CN114487597A - CZT frequency estimation method - Google Patents

CZT frequency estimation method Download PDF

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CN114487597A
CN114487597A CN202210123342.7A CN202210123342A CN114487597A CN 114487597 A CN114487597 A CN 114487597A CN 202210123342 A CN202210123342 A CN 202210123342A CN 114487597 A CN114487597 A CN 114487597A
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czt
frequency
spectral line
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Zhongke Shuiyan Jiangxi Technology Co ltd
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Abstract

The invention relates to a CZT frequency estimation method, which comprises the following steps: s1: sampling a signal to be estimated, performing N-point fast Fourier transform on the sampled discrete time domain signal to obtain an FFT (fast Fourier transform) spectrum function, and extracting a frequency index of the maximum spectral line value of the FFT spectrum function from the FFT spectrum function; s2: determining a frequency interval of CZT conversion according to the value of the frequency index, carrying out CZT conversion on the discrete time domain signal to obtain a CZT spectrum function, and extracting to obtain a maximum spectral line value of the CZT spectrum function and two left and right spectral line values of the CZT spectrum function; s3: calculating a maximum spectral line value and a relation parameter between a left spectral line value and a right spectral line value; s4: calculating to obtain an error according to the relation parameter; s5: and calculating to obtain the estimated frequency of the signal to be estimated according to the relation parameters and the error.

Description

CZT frequency estimation method
Technical Field
The invention relates to the field of frequency estimation, in particular to a CZT frequency estimation method.
Background
Frequency estimation of signals is a problem often present in engineering applications, and many scenarios require an accurate estimation of the frequency of the signal. For example, in a Frequency Modulated Continuous Wave (FMCW) radar ranging system, it is possible to obtain relevant range information from a difference frequency signal of a transmitted wave and a reflected wave, i.e., to obtain an estimated range by estimating the frequency of the difference frequency signal, so the accuracy of frequency estimation on the difference frequency signal directly affects the accuracy of the estimated range, and the accuracy of measurement of the frequency of the difference frequency signal directly determines the accuracy of ranging.
To improve the accuracy of frequency estimation, many frequency estimation algorithms are proposed. In addition to Fast Fourier Transform (FFT), there are various frequency transform-based methods applied to improve frequency accuracy, for example, a tap fourier transform (zoomft), a zero-padding method, a Chirp Z Transform (CZT), and the like, and there are also many frequency transform-based methods proposed. Zero-filling methods are commonly used to increase the discrete Fourier transformThe number of points in the transform (DFT) is improved, thereby improving the approximation of DFT to DFT, which reduces spike-fence effect and obtains the approximation of DTFT local peaks at a limited number of stations. In order to overcome the problem of large calculation amount of the zero padding method, the prior art further provides a new zero padding method, which utilizes non-integer cycles in the orthogonal signal of the DFT kernel to provide the same result as the zero padding method in the spectrum analysis, but greatly reduces the calculation amount. Furthermore, the DFT computation using non-integer parameters provides the possibility to develop a DFT with variable bin resolution, a narrow-band DFT and a bin interpolation algorithm. The prior art also proposes a method of obtaining a composite material having
Figure BDA0003498868630000011
The technology of the order root mean square error estimator obtains the frequency estimator by performing complex interpolation on three continuous Fourier coefficients, and the mean square error of the frequency estimator has the same order as the asymptotic variance (Cram er-Rao lower bound) of the maximum value on all frequencies in the sample capacity.
Although the accuracy of many frequency estimation methods is high, the frequency refinement methods such as CZT with high refinement multiple, interpolation method, and improvement method thereof are focused. Theoretically, as long as the degree of refining the spectrum is enough, a very accurate frequency estimation value can be obtained, but the higher the refining multiple is, the more complicated the calculation is, and the larger the calculation amount is, so how to ensure the accuracy of estimation while reducing the calculation complexity and the calculation amount is a problem that needs to be solved by those skilled in the art urgently.
Disclosure of Invention
The invention aims to provide a CZT frequency estimation method, which improves the estimation accuracy and reduces the calculation complexity and the calculation amount.
The invention provides a CZT frequency estimation method, which comprises the following steps of:
s1: sampling a signal to be estimated, performing N-point fast Fourier transform on the sampled discrete time domain signal to obtain an FFT spectrum function X (k), and extracting a maximum spectral line value X (k) of the FFT spectrum function from the FFT spectrum functionp) Frequency index k ofp；
S2: according to the frequency index kpThe value of (A) determines the frequency interval of CZT conversion, and CZT conversion is carried out on the discrete time domain signal to obtain CZT frequency spectrum function XCZT(k) Extracting the maximum spectral line value X of the CZT spectral function from the spectrumCZT(km) And the left and right spectral line values XCZT(km+1)、XCZT(km-1)；
S3: calculating a maximum spectral line value and a relation parameter mu between the left spectral line value and the right spectral line value;
s4: calculating an error delta according to the relation parameter mu;
s5: calculating to obtain the estimated frequency of the signal to be estimated according to the relation parameter mu and the error delta
Figure BDA0003498868630000024
Further, in the step S2, the frequency range of the CZT conversion is
Figure BDA0003498868630000022
Wherein f issQ is the size of the CZT transform interval for the sampling frequency.
Further, a maximum spectral line value X of the CZT spectral functionCZT(km) Satisfies the following relation:
Figure BDA0003498868630000023
Figure BDA0003498868630000031
wherein k ismJ is the frequency index where the amplitude of CZT is maximum, j represents an imaginary number unit, B is the bandwidth of the refined frequency interval, and M is the degree of refinement of the refined frequency interval.
Further, the left spectral line value of the maximum spectral line value of the CZT spectral function satisfies the following relation:
Figure BDA0003498868630000032
Figure BDA0003498868630000033
further, the right spectral line value of the maximum spectral line value of the CZT spectral function satisfies the following relation:
Figure BDA0003498868630000034
Figure BDA0003498868630000035
further, the maximum spectral line value of the CZT spectral function and a relation parameter μ between the two spectral line values on the left and right of the maximum spectral line value satisfy the following relation:
Figure BDA0003498868630000036
wherein, XCZT(km) Is the maximum spectral line value, X, of the CZT spectral functionCZT(km+1) right spectral line, X, which is the maximum spectral line value of the CZT spectral functionCZT(km-1) the left spectral line being the maximum spectral line value of the CZT spectral function.
Further, the error δ satisfies the following relation:
Figure BDA0003498868630000037
wherein q is the size of the CZT conversion interval, and M is the thinning degree of the thinning frequency interval.
Further, the relationship parameter μ satisfies the following relationship:
Figure BDA0003498868630000041
further, a maximum spectrum of the CZT spectrum functionEstimated frequency corresponding to line value
Figure BDA0003498868630000042
Satisfies the following relation:
Figure BDA0003498868630000043
according to the CZT frequency estimation method, the relation parameter mu is introduced by analyzing the information of the maximum spectral line and the left and right spectral lines of the frequency spectrum in the CZT, the magnitude of the error is directly estimated through the relation parameter mu, more accurate frequency estimation is obtained, under the condition of a certain signal-to-noise ratio, the estimation accuracy is improved, and the calculation complexity and the calculation amount are reduced.
Drawings
Fig. 1 is a flow chart of a CZT frequency estimation method according to an embodiment of the invention.
Detailed Description
The preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings.
In the current FMCW radar ranging system, the transmission signal t (t) can be represented as:
Figure BDA0003498868630000044
Figure BDA0003498868630000045
the received reflection signal r (t) is:
Figure BDA0003498868630000046
Figure BDA0003498868630000047
wherein, f0Is the initial frequency of the chirp wave,
Figure BDA0003498868630000048
is the slope of the chirp, B is the chirp bandwidth, T is the chirp period, r is the distance of the target from the radar, τ is the time delay caused by the distance r of the target from the radar,
Figure BDA0003498868630000049
c is the speed of the electromagnetic wave in air,
Figure BDA0003498868630000051
is an initial phase, atTo transmit gain, arTo receive gain, a0＝aratPath losses and losses due to reflections from objects are neglected in this system.
Mixing the transmission signal T (t) and the reflection signal R (t), and filtering high-frequency components to obtain a difference frequency signal x (t):
x(t)＝a0 exp{j2π(f0τ+Kτt)} (3)
at a sampling frequency fsSampling the difference frequency signal x (t) to obtain a discrete time domain signal x (n):
Figure BDA0003498868630000052
where n is a sampling point in the discrete time domain, a0＝arat，atTo transmit gain, arTo receive gain, f0Is the initial frequency of the chirp wave, fcτ is the time delay caused by the range r of the target relative to the radar, which is the frequency of the difference frequency signal.
By frequency f of difference frequency signalcObtaining a distance r of the target relative to the radar:
Figure BDA0003498868630000053
therefore, in order to obtain a highly accurate distance, it is necessary to obtain a highly accurate frequency of the difference frequency signal.
As shown in fig. 1, an embodiment of the present invention provides a CZT frequency estimation method, including the following steps:
s1: sampling a signal to be estimated, and performing N-point fast Fourier transform on the sampled discrete time domain signal X (N) to obtain a frequency spectrum function, thereby obtaining a maximum spectral line X (k)p) Frequency index k ofp(ii) a The signals to be estimated are difference frequency signals of a transmission signal T (t) and a reflection signal R (t) of a Frequency Modulated Continuous Wave (FMCW) radar ranging system.
According to the formula of the fast Fourier transform of N points, the FFT spectrum function X (k) is obtained as follows:
Figure BDA0003498868630000054
Figure BDA0003498868630000055
Figure BDA0003498868630000061
wherein N is the signal length, and may be 512, 1024 or more, and the frequency resolution of the N-point fast Fourier transform is
Figure BDA0003498868630000062
The maximum amplitude value of the FFT spectrum function is calculated to be used as the maximum spectral line value of the FFT spectrum function, and the frequency index at the maximum amplitude position of the FFT spectrum function is used as the frequency index k of the maximum spectral line value of the FFT spectrum functionpTo obtain the frequency of the frequency point where the amplitude is maximum
Figure BDA0003498868630000063
Figure BDA0003498868630000064
The frequency corresponding to the maximum amplitude position of the FFT spectrum function
Figure BDA0003498868630000065
For the center, the frequency range (f) of the CZT conversion is selected1，f2) For the interval (f) where the peak point is located1，f2) Thinning is carried out to obtain the thinned frequency f:
Figure BDA0003498868630000066
wherein, BCZTRepresentative interval (f)1，f2) Bandwidth of (d), M represents the interval (f)1，f2) Degree of refinement of (1).
S2: according to the frequency index kpDetermining a CZT thinning frequency interval and carrying out CZT conversion to obtain a CZT frequency spectrum function XCZT(k) Extracting the maximum spectral line value X from the obtained dataCZT(km) Left and right two spectral line values XCZT(km+1)、XCZT(km-1)；
CZT transformation is carried out on the discrete time domain signal, and the CZT frequency spectrum function X (z) of the discrete time domain signalk) The formula of (1) is:
Figure BDA0003498868630000067
wherein z isk＝AW-k，
Figure BDA0003498868630000068
θ0Is the initial sampling angle, phi0Is the angle between two adjacent sample points.
A0、W0The constant value in CZT conversion can be set according to actual conditions. In this example, A0＝1、W0Thus, the CZT spectral function x (zk) obtained after CZT transformation for the discrete time-domain signal x (n) is:
Figure BDA0003498868630000069
Figure BDA00034988686300000610
Figure BDA0003498868630000071
k＝0，...，M-1；
according to the frequency index k at the maximum amplitude of fast Fourier transformpDetermining a spatial range of the CZT conversion, which corresponds to a frequency range (f) of the CZT conversion1，f2) Comprises the following steps:
Figure BDA0003498868630000072
and q is the size of the CZT conversion interval and can be set according to the actual situation as long as the q is a positive integer. Corresponding to a frequency resolution of the CZT transform of
Figure BDA0003498868630000073
According to equation (10), corresponding to CZT spectrum function X obtained after CZT conversionCZT(k) Comprises the following steps:
Figure BDA0003498868630000074
Figure BDA0003498868630000075
finding CZT spectral function XCZT(k) Maximum value of amplitude of as maximum spectral line value XCZT(km) The resulting frequency index k at which the amplitude is maximummMaximum spectral line value X of CZT spectral functionCzT(km) The estimated frequency corresponding to the maximum spectral line value of the CZT spectral function can be obtained according to the frequency index
Figure BDA0003498868630000076
Figure BDA0003498868630000077
Due to the fence effect of the discrete signal, the maximum value of the spectral line (i.e. the maximum spectral line value) is difficult to be related to the frequency f to be measuredcCoincidence, i.e. frequency index corresponding to the maximum spectral line value of the spectral function of the FFT
Figure BDA0003498868630000078
Or frequency index corresponding to maximum spectral line value of CZT spectral function
Figure BDA0003498868630000079
All with the frequency f to be measuredcThere is a certain error. In this embodiment, the frequency f to be measuredcCan be expressed as:
Figure BDA00034988686300000710
wherein, k ispFor the frequency index where the fast fourier transform amplitude is largest,
Figure BDA00034988686300000711
selecting a refined frequency bandwidth, k, for CZTmFor the frequency index corresponding to the maximum amplitude of the CZT transform in the refinement interval, the error delta E [ -0.5, 0.5]Is the fractional part. Since τ is satisfied
Figure BDA00034988686300000712
Tau and f are obtained by combining formula (4)cThe relationship between them satisfies:
Figure BDA0003498868630000081
maximum spectral line value X of CZT frequency spectrum function obtained after corresponding CZT conversionCZT(km) Comprises the following steps:
Figure BDA0003498868630000082
Figure BDA0003498868630000083
left spectral line value XCZT(km-1) is:
Figure BDA0003498868630000084
Figure BDA0003498868630000085
right spectral line value XCZT(km+1) is:
Figure BDA0003498868630000086
Figure BDA0003498868630000087
s3: calculating a maximum spectral line value and a relation parameter mu between the left spectral line value and the right spectral line value;
Figure BDA0003498868630000088
substituting formulae (16), (17) and (18) into formula (19):
Figure BDA0003498868630000089
and (3) after simplification:
Figure BDA00034988686300000810
s4: calculating an error delta according to the relation parameter mu;
the relationship between μ and δ is derived from the formula:
Figure BDA0003498868630000091
when N is large and δ is small,
Figure BDA0003498868630000092
therefore, the temperature of the molten metal is controlled,
Figure BDA0003498868630000093
from this, an approximation of μ can be obtained:
Figure BDA0003498868630000094
the corresponding delta values can be estimated from the left and right two spectral lines by equations (22) and (24).
S5: calculating to obtain estimated frequency according to the relation parameter mu and the error delta
Figure BDA0003498868630000095
After the relation parameter mu and the error delta are determined, the corresponding estimated frequency can be obtained through the formula (14), and then the corresponding estimated distance value can be obtained according to the formula (5).
In order to verify the effect of the frequency estimation method of the present invention, N-point FFT, 16-time refined spectrum (range q is 2), 32-time refined spectrum CZT (2-32CZT, range q is 2), the CZT frequency estimation method of the present invention (2-32CZT +, 32-time refined spectrum CZT, range q is 2), zoomfft (frequency shift is set to 2000Hz according to data), zero-filling method (zero-filling point is set to 6 × N point because the effect of improving accuracy is not significant when the number of zero-filling points is small), rife are simulated under different snr conditions to obtain the estimated distance mean and variance of different FMCW radars.
The parameters in the simulation were as follows:
sampling rate fs92.7835e3Hz, 1024 sampling points, and the initial frequency f of the chirp wave0100KHz, 999.47055MHZ, chirp bandwidth B, chirp period
Figure BDA0003498868630000096
Slope of chirp
Figure BDA0003498868630000097
Initial phase
Figure BDA0003498868630000098
Transmission gain atWhen 1, the transmission signal is known from equation (1):
Figure BDA0003498868630000099
assuming that the distance between a measured static object and the FMCW system is r, and the speed of electromagnetic wave in air is 299709Km/s, the time delay caused by the distance r
Figure BDA0003498868630000101
Reception gain arNeglecting the path loss and the loss due to the object reflection, the received reflection signal is known from equation (2) as:
Figure BDA0003498868630000102
mixing T (t) and R (t), and filtering high-frequency components to obtain a difference frequency sampling signal, wherein the difference frequency sampling signal is as follows according to formula (4):
Figure BDA0003498868630000103
when the sampling signal is interfered by noise, the difference frequency sampling signal is:
Figure BDA0003498868630000104
w (n) represents noise, different signal-to-noise ratios have different influences on the final calculated distance, corresponding difference frequency sampling signals x (n) are obtained by setting different distances r and signal-to-noise ratios, the frequency is estimated by adopting different frequency estimation methods, and then the corresponding estimated distance is obtained according to the frequency.
It should be noted that, although the specific steps of the frequency estimation method of the present invention are described in the present embodiment by taking FMCW radar ranging as an example, the method is not limited to be applied to FMCW radar ranging, and can be extended to other frequency estimation applications.
In this embodiment, the actual distance value is 2.57m, the experiment is performed in the range of the snr [ -16dB, 16dB ], and the monte carlo experiments are independently performed 10000 times under different snrs, so that the obtained estimated distance mean is shown in table 1, and the estimated distance variance is shown in table 2:
TABLE 1 estimated mean distance (m) from simulation under different SNR conditions
Figure BDA0003498868630000105
Figure BDA0003498868630000111
TABLE 2 estimated distance variance (m) simulated under different SNR conditions
Figure BDA0003498868630000112
As can be seen from tables 1 and 2, when the signal-to-noise ratio is low (less than-15 dB), the signal is affected by the noise too much, resulting in that the accuracy of the estimation is affected by the noise more greatly. When the signal-to-noise ratio is larger than-15 dB, the performance of the CZT frequency estimation method (namely 2-32CZT +) is kept good all the time, and the fluctuation range is closer to the real distance under the condition that the estimated distance mean value is closer to the real distance value. Under the condition of a certain signal-to-noise ratio, compared with other methods, the CZT frequency estimation method disclosed by the invention improves the estimation accuracy, reduces the calculation complexity and the calculation amount, and has better estimation capability compared with other methods.
The above embodiments are merely preferred embodiments of the present invention, which are not intended to limit the scope of the present invention, and various changes may be made in the above embodiments of the present invention. All simple and equivalent changes and modifications made according to the claims and the content of the specification of the present application fall within the scope of the claims of the present patent application. The invention has not been described in detail in order to avoid obscuring the invention.

Claims (9)

1. A CZT frequency estimation method is characterized by comprising the following steps:
s1: sampling a signal to be estimated, performing N-point fast Fourier transform on the sampled discrete time domain signal to obtain an FFT spectrum function X (k), and extracting a maximum spectral line value X (k) of the FFT spectrum function from the FFT spectrum functionp) Frequency index k ofp；
S2: according to the frequency index kpThe value of (A) determines the frequency interval of CZT conversion, and CZT conversion is carried out on the discrete time domain signal to obtain CZT frequency spectrum function XCZT(k) Extracting the maximum spectral line value X of the CZT spectral function from the obtained dataCZT(km) And the left and right spectral line values XCZT(km+1)、XCZT(km-1)；
S3: calculating a maximum spectral line value and a relation parameter mu between the left spectral line value and the right spectral line value;
s4: calculating an error delta according to the relation parameter mu;
s5: calculating to obtain the estimated frequency of the signal to be estimated according to the relation parameter mu and the error delta
Figure FDA0003498868620000011
2. The CZT frequency estimation method according to claim 1, wherein in the step S2, the CZT conversion frequency interval range is
Figure FDA0003498868620000012
Wherein f issAnd q is the size of the CZT conversion interval for the sampling frequency.
3. The CZT frequency estimation method of claim 2, wherein a maximum spectral line value X of the CZT spectral functionCZT(km) Satisfies the following relation:
Figure FDA0003498868620000013
wherein k ismIs the frequency index where the amplitude of CZT is maximum, j represents an imaginary unit, B is the bandwidth of the refined frequency interval, and M is the degree of refinement of the refined frequency interval.
4. The CZT frequency estimation method according to claim 3, wherein a left spectral line value of a maximum spectral line value of the CZT spectral function satisfies the following relation:
Figure FDA0003498868620000014
5. the CZT frequency estimation method according to claim 4, characterized in that a right spectral line value of a maximum spectral line value of the CZT spectral function satisfies the following relation:
Figure FDA0003498868620000021
6. the CZT frequency estimation method according to claim 5, wherein a relationship parameter μ between a maximum spectral line value of the CZT spectral function and two spectral line values around the maximum spectral line value satisfies the following relationship:
Figure FDA0003498868620000022
wherein, XCZT(km) Is the maximum spectral line value, X, of the CZT spectral functionCZT(km+1) right spectral line, X, which is the maximum spectral line value of the CZT spectral functionCZT(km-1) the left spectral line being the maximum spectral line value of the CZT spectral function.
7. The CZT frequency estimation method of claim 6, wherein the error δ satisfies the following relation:
Figure FDA0003498868620000023
wherein q is the size of the CZT conversion interval, and M is the thinning degree of the thinning frequency interval.
8. The CZT frequency estimation method according to claim 6, wherein the relation parameter μ satisfies the following relation:
Figure FDA0003498868620000024
9. the CZT frequency estimation method of claim 2, wherein the estimated frequency corresponding to a maximum spectral line value of the CZT spectral function
Figure FDA0003498868620000025
Satisfies the following relation:
Figure FDA0003498868620000026
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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116032703A (en) * 2023-03-29 2023-04-28 中国人民解放军海军工程大学 Method and system for estimating number of signal code elements of transform domain communication system

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116032703A (en) * 2023-03-29 2023-04-28 中国人民解放军海军工程大学 Method and system for estimating number of signal code elements of transform domain communication system
CN116032703B (en) * 2023-03-29 2023-06-27 中国人民解放军海军工程大学 Method and system for estimating number of signal code elements of transform domain communication system

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