A discrete Fourier transform and difference frequency measurement method

Abstract

Due to the point limit of the discrete Fourier transform (DFT), the frequency measurements of a signal's line spectrum are limited to frequency bins. This paper proposes a sum-and-difference frequency measurement method to improve frequency measurement accuracy. This method includes two parts: establishing a normalized frequency deviation table and a frequency calculation method.

Description

Technical Field

[0001] The present invention belongs to the technical field of radar information processing, and in particular relates to a discrete Fourier transform and difference frequency measurement method. Background Art

[0002] Frequency estimation is a fundamental digital signal processing technique widely used across various industries. Applications include frequency estimation in power systems, modeling standard musical instrument signals in music, orthogonal frequency division multiplexing systems in communications, frequency analysis and fault diagnosis of vibration response signals from mechanical equipment, and estimation of target Doppler frequency in radar signal processing. The discrete Fourier transform (DFT) is a commonly used method for frequency estimation, but the number of DFT points limits frequency measurement accuracy. To improve frequency measurement accuracy, interpolation is a common method, but the increased computational complexity makes it difficult to meet practical requirements. Among currently used methods, weighting frequency points near the measured frequency by amplitude can significantly improve frequency measurement accuracy. This method requires minimal computation, but measurement accuracy varies significantly across the frequency range. With the increasing sophistication of applications, demand for frequency measurement accuracy is increasing across various industries, and currently used frequency measurement methods are struggling to meet these demands. Therefore, finding a frequency measurement method with higher accuracy and lower computational complexity is an urgent need. Summary of the Invention

[0003] The present invention aims to implement a discrete Fourier transform (DFT) and difference frequency measurement method to improve the accuracy of discrete Fourier transform (DFT) frequency measurements. Due to the point limit of the discrete Fourier transform (DFT), the frequency measurement value of a signal line spectrum is limited to frequency units. The present invention proposes a discrete Fourier transform and difference frequency measurement method to improve the frequency measurement accuracy. The invention includes two parts: establishing a normalized frequency deviation table and a frequency calculation method.

[0004] A discrete Fourier transform and difference frequency measurement method, the specific steps are as follows:

[0005] Step 1: Create a normalized frequency deviation table

[0006] Step 1.1, construct signal s(n)

[0007] Using the discrete single frequency signal of formula (1):

[0008]

[0009] Where A is the signal amplitude, j is the imaginary unit, f0 is the signal frequency, and T = 1 / f s is the sampling interval, f s is the sampling rate, n=0,1,…,L-1 is the sampling point index, L is the number of sampling points, is the normalized frequency, N is an integer power of 2 and N≥L.

[0010] Select a positive integer n0 such that n0 < N, and select the number of points M of the constructed table. For m = 0, 1, …, M - 1, respectively let

[0011]

[0012] and construct the signal s(n) according to Equation (1).

[0013] Step 1.2: Perform Steps 1.2.1 to 1.2.4 on each s(n) to obtain p(m) and the normalized frequency deviation q(m) required for tabulation;

[0014] Step 1.2.1: Use the left and right halves of s(n) to construct the signals s L (n), s R (n)

[0015] When L is even, denote

[0016]

[0017] When L is odd, denote

[0018]

[0019] Step 1.2.2: Window and zero-pad s L (n) and s R (n) respectively, and perform N-point DFT to obtain S L ))(k), S R ) (k)

[0020] Denote the window function as W(n), n = 0, 1, …, L - 1. Windowing and zero-padding s L (n) and s R (n) respectively gives:

[0021]

[0022]

[0023] Then perform N-point DFT on and respectively:

[0024]

[0025] where k = 0, 1, …, N - 1.

[0026] Step 1.2.3: Calculate S L (k) and S RThe sum and difference of (k) gives S + (k), S - (k)

[0027] Calculate S separately L (k) and S R The sum and difference of (k) are expressed as:

[0028] S + (k) = S L (k)+S R (k) (11)

[0029] S - (k) = S L (k)-S R (k) (12)

[0030] Step 1.2.4. Calculate S - (k) and S + (k) Absolute value of the imaginary part of the ratio p(m) and normalized frequency deviation q(m)

[0031] Due to symmetry, when k=n0, S - (k) and S + (k) is orthogonal. Thus, calculate S - (k) and S + (k) The absolute value of the imaginary part of the ratio p(m):

[0032] p(m)=|imag{S - (n0) / S + (n0)}| (13)

[0033] Record the normalized frequency deviation at this time:

[0034]

[0035] Step 1.3: After processing all m=0,1,…,M-1, set p=[p(0),p(1),…,p(M-1)] T and q=[q(0),q(1),…,q(M-1)] T , p and q are matched one by one, a frequency deviation table is made and stored for subsequent use.

[0036] Step 2: Frequency calculation method

[0037] Step 2.1: For the convenience of representation, assume that the discrete signal is:

[0038] s(n)=Aexp(j2πfnT)+v(nT) (15)

[0039] Where n = 0, 1, ..., L-1 is the sampling point index, L is the number of sampling points, f is the frequency to be estimated, and v(nT) is Gaussian white noise.

[0040] Step 2.2: Same as steps 1.2.1 to 1.2.3, construct S - (k) and S + (k).

[0041] Step 2.3, Search | S + (k)|, record its index as k0, calculate S at k0 - (k) and S + (k) Absolute value of the imaginary part of the ratio:

[0042]

[0043] Step 2.4, load the stored frequency deviation table, according to The two closest values ​​of p are selected and recorded as p(n1) and p(n2) respectively, and the corresponding frequency deviations are recorded as q(n1) and q(n2) respectively.

[0044] Step 2.5: Calculate the normalized frequency by linear interpolation

[0045]

[0046] In the formula, when imag{S - (k0) / S + (k0)}≥0, the ± in formula (17) is taken as +, when imag{S - (k0) / S + When (k0)}<0, ± in formula (17) is -.

[0047] Finally calculate the estimate of the signal frequency f

[0048]

[0049] The beneficial effects of the present invention are:

[0050] The method of the present invention has higher accuracy than the currently commonly used three-point amplitude weighting method.

[0051] Figure 1 The figure shows the root mean square error (RMSE) of normalized frequency measurements corresponding to different normalized frequency deviations on the whole scale, using the frequency deviation table of Example 1, when the signal-to-noise ratio is 6 dB. It can be seen that the accuracy of the three-point amplitude weighting method is only better than that of the method of the present invention when the deviation from the whole scale is large.

[0052] Figure 2 The RMSE values ​​for frequency measurements at different signal-to-noise ratios (SNRs) are shown using the frequency deviation table from Example 1, with a uniform distribution of signal frequencies between 1 MHz and 2 MHz. It can be seen that both methods perform poorly when the SNR is below -4 dB. When the SNR is greater than or equal to -4 dB, the RMSE of the method of the present invention is significantly lower than that of the three-point amplitude-weighted method, and gradually approaches zero as the SNR increases. These experimental results demonstrate that the method of the present invention has higher accuracy than the three-point amplitude-weighted method. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 Normalized frequency measurement RMSE under different normalized frequency deviations

[0054] Figure 2 Normalized frequency measurement RMSE at different SNRs DETAILED DESCRIPTION

[0055] The technical solutions provided by the present invention will be described in detail below with reference to specific embodiments. It should be understood that the following specific embodiments are only used to illustrate the present invention and are not used to limit the scope of the present invention.

[0056] Example 1: Creating and storing a frequency deviation table

[0057] (1) Discrete single-frequency signal used:

[0058]

[0059] Where n=0,1,…,116 is the sampling point index, L=117 is the number of sampling points, is the normalized frequency, f0 is the signal frequency, f s is the sampling rate, N=128.

[0060] (2) Select an integer n0 = 30, select the number of points in the table to be built as M = 61, and for m = 0, 1, ..., M-1, let the normalized frequency

[0061]

[0062] And construct the signal s(n) according to formula (19), execute (3) to (6) for each s(n) to obtain the p(m) and q(m) required for tabulation.

[0063] (3) Use the left and right halves of s(n) to construct the signal respectively. Since L = 117 is an odd number, we can get from step 3:

[0064]

[0065] (4) The window function uses a Hamming window, denoted as W(n), n = 0, 1, ..., 116, for s L (n) and s R (n) are windowed and filled with zeros to obtain:

[0066]

[0067] Then respectively and Perform N=128-point DFT:

[0068]

[0069] Where k = 0, 1,…, N-1.

[0070] (5) Calculate S respectively L (k) and S R The sum and difference of (k) are expressed as:

[0071] S + (k) = S L (k)+S R (k) (27)

[0072] S - (k) = S L (k)-S R (k) (28)

[0073] (6) Calculate S at k = n0 = 30 - (k) and S + (k) Absolute value of the imaginary part of the ratio:

[0074] p(m)=|imag{S - (n0) / S + (n0)}| (29)

[0075] Record the frequency deviation at this time:

[0076]

[0077] (7) After processing all frequency deviations, let p = [p(0), p(1), ..., p(M-1)] T and q=[q(0),q(1),…,q(M-1)] T , p and q are mapped one by one, and a frequency deviation table (see Table 1) is prepared and stored for subsequent use.

[0078] Table 161-point normalized frequency deviation table

[0079]

[0080] Example 2: Frequency Measurement Method Example

[0081] (1) Using discrete signals:

[0082] s(n)=exp(j2πfnT)+v(nT) (31)

[0083] Where, f = 1MHz, sampling rate f s =5MHz, T=1 / f s =0.2us, number of sampling points L=117, n=0,1,…,N-1, v(nT) is Gaussian white noise, and the signal-to-noise ratio is 6dB.

[0084] (2) Use the left and right halves of s(n) to construct the signal respectively. Since N is an odd number,

[0085]

[0086] (3) Using a Hamming window, denoted as W(n), n = 0, 1, ..., 116, for s L (n) and s R (n) are windowed and filled with zeros to obtain:

[0087]

[0088]

[0089] Then respectively and Perform N=128-point DFT:

[0090]

[0091] Where k = 0, 1,…, N-1.

[0092] (4) Calculate S respectively L (k) and S R The sum and difference of (k) are expressed as:

[0093] S + (k) = S L (k)+S R (k) (38)

[0094] S - (k) = S L (k)-S R (k) (39)

[0095] (5) Search | S + (k)|, its index is k=26, calculate S at k - (k) and S +(k) Absolute value of the imaginary part of the ratio:

[0096]

[0097] (6) Load the frequency deviation table stored in Example 1 (see Table 1), according to The value of selects the two closest values ​​in p. Among them, The two closest values ​​of p are selected and recorded as p(n1)=0.398565 and p(n2)=0.40745, and the corresponding frequency deviations q(n1)=0.408333 and q(n2)=0.416667 are recorded.

[0098] (7) Calculate the value of imag{S - (k1) / S + (k1)}<0, calculate the normalized frequency according to formula (18)

[0099]

[0100] Then, we can calculate the estimate of frequency f

[0101] The above description is only the best specific implementation method of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or replacements that can be easily thought of by any technician familiar with this technical field within the technical scope disclosed by the present invention should be covered by the scope of protection of the present invention.

[0102] The contents not described in detail in the specification of the present invention belong to the common knowledge of professionals in this field.

Claims

1. A discrete Fourier transform and difference frequency measurement method, characterized in that: The method steps are as follows: Step 1: Establish a normalized frequency deviation table Step 1.1: Construct a signal s(n), where n = 0, 1, …, L - 1 is the sampling point index and L is the number of sampling points; Step 1.2: For each s(n), execute Step 1.2.1 to Step 1.2.4 to obtain p(m) and the normalized frequency deviation q(m) required for tabulation, where m = 0, 1, …, M - 1 and M is the number of points in the constructed table; Step 1.2.

1. Use the left and right halves of s(n) to construct the signal s L (n), s R (n); Step 1.2.2, for s L (n) and s R (n) Add window and fill zero respectively, perform N-point DFT, and get S L (k), S R (k); Step 1.2.3, calculate S respectively L (k) and S R The sum and difference of (k) gives S + (k), S - (k); Step 1.2.

4. Calculate S - (k) and S + (k) the absolute value of the imaginary part of the ratio p(m) and the normalized frequency deviation q(m); Step 1.3: After processing all m=0,1,…,M-1, set p=[p(0),p(1),…,p(M-1)] T and q=[q(0),q(1),…,q(M-1)] T , match p and q one by one, make a frequency deviation table and store it; Step 2: Frequency calculation method Step 2.1: For convenience of representation, assume the discrete signal is: s(n) = Aexp(j2πfnT) + v(nT) (15) Where n = 0, 1, ..., L-1 is the sampling point index, L is the number of sampling points, A is the signal amplitude, j is the imaginary unit, and T = 1 / f s is the sampling interval, f s is the sampling rate, f is the frequency to be estimated, and v(nT) is Gaussian white noise; Step 2.2: Same as steps 1.2.1 to 1.2.3, construct S - (k) and S + (k); Step 2.3, Search | S + (k)|, record its index as k0, calculate S at k0 - (k) and S + (k) Absolute value of the imaginary part of the ratio: Step 2.4, load the stored frequency deviation table, according to Select the two closest values ​​of p in p, record them as p(n1) and p(n2), and record the corresponding frequency deviations, record them as q(n1) and q(n2); Step 2.5: Calculate the normalized frequency by linear interpolation Finally calculate the estimate of the signal frequency f where N is an integer power of 2 and N ≥ L.

2. The method according to claim 1, characterized in that In Step 1.1, use the discrete single-frequency signal of Equation (1): Where A is the signal amplitude, j is the imaginary unit, f0 is the signal frequency, and T = 1 / f s is the sampling interval, f s is the sampling rate, n=0,1,…,L-1 is the sampling point index, L is the number of sampling points, is the normalized frequency, N is an integer power of 2 and N ≥ L; Select a positive integer n0 such that n0 < N, and select the number of points M in the constructed table; for m = 0, 1, …, M - 1, respectively let and construct the signal s(n) according to Equation (1).

3. The method according to claim 1, characterized in that In Step 1.2.1, when L is even, denote When L is odd, denote 4. The method according to claim 1, wherein In step 1.2.2, let the window function be W(n), n=0,1,…,L-1, for s L (n) and s R (n) are windowed and filled with zeros to obtain: Then respectively and Perform N-point DFT: where k = 0, 1, …, N - 1.

5. The method according to claim 1, characterized in that In step 1.2.3, calculate S L (k) and S R The sum and difference of (k) are expressed as: S + (k)=S L (k)+S R (k) (11) S - (k)=S L (k)-S R (k) (12)。 6. The method according to claim 1, characterized in that In step 1.2.4, due to symmetry, when k = n0, S - (k) and S + (k) orthogonal; thus, calculate S - (k) and S + (k) The absolute value of the imaginary part of the ratio p(m): p(m)=|imag{S - (n0) / S + (n0)}| (13) Record the normalized frequency deviation at this time:

7. The method according to claim 1, characterized in that In Step 2.5, In the formula, when imag{S - (k0) / S + (k0)}≥0, the ± in formula (17) is taken as +, when imag{S - (k0) / S + When (k0)}<0, ± in formula (17) is -.

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