CN109709397B - Power grid harmonic asynchronous compressed sensing detection method with continuous Hanning window - Google Patents

Power grid harmonic asynchronous compressed sensing detection method with continuous Hanning window Download PDF

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CN109709397B
CN109709397B CN201811534864.6A CN201811534864A CN109709397B CN 109709397 B CN109709397 B CN 109709397B CN 201811534864 A CN201811534864 A CN 201811534864A CN 109709397 B CN109709397 B CN 109709397B
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CN109709397A (en
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刘嫣
汤伟
刘宝泉
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Shaanxi Xingjian Kinetic Energy Electric Technology Co.,Ltd.
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Shaanxi University of Science and Technology
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Abstract

A power grid harmonic non-synchronous compressed sensing detection method added with a continuous Hanning window comprises the following steps: deriving and defining a continuous Hanning window signal function; analyzing orthogonal decomposition models of power grid harmonic signals added with continuous Hanning windows under continuous Fourier transform and generalized Fourier series, and accordingly providing finite term orthogonal decomposition sum of continuous signals under discrete Fourier transformK-a sparse reference model; deducing an input and output observation model of the power grid harmonic signal with the continuous Hanning window under a random analog information converter; and correcting the formula for restoring the reconstructed signal into the original power grid harmonic signal. The method can effectively reduce the frequency spectrum leakage problem generated by the non-synchronization of the generation frequency of the pseudo-random sequence in the random analog information converter, the sampling frequency of the low-speed ADC and the fundamental wave frequency of the power grid, and further improve the efficiency of a signal reconstruction algorithm and the detection precision of the amplitude of each component of harmonic wave.

Description

Power grid harmonic asynchronous compressed sensing detection method with continuous Hanning window
Technical Field
The invention belongs to the technical field of harmonic detection, relates to power grid harmonic asynchronous sampling detection, and particularly relates to a power grid harmonic asynchronous compressed sensing detection method with a continuous Hanning window.
Background
The accurate detection of the harmonic signals is the basis and the premise for controlling the harmonic pollution of the power grid. As a new signal processing method, the Compressed Sensing (CS) theory breaks through the limitation of the Nyquist sampling theorem and provides a new idea for detecting harmonic signals of the power grid. If the signal to be detected is K-sparse under a certain basis function, the detection method utilizing the theory has the advantages of high-compression-ratio asynchronous observation and reconstruction.
An Analog-to-information converter (AIC) is a leading-edge technology that generalizes the CS theory to Analog signals. There are four current implementations of AIC: random Sampling (RS), Random filter (RS), Random Demodulation (RD), and Modulated bandwidth demodulator (MWC). Among them, the RD scheme (Sami Kirolos, Analog-to-information conversion video random modulation, Design, Applications, Integration and Software,2006IEEE Dallas/CAS Workshop on. IEEE,2007:71-74.) proposed by Ason Laska et al has a great pushing effect on the development of AIC. The scheme consists of a pseudo-random sequence, a low-pass filter and a low-speed ADC (analog-to-digital converter), can effectively convert an analog signal into a small number of 'finely-selected' observation signals, and is applied on the premise that the analog signal is spread in a finite term under a continuous basis function. The harmonic signal is used as a continuous periodic signal and meets the application premise of RD-AIC. At present, scholars at home and abroad preliminarily apply RD-AIC to the detection of electric energy quality disturbing signals, but do not consider the problem of spectrum leakage caused by the fact that the frequency of a harmonic signal generated by a pseudorandom sequence in the RD-AIC and the sampling frequency of a low-speed ADC are asynchronous with the fundamental frequency of a power grid, so that the sparsity of the signal in a frequency domain is poor, the time-frequency reconstruction effect of the signal is poor, and the detection precision is influenced.
In order to avoid the occurrence of the spectrum leakage phenomenon, the traditional detection method mainly adopts an analog phase-locked loop and a digital phase-locked loop, but the methods are all based on the Nyquist sampling theorem and are contrary to the purpose of applying the CS theory and the RD-AIC. Another method is to suppress the leaked spectrum by adding different window functions, wherein the Hanning window is easy to be widely applied due to small calculation amount and programming, however, the Hanning window is a discrete window function, the length of which depends on the Nyquist sampling frequency and cannot be directly added to the signal observed by the RD-AIC, and therefore, the leaked spectrum cannot be directly suppressed.
Disclosure of Invention
In order to effectively solve the problems that the sparsity of signals in a frequency domain is poor due to RD-AIC asynchronous sampling of power grid harmonics, and further the efficiency of a reconstruction algorithm is low and the detection precision is poor, the invention provides a power grid harmonic asynchronous compressed sensing detection method with a continuous Hanning window. The method organically combines the advantage that the RD-AIC can solve the analog signal compression observation with the advantage that the Hanning window can reduce the signal leakage frequency spectrum in the discrete domain, realizes the compression observation of the original harmonic signal on the basis of not changing the RD-AIC hardware parameters and the CS reconstruction algorithm, and simultaneously improves the detection precision of each component of the harmonic and the recovery speed of the original harmonic signal. In order to realize the purpose, the invention adopts the technical scheme that:
a power grid harmonic non-synchronous compressed sensing detection method added with a continuous Hanning window comprises the following steps:
step 1), deriving and defining a continuous Hanning window signal w (t);
the continuous Hanning window signal is defined as
Figure BDA0001906612720000021
Wherein
Figure BDA0001906612720000022
N＝T0/T，T0Is the analog output time of w (n), T ═ 1/f, f is the analog output frequency of w (n);
the derivation process is as follows:
discrete Hanning function
Figure BDA0001906612720000023
According to the discrete signal recovery theory, the continuous Hanning window signal can be expressed as:
w(t)＝w(n)*h(t) (2)
where h (t) is the unit impulse response of an ideal low-pass filter,
Figure BDA0001906612720000031
Figure BDA0001906612720000032
step 2) establishing a harmonic signal x added with a continuous Hanning window1(t)；
x1(t)＝x(t)w(t) (4)
Step 3) analyzing the harmonic signal x added with the continuous Hanning window1(t) a decomposition model under a continuous fourier transform and a generalized fourier series;
from the continuous Fourier transform, x1(t) in a continuous orthogonal baseUnder the function can be decomposed into
Figure BDA0001906612720000033
Since the spectrum prior information is unknown, equation (5) cannot directly reflect x1The information quantity and sparsity of (t) cannot reflect whether the application premise of RD-AIC is met, namely, any continuous signal can be subjected to finite term expansion on a continuous basis function;
then, x is known from the generalized Fourier series1(t) can be decomposed into
Figure BDA0001906612720000034
Formula (6) reflects x1(t) an approximate satisfaction of the formula RD-AIC finite term expansion in the generalized Fourier series, which can be regarded as x1(t) noisy models under a finite number of consecutive orthogonal bases of length N;
step 4) establishing a harmonic signal x added with a continuous Hanning window1(t) finite term decomposition under discrete fourier transform and K-sparse reference model;
if the length N of the signal reconstructed by CS meets the requirement of the Nyquist sampling theorem, the reconstructed original signal can recover a continuous signal without distortion. Based on this, a harmonic signal x with a continuous Hanning window is established1(t) finite decomposition under discrete fourier transform and sparse reference model;
let x1(T) at a Nyquist sampling frequency fs and a finite duration T0The discretization signal in is x1(N) having a length of N ═ T0fs. From x1(n) inverse discrete Fourier transform of x1(n) capable of limited term decomposition, as follows:
Figure BDA0001906612720000041
wherein the content of the first and second substances,
Figure BDA0001906612720000042
θ1＝[α1_0,α1_1,...,α1_N-1]T,
Figure BDA0001906612720000043
if theta1K in number different from 0, and K<<N, then x is considered1(n) spectral information is limited and sparse, satisfying the application premise of RD-AIC, and x reconstructed by CS is considered1(n) capable of recovering a continuous signal without distortion;
step 5) deducing harmonic signal x added with continuous Hanning window1(t), RD-AIC output observation signal y1(n) and reconstructed time-frequency signal θ'1(i)、x’1(n) the input-output mathematical model;
x is to be1(t) feeding RD-AIC, first with a pseudo-random sequence pc(t) multiplication by the following formula:
x2(t)＝x1(t)w(t) (8)
secondly, x is2(t) feeding a low pass filter with a unit impulse response of h' (t) as follows:
y1(t)＝x2(t)*h’(t) (9)
expansion (9) is given by:
Figure BDA0001906612720000044
x is to be1(t) the sparse reference model under inverse discrete fourier transform (7) is substituted into it as follows:
Figure BDA0001906612720000051
then y is put1(t) feeding into the low speed ADC, as follows:
Figure BDA0001906612720000052
order to
Figure BDA0001906612720000053
Then there are:
Figure BDA0001906612720000054
wherein, A is a sensing matrix which is composed of a pseudo-random sequence, a unit impulse response of a low-pass filter, a low-speed ADC and a discrete Fourier base;
finally for y1(m) performing CS reconstruction as follows:
Figure BDA0001906612720000055
by the equation (14), a harmonic spectrum vector θ of a continuous Hanning window is obtained1' (i) and obtaining an inverse transformation signal x ' from the formula (8) '1(n)；
Step 6), establishing a correction formula, and recovering an original harmonic discrete signal x' (n);
as can be seen from formula (4), x '(n) ═ x'1(n)/w (n), but because the amplitude of the discrete Hanning window function w (n) at the start and end times is 0, both ends cannot be directly solved. In order to effectively recover x' (n), a constant C close to 0 is introduced for correction, and the formula is as follows:
Figure BDA0001906612720000056
the invention has the advantages and positive effects that:
1) according to the method, the RD-AIC and the continuous Hanning window signal are combined, compared with the method only using the RD-AIC, the method not only can reduce the frequency spectrum leakage problem generated by the non-synchronization of the pseudo-random sequence generation frequency and the low-speed ADC sampling frequency in the RD-AIC and the power grid fundamental wave frequency, but also can improve the reconstruction efficiency and the detection precision of the observation signal on the basis of not changing the CS reconstruction algorithm;
2) the finite term decomposition of discrete Fourier transform and the K-sparse reference model of the continuous signal provided by the method can organically unify the observation of a CS theory aiming at the discrete sparse signal model and the observation of RD-AIC aiming at the continuous sparse signal model.
3) The method can also be popularized to harmonic signal detection in other fields. If the continuous Hanning window signal in the detection method is improved, the influence of the leakage frequency spectrum on the reconstruction algorithm and the signal detection precision can be further reduced.
Drawings
FIG. 1 is a schematic flow chart of the present invention;
FIG. 2 is a block diagram of the structure of the method of the present invention;
wherein, 1 is a power grid harmonic sensing circuit, 2 is a generated continuous Hanning window signal, 3 is RD-AIC, and 4 is reconstruction and detection;
FIG. 3 is a time frequency waveform under harmonic synchronous sampling, asynchronous sampling, and asynchronous plus Hanning window sampling;
FIG. 4 is a time-frequency waveform obtained by multiplying a pseudorandom sequence;
FIG. 5 is a time-frequency waveform after passing through a low-pass filter;
FIG. 6 is a time-frequency waveform through a low-speed ADC;
FIG. 7 is a reconstructed time-frequency waveform;
fig. 8 is the original harmonic signal recovered when the correction formula is added.
Fig. 9 shows the original harmonic signal recovered without the correction formula.
Detailed Description
The invention is further described with reference to the accompanying drawings and examples, and the research content of the invention provides theoretical basis and guidance for the concrete implementation of the method.
As shown in fig. 1, a power grid harmonic non-synchronous compressed sensing detection method with a continuous Hanning window includes the following steps:
step 1), deriving and defining a continuous Hanning window signal w (t);
the continuous Hanning window signal is defined as
Figure BDA0001906612720000061
Wherein
Figure BDA0001906612720000071
N＝T0/T，T0Is the analog output time of w (n), T ═ 1/f, f is the analog output frequency of w (n);
the derivation process is as follows:
discrete Hanning function
Figure BDA0001906612720000072
According to the discrete signal recovery theory, the continuous Hanning window signal can be expressed as:
w(t)＝w(n)*h(t) (2)
where h (t) is the unit impulse response of an ideal low-pass filter,
Figure BDA0001906612720000073
Figure BDA0001906612720000074
step 2) establishing a harmonic signal x added with a continuous Hanning window1(t)；
x1(t)＝x(t)w(t) (4)
Step 3) analyzing the harmonic signal x added with the continuous Hanning window1(t) a decomposition model under a continuous fourier transform and a generalized fourier series;
from the continuous Fourier transform, x1(t) is decomposable into
Figure BDA0001906612720000075
Because the spectrum prior information is unknown,the formula (5) does not directly reflect x1The information quantity and sparsity of (t) cannot reflect whether the application premise of RD-AIC is met, namely, any continuous signal can be subjected to finite term expansion on a continuous basis function;
then, x is known from the generalized Fourier series1(t) can be decomposed into
Figure BDA0001906612720000081
Formula (6) reflects x1(t) an approximate satisfaction of the formula RD-AIC finite term expansion in the generalized Fourier series, which can be regarded as x1(t) noisy models under a finite number of consecutive orthogonal bases of length N;
step 4) establishing a harmonic signal x added with a continuous Hanning window1(t) finite term decomposition under discrete fourier transform and K-sparse reference model;
if the length N of the signal reconstructed by CS meets the requirement of the Nyquist sampling theorem, the reconstructed original signal can recover a continuous signal without distortion. Based on this, a harmonic signal x with a continuous Hanning window is established1(t) finite decomposition under discrete fourier transform and sparse reference model;
let x1(T) at a Nyquist sampling frequency fs and a finite duration T0The discretization signal in is x1(N) having a length of N ═ T0fs. From x1(n) inverse discrete Fourier transform of x1(n) capable of limited term decomposition, as follows:
Figure BDA0001906612720000082
wherein the content of the first and second substances,
Figure BDA0001906612720000083
θ1＝[α1_0,α1_1,...,α1_N-1]T,
Figure BDA0001906612720000084
if theta1K in number different from 0, and K<<N, then x is considered1(n) spectral information is limited and sparse, satisfying the application premise of RD-AIC, and x reconstructed by CS is considered1(n) capable of recovering a continuous signal without distortion;
step 5) deducing harmonic signal x added with continuous Hanning window1(t), RD-AIC output observation signal y1(n) and reconstructed time-frequency signal θ'1(i)、x’1(n) the input-output mathematical model;
x is to be1(t) feeding RD-AIC, first with a pseudo-random sequence pc(t) multiplication by the following formula:
x2(t)＝x1(t)w(t) (8)
secondly, x is2(t) feeding a low pass filter with a unit impulse response of h' (t) as follows:
y1(t)＝x2(t)*h’(t) (9)
expansion (9) is given by:
Figure BDA0001906612720000091
x is to be1(t) the sparse reference model under inverse discrete fourier transform (7) is substituted into it as follows:
Figure BDA0001906612720000092
then y is put1(t) feeding into the low speed ADC, as follows:
Figure BDA0001906612720000093
order to
Figure BDA0001906612720000094
Then there are:
Figure BDA0001906612720000095
wherein, A is a sensing matrix which is composed of a pseudo-random sequence, a unit impulse response of a low-pass filter, a low-speed ADC and a discrete Fourier base;
finally for y1(m) performing CS reconstruction as follows:
Figure BDA0001906612720000096
by equation (14), a harmonic spectrum vector θ 'added to a continuous Hanning window is obtained'1(i) And then obtaining an inverse transform signal x 'from equation (8)'1(n)；
Step 6), establishing a correction formula, and recovering an original harmonic discrete signal x' (n);
as can be seen from formula (4), x '(n) ═ x'1(n)/w (n), but because the amplitude of the discrete Hanning window function w (n) at the start and end times is 0, both ends cannot be directly solved. In order to effectively recover x' (n), a constant C close to 0 is introduced for correction, and the formula is as follows:
Figure BDA0001906612720000101
as shown in fig. 2, the method of the present invention is implemented by the following steps:
step 1), a power grid harmonic signal x (t) acquired by a voltage or current sensor is multiplied by a generated continuous Hanning window signal w (t) to form a harmonic signal x added with a continuous Hanning window1(t)；
Step 2) mixing x1(t) feeding into RD-AIC. First with a pseudorandom sequence pc(t) multiplication to y1(t), then y1(t) obtaining the signal y 'by a low-pass filter'1(t), last y'1(t) sending to a low-speed ADC for sampling to obtain an observation signal y1(m)；
Step 3) utilizing a Sparsity adaptive matching pursuit algorithm (SAMP) to pair y1(m) reconstructing to obtain a discrete spectrum theta 'added with a continuous Hanning window harmonic signal'1(i)，θ’1(i) And taking the corresponding amplitude spectrum on the integral multiple of the fundamental frequency after the modulus value as the amplitude of each component of the harmonic wave.
Step 4) to theta'1(i) Performing inverse discrete Fourier transform to obtain harmonic reconstruction signal x 'added with continuous Hanning window'1(n)；
Step 5) utilizing the correction formula to x'1And (n) correcting and restoring the original harmonic signal x' (n).
2 examples were set up using the contents and implementation steps of the inventive study shown in FIGS. 1 and 2. The embodiment 1 provides a time frequency signal of a power grid harmonic signal under synchronous sampling, asynchronous sampling and non-synchronous Hanning window sampling, and obtains the advantage of the sampling of the asynchronous Hanning window; example 2 shows the time frequency waveform obtained from the RD-AIC observation process of the harmonic signal with the continuous Hanning window, the signal reconstructed by the SAMP algorithm, and the original harmonic signal recovered by the modified formula, and compares the obtained time frequency waveform with the original harmonic signal without the continuous Hanning window.
The normalized harmonic signal model of the power grid is set as
Figure BDA0001906612720000102
The power frequency f of the signal is 49.5Hz, and contains 2-6 harmonics, and the amplitudes of harmonic components are small. The pseudo-random sequence generation clock frequency in the RD-AIC is set to be 10kHz, the cut-off frequency of a low-pass filter is set to be 1kHz, the sampling frequency of a low-speed ADC is set to be 2kHz, and the observation duration T is set0The data compression ratio is 20% at 0.1s, the observation data length M is 200, and the reconstructed signal length N is 1000. The RD-AIC compressive sampling and SAMP reconstruction algorithms both run on Intel (R) core (TM) i7-4500U machines, with a software version of Matlab 7.1.
Example 1:
the signals of the power grid harmonic generated by the formula (16) under synchronous sampling, asynchronous sampling and asynchronous and continuous Hanning window sampling are subjected to discrete Fourier transform, 3 kinds of amplitude spectrums which are respectively obtained are shown in figure 3, and the amplitude and the relative error of each harmonic are shown in table 1.
TABLE 1
Figure BDA0001906612720000111
During synchronous sampling, the sampling frequency is set to be integral multiple of the power frequency, at the moment, the frequency spectrum leakage phenomenon cannot occur to each harmonic amplitude spectrum, the sparsity of a harmonic signal in a frequency domain is optimal, and each component amplitude can be accurately detected; when asynchronous sampling is carried out, because the sampling frequency is not integral multiple of the power frequency, the amplitude spectrum can generate a frequency spectrum leakage phenomenon due to non-whole period sampling, the sparsity of a harmonic signal is poor, and the relative error is large when the harmonic component with smaller amplitude is detected; when asynchronous and continuous Hanning window sampling is carried out, the sparsity of harmonic signals is improved along with the reduction of leakage frequency spectrum, and meanwhile, the relative error of the detection of harmonic components with small amplitude components is small.
In the embodiment, the power frequency of the power grid harmonic wave is set to be f equal to 49.5Hz, and when the RD-AIC observes the power grid harmonic wave signal, the pseudo-random sequence generation clock frequency and the low-speed sampling frequency are not integral multiples of the power frequency, so that the RD-AIC is asynchronous sampling, the sparsity of the harmonic wave signal in a frequency domain can be improved by adding a continuous Hanning window, and the CS reconstruction algorithm efficiency and the detection precision of each harmonic component amplitude can be improved on the premise of not changing the RD-AIC hardware parameter.
Example 2:
according to the specific implementation steps 1 and 2) in fig. 2, firstly, a power grid harmonic signal x with a continuous Hanning window is generated1(t) feeding RD-AIC and mixing with pseudo-random sequence p with frequency of 10kHzc(t) multiplying and outputting the signal y1The time-frequency waveform of (t) is shown in fig. 4 (1). For comparison, the harmonic signal x (t) without continuous Hanning window is fed into RD-AIC and mixed with pseudo-random sequence p with frequency of 10kHzc(t) the time-frequency waveforms of the multiplied output signals y (t) are shown in fig. 4 (2); secondly, mixing y1(t) and y (t) are fed to low pass filters, respectively, to output a signal y'1(t) and y' (t) whose time-frequency waveforms are shown in fig. 5(1), (2), respectively; finally y'1(t) and y' (t) are fed into the low-speed ADC to output an observation signal y1(m) and y (m), the time-frequency waveforms of which are shown in fig. 6(1), (2), respectively;
carrying out steps 3, 4) according to FIG. 21(m) and y (m) are reconstructed by SAMP algorithm to output frequency domain signal θ'1(i) And theta '(i), and inverse discrete Fourier transform is further performed on the signal to output a time domain signal x'1(n) and x' (n). Fig. 7(1) shows that after RD-AIC observation and SAMP reconstruction, the harmonic signal with the continuous Hanning window has better time-frequency signal reduction degree, and the sparsity adaptive estimation value K is 52, while after RD-AIC observation and SAMP reconstruction, the harmonic signal without the Hanning window has overlarge sparsity of the frequency domain signal, which results in failure of the SAMP algorithm. For comparison, the invention performs OMP algorithm reconstruction on y (m) under the same sparsity, and the time-frequency waveform is shown in FIG. 7 (2). For the amplitude spectra | θ 'in FIGS. 7(1.b) and 7(2. b)'1(i) The 2-6 harmonic components in |, | θ' (i) | are detected respectively, and the results are shown in table 2.
The detection results in table 2 show that the harmonic signals without the continuous Hanning window have larger measurement deviation of each harmonic after RD-AIC observation and reconstruction, and the harmonic signals with the continuous Hanning window have smaller measurement deviation of each harmonic after RD-AIC observation and reconstruction.
X 'reconstructed according to embodiment 5) in FIG. 2'1(n) is restored by the modified equation (15) to obtain the original harmonic signal x' (n), as shown in fig. 8. To illustrate the recovery effect of the correction equation, x'1(n) the original harmonic signal x' (n) is obtained without adding a correction formula, as shown in fig. 9. As can be seen from fig. 8 and 9, introducing a constant coefficient C close to 0 can effectively improve the recovery capability of x' (n) at both ends.
TABLE 2
Figure BDA0001906612720000131
The results of the 2 embodiments show that the method can effectively solve the problem of power grid harmonic signal spectrum leakage caused by asynchronous compressed sampling of RD-AIC on the basis of not changing RD-AIC hardware parameters and a reconstruction algorithm, and improve the efficiency of a CS reconstruction algorithm and the detection precision of each subharmonic component amplitude.
In order to further illustrate the superiority of the method in the application of power grid harmonic detection, the compression ratio of the method is set to be 20%, meanwhile, the method and the traditional harmonic measurement instrument are adopted to carry out 10 power frequency period sampling on the same harmonic, and the number of points after sampling is shown in table 3.
TABLE 3
Figure BDA0001906612720000132
In conclusion, the method can reduce the frequency spectrum leakage phenomenon caused by asynchronous sampling on the premise of meeting the requirement of RD-AIC asynchronous observation of the power grid harmonic signals, and further solves the problems of overlarge signal sparsity, low reconstruction algorithm efficiency and poor detection precision caused by frequency spectrum leakage. Meanwhile, the method can also be applied to harmonic signal detection in other fields.
Those skilled in the art will appreciate that the invention may be practiced without these specific details.

Claims (1)

1. A power grid harmonic non-synchronous compressed sensing detection method added with a continuous Hanning window is characterized by comprising the following steps:
step 1), deriving and defining a continuous Hanning window signal w (t);
the continuous Hanning window signal is defined as
Figure FDA0002827538120000011
Wherein
Figure FDA0002827538120000012
N＝T0/T，T0Is the analog output time of w (n), T ═ 1/f, f is the analog output frequency of w (n);
the derivation process is as follows:
discrete Hanning function
Figure FDA0002827538120000013
According to the discrete signal recovery theory, the continuous Hanning window signal can be expressed as:
w(t)＝w(n)*h(t) (2)
where h (t) is the unit impulse response of an ideal low-pass filter,
Figure FDA0002827538120000014
Figure FDA0002827538120000015
step 2) establishing a harmonic signal x added with a continuous Hanning window1(t)；
x1(t)＝x(t)w(t) (4)
Step 3) analyzing the harmonic signal x added with the continuous Hanning window1(t) a decomposition model under a continuous fourier transform and a generalized fourier series;
from the continuous Fourier transform, x1(t) is decomposable into
Figure FDA0002827538120000021
Since the spectrum prior information is unknown, equation (5) cannot directly reflect x1The information quantity and sparsity of (t) cannot reflect whether the application premise of RD-AIC is met, namely, any continuous signal can be subjected to finite term expansion on a continuous basis function;
then, x is known from the generalized Fourier series1(t) can be decomposed into
Figure FDA0002827538120000022
Formula (6) reflects x1(t) an approximate satisfaction of the formula RD-AIC finite term expansion in the generalized Fourier series, which can be regarded as x1(t) noisy models under a finite number of consecutive orthogonal bases of length N;
step 4) establishing a harmonic signal x added with a continuous Hanning window1(t) finite term decomposition under discrete fourier transform and K-sparse reference model;
if the signal length N reconstructed by CS meets the requirement of Nyquist sampling theorem, the reconstructed original signal can recover a continuous signal without distortion, and based on the continuous signal, a harmonic signal x added with a continuous Hanning window is established1(t) finite decomposition under discrete fourier transform and sparse reference model;
let x1(T) at a Nyquist sampling frequency fs and a finite duration T0The discretization signal in is x1(N) having a length of N ═ T0fsFrom x1(n) inverse discrete Fourier transform of x1(n) capable of limited term decomposition, as follows:
Figure FDA0002827538120000023
wherein the content of the first and second substances,
Figure FDA0002827538120000024
θ1＝[α1_0,α1_1,...,α1_N-1]T,
Figure FDA0002827538120000025
if theta1K in number different from 0, and K<<N, then x is considered1(n) has limited and sparse spectral information, satisfies the application premise of RD-AIC, and is considered to be reconstructed by CSX is out1(n) capable of recovering a continuous signal without distortion;
step 5) deducing harmonic signal x added with continuous Hanning window1(t), RD-AIC output observation signal y1(n) and reconstructed time-frequency signal θ'1(i)、x’1(n) the input-output mathematical model;
x is to be1(t) feeding RD-AIC, first with a pseudo-random sequence pc(t) multiplication by the following formula:
x2(t)＝x1(t)pc(t) (8)
secondly, x is2(t) feeding a low pass filter with a unit impulse response of h' (t) as follows:
y1(t)＝x2(t)*h’(t) (9)
expansion (9) is given by:
Figure FDA0002827538120000031
by substituting formula (10) into formula (7), it is possible to obtain:
Figure FDA0002827538120000032
then y is put1(t) feeding into the low speed ADC, as follows:
Figure FDA0002827538120000033
order to
Figure FDA0002827538120000034
Then there are:
Figure FDA0002827538120000035
wherein, A is a sensing matrix which is composed of a pseudo-random sequence, a unit impulse response of a low-pass filter, a low-speed ADC and a discrete Fourier base;
finally for y1(m) performing CS reconstruction as follows:
Figure FDA0002827538120000036
by equation (14), a harmonic spectrum vector θ 'added to a continuous Hanning window is obtained'1(i) And then obtaining an inverse transform signal x 'from equation (8)'1(n)；
Step 6), establishing a correction formula, and recovering an original harmonic discrete signal x' (n);
as can be seen from formula (4), x '(n) ═ x'1(n)/w (n), however, since the amplitudes of the discrete Hanning window function w (n) at the starting and ending time are both 0, i.e. both ends cannot be directly solved, in order to effectively recover x' (n), a constant C close to 0 is introduced for correction, and the formula is as follows:
Figure FDA0002827538120000041
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