CN112000918A - FFT-based frequency estimation interpolation compensation method - Google Patents

FFT-based frequency estimation interpolation compensation method Download PDF

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CN112000918A
CN112000918A CN202010731155.8A CN202010731155A CN112000918A CN 112000918 A CN112000918 A CN 112000918A CN 202010731155 A CN202010731155 A CN 202010731155A CN 112000918 A CN112000918 A CN 112000918A
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CN112000918B (en
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史学森
王永庆
申宇瑶
刘东磊
陈岩
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Beijing Institute of Technology BIT
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    • G06F17/10—Complex mathematical operations
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    • G06F17/141—Discrete Fourier transforms
    • G06F17/142—Fast Fourier transforms, e.g. using a Cooley-Tukey type algorithm
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    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
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    • G05B13/0265—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric the criterion being a learning criterion
    • G05B13/0275—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric the criterion being a learning criterion using fuzzy logic only
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Abstract

The invention provides a frequency estimation interpolation compensation method based on FFT coefficients. The method uses phase information to estimate interpolation direction, uses side lobe coefficient to perform weighted correction on correction value calculated by main lobe coefficient, and uses weight factor generated by fuzzy controller. The method effectively improves the frequency estimation precision under the condition of lower carrier-to-noise ratio. Meanwhile, the estimation performance of the method is less influenced by the initial frequency difference, and the estimation performance is more stable.

Description

FFT-based frequency estimation interpolation compensation method
Technical Field
The invention belongs to the technical field of aerospace measurement and control communication, and particularly relates to a frequency estimation interpolation compensation method based on FFT (fast Fourier transform).
Background
In the aerospace measurement and control signal synchronization process, a frequency estimation technology is generally used for extracting carrier frequency. Because the space flight measurement and control signal has a low signal-to-noise ratio characteristic, frequency estimation is often completed by using an FFT-based frequency estimation method. In the frequency estimation algorithm based on FFT, the frequency point corresponding to the peak spectral line of the FFT is used as the frequency estimation value, if the actual frequency does not fall on the quantization frequency point, the FFT frequency resolution will have a large influence on the frequency estimation precision, and the maximum estimation error which may be generated is half of the FFT frequency resolution. In contrast, the method of increasing the number of FFT points can improve the resolution to some extent, thereby improving the estimation accuracy, but this will inevitably result in an increase in the computational complexity. In practical application, a mode of combining rough estimation and fine estimation is adopted for frequency estimation, namely, a frequency point corresponding to an FFT peak value is used as a rough estimation value, and then an initial frequency difference generated by the rough estimation value is corrected by using a frequency interpolation compensation method to obtain a fine estimation value.
The basic idea of the frequency interpolation compensation method is to estimate a frequency difference correction coefficient by using the amplitude relationship between a peak spectral line and two spectral lines to correct the initial frequency difference, wherein the frequency difference correction coefficient comprises an interpolation direction and a correction value. The conventional interpolation compensation algorithm is influenced by initial frequency difference and noise, and the effect of correcting the frequency difference is not ideal. When the actual frequency is close to the quantized frequency point and the initial frequency difference is small, the interpolation direction is estimated by using the spectral line amplitude, and interpolation direction errors are prone to occurring. In comparison, the interpolation direction is estimated by using the phase information, so that the accuracy of the estimation of the interpolation direction can be effectively improved. However, when the carrier-to-noise ratio is low, the conventional algorithm for estimating the interpolation direction using the phase information still has a problem that the accuracy of estimation of the interpolation direction is low. In addition, the correction value calculated by using the spectral line amplitude is seriously influenced by noise, so that the calculated correction value is greatly deviated from the initial frequency difference, and the final estimation precision is influenced.
Disclosure of Invention
In view of the above, the present invention provides a frequency estimation interpolation compensation method based on FFT, which can improve the frequency estimation accuracy.
A frequency estimation interpolation compensation method comprises the following steps:
step 1, interpolation direction estimation, specifically:
(1) when alpha isnαp<At 0, the interpolation direction is:
Figure BDA0002603185110000021
(2) when alpha isnαp>At 0, the interpolation direction is:
Figure BDA0002603185110000022
wherein the content of the first and second substances,
Figure BDA0002603185110000023
n＝k0-1 and p ═ k0+1, X (-) denotes the FFT-transformed signal of the input signal, k0Represents the maximum peak value X (k)0) The sequence number of the corresponding frequency spectrum,
Figure BDA0002603185110000028
is X (k)0) Conjugation of (1);
step 2, calculating a final frequency difference correction value:
Figure BDA0002603185110000024
wherein, V0Represents X (k)0) Modulus value of (V)nAnd VpAre respectively X (k)0) Modulus, Δ V, of the two spectral linesn＝Vn/V0，ΔVp＝Vp/V0F is the resolution of the Fourier transform; β represents a weighting factor;
and step 3, the final frequency estimation value is as follows:
Figure BDA0002603185110000025
wherein the content of the first and second substances,
Figure BDA0002603185110000029
the frequency point corresponding to the peak value is shown,FFT＝d。
preferably, the weight factor β is calculated as follows:
step 21, when d is equal to-1, setting input variable r of fuzzy control1And r2Comprises the following steps:
Figure BDA0002603185110000026
when d is +1, the input variables of the fuzzy controller are set as follows:
Figure BDA0002603185110000027
variable r1Is U e [0,1 ∈]，r1Has a fuzzy domain of Nu∈[0,1](ii) a For variable r1Fuzzification is carried out, and the four fuzzy subsets ZE, PS, PM and PL are mapped respectively;
for variable r2The theory domain of physics is U is equal to [0,3 ]]，r2Has a fuzzy domain of Nu∈[0,3]. For variable r2Fuzzification is carried out, and the four fuzzy subsets ZE, PS, PM and PL are mapped respectively;
the output variable of the fuzzy controller is beta, and the fuzzy domain of the fuzzy controller is set to be Nu∈[0.5,1.3]Mapping to four fuzzy subsets ZE, PS, PM and PL, respectively;
scaling factor k for transformation of output variable from ambiguity domain to physics domain u1, so the output variable has a physical range of U ═ 0.5,1.3]。
Step 22, fuzzy control rules of fuzzy inference, as shown in table 1:
table 1: fuzzy control rule
Figure BDA0002603185110000031
Step 23, according to the actually obtained input variable r1And r2And based on an area center method, performing deblurring on the output variable to obtain the value of the output variable beta.
Preferably, changeQuantity r1After fuzzification, the membership functions respectively mapped to the four fuzzy subsets ZE, PS, PM and PL are as follows:
membership functions for ZE are:
Figure BDA0002603185110000032
the membership function for PS is:
Figure BDA0002603185110000033
the membership function of PM is:
Figure BDA0002603185110000034
the membership function of PL is:
Figure BDA0002603185110000035
variable r2After fuzzification, the membership functions respectively mapped to the four fuzzy subsets ZE, PS, PM and PL are as follows:
Figure BDA0002603185110000041
the membership function for PS is:
Figure BDA0002603185110000042
the membership function of PM is:
Figure BDA0002603185110000043
the membership function of PL is:
Figure BDA0002603185110000044
the output variables are membership functions of the four fuzzy subsets ZE, PS, PM and PL to which they are respectively mapped:
membership functions for ZE are:
Figure BDA0002603185110000045
the membership function for PS is:
Figure BDA0002603185110000046
the membership function of PM is:
Figure BDA0002603185110000047
the membership function of PL is:
Figure BDA0002603185110000051
the invention has the following beneficial effects:
the invention provides a frequency estimation interpolation compensation method based on FFT coefficients. The method uses phase information to estimate interpolation direction, uses side lobe coefficient to perform weighted correction on correction value calculated by main lobe coefficient, and uses weight factor generated by fuzzy controller. The method effectively improves the frequency estimation precision under the condition of lower carrier-to-noise ratio. Meanwhile, the estimation performance of the method is less influenced by the initial frequency difference, and the estimation performance is more stable.
Drawings
FIG. 1 is a flow chart of a frequency estimation algorithm based on interpolation compensation according to the present invention;
FIG. 2 shows X (r) and X (k) without noise0) A phase difference therebetween;
FIG. 3 shows X (r) and X (k) in a noisy environment0) Phase difference therebetween (Δ f ═ 0.25 f);
FIG. 4 shows X (r) and X (k) in a noisy environment0) Phase difference therebetween (Δ f ═ 0.125 f);
FIG. 5 is a schematic diagram of a correction value estimation algorithm based on side lobe coefficient weighting correction;
FIG. 6(a) and FIG. 6(b) show the input variable r1And r2Graph of membership function of;
FIG. 7 is a graph of output variable membership function.
Detailed Description
The invention is described in detail below by way of example with reference to the accompanying drawings.
The invention provides a frequency estimation interpolation compensation method based on phase coefficient comparison and side lobe coefficient weighting correction. On the basis of the frequency estimation method based on FFT, the method utilizes the FFT coefficient to estimate the frequency difference correction coefficient to correct the initial frequency difference so as to achieve the purpose of improving the frequency estimation precision, and a frequency estimation principle block diagram based on interpolation compensation is shown in figure 1.
Let the signal model be
x(n)＝s(n)+w(n) (1.1)
Figure BDA0002603185110000052
Where s (n) is a useful signal, n represents a quantized sampling time, and w (n) is gaussian noise. A is the signal amplitude, f0Is the signal frequency, fsIs the sampling frequency (the sampling frequency satisfies the Nquist sampling theorem, i.e. f0/fs∈(0,0.5])，
Figure BDA0002603185110000061
Is the unknown carrier phase of the signal. Without considering noise, the N-point FFT of the input signal is
Figure BDA0002603185110000062
The FFT resolution is f (f ═ f)s/N)，k0Represents the maximum peak value X (k)0) Corresponding to the serial number of the frequency spectrum of which the corresponding frequency is
Figure BDA0002603185110000063
The initial frequency difference is:
Figure BDA0002603185110000064
the frequency estimation accuracy is affected by the FFT resolution, and the maximum initial frequency difference that may be generated is Δ f ═ f/2, and in order to improve the estimation accuracy, the initial frequency difference needs to be corrected by interpolation compensation. The interpolation compensation algorithm utilizes the frequency difference correction coefficient estimated by the FFT coefficient to beFFT(i.e. the initial frequency offset estimate
Figure BDA0002603185110000065
) And thereby obtain a final frequency estimate of
Figure BDA0002603185110000066
The frequency estimation interpolation compensation method provided by the invention comprises two parts: the estimation of the interpolation direction d and the estimation of the correction value, so that the frequency offset correction factor is calculated by the formulaFFT＝d。
1. Interpolation direction estimation
The invention provides an interpolation direction estimation method based on phase coefficient comparison. By the symbol alpharExpressing the phase coefficient, and the calculation formula is as follows:
Figure BDA0002603185110000067
wherein r can be n ═ k0-1 and p ═ k0+1,
Figure BDA0002603185110000068
Is X (k)0) Conjugation of (1). Alpha is alpharIs a function of cos (Δ φ)r) I.e.:
αr＝f[cos(Δφr)] (1.7)
wherein, isrIs X (r) and X (k)0) The phase difference between them. When in a noise-free condition, as shown in FIG. 2, Δ φrAnd alpharThe relationship of (1) is:
Figure BDA0002603185110000069
wherein phi isrAnd phik0Are respectively X (r) and X (k)0) The phase of (c). From the phase characteristics of the Fourier transform of the sinusoidal signal, αr<0 denotes that X (r) is located within the main lobe, αr≧ 0 means that X (r) is located within the sidelobe and has αnαp<0. In this case, α can be utilizedrThe interpolation direction is estimated. However, in practice, it is influenced by noise, Δ φrNot equal to 0 or pi but fluctuates around 0 or pi as shown in fig. 3 and 4. The fluctuation caused by the noise may be alphanαp>The case of 0 occurs if alpha is directly utilizedrThe interpolation direction estimation is carried out on the symbol estimation, and the estimation direction error is likely to occur. Further, the phase difference when the initial frequency difference Δ f is 0.125f is larger in fluctuation than the phase difference when the initial frequency difference Δ f is 0.25f, and it is found that the problem of interpolation direction error is more likely to occur as the actual frequency is closer to the quantization bin.
As can be seen from FIGS. 3 and 4, when affected by noise, Δ φrFluctuates around 0 and pi, even in the case of large fluctuations, Δ φpStill ratio delta phinWill probably approach pi. According to the curve characteristic of the cos (-) function, αnAnd alphapIs alpha with large probabilityn>αpThus using αnAnd alphapIs possible to estimate the interpolation direction. In the same way, when XnLocated in the main lobe, XpAt side lobe, αn<αpIs more probableThe correct interpolation direction is then d-1. It can be seen that the Quinn algorithm utilizes alphanOr alphapOn the basis of judging the interpolation direction by positive and negative, when the phase coefficients have the same sign, alpha is utilizednAnd alphapThe interpolation direction is estimated according to the magnitude relation of the interpolation direction, so that the accuracy of the estimation of the interpolation direction is effectively improved.
In summary, the interpolation direction estimation method based on the phase coefficient comparison comprises
(1) When alpha isnαp<At 0, the interpolation direction is
Figure BDA0002603185110000071
(2) When alpha isnαp>At 0, the interpolation direction is
Figure BDA0002603185110000072
2. Correction value estimation
The invention provides a correction value estimation method based on sidelobe coefficient weighting correction. The method is directed to correction values calculated from the main lobe coefficients1Possibly has the problem of larger error, and is calculated by utilizing FFT sidelobe coefficient2To pair1And performing weighting correction to obtain a final correction value, wherein a weighting factor for weighting is generated by a fuzzy controller, and the implementation principle of the method is shown in fig. 5.
Let V0Represents X (k)0) Modulus value of (V)nAnd VpAre respectively X (k)0) The modulus of the two spectral lines, i.e. left and right
Figure BDA0002603185110000073
The discussion will be given by way of example of d ═ 1, where VnThe next largest value within the main lobe. Definition of Δ VnIs the proportionality coefficient of the sub-maximum spectral line amplitude and peak value, called the main lobe coefficient, and is DeltaVn＝Vn/V0. At this time, the correction value calculated from the main lobe coefficient1Is taken asnAnd is provided with
Figure BDA0002603185110000081
Under the conditions of no noise and correct interpolation direction, the frequency difference correction value calculated by the above formula is close to the initial frequency difference, and the method has good estimation performance.
However, under the influence of noise, Δ VnThe actual value of (a) may deviate from the ideal value, reducing the estimation accuracy. For this, we introduce the FFT spectral line amplitude V of the side lobe on the other side of the next largest valuepAnd calculating to obtain a side lobe coefficient delta Vp＝Vp/V0Correction values calculated using side lobe coefficients2Is taken aspThe calculation formula is as follows:
Figure BDA0002603185110000082
by usingpTo pairnThe weight factor for weight correction is represented by beta, and the obtained frequency difference correction value is
＝βn+(1-β)p (1.14)
It can be seen that this is a function of β. For the same reason, when d is +1, VpIs the second largest value of the main lobe, and the coefficient of the main lobe is delta Vp，VnThe corresponding side lobe coefficient is delta V for the side lobe spectral line amplitudenThen usenTo pairpAnd carrying out weighted correction.
In summary, the final frequency difference correction value is
Figure BDA0002603185110000083
The weight factor β determines the degree of correction of the correction value calculated from the main lobe coefficient by the side lobe coefficient, and directly affects the estimation performance of the final frequency offset correction value, so the calculation of the weight factor is the key of the frequency offset correction of the interpolation compensation algorithm, and needs to be discussed in detail.
Under the influence of noise, the distribution rule of the FFT spectrum changes, and the state deviates from the state under the ideal noise-free condition. Since such a change has uncertainty and is unpredictable, it is considered to perform analysis using the fuzzy control theory and to generate a weight factor for weight correction from the analysis result. A dual input-single output Mamdani type fuzzy controller is used to generate the weighting factors. The core of the weight factor calculation method based on fuzzy control is the design of a fuzzy controller, which is mainly divided into three parts: fuzzification, fuzzy reasoning and deblurring.
(a) Fuzzification
According to Vn、VpAnd V0Constructing two variables to represent the relationship between the FFT spectrum amplitudes along with the change rule of the initial frequency difference, and discussing d as-1, wherein the two constructed variables are respectively
Variable 1: Δ VnThe ratio of the next largest spectral line to the peak, i.e. the main lobe coefficient as defined above;
variable 2: vn/VpThe ratio of the next largest spectral line amplitude to the sidelobe spectral line amplitude is shown.
Under the ideal condition of no noise, the position relation between the actual frequency and the quantized frequency point can be judged by selecting one of the two variables. Under the influence of noise, firstly, the noise is divided into delta VnThe value of (a) is taken as a main judgment condition, and the distance between the actual frequency and the quantized frequency point is roughly judged. Then again with Vn/VpAnd as an auxiliary judgment condition, the condition that the FFT spectrum distribution rule is influenced by noise is presumed, and finally, the value of the weight factor is deduced by combining the values of the two variables. In summary, when d is-1, the input variable r of the fuzzy control is set1And r2Is composed of
Figure BDA0002603185110000091
Similarly, when d is +1, the input variable of the fuzzy controller is set as
Figure BDA0002603185110000092
The determination of the input variable physics discourse domain comes from the analysis of the actual simulation data. In the absence of noise, variable r1Has a variation range of r1∈[0,1](ii) a Whereas under the influence of noise, the variable r1May be greater than 1, provided that r is greater than 11>At 1 time, take r 11, so the variable r1Is U e [0,1 ∈]The conversion coefficient from the physical variable to the fuzzy variable is chosen to be 1, so r1Has a fuzzy domain of Nu∈[0,1]. For variable r1Fuzzification is performed, and mapping is performed to four fuzzy subsets ZE, PS, PM and PL.
Membership function of ZE as
Figure BDA0002603185110000093
The membership function for PS is:
Figure BDA0002603185110000094
the membership function of PM is:
Figure BDA0002603185110000095
the membership function of PL is:
Figure BDA0002603185110000096
for variable r2Its variation range is large, and in principle it can be taken as r2E [0, + ∞)) by analysis of simulation data when r is2When the value is near 3, V can be reflectednAnd VpIs onIs, therefore, defined as r2>At 3 time, get r2Is 3, so the variable r2Is U e [0,3 ∈]Selecting 1 from the conversion coefficient from the physical variable to the fuzzy variable to obtain r2Has a fuzzy domain of Nu∈[0,3]. For variable r2Fuzzification is performed, and mapping is performed to four fuzzy subsets ZE, PS, PM and PL.
Membership functions for ZE are:
Figure BDA0002603185110000101
the membership function for PS is:
Figure BDA0002603185110000102
the membership function of PM is:
Figure BDA0002603185110000103
the membership function of PL is:
Figure BDA0002603185110000104
the output variable of the fuzzy controller is beta, and the fuzzy domain of the fuzzy controller is set to be Nu∈[0.5,1.3]Mapping to four fuzzy subsets ZE, PS, PM and PL, respectively.
Membership functions for ZE are:
Figure BDA0002603185110000105
the membership function for PS is:
Figure BDA0002603185110000106
the membership function of PM is:
Figure BDA0002603185110000111
the membership function of PL is:
Figure BDA0002603185110000112
scaling factor k for transformation of output variable from ambiguity domain to physics domain u1, so the output variable has a physical range of U ═ 0.5,1.3]。
(b) Fuzzy inference
Fuzzy inference is the theoretical basis of fuzzy controller design, and refers to a process of deducing a possibly inaccurate conclusion from an imprecise premise according to a fuzzy control rule, that is, the fuzzy inference refers to a process of deducing a fuzzy output variable from a fuzzy input variable through a certain inference method according to the fuzzy control rule. The fuzzy control rules for fuzzy reasoning are summarized and summarized by analyzing experimental test data, as shown in table 1.
Table 1: fuzzy control rule
Figure BDA0002603185110000113
(c) Deblurring
Deblurring is the process of equating a fuzzy set output through fuzzy inference to a distinct value, also called as sharpening. And performing deblurring processing by adopting an area center method. The area-centric method is to find the fuzzy set membership function curve and the center of the area surrounded by the abscissa, and then to take the abscissa of the center as the output value. The area center method has the following calculation principle:
Figure BDA0002603185110000114
wherein U (u) is the domain of ambiguityNuA membership function of the fuzzy set U.
In summary, the above description is only a preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (3)

1. A method for compensating for interpolation of frequency estimates, comprising the steps of:
step 1, interpolation direction estimation, specifically:
(1) when alpha isnαp<At 0, the interpolation direction is:
Figure FDA0002603185100000011
(2) when alpha isnαp>At 0, the interpolation direction is:
Figure FDA0002603185100000012
wherein the content of the first and second substances,
Figure FDA0002603185100000013
n＝k0-1 and p ═ k0+1, X (-) denotes the FFT-transformed signal of the input signal, k0Represents the maximum peak value X (k)0) The sequence number of the corresponding frequency spectrum,
Figure FDA0002603185100000014
is X (k)0) Conjugation of (1);
step 2, calculating a final frequency difference correction value:
Figure FDA0002603185100000015
wherein, V0Represents X (k)0) Modulus value of (V)nAnd VpAre respectively X (k)0) Modulus, Δ V, of the two spectral linesn＝Vn/V0，ΔVp＝Vp/V0F is the resolution of the Fourier transform; β represents a weighting factor;
and step 3, the final frequency estimation value is as follows:
Figure FDA0002603185100000016
wherein the content of the first and second substances,
Figure FDA0002603185100000017
the frequency point corresponding to the peak value is shown,FFT＝d。
2. the method of claim 1, wherein the weighting factor β is calculated by:
step 21, when d is equal to-1, setting input variable r of fuzzy control1And r2Comprises the following steps:
Figure FDA0002603185100000018
when d is +1, the input variables of the fuzzy controller are set as follows:
Figure FDA0002603185100000019
variable r1Is U e [0,1 ∈]，r1Has a fuzzy domain of Nu∈[0,1](ii) a For variable r1Fuzzification is carried out, and the four fuzzy subsets ZE, PS, PM and PL are mapped respectively;
for variable r2The theory domain of physics is U is equal to [0,3 ]]，r2Has a fuzzy domain of Nu∈[0,3]. For variable r2The mixture is subjected to fuzzification and then treated,mapping to four fuzzy subsets ZE, PS, PM and PL respectively;
the output variable of the fuzzy controller is beta, and the fuzzy domain of the fuzzy controller is set to be Nu∈[0.5,1.3]Mapping to four fuzzy subsets ZE, PS, PM and PL, respectively;
scaling factor k for transformation of output variable from ambiguity domain to physics domainu1, so the output variable has a physical range of U ═ 0.5,1.3]。
Step 22, fuzzy control rules of fuzzy inference, as shown in table 1:
table 1: fuzzy control rule
Figure FDA0002603185100000021
Step 23, according to the actually obtained input variable r1And r2And based on an area center method, performing deblurring on the output variable to obtain the value of the output variable beta.
3. The method of claim 2, wherein the interpolation compensation is performed by a frequency estimation device,
variable r1After fuzzification, the membership functions respectively mapped to the four fuzzy subsets ZE, PS, PM and PL are as follows:
membership functions for ZE are:
Figure FDA0002603185100000022
the membership function for PS is:
Figure FDA0002603185100000023
the membership function of PM is:
Figure FDA0002603185100000024
the membership function of PL is:
Figure FDA0002603185100000031
variable r2After fuzzification, the membership functions respectively mapped to the four fuzzy subsets ZE, PS, PM and PL are as follows:
Figure FDA0002603185100000032
the membership function for PS is:
Figure FDA0002603185100000033
the membership function of PM is:
Figure FDA0002603185100000034
the membership function of PL is:
Figure FDA0002603185100000035
the output variables are membership functions of the four fuzzy subsets ZE, PS, PM and PL to which they are respectively mapped:
membership functions for ZE are:
Figure FDA0002603185100000036
the membership function for PS is:
Figure FDA0002603185100000037
the membership function of PM is:
Figure FDA0002603185100000041
the membership function of PL is:
Figure FDA0002603185100000042
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