CN113541648B - Optimization method based on frequency domain filtering - Google Patents
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Abstract
The invention discloses an optimization method based on frequency domain filtering, which specifically comprises the following steps: performing time domain windowing on a time domain long sequence signal x (n) to be filtered to obtain a time domain windowed segmented signal x 1(n),x2(n),...,xDA (n); according to the filtering requirement of a user, a filter function is designed by itself, and frequency domain windowing processing is carried out on signals to obtain a time domain filter function h (n); performing FFT calculation linear convolution processing on the acquired segmented signal x 1(n),x2(n),...,xDA (n) and a time domain filter function h (n) to obtain a filtered segmented signal y 1(n),y2(n),...,yDA (n); and performing overlap-add-and-restore signal processing on the segmented signal y 1(n),y2(n),...,yDA (n) to obtain a filtered time domain output signal y (n).
Description
Technical Field
The invention relates to the technical field of signal processing, in particular to an optimization method based on frequency domain filtering.
Background
In practical communication systems, the detected signal is a time domain signal, and sometimes the signal and its difficulties are analyzed from the time domain, requiring conversion to the frequency domain for analysis. A filtering method can be designed on the frequency domain, and unnecessary frequency components in the time domain are filtered out to obtain a filtered signal. In practice, the interference signal in the signal can be removed through frequency domain filtering. The frequency domain filtering method is to transform the original signal to be filtered into the frequency domain by utilizing the FFT technology, filter the part to be filtered by windowing according to the filtering requirement, and obtain the filtered signal after the inverse fast Fourier transform.
At present, the conventional frequency domain filtering is to perform FFT on a signal to the frequency domain, and increase or decrease the value of the point number corresponding to the frequency range to be filtered by a corresponding multiple. The conventional frequency domain filtering is equivalent to truncation because the frequency domain directly operates on the frequency response, so that the time domain signal can be stretched. Because the actual signals are generally time domain long sequence signals, in order to realize real-time processing of computer processing data, the long sequence signals are segmented, and direct truncation can lead to the widening phenomenon of frequency domain signals, thereby affecting the filtering of the frequency domain. Moreover, the filter designed by the conventional frequency domain filtering is affected by the transition band, so that the filtering effect at a specific frequency is poor. Meanwhile, conventional filtering is troublesome for complex filter designs. Therefore, the traditional frequency domain filtering has certain defects such as low precision, poor flexibility, low efficiency and the like.
The prior art related to the present invention is as follows:
The first technical scheme is as follows:
The "time domain end effect and avoiding method generated by frequency domain filtering" [1] published in Wang Yan, literature [1], refers to the disadvantage of conventional frequency domain filtering, that is, the time domain signal after filtering is truncated at one end, and the truncated portion appears at the other end, which may generate serious distortion. Two optimized filtering methods are provided, the first method is zero filling at the beginning end and the end of the time domain signal, and the second method is widening the frequency spectrum of the filter.
The scheme provides two processing methods, and for the first method, zero padding at the starting end can affect an original signal, and a series of processing needs to be carried out on an output signal of the original signal, so that the original signal is troublesome. And the zero-filling operation does not make clear calculation, if the zero filling is too much, the calculation is complex, and if the zero filling is too little, the distortion cannot be effectively avoided. For the second method, the filter spectrum is widened, that is, a relatively slowly varying window function is used, only the low-pass filter spectrum of the widened transition band is provided, the complex filtering condition in the actual filtering process is not considered, only the low frequency can be filtered, and any frequency component cannot be filtered. Furthermore, both methods will have a huge computational effort for processing time-domain long sequence signals.
The second technical scheme is as follows:
The 'real-time frequency domain filtering interference suppression compensation technical research' [2] published in Liu Yanliang is document [2], and the defects of frequency domain filtering are overcome by adopting a method of time domain windowing and overlapping treatment. Time domain windowing requires a window function with low side lobes to smooth discontinuities at block edges to reduce spectral leakage. The overlapping process compensates for the processing gain loss due to windowing and enhances the correlation of the signals. One channel for processing signals needs to be added, and two channels of signals have delay of N/2 point length. The method comprises the steps of carrying out segmentation processing on two paths of signals, carrying out frequency domain filtering on each path of signals with the length of N, directly setting corresponding points of the frequency domain to zero by utilizing threshold setting, then carrying out inverse transformation to obtain two paths of filtered signals, only reserving middle N/2 sample points on each path of signals, discarding N/4 sample points before and after each path of signals, splicing the corresponding two paths of N/2 sample point signals into N point signals, and splicing each path of signals into output signals.
When the scheme is used for splicing, one channel for processing delay signals is added, so that the calculated amount is increased by 1 time, and the running speed of the system is greatly reduced. Meanwhile, when frequency domain filtering is carried out, the direct zeroing filtering corresponds to the frequency domain truncation, so that the time domain signal is widened, and the filtered signal is distorted.
The literature relevant to the invention is as follows:
[1] Wang Yan time domain end effects generated by frequency domain filtering and avoidance method [ J ]. Scientific field of view 2020 (22): 7-10.
[2] Liu Yanliang, liu, kong Junhui. Real-time frequency domain filter interference suppression compensation techniques research [ J ]. Information communication, 2013 (05): 19-21.
Disclosure of Invention
According to the problems existing in the prior art, the invention discloses an optimization method based on frequency domain filtering, which specifically comprises the following steps: a time domain windowing process, a frequency domain windowing process, an FFT calculation linear convolution process and an overlap-add method signal restoration process.
The input of the time domain windowing processing is a time domain long sequence signal x (n) which needs to be filtered, and the output of the time domain long sequence signal x (n) is a segmented signal x 1(n),x2(n),…,xDA (n) after the time domain windowing; the frequency domain windowing processing can design a filter function according to the filtering requirement of a user and output the filter function as a designed time domain filter function h (n); the two input ends of the FFT calculation linear convolution are the time domain windowing processing output end and the frequency domain windowing processing output end respectively, and the output ends are the filtered segmented signals y 1(n),y2(n),…,yDA (n); the input of the splice addition method restoring signal end is an output end of FFT calculation linear convolution, and the output of the FFT calculation linear convolution is a filtered time domain output signal y (n).
The time domain windowing processing process comprises the following steps: a window function for preventing the system from generating time domain truncation is designed, q-point overlapping pretreatment is firstly carried out on the original long sequence signal in a segmented mode, and windowing treatment is carried out on each segment of signal through the designed time domain window function, so that segmented signal x 1(n),x2(n),…,xDA (n) after time domain windowing is obtained.
The q-point overlapping preprocessing of the time domain windowing processing, namely, the first q-point data of the input signal x (N) is firstly imported as the initial data of the data, and then the data with the data point of p is imported from the input signal x (N) as the last p-point data of the section, so that the data section needing to be processed for the first time is obtained, and the length is N=q+p; and when the subsequent data segment processing is carried out, taking the rear q point of the previous segment as the front q point of the data of the present segment, reading the p point data of the input signal x (N) as the rear p point data of the present segment, namely obtaining the data segment which is required to be processed at the kth time, and repeating the above operation when the length is still N=q+p point.
The q-point overlapping preprocessing of the time domain windowing processing can obtain data segments of DA segments entering subsequent processing, and each data segment of the data has the following characteristics: firstly, the front q point data of each section and the rear q point data of the last section of data are repeated data; the p-point data after each segment is p data points continuously read in from an input signal x (N), the length of each segment is N points, and the segmented data is used as input to be processed such as frequency domain windowing; third, the repeated data with the length of q and the input data of p points share n=p+q points.
The window function of the time domain windowing process needs to meet two conditions of time domain windowing, namely, the first condition is that a window function with a transition zone being more slowly changed is needed; the second condition is that slight distortion caused by time domain windowing on an input signal is compensated by an overlapped splicing algorithm, and the influence caused by time domain windowing can be compensated by the overlapped splicing algorithm in actual operation. Therefore, the invention takes the time domain window function to meet the condition that the transition zone of the previous window function and the transition zone function of the next window function are added to be 1, so that the signal before windowing and the signal after overlapping and splicing are almost unchanged, and the distortion degree of the signal after splicing is obviously reduced.
In the design of the window function of the time domain windowing process, in order that the overlap-add method restore signal can restore the original data, the transition zone which also needs to meet the windowing function is q point, and meanwhile, the number of overlap points for the subsequent overlap-add method is q point. Therefore, the window function required by the overlap-add method is designed, and the time domain window function is designed as
The frequency domain windowing processing process comprises the following steps: the frequency domain windowing process is to design an ideal filter function meeting the system requirement on the frequency domain in practice, and then window the ideal filter function to become a realizable filter.
The frequency domain windowing processing can be used for designing an achievable high-performance filter by itself, firstly, according to the filtering requirement of a system, frequency preprocessing is carried out on a frequency range, then the frequency response function H d(ejw of an ideal filter meeting the requirement is obtained, and the time domain function of the ideal filter can be calculated to be H d (n) through IFFT. The type of window function w (n) may be determined by the minimum stop band attenuation required by the system, and the magnitude of the window function order M may be determined by the excess bandwidth required by the system, whereby the window function w (n) may be uniquely determined. Thus, the time domain expression of the actual filter function can be calculated as
h(n)＝hd(n)w(n)，n＝0,1,…,N-1 (2)
The frequency preprocessing of the frequency domain windowing is to cope with complex filtering requirements of a system, and the invention takes g sections for discussion, and is specifically shown in table 1.
Table 1 system filtering requirement frequency range table
The transition zone allowed by the system should meet all transition zone requirements, so the transition zone with the smallest frequency in the transition zones should be taken as
△f＝min{f21-f12,f31-f32,…,f1-f(g-1)2} (3)
The input frequencies are preprocessed to prevent transition bands from occurring in the desired frequency range, affecting filter performance. The frequencies of Table 1 were subjected to the following preprocessing commands
The transition band of the filter designed by the frequency is concentrated near f i0, and the frequency range of the transition band is delta f, and the transition band is the smallest frequency range in all the frequency ranges, so that the set of the transition bands meets the requirement
The time domain function of the ideal filter of the frequency domain windowing treatment can write out the ideal frequency response function as follows according to the filtering requirement of the system
Then find the ideal time domain expression
Where τ is the shift necessary for the linear phase, τ=m/2 should be satisfied.
Determination of the window function of the frequency domain windowing process the present invention provides several common window function basic parameters, as shown in table 2, by first determining the selected window function type based on the minimum stop band attenuation required by the system.
Table 2 common window function basic parameter table
| Window function | -60DB transition bandwidth c value (2 pi/M) | Stopband minimum attenuation (dB) |
| Rectangular window | 1.998 | -21 |
| Buterglide | 3.830 | -25 |
| Hanning window | 3.977 | -44 |
| Hamming window | 3.950 | -53 |
| Blackman window | 5.525 | -74 |
In the present invention, the transition bandwidth is taken as the bandwidth where the main lobe normalized amplitude drops to-60 dB, as shown in table 2. Based on the system-required stopband attenuation, the type of window function w (n) and the value of-60 dB transition bandwidth corresponding to the window function can be determined by looking up Table 2
Where c is the value of the-60 dB transition bandwidth corresponding to the selected window function in Table 2, in units of 2π/M. From equation (2), the transition band B of the designed filter function should be smaller than the transition bandwidth Δf allowed by the system, i.e. the digital frequency of the transition band should satisfy
M is the order of the filter, from which M can be calculated to satisfy
Wherein M is the integer of the order of the filter. The type and c-value of the window function can be determined by the minimum attenuation of the stop band required by the system, and the order of the window function can be determined by the formula (10), so that the time domain expression w (n) of the window function can be uniquely determined. The time domain expression h d (n) of the ideal filter can be obtained by substituting M into the expression (7), and the time domain expression h (n) of the actual filter can be obtained by substituting w (n) and h d (n) into the expression (2).
The FFT calculates the linear convolution process as
The FFT calculation linear convolution is that the system performs FFT conversion on a time domain windowed signal x 1(n),x2(n),…,xDA (n) of a time domain long sequence signal segment, multiplies the time domain windowed signal x 1(n),x2(n),…,xDA (n) by FFT conversion H (e jw) of H (n) respectively, and performs IFFT inverse conversion on the obtained result to obtain a filtered signal. To require a linear convolution of x k (N) of N-point sequence length and h (N) of M-point sequence length, the length L of the output y k (N) is required to satisfy
At the same time that L is greater than or equal to N+M-1 (11), the FFT operation needs to satisfy the multiple of the number of points to be the integer power of 2, so L must be taken to satisfy
L＝2k≥N+M-1 (12)
The FFT calculates the linear convolution, and uses the linear convolution of x k (N) of the N-point sequence length and h (N) of the M-point sequence length to obtain a signal of n+m-1-point length, and since the system is zero-padded to a multiple of an integer power of 2, the signal is zero-padded data 0 value after the tail of the output signal y k (N), and therefore the first n+m-1 data of y k (N) is needed to be taken as an output result.
The processing process of the cascade-connected addition method for restoring the signal is as follows
The overlap-add method is to overlap-add q-point overlapped data in time domain windowing, and the effect is to restore the filtered signal without distortion as much as possible. When the time domain windowing is performed, a data overlapping processing method is adopted, and each segment of input signal x k (N) is subjected to N-point windowing processing, so that y k (N) of N+M-1 points obtained after convolution is equivalent to the broadening of a transition zone of a time domain window function, namely the number of effective data points is unchanged, N-2q+1 points are required, and the number of transition zone points is widened from q-1 points to H points.
Assuming that the number of the front and rear transition band portions of y k (n) is H, according to the discussion of the time domain windowing portion, the time domain windowing portion x k (n) needs to satisfy the q-point data overlap to restore the original signal. Y k (n) also needs to be satisfied that there is a q point overlap in the transition band portion, and the remaining H-q points are added to the data points to prevent truncation from affecting the filtering performance. The expression of the transition zone can be obtained as
The total overlapping part point number of each piece of data is
F＝q+H-q+H-q＝2H-q＝M-1+q (14)
From this, it is possible to obtain a splice length F, i.e., N+M-1>M-1+q, of N, M and q which must satisfy a data length N+M-1 of more than 2 times, and to reduce it to a value which is required to satisfy
p≥M-1+q (15)
The overlap-add method restores the signals, namely, each segment of signal x k (N) and h (N) are subjected to linear convolution according to the FFT method to obtain an output signal y k (N) of N+M-1 point. The back F term of the signal y k (n) of each segment is correspondingly added with the front F term of the signal y k+1 (n) of the next segment, and then the results y (n) after convolution can be obtained by arranging the results in sequence.
Due to the adoption of the technical scheme, the optimization method based on the frequency domain filtering has the following beneficial effects:
1. When the long sequence convolution is carried out, time domain windowing processing and subsequent overlap-add processing are carried out on the long sequence, when the time domain windowing processing is carried out, instead of directly windowing signals, preprocessing of overlapping q points is carried out on the signals, thus the segmented signals are subjected to time domain q point transition zone windowing, and then the original signals are restored through q point overlap-add. This counteracts the truncation effect caused by conventional FFT filtering and prevents signal distortion.
2. When the frequency domain filter function is designed, preprocessing is carried out on the input frequency, namely, the center frequency of a frequency range which is not limited by a user is taken for frequency domain windowing, and the minimum transition band is obtained, so that the transition band of the designed filter is strictly limited outside the frequency range specified by the user. Meanwhile, parameters are set by only inputting a few sections of preset frequency ranges and minimum stop band attenuation, the invention can automatically select window function types and calculate the minimum window function orders, design the window function with the minimum orders, and then obtain the filter function which is output subsequently.
3. When the invention uses the overlap-add algorithm to combine signals, in order to meet the q-point overlap of effective data points strictly according to time domain windowing analysis, the transition zone change of the time domain window function after frequency domain windowing is analyzed, the components of the points of each segment signal after filtering are obtained, and the integral signals of the segment signals are subjected to F-point overlap-add.
Drawings
In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings that are required to be used in the embodiments or the description of the prior art will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments described in the present application, and other drawings may be obtained according to the drawings without inventive effort to those skilled in the art.
Fig. 1 is a general block diagram of a frequency domain filtering improvement method in the present invention.
Fig. 2 is a schematic diagram of the overlap-add method of the present invention.
Fig. 3 is a window function diagram of time-domain windowing in accordance with the present invention.
Fig. 4 is a block diagram of a linear convolution of FFT computation in the present invention.
FIG. 5 is a graph showing the amplitude-frequency response of an ideal filter versus an actual filter in the test of the present invention.
FIG. 6 is a graph showing the amplitude-frequency response of the ideal filter to the actual filter in the test of the present invention, in a range from 1200Hz to 1600 Hz.
Detailed Description
In order to make the technical scheme and advantages of the present invention more clear, the technical scheme in the embodiment of the present invention is clearly and completely described below with reference to the accompanying drawings in the embodiment of the present invention:
the optimization method based on frequency domain filtering shown in fig. 1 specifically comprises the following steps:
Time domain windowing processing, frequency domain windowing processing, FFT (fast Fourier transform) calculation linear convolution processing and overlap-add signal restoration processing. The input signal of the device is a time domain long sequence signal x (n) which needs to be filtered, the window function of the frequency domain windowing can be designed according to the filtering requirement, and the output signal y (n) is a filtered time domain output signal.
1. The specific mode of the time domain windowing processing process is as follows:
when the traditional frequency domain filtering is carried out on the long-sequence signal, the requirement on the memory capacity of a computer is overlarge, filtering can be realized only when the long-sequence signal is completely input into the computer, and 'real-time processing' of the signal can not be realized, so that the long-sequence signal is required to be truncated, and then the filtered signal is spliced and restored after the filtering treatment.
The long sequence signal truncates the signal in the time domain and requires the use of a correct window function to reduce the impact on the signal. The window function of time domain windowing should satisfy two conditions:
(1) A window function with a more gradual transition zone is required. Direct truncation leads to broadening of the frequency domain spectrum and causes distortion of the filtered signal, so that a window function with a relatively slow transition band is needed to be used, and the influence of truncation on the signal is reduced.
(2) Although the time domain windowing can reduce spectrum leakage, the input signal is slightly distorted, and the influence caused by the time domain windowing can be compensated through an overlapped splicing algorithm in actual operation.
After time-domain windowing, the signal needs to be restored using an overlap-add method. The principle of the overlap-add algorithm is shown in fig. 2, the original long-sequence signal is segmented through time domain windowing, and after the post-transition zone of the previous segment signal and the pre-transition zone of the next segment signal are overlapped and added, if the amplitude value after the transition zone addition is ensured to be equal to the amplitude value of the original signal, that is, the transition zone meeting the previous window function and the transition zone function of the next window function are added to be 1, the signal before windowing and the signal after overlapping and splicing are almost unchanged, and the distortion degree of the signal after splicing is obviously reduced.
In the invention, in order to restore the original signal without distortion, the window function needs to meet the two conditions of the time domain windowing, the designed time domain window function w 0 (n) is used for intercepting the data, and then the signal is restored by the overlap-add method. While using the overlap-add method to restore the signal, we need to do the pre-processing of the overlap segments of q points on the original data.
Preprocessing overlapped segments is carried out, wherein the first segment takes front q point data as initial data points, then p points of each data segment are p point data imported from an input signal x (N), the front q point is rear q point data of the previous data segment, and the length of each data segment is N=q+p points. From this, a data segment is obtained which enters the DA segment for subsequent processing, each data segment of this data having the following characteristics:
(1) The front q point data of each section and the rear q point data of the last section of data are repeated data;
(2) The p-point data after each segment is p data points continuously read in from an input signal x (N), the length of each segment is N points, and the segmented data is used as input to be processed such as frequency domain windowing;
(3) Each segment of the repeated data with the length of q and the input data of p points share n=p+q points.
The data segment of the DA segment can be obtained by the q-point overlapped segmentation preprocessing method, each segment of data with the length of N is multiplied by a predesigned window function w 0 (N) to be subjected to time domain windowing, and the data segment of the DA segment is x 1(n),x2(n),…,xDA (N), and is segmented after time domain windowing for subsequent frequency domain windowing and the like.
In order to restore the original data by using the overlap-add method, the window function needs to satisfy both the above two conditions, and the transition zone satisfying the window function needs to be q point. Therefore, the window function required by the overlap-add method is designed, and the time domain window function is designed as
The front q point and the rear q point of each piece of data are multiplied by the front transition zone and the rear transition zone of the window function respectively, and the transition zone function is overlapped to be 1 because the data of the front q point and the rear q point are the same. Therefore, the overlap-add method is used for overlapping the rear q point of the front section after windowing with the front q point of the rear section, so that q point data corresponding to the original signal can be restored; the intermediate p-q points are windowed and are multiplied by 1, and the data segments generated by the data overlap processing method are identical to the original data, so that the time domain windowed signals can be restored to the original signals by using an overlap-add method through the overlap of q points. The principle of the overlap-add method of time-domain windowing is shown in fig. 3.
2. The specific mode of the frequency domain windowing processing process is as follows:
Because the filtering requirements of the system are complex and changeable, signals in different frequency ranges need to be processed differently, and therefore different filters need to be designed on different frequencies. The invention takes g section for discussion, when the frequency amplitude response of the system is set at f 11Hz～f12 Hz, the frequency amplitude response of the system is a 1 times of the original frequency response of the system; at f 21Hz～f22 Hz, the frequency amplitude response of the system is 2 times that of the original system; and so on, at f g1Hz～fg2 Hz, the frequency amplitude response of the system is a g times that of the original system; specifically, the results are shown in Table 1.
Table 1 system filtering requirement frequency range table
The number g of the frequency ranges can be automatically increased according to specific filtering requirements, and the set amplitude attenuation coefficient value A g is expressed by taking decibel values according to the set amplitude attenuation coefficient for the more convenient larger fluctuation range of frequency attenuation, and the conversion relation formula is that
dB＝20log10(A) (2)
It is observed that the system allows excessive banding of the above-mentioned undesirable frequency range, i.e., f 21-f12,f31-f22,…,fg1-f(g-1)2. When the frequency domain windowing is performed, the transition zone allowed by the system should meet all transition zone requirements, so that the transition zone with the minimum frequency in the transition zones is adopted as Deltaf for determining the order of the filter.
In order to meet the stringent conditions in the system over the user specified frequency range, the frequency range of the transition band cannot appear in the required frequency range, i.e. f 11Hz～f12Hz,f21Hz～f22Hz,…,fg1Hz～fg2 Hz, which requires pre-processing of the designed filter frequency range to prevent the transition band from appearing in the required frequency range, affecting the filter performance. When designing the filter, it can be found that the transition zone occurs near the sudden frequency of the filter and is uniformly distributed on both sides of the frequency, so the frequency is taken as
The transition band of the filter designed by the frequency is concentrated near f i0, and the frequency range of the transition band is delta f, and the transition band is the smallest frequency range in all the frequency ranges, so that the set of the transition bands meets the requirement
I.e. such that the transition band is limited outside the frequency range specified by the user, system performance is guaranteed.
The ideal frequency response function can be written according to the filtering requirement of the system
Wherein w is the digital frequency corresponding to the system frequency f,F s is the sampling frequency of the system. From the formula
Can find ideal time domain expression
Where τ is the shift necessary for the linear phase, τ=m/2 should be satisfied.
The present invention provides several common window function basic parameters, as shown in table 2, depending on the minimum stop band attenuation required by the system to determine the type of window function selected.
Table 2 common window function basic parameter table
| Window function | -60DB transition bandwidth c value (2 pi/M) | Stopband minimum attenuation (dB) |
| Rectangular window | 1.998 | -21 |
| Buterglide | 3.830 | -25 |
| Hanning window | 3.977 | -44 |
| Hamming window | 3.950 | -53 |
| Blackman window | 5.525 | -74 |
The traditional transition bandwidth takes the bandwidth that the amplitude of the main lobe normalization is reduced to-3 dB, and the bandwidth between two zero crossing points of the main lobe is also taken. In the invention, the transition bandwidth is taken as the bandwidth of which the main lobe normalized amplitude is reduced to-60 dB, as shown in a table 2, so that the accuracy of the system can be improved. Based on the system-required stopband attenuation, the type of window function w (n) and the value of-60 dB transition bandwidth corresponding to the window function can be determined by looking up Table 2
Where c is the value of the-60 dB transition bandwidth corresponding to the selected window function in Table 2, in units of 2π/M. From equation (1), the actual filter function is equal to the multiplication of the ideal filter function and the window function in the time domain, that is, the frequency response of the actual filter function is equal to the convolution of the frequency response of the ideal filter function and the frequency response of the window function in the frequency domain, so that the transition band formed by the convolution of the frequency response of the actual filter function and the frequency response of the window function appears, and the width of the transition band is equal to the bandwidth B of the window function. The above requirement results in the allowable transition band width of the system being Deltaf, the designed transition band B of the filter function should be smaller than the allowable transition band width Deltaf, i.e. the digital frequency of the transition band should satisfy
M is the order of the filter, from which M can be calculated to satisfy
Wherein M is the integer of the order of the filter. The type and c value of the window function can be determined by the minimum attenuation of the stop band required by the system, and the minimum order of the window function can be determined by the formula (10), so that the time domain expression w (n) of the window function can be uniquely determined. Bringing M into equation (7) can obtain the time domain expression h d (n) of the ideal filter according to the equation
H (N) =h d (N) w (N), n=0, 1, …, N-1 (11) can be used to calculate the time domain expression H (N) of the actual filter, and the time domain expression H (e jw) of the actual filter function can be calculated by FFT transformation.
3. The specific way of the FFT calculation linear convolution process is:
The specific operation steps are as follows: firstly, taking a section of signal x k (N) of a time domain long sequence signal after time domain windowing, and zero padding to L points after x k (N) of the N point sequence length and h (N) tail of the M point sequence length, so that FFT operation can be facilitated, and the fence effect can be reduced. And performing FFT conversion on the sequences X 'k (n) and H' (n) subjected to zero padding by L points, multiplying corresponding points of X 'k (k) and H' (k) to obtain Y 'k (k), and performing IFFT inverse conversion on Y' k (k) to obtain a filtered signal Y k (n). Since the linear convolution of x k (N) of the N-point sequence length and h (N) of the M-point sequence length can obtain a signal of n+m-1-point length, only the first n+m-1-point length is a useful signal in the L-point length of the output signal y 1 (N), and the last n+m-1 data of y 1 (N) needs to be taken as an output result because the last n+m-1-point length is zero padding data 0 value. The functional block diagram is shown in fig. 4. The filtered output signal y 1(n),y2(n),…,yDA (N) at the n+m-1 point can be obtained by repeating the above operations.
4. The specific mode of the process of the superposition addition method for restoring the signal is as follows:
The whole system is equivalent to the linear convolution of the input long sequence signal and h (n) in the time domain, and when the long sequence signal and the long sequence signal are convolved, in order to meet the requirement of processing the signal in real time, the convolution of the long sequence and h (n) can be obtained by using an overlap-add method, and the long sequence is combined into a convolved sequence after being subjected to convolution operation in a segmented mode.
In the invention, the time domain long sequence signal is segmented, the length of each segment is N points, and the segments are divided into DA segments. Each segment of data is subjected to FFT (fast Fourier transform) calculation and linear convolution, and each segment of data y k (N) is the convolution of N point data x k (N) and h (N) after the input data is subjected to time domain windowing segmentation, and the length is N+M-1 points. In the invention, when the time domain windowing is performed, a data overlapping processing method is adopted, and each segment of input signal x k (N) is subjected to N-point windowing processing, so that y k (N) of N+M-1 points obtained after convolution is equivalent to the broadening of a transition zone of a time domain window function, namely the number of effective data points is unchanged, N-2q+1 points are needed, and the number of the transition zone points is widened from q-1 points to H points.
Assuming that the number of the front and rear transition band portions of y k (n) is H, according to the discussion of the time domain windowing portion, the time domain windowing portion x k (n) needs to satisfy the q-point data overlap to restore the original signal. Y k (n) also needs to be satisfied that there is a q point overlap in the transition band portion, and the remaining H-q points are added to the data points to prevent truncation from affecting the filtering performance. The expression of the transition zone can be obtained as
The overlapping portion can thus be obtained as three portions:
(1) The first part is q point data which are overlapped when the upper end part of the rear transition zone close to the effective data point of the section is overlapped with the upper end part of the front transition zone close to the effective data point of the next section, namely the time domain windowing is needed;
(2) The second part is that the lower end part of the rear transition zone of the section, which is close to the tail data, is overlapped with the effective data point of the next section, namely the redundant H-q points of the transition zone of the section are accumulated on the data point of the next section;
(3) The third part is the data that the effective data point of the section overlaps with the lower end part of the front transition zone of the next section, which is close to the initial data, namely the next section excessively carries redundant H-q points and is accumulated on the section;
So that the total overlapping part point number of each piece of data can be obtained as
F＝q+H-q+H-q＝2H-q＝M-1+q (16)
From this, it is possible to obtain a splice length F, i.e., N+M-1>M-1+q, of N, M and q which must satisfy a data length N+M-1 of more than 2 times, and to reduce it to a value which is required to satisfy
p≥M-1+q (17)
The specific method comprises the following steps: and (3) carrying out linear convolution on each segment of signal x k (N) and h (N) according to an FFT (fast Fourier transform) linear convolution method to obtain an output signal y k (N) of an N+M-1 point. The back F term of the signal y k (n) of each segment is correspondingly added with the front F term of the signal y k+1 (n) of the next segment, and then the results y (n) after convolution can be obtained by arranging the results in sequence.
The formula of the overlap-add method is
Where k is the number of segments, k=1, 2,3 …, DA; n is the number of sequence points of each segment after segmentation; f is the number of overlapping portions of each piece of data. According to this method, the final filtered time-domain signal y (n) is determined.
In order to verify the effectiveness of the invention, it is necessary to detect whether the filtering effect of the designed filter for all frequencies meets the requirements, so that a chirp signal LMF is used as an input signal of the system through the whole system. In the present invention, taking the sampling frequency f s =8000 Hz, the system filtering requires that the minimum stop band attenuation allowed by the reference value f11＝0Hz,f12＝500Hz,f21＝1200Hz,f22＝1600Hz,f31＝2000Hz,f32＝2500Hz,f31＝3000Hz,f32＝4000Hz,A1＝-82,A2＝-73,A3＝-76,A4＝-69. be-44 dB. The transition band Δf allowed by the system can be calculated to be 400Hz, so that the number of data points m=64 can be obtained, and the reference values of other data points are n=128, l=256, q=32, p=96, and f=95. The segment of the segmented signal of the time domain input signal is x 1(n),x2(n),…,xH (n) obtained by the window function w 0 (n) of the time domain windowing, and is subjected to FFT transformation to the frequency domain signal. The method for calculating the linear convolution by utilizing the FFT is multiplied by H (e jw) respectively for filtering, the obtained filtered time domain signal is y 1(n),y2 (n), …, y (n), and finally the final filtered time domain signal y (n) is obtained by utilizing the overlap-add method of the F point.
And drawing a frequency response diagram of each frequency change according to the output filtered time domain signal y (n), as shown in fig. 5, wherein a dotted line represents the frequency response of an ideal filter preset by a user, and a solid line represents a filter frequency response curve actually designed by the system. In the required frequency range, namely 0Hz-500Hz,1200 Hz-5000 Hz,2000Hz-2500Hz,3000Hz-4000Hz, the amplitude-frequency response of the actual filter almost coincides with that of the ideal filter, the effect diagram of 1200Hz-1600Hz is shown in figure 6, and as can be observed from figure 6, the designed actual filter can completely replace the preset ideal filter, meets the basic filtering requirement of users and has good filtering performance.
The foregoing is only a preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art, who is within the scope of the present invention, should make equivalent substitutions or modifications according to the technical scheme of the present invention and the inventive concept thereof, and should be covered by the scope of the present invention.
In the technical scheme of the invention, the following alternatives can also fulfill the aim of the invention:
(1) In the invention, when the time domain window function is redesigned, the Hanning window transition zone is used as a basis to design the time domain window function, and other window designs meeting the requirement that the front transition zone and the rear transition zone are added to obtain 1 can be used.
(2) In the invention, when the frequency domain filter function is designed, the window function is adopted to calculate the order of the filter by adopting the bandwidth of-60 dB, and the definition calculation of other bandwidths can also be adopted.
(3) In the invention, the FFT fast algorithm is adopted to calculate the linear convolution, and the calculation of the linear convolution can also be directly calculated in the time domain, namely, the calculation of the linear convolution is carried out by adopting turnover shift through a program.
(4) In the invention, when the overlap-add method is used, the transition zone is partially overlapped and q points are accumulated, other H-q points are accumulated to the data points, or the transition zone is partially overlapped and q points are accumulated, and other H-q point data are omitted.
The foregoing is only a preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art, who is within the scope of the present invention, should make equivalent substitutions or modifications according to the technical scheme of the present invention and the inventive concept thereof, and should be covered by the scope of the present invention.
Claims (1)
1. An optimization method based on frequency domain filtering is characterized by comprising the following steps:
performing time domain windowing on a time domain long sequence signal x (n) to be filtered to obtain a time domain windowed segmented signal x 1(n),x2(n),…,xDA (n);
according to the filtering requirement of a user, an ideal filter function is designed by itself, and the ideal filter function is subjected to frequency domain windowing processing through a window function to obtain a time domain filter function h (n);
Performing FFT calculation linear convolution processing on the acquired segmented signal x 1(n),x2(n),…,xDA (n) and a time domain filter function h (n) to obtain a filtered segmented signal y 1(n),y2(n),…,yDA (n);
performing overlap-add-restore signal processing on the segmented signal y 1(n),y2(n),…,yDA (n) to obtain a filtered time domain output signal y (n);
Firstly, carrying out q-point overlapping pretreatment segmentation treatment on an original long sequence signal, determining the length of a transition zone according to q points, designing a window function w 0 (n) for preventing a system from generating time domain truncation, and carrying out windowing treatment on each segment of signal through the designed time domain window function w 0 (n) to obtain a segmented signal x 1(n),x2(n),…,xDA (n) after time domain windowing;
According to the filtering requirement, firstly, frequency preprocessing is carried out on a preset frequency range, then, the frequency response function H d(ejw of an ideal filter meeting the requirement is obtained, the time domain function of the ideal filter can be calculated to be H d (n) through IFFT, the type of a window function w (n) is determined according to the minimum stop band attenuation actually required, the size of a window function order M is determined according to the excessive bandwidth actually required, and therefore, the window function w (n) is uniquely determined, and the actual filter function is obtained by windowing the ideal filter function, namely, the multiplication of H d (n) and w (n) in the time domain;
The FFT calculation linear convolution processing process comprises the following steps: performing FFT conversion on a time domain windowed signal x 1(n),x2(n),…,xDA (N) of a time domain long sequence signal segment, multiplying the time domain windowed signal x 1(n),x2(n),…,xDA (N) by FFT conversion H (e jw) of a time domain filter function H (N) respectively, performing IFFT inverse conversion on the obtained result, and taking an effective point of an output result, namely taking the first N+M-1 data of y k (N) as the output result, thereby obtaining a filtered signal;
The processing process of the cascade addition method for restoring the signals comprises the following steps: obtaining the effective data points of each segmented signal y k (N) of the signal as N-2q+1, the transition zone points as H points, and y k (N) needs to be overlapped with q points at the transition zone part, accumulating the rest H-q points on the data points, calculating the total overlapped part points of the transition zone expression and each segment of data, performing overlap addition of F points on the signal, namely correspondingly adding the back F item of the signal y k (N) of each segment with the front F item of the signal y k+1 (N) of the next segment, and arranging according to the sequence to obtain a convolved result y (N);
The q-point overlapping preprocessing process of the time domain windowing processing comprises the following steps: leading the front q point data of the input signal x (N) as initial data of the data for the first time, and then leading the data with the data point of p as the rear p point data of the section from the input signal x (N) to obtain a data section which needs to be processed for the first time, wherein the length of the data section is N=q+p points; when the next data segment is processed, the rear q point of the previous segment is taken as the front q point of the data of the present segment, the p point data of the input signal x (N) is read as the rear p point data of the present segment, the data segment which is required to be processed at the kth time is obtained, the length is still N=q+p point, and the above operation is repeated;
q-point overlapping preprocessing of the time domain windowing processing is carried out, a data segment of a DA segment entering subsequent processing is obtained, and each data segment of the data has the following characteristics: the front q point data of each section and the rear q point data of the last section of data are repeated data; the p-point data after each segment is p data points continuously read in from an input signal x (N), the length of each segment is N points, and the segmented data is taken as input to be subjected to frequency domain windowing; repeating data with the length of q and input data of p points share N=p+q points;
the window function of the time domain windowing process needs to meet two conditions of time domain windowing, a window function with a transition zone which is gradually changed is needed, slight distortion of the time domain windowing on an input signal is compensated through an overlapped splicing algorithm, and the influence of the time domain windowing is compensated through the overlapped splicing algorithm in actual operation.
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