CN104931777B - A kind of signal frequency measuring method based on two DFT plural number spectral lines - Google Patents
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Abstract
The present invention relates to a kind of signal frequency measuring method based on two DFT plural number spectral lines, belong to signal parameter field of measuring technique.It is a feature of the present invention that its processing step includes：Sampled signal carries out DFT transform after windowing process, in the range of respective frequencies highest and time high spectral line are searched for by comparing spectral line amplitude or real part imaginary part absolute value sum, intermediate parameters are asked in computing based on two spectral line real part imaginary part absolute value sums, frequency deviation parameter is obtained by solving equation or inverse function formula or approximating polynomial again, finally measures signal frequency.The present invention avoids multiplication and square root when searching for highest and time high two spectral lines and calculating intermediate parameters λ, simplifies the realization of signal frequency Measurement Algorithm, shortens calculating speed.
Description
Technical Field
The invention relates to a signal frequency measurement method based on two DFT complex spectral lines, and belongs to the technical field of signal parameter measurement.
Background
Currently, a method of analyzing a frequency signal based on a discrete fourier transform DFT or a fast algorithm FFT thereof has been widely used. However, DFT has a barrier effect, i.e., the actual signal frequency does not necessarily fall on a discrete spectral line, and thus an interpolation algorithm needs to be used to estimate the frequency of the actual signal. An article, "improved algorithm for performing harmonic analysis of power system by applying FFT", published in the book 23 of the Chinese electro-mechanical engineering journal of 2003, provides a method for measuring signal frequency by interpolation by selecting two spectral lines, i.e., the highest and the second highest, of amplitude after windowing Fourier transform of an input discrete signal. If the discrete frequency serial numbers of the two spectral lines respectively correspond to k 1 And k 2 ＝k 1 +1, the position k corresponding to the actual signal frequency 0 Satisfy k 1 ≤k 0 ≤k 2 . Introducing an auxiliary parameter alpha = k 0 -k 1 0.5, ignoring other signal interference, the value range of α is [ -0.5,0.5]. Thus, the amplitude Y (k) of the two discrete spectral lines 1 ) I and Y (k) 2 ) I satisfies:
when N is large, the above formula can be simplified to be λ = g (α), and its inverse function is denoted as α = g -1 (lambda). The article further proposes to calculate using a polynomial approximation method
The existing methods are deficient in performing top and bottom spectral line searches based on spectral line amplitude. Obtaining the spectral line amplitude requires calculating the sum of the squares of the real and imaginary parts and then squaring, which is computationally expensive.
Disclosure of Invention
The invention aims to provide a signal frequency measuring method based on two DFT complex spectral lines, which is used for solving the problem of large calculation amount in the prior art.
In order to achieve the above purpose, the scheme of the invention comprises:
a signal frequency measurement method based on two DFT complex spectral lines is characterized by comprising the following steps:
step (1): carrying out windowing processing on a sampling signal x (n) with a sampling rate of FS and a sampling point of continuous interception to obtain a windowed signal y (n), wherein the windowing processing formula is as follows:
y(n)＝x(n)·w(n)，
wherein w (N) is a window function sequence of N points, N =0 (N-1);
step (2): performing Discrete Fourier Transform (DFT) on the windowed signal Y (N) to obtain a discrete spectrum Y (k), wherein the discrete frequency index k =0 (N-1);
and (3): searching a spectral line with the maximum (| Re (Y (k)) + | Im (Y (k)) |) in a discrete frequency number range [ ks, ke ] corresponding to a set frequency range as the highest spectral line, comparing (| Re (Y (k)) + | Im (Y (k)) |) values of spectral lines on two sides of the spectral line, selecting the largest spectral line as the next highest spectral line, and recording discrete frequency numbers k1 and k2 of the two spectral lines, wherein 0 & ltks ks & lt ke & lt (N/2), k2= k1+1, re (Z) is the real part of a complex number Z, and Im (Z) is the imaginary part of the complex number Z;
and (4): calculating an intermediate parameter lambda according to the two spectral lines Y (k 1) and Y (k 2) corresponding to k1 and k 2:
and (5): solving the frequency offset parameter α in the following equation:
wherein W (ω) is the result of a discrete-time fourier transform DTFT of the window function W (N) and the normalized angular frequency ω =2 π f/FS =2 π k/N;
and (6): and calculating the frequency f alpha of the measured signal according to the frequency offset parameter alpha, wherein the calculation formula is as follows:
f α ＝(k 1 +0.5+α)·Fs/N。
the step (3) of searching the highest and second highest spectral lines comprises the following steps: in a discrete frequency number range [ ks, ke ] corresponding to a set frequency range, searching a spectral line with the maximum | Y (k) | as the highest spectral line, comparing the values of the spectral lines | Y (k) | on two sides of the maximum spectral line, selecting the maximum spectral line as the next highest spectral line, and recording discrete frequency numbers k1 and k2 of the two spectral lines, wherein 0 & lt ks & lt ke & lt (N/2), and k2= k1+1.
The step (5) calculates the frequency offset parameter α by using an inverse function formula of the relation function λ = g (α) between the measured value λ of the intermediate parameter and the frequency offset parameter α, where the inverse function is:
α＝g -1 (λ)。
the step (5) adopts an inverse function g of a relation function lambda = g (alpha) of an intermediate parameter measured value lambda and a frequency deviation parameter alpha -1 (λ) calculating a frequency offset parameter α by an approximating polynomial equation:
where M is the highest degree of approximation to the polynomial and am (M = 0:M) is the coefficient of the mth term λ M of the polynomial.
The design principle of the signal frequency measuring method is as follows: assuming a frequency f 0 And obtaining a single frequency signal x (t) with the amplitude of A and the initial phase of theta after analog-to-digital conversion with the sampling rate of Fs, wherein the single frequency signal x (t) is in the following form:
if the time domain form of the added window function is W (n), and the continuous frequency spectrum obtained by the Discrete Time Fourier Transform (DTFT) is W (omega), the negative frequency point-f is ignored 0 Side lobe influence of treatment frequency peak, at positive frequency point f 0 The nearby continuous spectrum function can be expressed as:
the above formula is used for discrete sampling, and the expression of discrete fourier transform DFT can be obtained as follows:
wherein the discrete frequency interval is Δ F = F S /N。
Cosine window functions are the most common type of window functions used by DFT. The unified time domain form corresponding to the cosine window function is:
the result of the discrete time fourier transform DTFT of the cosine window w (n) is:
wherein:
within the main lobe of the DTFT spectrum curve of the signal, and when N is large, the approximation is:
when in useIn time, the above formula takes equal sign. According to the coefficients of the common window function, in the main lobe-H<k&Within lt, two adjacent spectral lines W (omega) andthe phase difference of (a) is approximately pi; and corresponds to H<k&W (omega) and within sidelobe of lt/2Close to the same phase. Thus, when the spectral line Y (k) 1 ) And Y (k) 2 ) When the phase difference of (a) is 0 or pi, there are:
therefore, the searching of the highest and the second highest spectral lines and the calculation of the intermediate parameter lambda can be realized by directly using the sum of absolute values of a real part and an imaginary part of the spectral line without calculating the spectral line amplitude. Further, the actual signal frequency is obtained. The method is designed based on the principle, omits multiplication and evolution operation, simplifies the realization of a signal frequency measurement algorithm and shortens the calculation speed.
Drawings
Fig. 1 is a calculation process diagram of a signal frequency measurement method based on two DFT complex spectral lines according to an embodiment of the invention.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings.
Two embodiments are provided below, which are applied to the measurement of frequency signals around 50 Hz.
Example 1
The method comprises the following specific steps:
step (1): continuously intercepting a signal x (N) with a sampling rate of Fs =1500Hz and N =512 points, and performing windowing processing to obtain a windowed signal y (N), wherein the windowing processing formula is as follows:
y(n)＝x(n)·w(n)，
where w (N) selects a Hanning window function sequence of N =512 points, i.e.:
step (2): performing Discrete Fourier Transform (DFT) on the windowed signal Y (N) to obtain a discrete spectrum Y (k), wherein the discrete frequency index k =0 (N-1);
and (3): in the discrete frequency number range 15,23]I.e. corresponding to the frequency [43.945,67.383]In the range of Hz, the spectral line with the maximum Y (k) is searched as the highest spectral line, the values of the spectral lines Y (k) on the two sides of the maximum spectral line are compared, the maximum spectral line is selected as the next highest spectral line, and the discrete frequency serial number k of the two spectral lines is recorded 1 And k 2 ＝k 1 +1；
And (4): according to k 1 And k 2 Corresponding two spectral lines Y (k) 1 ) And Y (k) 2 ) Calculating an intermediate parameter lambda:
and (5): solving the frequency offset parameter α in the following equation:
thus, the function formula for calculating the frequency offset parameter α is:
α＝1.5λ，
and (6): calculating the frequency f of the measured signal according to the frequency deviation parameter alpha α The calculation formula is as follows:
f α ＝(k 1 +0.5+α)·Fs/N。
example 2
Step (1): continuously intercepting a signal x (N) with a sampling rate of Fs =1500Hz and N =512 points, and performing windowing processing to obtain a windowed signal y (N), wherein the windowing processing formula is as follows:
y(n)＝x(n)·w(n)，
where w (N) selects the BlackMan (BlackMan) window function sequence of N =512 points, i.e.:
step (2): performing Discrete Fourier Transform (DFT) on the windowed signal Y (N) to obtain a discrete spectrum Y (k), wherein the discrete frequency index k =0 (N-1);
and (3): in the discrete frequency number range 15,23]I.e. corresponding to the frequency [43.945,67.383]In the range of Hz, searching the spectral line with the maximum (| Re (Y (k)) | + | Im (Y (k)) |) as the highest spectral line, comparing the (| Re (Y (k)) | + | Im (Y (k)) |) values of the spectral lines at two sides of the maximum spectral line, selecting the maximum spectral line as the next highest spectral line, and recording the discrete frequency serial numbers k of the two spectral lines 1 And k 2 ＝k 1 +1；
And (4): according to k 1 And k 2 Corresponding two spectral lines Y (k) 1 ) And Y (k) 2 ) Calculating an intermediate parameter lambda:
and (5): adopting an inverse function g of a relation function lambda = g (alpha) of an intermediate parameter measured value lambda and a frequency deviation parameter alpha -1 (λ) calculating the frequency offset parameter α using an approximation polynomial equation of up to 7 degrees, the form being:
α≈1.87500108·λ+0.24171091·λ 3 +0.10443757·λ 5 +0.06718760·λ 7 ，
and (6): calculating the frequency f of the measured signal according to the frequency deviation parameter alpha α The calculation formula is as follows:
f α ＝(k 1 +0.5+α)·Fs/N。
according to the first and second embodiments, the same set of simulation test data is input to verify the calculation results of the two embodiments, respectively. The input signal x (n) is the fundamental frequency f 1 A signal at 50.1Hz containing 2 to 9 harmonics in the form of:
wherein, the amplitude of fundamental wave and each harmonic is respectively: 1,0.02,0.1,0.01,0.05,0.0,0.02,0.0,0.01; the initial phases are-23.1 °,115.6 °,59.3 °,52.4 °,123.8 °,161.8 °, -31.8 °,119.9 °, -63.7 °, respectively.
In the first embodiment, the calculation results of the 9 spectral lines in the discrete frequency number ranges 15 to 23 are sequentially as follows: 0.152783905 jj 1.76229599,4.70210906+ j54.2254920, -10.9858392-j126.688437,6.36734959+ j73.4295187, -0.221526634-j2.55345712, -0.0512202109-j0.589197696, -0.0199885230-j0.228698046, -0.00996303987-j0.112669252.No matter the amplitude is compared or the sum of absolute values of real part and imaginary part is adopted, the corresponding k of the highest and second highest spectral lines can be obtained 1 ＝17，k 2 =18. Then, the calculation result of the intermediate parameter is λ = -0.26613833, the frequency offset parameter α =1.5 λ = -0.39920750, and the final measured signal frequency is 50.099978Hz with a measurement error of-0.000044%.
In the second embodiment, the results of the calculation of the 9 spectral lines in the discrete frequency number ranges 15 to 23 are sequentially: -0.63151373-j7.27542732,4.87561219+ j56.23242377, -9.38422261-j108.21320679,6.10560557+ j70.41558734, -1.11464837-j12.84931264,0.00734999+ j0.0889938, -0.00022265+ j0.00101658,0.00048733+ j0.00853355. No matter the amplitude is compared or the sum of absolute values of real part and imaginary part is adopted, the corresponding k of the highest and second highest spectral lines can be obtained 1 ＝17，k 2 =18. Then, the intermediate parameter calculation result is λ = -0.21160379. After the approximation polynomial is substituted, a frequency offset parameter alpha = -0.39909308 is obtained, the finally measured signal frequency is 50.100313Hz, and the measurement error is 0.00063%.
The specific embodiments are given above, but the present invention is not limited to the described embodiments. The basic idea of the present invention lies in the above basic scheme, and it is obvious to those skilled in the art that no creative effort is needed to design various modified models, formulas and parameters according to the teaching of the present invention. Variations, modifications, substitutions and alterations may be made to the embodiments without departing from the principles and spirit of the invention, and still fall within the scope of the invention.
Claims (4)
1. A signal frequency measurement method based on two DFT complex spectral lines is characterized by comprising the following steps:
step (1): carrying out windowing processing on a sampling signal x (n) with a sampling rate of FS and a sampling point of continuous interception to obtain a windowed signal y (n), wherein the windowing processing formula is as follows:
y(n)＝x(n)·w(n)，
wherein w (N) is a window function sequence of N points, N =0 (N-1);
step (2): performing Discrete Fourier Transform (DFT) on the windowed signal Y (N) to obtain a discrete spectrum Y (k), wherein the discrete frequency index k =0 (N-1);
and (3): searching a spectral line with the maximum (| Re (Y (k)) + | Im (Y (k)) |) in a discrete frequency number range [ ks, ke ] corresponding to a set frequency range as the highest spectral line, comparing (| Re (Y (k)) + | Im (Y (k)) |) values of spectral lines on two sides of the spectral line, selecting the largest spectral line as the next highest spectral line, and recording discrete frequency numbers k1 and k2 of the two spectral lines, wherein 0 & ltks ks & lt ke & lt (N/2), k2= k1+1, re (Z) is the real part of a complex number Z, and Im (Z) is the imaginary part of the complex number Z;
and (4): calculating an intermediate parameter lambda according to the two spectral lines Y (k 1) and Y (k 2) corresponding to k1 and k 2:
and (5): solving the frequency offset parameter α in the following equation:
wherein W (ω) is the result of a discrete-time fourier transform DTFT of the window function W (N) and the normalized angular frequency ω =2 π f/FS =2 π k/N;
and (6): and calculating the frequency f alpha of the measured signal according to the frequency offset parameter alpha, wherein the calculation formula is as follows:
f α ＝(k 1 +0.5+α)·Fs/N。
2. a method for measuring signal frequency based on two DFT complex spectral lines as claimed in claim 1, wherein: the step (3) of searching the highest and second highest spectral lines comprises the following steps: in a discrete frequency number range [ ks, ke ] corresponding to a set frequency range, searching a spectral line with the maximum | Y (k) | as the highest spectral line, comparing the values of the spectral lines | Y (k) | on two sides of the maximum spectral line, selecting the maximum spectral line as the next highest spectral line, and recording discrete frequency numbers k1 and k2 of the two spectral lines, wherein 0 & lt ks & lt ke & lt (N/2), and k2= k1+1.
3. A method for measuring a signal frequency based on two DFT complex spectral lines as claimed in claim 1, wherein: the step (5) calculates the frequency offset parameter α by using an inverse function formula of the relation function λ = g (α) between the measured value λ of the intermediate parameter and the frequency offset parameter α, where the inverse function is:
α＝g -1 (λ)。
4. a method for measuring signal frequency based on two DFT complex spectral lines as claimed in claim 3, wherein: the step (5) adopts an inverse function g of a relation function lambda = g (alpha) of an intermediate parameter measured value lambda and a frequency deviation parameter alpha -1 (λ) calculating a frequency offset parameter α by an approximating polynomial equation:
where M is the highest degree of approximation to the polynomial and am (M = 0:M) is the coefficient of the mth term λ M of the polynomial.
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| CN106911606A (en) * | 2017-03-06 | 2017-06-30 | 金陵科技学院 | A kind of frequency capture method of π/4 DQPSK |
| CN113341224B (en) * | 2021-06-08 | 2022-05-24 | 国网湖南省电力有限公司 | Method and device for measuring low-frequency oscillation signal of power system |
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