mixing signal is a signal obtained by mixing an initial signal and a random sequence signal, and the mixer is connected with an analog low-pass filter;
superposing the mixing signal and the weighted signal to obtain a synthesized signal;
delaying the synthesized signal for a unit time, and delaying the weighted signal for a unit time;
weighting the synthesized signal delayed by unit time and the weighted signal delayed by unit time by using a periodic cyclic matrix to obtain the weighted signal; the parameters of the periodic cyclic matrix are obtained according to the coefficients of the frequency domain impulse response function of the analog low-pass filter, and the coefficients of the frequency domain impulse response function meet the following conditions: any two columns of the measurement matrix obtained according to the frequency domain impulse response function and the sampling function are irrelevant;
inputting the combined signal delayed by the unit time or the weighted signal delayed by the unit time to a sampler connected to the analog low pass filter through the communication port.
11. An analog low-pass filter according to claim 10, characterized by a frequency-domain impulse response function H at time jj(z) is:
wherein i is a time number, T is a period of the periodic cyclic matrix, N is the number of row vectors or column vectors of A (T) 'lambda'k,i,jK characteristic value of Ψ (T,0), λ "k,i,jIs the k-th characteristic value, λ'k,i,jThe kth eigenvalue of Ψ (i-1, j);
where A (t) is a periodic circulant matrix.
12. The analog low-pass filter according to claim 11, wherein any two columns of uncorrelation of the measurement matrix are specifically expressed by the following formula:
wherein F phii＝D(z)Hi(z)，Hi(z) is the frequency domain impulse response function at time i, and D (z) is the sampling function; phiiIs D (z) Hi(z) a function after Fourier transform, namely a measurement matrix corresponding to the time i, wherein F is Fourier transform; f phij＝D(z)Hj(z)，ΦjIs D (z) Hj(z) a function after Fourier transform, namely a measurement matrix corresponding to the moment j, wherein F is Fourier transform; (F.phi.)i)HTo F phiiFind the conjugate transpose, | | F Φi||2To find F phii2 norm of the modulus, | F Φj||2To find F phijIs a 2 norm of a norm of an allowable waveDynamic value, said VlFor measuring the matrix F phijAny two columns of.
13. The analog low pass filter of claim 10, wherein the conditions further comprise: the poles of the system are within the unit circle.
14. The analog low pass filter of claim 10, wherein the conditions further comprise: the system pole is greater than or equal to a preset value.
15. The analog low pass filter of claim 10, wherein the conditions further comprise: and the parameters of the periodic cyclic matrix are in a preset range.
16. A method of filtering, the method comprising:
the analog low-pass filter receives a mixing signal from the mixer, wherein the mixing signal is a signal obtained by mixing an initial signal and a random sequence signal, and the mixer is connected with the analog low-pass filter;
the analog low-pass filter superposes the mixing signal and the weighted signal to obtain a synthesized signal;
the analog low-pass filter delays the synthesized signal by unit time and delays the weighted signal by unit time;
the analog low-pass filter uses a periodic cyclic matrix to perform weighting processing on the synthesized signal delayed by unit time and the weighted signal delayed by unit time to obtain the weighted signal; the parameters of the periodic cyclic matrix are obtained according to the coefficients of the frequency domain impulse response function of the analog low-pass filter, and the coefficients of the frequency domain impulse response function meet the following conditions: any two columns of the measurement matrix obtained according to the frequency domain impulse response function and the sampling function are irrelevant;
the analog low-pass filter inputs the combined signal delayed by a unit time or the weighted signal delayed by a unit time to a sampler connected to the analog low-pass filter.
17. The filtering method according to claim 16, wherein the frequency-domain impulse response function H at time jj(z) is:
wherein i is a time number, T is a period of the periodic cyclic matrix, N is the number of row vectors or column vectors of A (T) 'lambda'k,i,jK characteristic value of Ψ (T,0), λ "k,i,jIs the k-th characteristic value, λ'k,i,jThe kth eigenvalue of Ψ (i-1, j);
where A (t) is a periodic circulant matrix.
18. The filtering method according to claim 17, wherein any two columns of uncorrelation of the measurement matrix are specifically represented by the following formula:
wherein F phii＝D(z)Hi(z)，Hi(z) is the frequency domain impulse response function at time i, and D (z) is the sampling function; phiiIs D (z) Hi(z) a function after Fourier transform, namely a measurement matrix corresponding to the time i, wherein F is Fourier transform; f phij＝D(z)Hj(z)，ΦjIs D (z) Hj(z) after Fourier transformThe function, i.e. the measurement matrix corresponding to time j, F is the fourier transform; (F.phi.)i)HTo F phiiFind the conjugate transpose, | | F Φi||2To find F phii2 norm of the modulus, | F Φj||2To find F phijIs the allowable fluctuation value, said V is the 2 norm of the modulus oflFor measuring the matrix F phijAny two columns of.
19. The filtering method according to claim 16, wherein the condition further comprises: the poles of the system are within the unit circle.
20. The filtering method according to claim 16, wherein the condition further comprises: the system pole is greater than or equal to a preset value.
21. The filtering method according to claim 16, wherein the condition further comprises: and the parameters of the periodic cyclic matrix are in a preset range.
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