CN107315714B - Deconvolution power spectrum estimation method - Google Patents

Deconvolution power spectrum estimation method Download PDF

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CN107315714B
CN107315714B CN201710502440.0A CN201710502440A CN107315714B CN 107315714 B CN107315714 B CN 107315714B CN 201710502440 A CN201710502440 A CN 201710502440A CN 107315714 B CN107315714 B CN 107315714B
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朴胜春
郭微
宋扬
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Abstract

The invention provides a deconvolution power spectrum estimation method. (1) Preprocessing the data sample, and performing power spectrum estimation on the preprocessed data sample; (2) carrying out deconvolution operation on the power spectrum of the power spectrum homowindow function of the data sample by using a deconvolution algorithm; (3) and selecting proper parameters for deconvolution operation, selecting iteration times, and then obtaining an estimation value of a true power spectrum of the signal through iteration convergence. The invention overcomes the problems of frequency spectrum leakage and lower frequency resolution in power spectrum estimation caused by limited data sample length. The invention can obviously improve the frequency resolution of power spectrum estimation under the condition of short data length, and can effectively inhibit the influence of side lobes caused by limited data length while realizing high-resolution power spectrum estimation, thereby obtaining extra signal processing gain, which has important significance for weak signal detection under the background of strong interference.

Description

Deconvolution power spectrum estimation method
Technical Field
The invention relates to a signal processing method, in particular to a signal spectrum estimation method.
Background
Spectral analysis techniques, including spectral analysis and power spectrum estimation, are the most common means of performing frequency domain analysis on signals. The power spectrum of a signal is the distribution of signal energy with respect to all frequencies that make up the signal. As a basic theory of signal spectrum analysis, fourier transform was initially defined as a type of global transform performed in an infinite time domain and a frequency domain, decomposing a time-domain signal into sinusoidal signals of different frequencies. In order to process discrete sequences, discrete-time fourier transforms and discrete fourier transforms have been developed. The Fourier transform can be applied to power spectrum estimation, and can also be extended to the field of time-frequency characteristic analysis of signals, such as short-time Fourier transform.
However, in practical applications, fourier transforms are typically used to compute a signal spectrum or power spectrum for a limited number of samples. These finite length samples are the product of the discrete sampled signal and a window function. This results in a spectral resolution that is limited by the length of the data sample, while the spectral leakage due to the side lobes of the window function is also non-negligible. Therefore, the influence of the limited sampling length of the signal on the spectrum estimation result has been a problem to be solved in the field of signal processing.
With the continuous development of signal processing technology, many methods for estimating a signal spectrum with good composition and high frequency resolution are proposed. For example, a frequency refinement technique, Zoom-FFT, can be used to perform high resolution processing on a localized portion of the spectrum. In addition, modern spectral estimation methods based on parameter models, such as those based on models such as AR and ARMA, which are proposed in succession, can obtain high-resolution spectral estimation results. The frequency resolution of the spectral estimation result can also be greatly improved by applying a Minimum Variance Distortionless Response (MVDR) method.
The above methods and other high-frequency-resolution signal spectrum estimation methods proposed by the scholars can indeed improve the resolution of the signal spectrum estimation result, but the sampling length of the signal is still one of the main factors influencing the processing performance of the high-resolution spectrum estimation methods. And other factors, such as signal-to-noise ratio of the signal, model matching degree of the model method, etc., all affect the resolution and accuracy of the spectrum estimation result. Therefore, it is necessary to research a high-resolution spectrum estimation technique with strong stability, moderate calculation amount, and capability of overcoming the limitation on frequency resolution caused by the limited signal sampling length.
Disclosure of Invention
The invention aims to provide a deconvolution power spectrum estimation method which can remarkably improve the frequency resolution of power spectrum estimation under the condition of short data length and can effectively inhibit the influence of side lobes caused by limited data length.
The purpose of the invention is realized as follows:
(1) preprocessing the data sample, and performing power spectrum estimation on the preprocessed data sample;
(2) carrying out deconvolution operation on the power spectrum of the power spectrum homowindow function of the data sample by using a deconvolution algorithm;
(3) and selecting proper parameters for deconvolution operation, selecting iteration times, and then obtaining an estimation value of a true power spectrum of the signal through iteration convergence.
The deconvolution power spectrum estimation method specifically comprises the following steps:
the preprocessing of the data samples is to select a window function and multiply the window function and the data samples in a time domain; the power spectrum estimation is carried out by adopting a periodogram method, calculating the mode square of the Fourier transform result of the data sample and the window function as the result of the power spectrum estimation of the data sample and the window function, namely solving
Figure GDA0002436463950000021
Pw(f)＝|W(f)|2Wherein X isN(f) And W (f) are data samples xN(n) and the Fourier transform of the window function w (n);
the deconvolution operation of the power spectrum homowindow function of the data samples by using a deconvolution algorithm is to use a Lucy-Richardson deconvolution algorithm to calculate the power spectrum of the data samples obtained in the step (1)
Figure GDA0002436463950000022
Sum window function power spectrum Pw(f) Respectively processing the sequences into discrete sequences with elements and 1;
the estimating of the true power spectrum of the signal obtained through iterative convergence specifically includes: treating the product obtained in the step (2)
Figure GDA0002436463950000023
And Pw(f) Substituting into the formula for r (x) and h (x), respectively
Figure GDA0002436463950000024
In the method, the iteration number is selected, then an estimation value of a true power spectrum P (f) of a signal is obtained through iteration convergence, s (x) is a signal to be obtained through deconvolution operation, and r represents the iteration number.
The invention provides a power spectrum estimation method, which solves the problems of frequency spectrum leakage and low frequency resolution in power spectrum estimation caused by limited data sample length. Although the existing signal processing technology can perform high-resolution power spectrum estimation, the limited data length is still a main factor influencing the power spectrum estimation effect, and other factors such as the signal-to-noise ratio and the matching degree of the processing technology and the signal characteristics also influence the resolution and the accuracy of the spectrum estimation result. The invention breaks through the Rayleigh limit of time-frequency resolution in the traditional Fourier analysis and can quickly and stably estimate the power spectrum with high resolution. Compared with other power spectrum estimation technologies, the method can obviously improve the frequency resolution of power spectrum estimation under the condition of short data length, and can effectively inhibit the influence of side lobes caused by limited data length while realizing high-resolution power spectrum estimation, thereby obtaining extra signal processing gain, which has important significance for weak signal detection under the background of strong interference.
The principle on which the invention is based is as follows:
for power spectrum estimation of a finite length of received data, the power spectrum estimation result is usually related to the data itself and the window function. Because in practical applications, whether the received data is in a continuous or discrete form, it is inevitably affected by the window function during data acquisition and signal processing. For example, a finite length of data, even if no window function is used, is itself equivalent to multiplying in the time domain by a rectangular window of equal length. To illustrate the relationship of the window function to the finite length of the received data, assume N point long data samples xN(N) is equal to the product of the discrete-time signal x (N) and the N-point window function w (N), i.e. has
xN(n)＝x(n)w(n) (1)
Based on the derivation of the classical spectrum estimation theory, a data sample x can be obtainedN(n) power spectrum
Figure GDA0002436463950000031
Power spectrum P (f) of infinite-length discrete signal x (n) and power spectrum P of window function w (n)w(f) In a relationship of
Figure GDA0002436463950000032
Wherein, the symbol
Figure GDA0002436463950000033
Representing convolution operations
The convolution relation in equation (2) indicates that the power spectrum of the data sample
Figure GDA0002436463950000034
Medium relative window function power spectrum Pw(f) The power spectrum p (f) of the infinite signal can be obtained by performing a deconvolution (deconvolution) operation. P (f) obtained at this time is the true power spectrum of the signal for an ideally infinite data sample.
There are many algorithms that can achieve the goal of deconvolution, and the most common Lucy-Richardson deconvolution algorithm is used as an example here. Suppose a one-dimensional signal is deconvoluted using the Lucy-Richardson deconvolution algorithm, the signal model being
Figure GDA0002436463950000035
Where r (x) and h (x) are known and s (x) is the signal to be found by the deconvolution operation. To utilize the Lucy-Richardson deconvolution algorithm, the signal model of equation (3) should satisfy the following condition: r (x), h (x), and s (x) must be non-negative; the integral of each of r (x) and h (x) is 1.
The Lucy-Richardson deconvolution algorithm is an iterative algorithm, and the convergence degree of the algorithm can be displayed by utilizing a discriminant function which is
Figure GDA0002436463950000036
Wherein p (x) and q (x) are both non-negative numbers and have an integral of 1.
For the signal model of formula (4), let p (x) r (x),
Figure GDA0002436463950000037
substitution discriminant is
Figure GDA0002436463950000038
By selecting an appropriate s (x), the discriminant can be minimized. The function that can be used to achieve the minimum is expressed as
Figure GDA0002436463950000039
The iterative function of s (x) derived by equation (6) is
Figure GDA00024364639500000310
Where r represents the number of iterations. Obtained after iteration of formula (7)
Figure GDA0002436463950000041
I.e. the estimation result of s (x) after the deconvolution calculation.
And (3) corresponding the convolution formula (2) of the power spectrum to the signal model (3), and applying the iteration formula (7) to obtain the true power spectrum P (f) of the signal.
On the premise that the signal is a sinusoidal signal and the noise is random white noise, after perfect deconvolution, the gain of the signal-to-noise ratio can be obtained
Figure GDA0002436463950000042
Wherein, Pw(f) And
Figure GDA0002436463950000043
respectively, the power spectrum of the window function before and after deconvolution. In the ideal case of the water-cooled turbine,
Figure GDA0002436463950000044
tending to a delta function. (8) The equation illustrates that the present invention is able to achieve additional signal processing gain by suppressing side lobes.
The invention has the following advantages:
1. compared with the traditional Fourier theory, the method can improve the frequency resolution of power spectrum estimation under the same data sample length, and breaks through the Rayleigh limit of the frequency resolution when the traditional Fourier theory is adopted for time-frequency analysis.
2. The invention can effectively inhibit the influence of the side lobe caused by limited data length, and obtains extra processing gain by inhibiting the side lobe. Therefore, the method has great significance for the detection of weak signals and the estimation of high-resolution power spectrum.
3. The method is less influenced by the sample data length and the signal-to-noise ratio of the sample in the aspect of power spectrum estimation stability, and compared with the existing high-resolution power spectrum estimation technology, the method can realize high-resolution power spectrum estimation under the conditions of shorter sample data length and lower signal-to-noise ratio.
4. The method has the advantages of simple operation, small calculated amount and high calculating speed, can meet the requirements on real-time high-resolution power spectrum estimation in engineering practice, and has wide application range.
Drawings
FIG. 1 is a flowchart of the process of the present invention;
FIG. 2 shows the result of a power spectrum preprocessing of the limited-length received data;
FIG. 3 shows the result of a window function power spectrum preprocessing;
FIG. 4 is a frequency resolution versus iteration number for deconvolution spectrum estimation using various window functions;
FIG. 5 is a graph of the effect of data length on the frequency resolution improvement of the deconvolution spectrum estimation method;
FIG. 6 is a graph of the effect of data length and iteration number on signal-to-noise ratio gain;
FIG. 7 deconvolution power spectrum estimation results (solid line); classical power spectrum estimation results (dashed line);
fig. 8 a-8 b experimental data processing results, fig. 8a is a classical spectrum estimation processing result, and fig. 8b is a deconvolution spectrum estimation result.
Detailed Description
The deconvolution power spectrum estimation method specifically comprises the following steps:
(1) selecting a proper window function, preprocessing the data sample (i.e. multiplying the window function and the data sample in time domain), and performing preprocessing on the data sampleAnd estimating a power spectrum. The spectrum estimation method has various types, in order to meet the application condition of the selected Lucy-Richardson deconvolution algorithm, a periodogram method in a classical spectrum estimation method is adopted, and the mode square of Fourier transform results of data samples and window functions is calculated as the power spectrum estimation results of the data samples and the window functions, namely the power spectrum estimation results are obtained
Figure GDA0002436463950000051
Pw(f)＝|W(f)|2Wherein X isN(f) And W (f) are data samples xN(n) and the Fourier transform of the window function w (n).
(2) And carrying out deconvolution operation on the power spectrum of the data sample and the power spectrum of the window function by using a deconvolution algorithm. If Lucy-Richardson deconvolution algorithm is used, the power spectrum of the data sample obtained in the step (1) is used
Figure GDA0002436463950000052
Sum window function power spectrum Pw(f) The algorithm has satisfied the precondition that the deconvolution term is non-negative. Only need to be combined with
Figure GDA0002436463950000053
And Pw(f) The sequence of (1) is processed into a discrete sequence of elements and 1, respectively.
(3) Suitable parameters are selected for the deconvolution operation. In the deconvolution power spectrum estimation operation provided by the invention, the (2) processed signal is processed
Figure GDA0002436463950000054
And Pw(f) Substituting r (x) and h (x) into the formula (7), selecting proper iteration times, and then obtaining an estimation value of a real power spectrum P (f) of the signal through iterative convergence.
The invention is described in more detail below by way of example.
(1) Selecting a proper window function, preprocessing the data sample (i.e. multiplying the window function and the data sample in the time domain), and performing power spectrum estimation on the preprocessed data sample. To satisfy the selected Lucy-Richardson deconvolution algorithmThe method adopts a periodogram method in a classical spectrum estimation method, calculates the mode square of the discrete Fourier transform result of the data sample and the window function as the result of the power spectrum of the two, namely, obtains
Figure GDA0002436463950000055
Pw(f)＝|W(f)|2Wherein X isN(f) And W (f) are data samples xN(n) and the Fourier transform of the window function w (n).
(2) And carrying out deconvolution operation on the power spectrum of the data sample and the power spectrum of the window function by using a deconvolution algorithm. If Lucy-Richardson deconvolution algorithm is used, the power spectrum of the data sample obtained in the step (1) is used
Figure GDA0002436463950000056
Power spectrum P of the sum window functionw(f) The algorithm has satisfied the precondition that the deconvolution term is non-negative. Only need to be combined with
Figure GDA0002436463950000057
And Pw(f) The sequence of (1) is processed into a discrete sequence of elements and 1, respectively. Suppose there is a segment of received data of length 1s, which includes two frequencies f1500Hz and f2505Hz, SNR10dB and SNR2A sinusoidal signal of 25 dB. The noise is random white noise, and the sampling rate is fs10 kHz. Deconvolution spectrum estimation is performed using a rectangular window. As shown in fig. 2 and 3, the data samples and the window function are the result of preprocessing the power spectra of the data samples and the window function using a deconvolution algorithm, respectively.
For the Lucy-Richardson deconvolution algorithm used, the frequency resolution of the resulting deconvolution spectrum estimation result is approximately the same regardless of which window function is used, as long as a sufficient number of iterations is provided. As shown in fig. 4, four window functions are used to perform deconvolution spectrum estimation at different iterations, and the corresponding main peak width of the signal is calculated to measure the frequency resolution at that time. FIG. 4 shows that, when the number of iterations reaches 400 or more, the widths of the main peaks (-3dB widths) of the deconvolution spectrum estimation results using these window functions are substantially equal; however, the rectangular window precedes several other window functions to achieve the maximum frequency resolution that can be provided. Thus, if a smaller number of iterations is used in order to increase the computation speed, a window function can be selected that achieves the convergence state faster.
The improvement of the frequency resolution of the present invention is independent of the data length. In fig. 5, power spectrum estimation is performed on data of different lengths using classical spectrum estimation and deconvolution spectrum estimation, respectively, and the signal main peak width is calculated. The frequency resolution of the power spectrum estimation result obtained by the two methods is increased along with the increase of the length of the data sample, but the ratio of the main peak width of the signal of the two methods in different data lengths is basically unchanged, and the main peak width of the signal of the deconvolution spectrum estimation method accounts for about 35% of that of the classical spectrum estimation method. This shows that the frequency resolution of the classical spectrum estimation method can be stably improved by about 2 times.
Signal to noise ratio gain
Figure GDA0002436463950000061
Can be used for explaining the suppression of the side lobe by the deconvolution spectrum estimation. The larger the signal-to-noise ratio gain value is, the more thorough the sidelobe suppression is shown.
Fig. 6 shows the effect of iteration number and data length on the signal-to-noise ratio gain. When the iteration times are fixed, the increase of the data length has little influence on the signal-to-noise ratio gain. However, increasing the number of iterations has a great effect on improving the signal-to-noise ratio gain for the same data length.
Therefore, the power spectrum estimation method and the deconvolution algorithm can be reasonably selected according to actual requirements in application, and relevant parameters of the algorithm are adjusted to obtain an ideal spectrum estimation result.
(3) The appropriate parameters are selected for the deconvolution operation. For example, the Lucy-Richardson deconvolution algorithm is used, and the influence of the iteration number, the data length and the window function type on the algorithm is analyzed in detail in (2). Therefore, the product after the treatment (2)
Figure GDA0002436463950000062
And Pw(f) Substituting r (x) and h (x) into the formula (7), selecting proper iteration times, and then obtaining the real signal power spectrum P (f) estimation through iterative convergence. Fig. 7 is a power spectrum estimation result obtained by deconvolving the data samples and the window function power spectrum of fig. 2 and 3 by using the Lucy-Richardson deconvolution algorithm. From the results of FIG. 7, it can be seen that the frequency is f1500Hz and f2The spectral peaks of both signals at 505Hz can be detected, but the power spectrum resolution using deconvolution spectral estimation is higher and the side lobes can all be suppressed below-50 dB. Fig. 8 shows the processing results of the experimental data of a sea test. Fig. 8a shows the result of processing using the classical spectral estimation method, and fig. 8b shows the result of processing without using the deconvolution spectral estimation method. In fig. 8b, compared to fig. 8a, the deconvolution spectrum estimation method can greatly improve the frequency resolution of the CW signal, can enhance the weak CW signal, and can detect a weak moving object near 63Hz, which is not detected in fig. 8 (a).

Claims (1)

1. A deconvolution power spectrum estimation method is characterized by comprising the following steps:
(1) preprocessing the data sample, and performing power spectrum estimation on the preprocessed data sample;
(2) carrying out deconvolution operation on the power spectrum of the power spectrum homowindow function of the data sample by using a deconvolution algorithm;
(3) selecting proper parameters for deconvolution operation, selecting iteration times, and then obtaining an estimated value of a true power spectrum of a signal through iterative convergence;
the preprocessing of the data samples is to select a window function and multiply the window function and the data samples in a time domain; the power spectrum estimation is carried out by adopting a periodogram method, calculating the mode square of the Fourier transform result of the data sample and the window function as the result of the power spectrum estimation of the data sample and the window function, namely solving
Figure FDA0002436463940000011
Pw(f)＝|W(f)|2Wherein X isN(f) And W (f) are data samples xN(n) and the Fourier transform of the window function w (n);
the deconvolution operation of the power spectrum homowindow function of the data samples by using a deconvolution algorithm is to use a Lucy-Richardson deconvolution algorithm to calculate the power spectrum of the data samples obtained in the step (1)
Figure FDA0002436463940000012
Sum window function power spectrum Pw(f) Respectively processing the sequences into discrete sequences with elements and 1;
the estimating of the true power spectrum of the signal obtained through iterative convergence specifically includes: treating the product obtained in the step (2)
Figure FDA0002436463940000013
And Pw(f) Substituting into the formula for r (x) and h (x), respectively
Figure FDA0002436463940000014
In the method, the iteration number is selected, then an estimation value of a true power spectrum P (f) of a signal is obtained through iteration convergence, s (x) is a signal to be obtained through deconvolution operation, and r represents the iteration number.
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