A method of CZT frequency estimation

Abstract

The present application relates to a kind of CZT frequency estimation methods, comprising the following steps: S1: the signal to be estimated is sampled, and the maximum spectral line value of the FFT spectrum function is extracted from the frequency index of the FFT spectrum function obtained by the N-point fast Fourier transform of the discrete time domain signal after sampling;S2: the frequency interval of the CZT transform is determined according to the value of the frequency index, and the maximum spectral line value of the CZT spectrum function is extracted from the CZT spectrum function obtained by the discrete time domain signal and the left and right two spectral line values of the CZT spectrum function;S3: the relationship parameter between the maximum spectral line value and the left and right two spectral line values is calculated;S4: error is calculated from the relationship parameter;S5: the estimated frequency of the signal to be estimated is calculated according to the relationship parameter and error.

Description

Technical Field

[0001] This invention relates to the field of frequency estimation, and more specifically to a CZT frequency estimation method. Background Technology

[0002] Signal frequency estimation is a common problem in engineering applications, and many scenarios require accurate estimation of signal frequencies. For example, in frequency modulated continuous wave (FMCW) radar ranging systems, relevant distance information can be obtained based on the difference frequency signal between the transmitted and reflected waves. That is, the estimated distance is obtained by estimating the frequency of the difference frequency signal. Therefore, the accuracy of the difference frequency signal frequency estimation directly affects the accuracy of the estimated distance, and the measurement accuracy of the difference frequency signal frequency directly determines the accuracy of the ranging.

[0003] To improve the accuracy of frequency estimation, numerous frequency estimation algorithms have been proposed. Based on the Fast Fourier Transform (FFT), various frequency transform-based methods have been applied to improve frequency accuracy, such as the selected-band Fourier Transform (ZoomFFT), zero-padding, and linear frequency modulated Z-transform (CZT). Many other methods based on these frequency transforms have also been proposed. Zero-padding is typically used to increase the number of points in the Discrete Fourier Transform (DFT), thereby improving the DFT approximation of the DFT. This method reduces the spike fence effect and obtains an approximation of the local peaks of the DTFT at a finite number of measurement points. To overcome the computational complexity of the zero-padding method, a new zero-padding method has been proposed, utilizing the non-integer number of periods in the orthogonal signal kernel of the DFT to provide the same results as the zero-padding method in spectral analysis, but with significantly reduced computational complexity. Furthermore, using non-integer parameters for DFT calculation makes it possible to develop DFTs with variable bin resolution, narrowband DFT, and bin interpolation algorithms. Existing techniques have also proposed methods to obtain... The technique of the root mean square error estimator obtains an estimate of the frequency by complex interpolating three consecutive Fourier coefficients. Its mean square error has the same order as the asymptotic variance (Cramér-Rao lower bound) of the maximum value at all frequencies in the sample size. The advantage of this new estimator is its computational simplicity.

[0004] While many of the aforementioned frequency estimation methods offer high accuracy, they all focus on frequency refinement methods, such as CZT with high refinement factors, interpolation methods, and their improvements. Theoretically, as long as the spectrum is refined sufficiently, a very accurate frequency estimate can be obtained. However, the higher the refinement factor, the more complex and computationally intensive the calculation becomes. Therefore, how to reduce computational complexity and workload while ensuring the accuracy of the estimation is a problem that urgently needs to be solved by those skilled in the art. Summary of the Invention

[0005] The purpose of this invention is to provide a CZT frequency estimation method that improves estimation accuracy while reducing computational complexity and computational load.

[0006] This invention provides a CZT frequency estimation method, comprising the following steps:

[0007] S1: Sample the signal to be estimated, and perform an N-point Fast Fourier Transform on the sampled discrete-time signal to obtain the FFT spectrum function X(k). Extract the maximum spectral line value X(k) of the FFT spectrum function from it. p Frequency index k p ;

[0008] S2: Based on frequency index k p The value determines the frequency range of the CZT transform, and the CZT spectrum function X is obtained by performing the CZT transform on the discrete-time domain signal. CZT (k) is used to extract the maximum spectral line value X of the CZT spectral function. CZT (k m ) and the values ​​X of its two spectral lines to the left and right CZT (k m +1)X CZT (k m -1);

[0009] S3: Calculate the maximum spectral line value and the relationship parameter μ between the values ​​of the two spectral lines to its left and right;

[0010] S4: The error δ is calculated from the relational parameter μ;

[0011] S5: Calculate the estimated frequency of the signal to be estimated based on the relational parameter μ and the error δ.

[0012] Furthermore, in step S2, the frequency range of the CZT transform is [missing information]. Among them, f s q is the sampling frequency, and q is the size of the CZT transform interval.

[0013] Furthermore, the maximum spectral line value X of the CZT spectral function CZT (k m The following relationship is satisfied:

[0014] Where, k m Here, j represents the frequency index at the point where the amplitude of CZT is maximum, B is the bandwidth of the refined frequency range, and M is the degree of refinement of the refined frequency range.

[0015] Furthermore, the left spectral line value of the maximum spectral line value of the CZT spectral function satisfies the following relationship:

[0016]

[0017] Furthermore, the right spectral line value of the maximum spectral line value of the CZT spectral function satisfies the following relationship:

[0018]

[0019] Furthermore, the parameter μ relating the maximum spectral line value of the CZT spectral function to the values ​​of its two left and right spectral lines satisfies the following equation:

[0020]

[0021] Among them, X CZT (k m X is the maximum spectral line value of the CZT spectral function. CZT (k m +1) is the right spectral line of the maximum spectral value of the CZT spectral function, X CZT (k m -1) is the left spectral line of the maximum spectral value of the CZT spectral function.

[0022] Furthermore, the error δ satisfies the following relationship:

[0023]

[0024] Where q is the size of the CZT transform interval, and M is the degree of refinement of the refined frequency interval.

[0025] Furthermore, the relational parameter μ satisfies the following relation:

[0026]

[0027] Furthermore, the estimated frequency corresponding to the maximum spectral line value of the CZT spectral function. The following relationship must be satisfied:

[0028]

[0029] The CZT frequency estimation method of the present invention introduces a relational parameter μ by analyzing the information of the maximum spectral line and the two spectral lines to its left and right in the CZT spectrum. The magnitude of the error is directly estimated by the relational parameter μ, resulting in a more accurate frequency estimate. Under certain signal-to-noise ratio conditions, this method improves the estimation accuracy and reduces the computational complexity and amount of computation. Attached Figure Description

[0030] Figure 1 This is a flowchart of the CZT frequency estimation method according to an embodiment of the present invention. Detailed Implementation

[0031] The preferred embodiments of the present invention are given below with reference to the accompanying drawings and described in detail.

[0032] In the current FMCW radar ranging system, the transmitted signal T(t) can be expressed as:

[0033]

[0034] The received reflected signal R(t) is:

[0035]

[0036] Where f0 is the initial frequency of the linear frequency modulated wave. Let be the slope of the linear frequency modulated (LFM) wave, B be the bandwidth of the LFM wave, T be the period of the LFM wave, r be the distance between the target and the radar, and τ be the time delay caused by the distance r between the target and the radar. c is the speed of electromagnetic waves in air. For the initial phase, a t For transmission gain, a r For receiving gain, a0 = a r a t In this system, path loss and loss caused by object reflection are ignored.

[0037] After mixing the transmitted signal T(t) and the reflected signal R(t), the high-frequency components are filtered out to obtain the difference frequency signal x(t):

[0038] x(t)=a0 exp{j2π(f0τ+Kτt)} (3)

[0039] With sampling frequency f s By sampling the difference frequency signal x(t), we obtain the discrete time domain signal x(n):

[0040]

[0041] Where n is the sampling point in the discrete time domain, and a0 = a r a t a t For transmission gain, a r For receiving gain, f0 is the initial frequency of the linear frequency modulated wave, f c Let τ be the frequency of the difference frequency signal, and τ be the time delay caused by the distance r of the target relative to the radar.

[0042] Through the frequency f of the difference frequency signal c =Kr gives the target's distance r relative to the radar:

[0043]

[0044] Therefore, in order to obtain a distance with high accuracy, it is necessary to obtain the frequency of the difference frequency signal with high accuracy.

[0045] like Figure 1 As shown, this embodiment of the invention provides a CZT frequency estimation method, including the following steps:

[0046] S1: Sample the signal to be estimated, and perform an N-point Fast Fourier Transform on the sampled discrete-time signal x(n) to obtain the spectrum function, and then obtain the maximum spectral line X(k). p Frequency index k p The signal to be estimated refers to the difference frequency signal between the transmitted signal T(t) and the reflected signal R(t) of the frequency modulated continuous wave (FMCW) radar ranging system.

[0047] Based on the N-point Fast Fourier Transform formula, the FFT spectrum function X(k) is obtained as follows:

[0048]

[0049] Where N is the signal length, which can be 512, 1024, or larger, and the frequency resolution of the N-point Fast Fourier Transform is... Find the maximum amplitude of the FFT spectrum function as the maximum spectral line value of the FFT spectrum function, and take the frequency index of the point where the amplitude of the FFT spectrum function is maximum as the frequency index k of the maximum spectral line value of the FFT spectrum function. p This allows us to determine the frequency of the point where the amplitude is at its maximum.

[0050]

[0051] The frequency corresponding to the maximum amplitude of the FFT spectrum function Centered on the CZT transform, the frequency range (f1, f2) is selected, and the interval (f1, f2) containing the peak point is refined to obtain the refined frequency f:

[0052]

[0053] Among them, B CZT M represents the bandwidth of the interval (f1, f2), and M represents the refinement of the interval (f1, f2).

[0054] S2: Based on frequency index k pThe value determines the refined frequency range of CZT, and the CZT spectrum function X is obtained by performing CZT transform. CZT (k), from which the maximum spectral line value X is extracted. CZT (k m The values ​​of the left and right spectral lines X CZT (k m +1)X CZT (k m -1);

[0055] The CZT transform of a discrete-time signal yields its CZT spectrum function X(z). k The formula for ) is:

[0056]

[0057] Where z k =AW -k , θ0 is the initial sampling angle, and φ0 is the angle between two adjacent sampling points.

[0058] A0 and W0 are constant values ​​in the CZT transform and can be set according to the actual situation. In this embodiment, A0 = 1 and W0 = 1, therefore, the CZT spectrum function X(zk) obtained after the discrete-time domain signal x(n) undergoes the CZT transform is:

[0059]

[0060] k = 0, ..., M-1;

[0061] Based on the frequency index k where the amplitude of the Fast Fourier Transform is maximum p The spatial range of the CZT transform is determined, and the corresponding frequency range (f1, f2) of the CZT transform is:

[0062]

[0063] Where q is the size of the CZT transform interval, which can be set according to the actual situation, as long as it is a positive integer. The corresponding frequency resolution of the CZT transform is... According to equation (10), the corresponding CZT spectrum function X obtained after CZT transformation is... CZT (k) is:

[0064]

[0065] Find the CZT spectrum function X. CZT The maximum amplitude of (k) is taken as the maximum spectral value X. CZT (k mThe frequency index k at the point of maximum amplitude is obtained. m X is the maximum spectral line value of the CZT spectral function. CzT (k m The frequency index of the CZT spectral function can be used to derive the estimated frequency corresponding to the maximum spectral line value.

[0066]

[0067] Due to the picket fence effect in discrete signals, the maximum value of the spectral line (i.e., the maximum spectral value) is difficult to find relative to the frequency f to be measured. c Overlap, meaning that regardless of the frequency index corresponding to the maximum spectral line value of the FFT spectrum function. Or the frequency index corresponding to the maximum spectral line value of the CZT spectral function. Both are related to the frequency f to be measured c There is a certain degree of error. In this embodiment, the frequency to be measured, f... c It can be represented as:

[0068]

[0069] Among them, k p This is the frequency index of the point where the Fast Fourier Transform amplitude is maximized. To select a refined frequency bandwidth for CZT, k m For the frequency index corresponding to the maximum amplitude of the CZT transform in the refined interval, the error δ∈[-0.5, 0.5] is the fractional part. Since τ satisfies... Combining equation (4), we can obtain τ and f c The relationship between them satisfies:

[0070]

[0071] The maximum spectral line value X of the CZT spectrum function obtained after the CZT transform. CZT (k m )for:

[0072]

[0073] Left spectral line value X CZT (k m -1) is:

[0074]

[0075] Right spectral line value X CZT (k m +1) is:

[0076]

[0077] S3: Calculate the maximum spectral line value and the relationship parameter μ between the values ​​of the two spectral lines to its left and right;

[0078]

[0079] Substituting equations (16), (17), and (18) into equation (19), we get:

[0080]

[0081] After simplification, we get:

[0082]

[0083] S4: The error δ is calculated from the relational parameter μ;

[0084] The relationship between μ and δ can be derived from the formula as follows:

[0085]

[0086] When N is large and δ is small therefore,

[0087]

[0088] Therefore, the approximate value of μ can be obtained as:

[0089]

[0090] The corresponding δ values ​​can be estimated from the left and right spectral lines using equations (22) and (24).

[0091] S5: The estimated frequency is calculated based on the relational parameter μ and the error δ.

[0092] Once the relational parameters μ and error δ are determined, the corresponding estimated frequency can be obtained through equation (14), and then the corresponding estimated distance value can be obtained according to equation (5).

[0093] To verify the effectiveness of the frequency estimation method of this invention, simulations will be performed below using N-point FFT transform, 16-fold refined spectrum (range q=2), 32-fold refined spectrum CZT (2-32CZT, range q=2), the CZT frequency estimation method of this invention (2-32CZT+, 32-fold refined spectrum CZT, range q=2), zoomfft (frequency shift set to 2000Hz based on data), zero-padding method (since the effect of improving accuracy is not significant when the number of zero-padding points is small, the number of zero-padding points is set to 6*N points), and RIFE under different signal-to-noise ratios to obtain the mean and variance of the estimated range of different FMCW radars.

[0094] The simulation parameters are as follows:

[0095] Sampling rate f s = 92.7835e3Hz, number of sampling points N = 1024, initial frequency f0 = 100kHz, frequency modulation bandwidth B = 999.47055MHz, period of linear frequency modulation wave The slope of a linear frequency modulated wave initial phase Send gain a t =1, then from equation (1), the transmitted signal is:

[0096]

[0097] Assuming the distance between the stationary object being measured and the FMCW system is r, and the speed of electromagnetic waves in air is c = 299709 km / s, then the time delay caused by the distance r... Receive gain a r =1, ignoring path loss and loss caused by object reflection, the received reflected signal can be obtained from equation (2):

[0098]

[0099] After mixing T(t) and R(t) and filtering out the high-frequency components, the difference frequency sampling signal is obtained. From equation (4), the difference frequency sampling signal is:

[0100]

[0101] When the sampled signal is affected by noise, the difference frequency sampled signal is:

[0102]

[0103] w(n) represents noise. Different signal-to-noise ratios have different effects on the final calculated distance. By setting different distances r and signal-to-noise ratios, the corresponding difference frequency sampling signal x(n) is obtained, and different frequency estimation methods are used to estimate the frequency. Then, the corresponding estimated distance is obtained based on the frequency.

[0104] It should be noted that although this embodiment uses FMCW radar ranging as an example to illustrate the specific steps of the frequency estimation method of the present invention, it is not limited to FMCW radar ranging and can be extended to other frequency estimation applications.

[0105] In this embodiment, the actual distance value is r = 2.57m. Experiments were conducted within a signal-to-noise ratio range of [-16dB, 16dB]. 10,000 Monte Carlo experiments were performed independently under different signal-to-noise ratios. The mean estimated distance is shown in Table 1, and the variance of the estimated distance is shown in Table 2.

[0106] Table 1. Mean estimated distances (m) obtained from simulations under different signal-to-noise ratios.

[0107]

[0108]

[0109] Table 2. Estimated distance variance (m) obtained from simulations under different signal-to-noise ratios.

[0110]

[0111] As shown in Tables 1 and 2, when the signal-to-noise ratio (SNR) is low (less than -15dB), the signal is significantly affected by noise, resulting in a substantial impact on the estimation accuracy. When the SNR is greater than -15dB, the performance of the CZT frequency estimation method (i.e., 2-32CZT+) of this invention remains consistently good, and its fluctuation range is also close to the true distance while maintaining a relatively close approximation between the estimated average distance and the true distance. Under certain SNR conditions, compared with other methods, the CZT frequency estimation method of this invention improves the estimation accuracy while reducing computational complexity and computational load, demonstrating superior estimation capabilities compared to other methods.

[0112] The above description is merely a preferred embodiment of the present invention and is not intended to limit the scope of the invention. Various variations can be made to the above embodiments of the present invention. That is, all simple and equivalent changes and modifications made based on the claims and description of this invention fall within the protection scope of the claims of this patent. All aspects not described in detail in this invention are conventional technical content.

Claims

1. A method of CZT frequency estimation, characterized by, The method comprises the following steps: S1: sampling the to-be-estimated signal, and performing N-point fast Fourier transform on the sampled discrete time domain signal to obtain an FFT spectrum function X(k), from which a frequency index k of a maximum spectrum line value X(k p ) of the FFT spectrum function is extracted p ; S2: determine the frequency interval of the CZT transform according to the value of the frequency index k p , and make CZT transform on the discrete time domain signal to obtain the CZT spectrum function X CZT (k), from which the maximum spectrum line value X CZT (k m ) and its left and right two spectrum line values X CZT (k m +1), X CZT (k m -1) of the CZT spectrum function are extracted. S3: calculating a relationship parameter μ between the maximum spectral line value and the spectral line values on the left and right of the maximum spectral line value; S4: calculating an error δ from the relationship parameter μ; S5: Calculate the estimated frequency of the signal to be estimated according to the relationship parameter μ and the error δ The relationship parameter μ between the maximum spectral line value of the CZT spectrum function and the spectral line values on the left and right of the maximum spectral line value satisfies the following relationship formula: where X(k) is the maximum spectral line value of the CZT spectrum function, CZT (k m ) is the right spectral line of the maximum spectral line value of the CZT spectrum function, CZT (k m +1) is the left spectral line of the maximum spectral line value of the CZT spectrum function, CZT (k m -1) is the left spectral line of the maximum spectral line value of the CZT spectrum function. The error δ satisfies the following relationship formula: Wherein, q is the size of the CZT transformation interval, and M is the refinement degree of the refined frequency interval; the estimated frequency corresponding to the maximum spectral line value of the CZT spectral function satisfies the following relationship:

2. The CZT frequency estimation method of claim 1, wherein, In the step S2, the frequency interval range of the CZT transformation is where f s is the sampling frequency and q is the size of the CZT transformation interval.

3. The CZT frequency estimation method of claim 2, wherein, The maximum spectral line value X of the CZT spectral function CZT (k m ) satisfies the following relationship: where k m is the frequency index of the maximum amplitude of the CZT, j represents the imaginary unit, B is the bandwidth of the refined frequency interval, and M is the refinement degree of the refined frequency interval.

4. The CZT frequency estimation method of claim 3, wherein, The left spectral line value of the maximum spectral line value of the CZT spectrum function satisfies the following relationship formula:

5. The CZT frequency estimation method of claim 4, wherein, The right spectral line value of the maximum spectral line value of the CZT spectrum function satisfies the following relationship formula:

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