CN110210348B - New frequency estimation algorithm based on different time and different frequency - Google Patents

New frequency estimation algorithm based on different time and different frequency Download PDF

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CN110210348B
CN110210348B CN201910427393.7A CN201910427393A CN110210348B CN 110210348 B CN110210348 B CN 110210348B CN 201910427393 A CN201910427393 A CN 201910427393A CN 110210348 B CN110210348 B CN 110210348B
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王开
冯丙乾
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Southeast University
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Abstract

The invention discloses a new frequency estimation algorithm based on different time and different frequency. The algorithm establishes a linear expression of any two spectral lines of the sinusoidal signal, and fully considers the problem of spectrum leakage caused by positive and negative frequencies, so that the algorithm has extremely high estimation accuracy. In this chapter we will derive the final frequency estimate expression and then simulate the simulation analysis and compare it with other algorithms including CLS-SDFT algorithm, TSCW algorithm, ipDFT algorithm with extra string windowing, ipDFT algorithm based on MSD windowing and IpDFT algorithm based on MDW windowing etc. The comparison is divided into the case of a noiseless cosine signal, a noisy cosine signal and a noisy harmonic signal. According to the method, the linear expression of any two spectral lines of the sinusoidal signal is established, and the problem of spectrum leakage caused by positive and negative frequencies is fully considered, so that the method has extremely high estimation accuracy.

Description

New frequency estimation algorithm based on different time and different frequency
Technical Field
The invention relates to the technical field of frequency operation, in particular to a new frequency estimation algorithm based on different time and different frequency.
Background
Frequency is the number of times a periodic change is made per unit time and is essentially an overall characteristic that characterizes the signal over a period of time. For sinusoidal signals, the frequency is the inverse of the period and its physical meaning is well defined. For a non-sinusoidal signal, the non-sinusoidal signal can be decomposed into a sum of a plurality of sinusoidal signals, and each signal has its own frequency definition, namely, the problem of 'hiding periodicity of the signal' proposed by relevant scholars in the nineteenth century is also led to the study of fourier series and fourier transform theory by the scholars.
Frequency estimation was originally derived in the 30 s of the last century, when it was mainly applied to radar and sonar technology. The frequency estimation of the sinusoidal signals is the key of radar and sonar technology, and the accurate positioning of the radar and the sonar is directly affected. In recent years, frequency estimation has achieved a great deal of research effort. Not only theoretically, but also in practice, has extremely important research value. Frequency estimation was originally applied in the military field, and later has been developed as an independent research field and is widely used in various engineering fields. Such as speech processing and recognition, communications, power systems, biomedical, detection theory, etc. Because these fields all suffer from the technical difficulty of accurately estimating the frequency of a sinusoidal signal in the presence of noise.
In the field of speech processing closely related to people, formants are one of the basic parameters characterizing speech signals, and play an important role in speech signal synthesis, speech recognition, speech coding, and the like. Formants represent the most direct source of pronunciation information and people utilize formant information in speech perception. Therefore, formant frequency estimation studies have an extremely important role in speech signal processing. In 1986, mcaulay and Quatieri proposed a sinusoidal model of the complete audio/speech signal, thereby attracting research interest to numerous scholars. Similarly, a sinusoidal model considers a speech signal to be a weighted combination of a set of sinusoidal signals. The sinusoidal coding method requires that the recoding end extracts the amplitude, frequency and phase of a set of sinusoidal signals, and the receiving end re-synthesizes the voice signals according to the above information. Extracting sinusoidal components from a speech signal involves the problem of frequency estimation of a multi-frequency real signal.
Disclosure of Invention
In order to solve the above problems, the present invention provides a new frequency estimation algorithm based on different time and different frequency, by deriving the final frequency estimation expression, then simulating the simulation analysis, and comparing with other algorithms, including CLS-SDFT algorithm, TSCW algorithm, ipDFT algorithm with redundant string windowing, ipDFT algorithm based on MSD windowing, ipDFT algorithm based on MDW windowing, etc. The comparison condition is divided into a noiseless cosine signal, a noisy cosine signal and a noisy harmonic signal, and the invention provides a new frequency estimation algorithm based on different time and different frequency, which comprises the following steps:
step 1: collecting N-point discrete time signals s (N);
step 2: acquiring N-point DFT (discrete Fourier transform) sequence of signal S (N), denoted S k (m)，
Step 3: considerBecause of S' k (m) is plural, so we can obtain S' k (m)＝S′ k,R (m)+jS′ k,I The form of (m);
step 4: obtaining k in the frequency spectrum 1 ，k 2 ，m 1 ，m 2 At spectral values, i.eS k2 (m 1 )
And S is k2 (m 2 ) Wherein k is 1 ，k 2 The value of k can be obtained by coarse estimation 2 ＝k 0 Is l 0 Integer parts, i.e. by finding the position of the peak of the signal spectrumObtaining k 2 The value of k 1 ＝k 0 -1；
Step 5: based on an interpolated fourier transform, usingS k2 (m 1 ) And S is k2 (m 2 ) The spectral values are estimates of the signal frequency and damping factor.
Further improvements of the present invention, step 51: by aligningS k2 (m 1 ) And S is k2 (m 2 ) The expression is subjected to a series of simplified operations to obtain a unitary system of cubic equations
acos 3 (ω 0 )+bcos 2 (ω 0 )+ccos(ω 0 )+d＝0 (1)
The value of each coefficient in the formula (1) can be obtained by the following formula
Step 52: solving a unitary tertiary equation set shown in the formula (1) to obtain
Wherein the method comprises the steps of
Step 53: the signal frequency omega can be finally obtained by the formula (3) 0 Is the estimated value of (1)
Working principle: in the present invention, different time and frequency methods are utilized to estimate the frequency of the signal. Due to the existence of Fast Fourier Transform (FFT), the algorithm has small calculated amount and high efficiency, and solves the problem of the fence effect existing in Discrete Fourier Transform (DFT) through interpolation. By taking the positive frequency and the negative frequency of the signal into consideration for calculation, the problem of DFT spectrum leakage is solved, and the accuracy of frequency estimation is improved, especially when the acquired signal time is short. The stability, the anti-noise performance and the computational complexity of the algorithm are superior to those of the existing similar frequency estimation algorithm.
The beneficial effects are that: from simulation, we can see that in the signal modelThe cosine signal comprises the conditions of noise or no noise, fixed phase or random phase, and the signal model is the harmonic signal, so that better effect performance is obtained, the overall performance is superior to other algorithms, the precision is higher, and the stability is better. In the case of cosine signals, however, when l 0 ∈[1,2]When the method is used, a better performance effect can be obtained; in the case of harmonic signals, when 0 ∈[0.5,2]The proposed algorithm performs best when.
Drawings
Fig. 1 is a graph of mean square error of signal frequency and damping factor estimate as signal frequency varies under a noiseless cosine random phase condition. The mean square error MSE of the frequency estimation value is plotted as a function of l 0 Is a variation of (2).
Fig. 2 is a mean square error plot of signal frequency and damping factor estimate as the signal damping factor varies with the fixed phase of the noisy cosine signal. Wherein graph (a) is the mean square error MSE of the frequency estimate as a function of l 0 Graph (b) shows the variation of the mean square error MSE of the damping factor estimation value with the SNR.
Fig. 3 is a graph of the mean square error of the signal frequency and damping factor estimate as the signal frequency varies with random phase of the noisy cosine signal. Wherein graph (a) is the mean square error MSE of the frequency estimate as a function of l 0 Graph (b) shows the variation of the mean square error MSE of the damping factor estimation value with the SNR.
Fig. 4 is a graph of the mean square error of the signal frequency and damping factor estimate as the signal damping factor varies with random phase of the noisy harmonic signal. Wherein graph (a) is the mean square error MSE of the frequency estimate as a function of l 0 Graph (b) shows the variation of the mean square error MSE of the damping factor estimation value with the SNR.
Detailed Description
The invention is described in further detail below with reference to the attached drawings and detailed description:
the invention provides a new frequency estimation algorithm based on different time and different frequency, which is characterized in that the final frequency estimation expression is deduced, then simulation analysis is performed, and the new frequency estimation algorithm is compared with other algorithms, including a CLS-SDFT algorithm, a TSCW algorithm, an IPDFT algorithm with redundant string windowing, an IPDFT algorithm based on MSD windowing, an IPDFT algorithm based on MDW windowing and the like. The comparison is divided into the case of a noiseless cosine signal, a noisy cosine signal and a noisy harmonic signal.
The discrete time expression of the sine signal is
From the above equation, where A is the signal amplitude, ω 0 ＝2πl 0 /N＝2π(k 0 +δ 0 ) N is the signal frequency, l 0 Representing the number of cycles, k, of the acquired sinusoidal signal 0 Is an integer part, delta 0 (||δ 0 I.ltoreq.0.5) is its fractional part, N is the number of sampling points,representing the initial phase of the signal. The signal N point DFT expression is
Can be consideredBecause of S' k (m) is a complex number, we can combine S' k (m) rewriting as S' k (m)＝S′ k,R (m)+jS′ k,I Form (m), wherein:
wherein the method comprises the steps of
Then we can get:
the derivation of the above equation can result in:
when we obtain any two frequency components at any instant:
it is then possible to obtain:
of the two formulas, only cos (ω) 0 ) This one unknowns can thus be solved by composing a system of unitary cubic equations.
Wherein the values of the coefficients are respectively
The signal frequency omega can be finally obtained through a unitary tertiary equation formula 0 Is the estimated value of (1)
The estimation performance of the method is verified through a simulation experiment, and the estimation condition of the algorithm is reflected by using a performance index of Mean Square Error (MSE). Setting signal amplitude a=1 in the simulation, signal initial phaseRandomly within [0,2 pi), sampling points and DFT points n=128.
Firstly, the estimation condition of the algorithm under the condition of the noiseless cosine random phase is analyzed. Fig. 1 is a simulation result when the signal model is a cosine signal and the phase is random, and in an ideal case without noise. FIG. 1 shows MSE is l 0 Is a function of (2). Where n=128. As can be seen from the figure, the proposed algorithm has frequency estimation singularities, and a glitch phenomenon occurs, resulting in poor frequency estimation, such as two glitches in the figure. The proposed algorithm performs worse than CLS-SDFT but better than others.
As shown in fig. 2, which is an estimation case in the case of a fixed phase of the noisy cosine signal, (a) represents that MSE is l 0 Is a function of (2). We set n=128, snr=40 dB. The simulation result has frequency estimation singular points, a burr phenomenon occurs, the frequency estimation effect is poor, for example, two burr points exist in the figure, and the fluctuation is large. By comparison with other algorithms, it can be seen that the proposed algorithm performs better than the other algorithms, but is inferior to the CLS-SDFT algorithm at individual points. Because of the burr, the individual points are less effective, as can be seen from the figure, at l 0 ∈[0.5,1]When the overall performance of the proposed algorithm is better than other algorithms, but inferior to the CLS-SDFT algorithm. When l 0 ∈[1,2]The proposed algorithm performs best, about-90 dB. But when l 0 ∈[2,2.5]The performance of the proposed algorithm is poor. (b) Representing MSE as a function of SNRNumber, n=128, l at this time 0 =1.22. Even though the SNR is small, the simulation results are still very ideal. We can also observe that the simulation results are very close to the theoretical analysis. As can be seen from the figure, the proposed algorithm works when the SNR is at [0,40 ]]In dB, the overall performance is better than other algorithms.
The random phase variation of the noisy cosine signal is shown in fig. 3. (a) MSE and l are shown for n=128, snr=40 dB 0 Is a relationship of (3). The simulation result has frequency estimation singular points, a burr phenomenon occurs, the frequency estimation effect is poor, for example, two burr points exist in the figure, and the fluctuation is large. By comparison with other algorithms, it can be seen that the proposed algorithm performs better than the other algorithms, but is inferior to the CLS-SDFT algorithm at individual points. Because of the burr, the individual points are less effective, as can be seen from the figure, at l 0 ∈[0.5,1]When the overall performance of the proposed algorithm is better than other algorithms, but inferior to the CLS-SDFT algorithm. When l 0 ∈[1,2]The proposed algorithm performs best, about-90 dB. But when l 0 ∈[2,2.5]The performance of the proposed algorithm is poor. (b) Indicating that MSE is a function of SNR, where n=128, l 0 =1.22. Even though the SNR is small, the simulation results are still very ideal. We can also observe that the simulation results are very close to the theoretical analysis. As can be seen from the figure, the proposed algorithm works when the SNR is at [0,40 ]]In dB, the overall performance is better than other algorithms.
The variation of the random phase performance of the noisy harmonic signal is shown in fig. 4, (a) shows the performance of the algorithm corresponding to snr=40 dB. Notably, when l 0 ∈[0.5,2]The performance of the proposed algorithm is superior to the other algorithms when l 0 ∈[2,2.5]When the performance is worse than other algorithms. So when l 0 When the method is very small, the method has better performance and higher precision. (b) Indicating that MSE is a function of SNR, where n=128, l 0 =1.22. When the SNR is [0,40 ]]dB, step size of 2dB, as shown in (b), the proposed algorithm performance and WIPDFT method are the most accurate, most effective, and more effective than other algorithmsIt is desirable that the performance of the proposed algorithm is very stable and the fluctuations are very small, and that the proposed algorithm is applied when the SNR is [0,40 ]]When dB, the average MSE of the whole algorithm is about-55 dB, other algorithms are larger than-55 dB, and the accuracy of the algorithm with the effect performance not provided is high.
The above description is only of the preferred embodiment of the present invention, and is not intended to limit the present invention in any other way, but is intended to cover any modifications or equivalent variations according to the technical spirit of the present invention, which fall within the scope of the present invention as defined by the appended claims.

Claims (1)

1. A new frequency estimation algorithm based on different time and different frequency, comprising the steps of:
step 1: collecting N-point discrete time signals s (N);
step 2: acquiring an N-point DFT discrete Fourier transform sequence of a signal S (N), and recording the sequence as S k (m)，
Step 3: considerBecause of S k ' (m) is plural, so S is obtained k ′(m)＝S k ′, R (m)+jS k ′, I The form of (m);
step 4: obtaining k in the frequency spectrum 1 ，k 2 ，m 1 ，m 2 At spectral values, i.eS k2 (m 1 )
And S is k2 (m 2 ) Wherein k is 1 ，k 2 The value of (k) is obtained by coarse estimation 2 ＝k 0 Is l 0 Integer parts, i.e. by finding the position of the peak of the signal spectrumObtaining k 2 The value of k 1 ＝k 0 -1；
Step 5: based on an interpolated fourier transform, usingS k2 (m 1 ) And S is k2 (m 2 ) An estimated value of the frequency of the signal and the damping factor;
step 51: by aligningS k2 (m 1 ) And S is k2 (m 2 ) The expression performs a series of simplified operations to obtain a unitary system of cubic equations
acos 3 (ω 0 )+bcos 2 (ω 0 )+ccos(ω 0 )+d＝0 (1)
The value of each coefficient in the formula (1) is obtained by the following formula
Step 52: solving a unitary tertiary equation set shown in the formula (1) to obtain
Wherein the method comprises the steps of
Step 53: finally obtaining the signal frequency omega through the formula (3) 0 Is the estimated value of (1)
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Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US8150643B1 (en) * 2004-12-21 2012-04-03 Battelle Energy Alliance, Llc Method of detecting system function by measuring frequency response
CN107085140A (en) * 2017-04-25 2017-08-22 东南大学 Nonequilibrium system frequency estimating methods based on improved SmartDFT algorithms
CN108020721A (en) * 2017-12-05 2018-05-11 南京福致通电气自动化有限公司 A kind of frequency estimating methods of the non-equilibrium electric system based on IpDFT

Patent Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US8150643B1 (en) * 2004-12-21 2012-04-03 Battelle Energy Alliance, Llc Method of detecting system function by measuring frequency response
CN107085140A (en) * 2017-04-25 2017-08-22 东南大学 Nonequilibrium system frequency estimating methods based on improved SmartDFT algorithms
CN108020721A (en) * 2017-12-05 2018-05-11 南京福致通电气自动化有限公司 A kind of frequency estimating methods of the non-equilibrium electric system based on IpDFT

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