Non-cooperative signal fractional order time difference estimation method based on band-limited signal reconstruction

Abstract

The invention belongs to the technical field of non-cooperative signal processing, and particularly relates to a non-cooperative signal fractional order time difference estimation method based on band-limited signal reconstruction. According to the method, signal integer sampling point deviation estimation, namely coarse time difference estimation, is completed firstly, and then fractional sampling point deviation estimation, namely fine time difference estimation, is completed. Fractional order time difference estimation performance evaluation of the method under different signal-to-noise ratios and different interpolation points is achieved through Monte Carlo simulation, and good fractional order time difference estimation can be achieved through the method.

Description

Technical Field

[0001] The invention belongs to the technical field of non-cooperative signal processing, and in particular relates to a non-cooperative signal fractional-order time difference estimation method based on band-limited signal reconstruction. Background Art

[0002] Signal time difference refers to the time delay difference between the same source signals received on different paths. In the field of signal processing, signal time difference is an important concept, which is crucial to understanding the propagation and reception characteristics of signals. Under the influence of signal sampling rate and noise, the accuracy of the time difference estimation between signals will be reduced. Summary of the invention

[0003] In view of the above problems, the present invention proposes a method for estimating fractional time difference of non-cooperative signals based on band-limited signal reconstruction. The method of the present invention first completes the signal integer sampling point deviation estimation, that is, the coarse time difference estimation, and then completes the fractional sampling point deviation estimation, that is, the fine time difference estimation. Finally, Monte Carlo simulation is used to implement the performance evaluation of fractional time difference estimation under different signal-to-noise ratios and different interpolation points.

[0004] The technical solution adopted by the present invention is:

[0005] A non-cooperative signal fractional-order time difference estimation method based on band-limited signal reconstruction comprises the following steps:

[0006] S1. Receive the source signal s[n] through two receivers and obtain the received signal x 1 [n] and x 2 [n]:

[0007] x 1 [n]=s[n]+q 1 [n]

[0008] x 2 [n]=s[nD]+q 2 [n]

[0009] Among them, q 1 [n], q 2 [n] is zero-mean Gaussian white noise and is uncorrelated with the source signal s[n], x 1 [n] is the reference signal observation, x 2 [n] is the delayed signal observation, n = 1, 2, ..., N, N is the number of signal samples, D is x 1 [n] and x 2 [n] The time difference of integer multiples of the sampling interval;

[0010] Calculate the signal x 1 [n] and x 2 The cross-correlation function of [n] is:

[0011]

[0012] Where E{·} represents averaging, * represents conjugation, and m represents time offset. 1 [n] and x 2 Substituting the expression of [n] into the cross-correlation function and simplifying it, we get:

[0013]

[0014] It can be obtained that when m = D, the correlation function Get the maximum value and estimate it using time average:

[0015]

[0016] Among them, 2M+1 is the order of cross-correlation, M≤N-1; By performing a peak search, we can obtain the integer-order time difference between the two signals;

[0017] S2. After obtaining the integer-order time difference, perform sinc function interpolation based on the correlation function value calculated near the correlation peak, and search for fractional-order time difference within the range of 0.5 around the correlation peak. Once the search peak is found, the fractional-order time difference estimation can be obtained. Specifically:

[0018] Perform sinc interpolation near the correlation peak, expressed as:

[0019]

[0020] Among them, 2W is the bandwidth of the signal, R 0 (k) is the cross-correlation function obtained in S1, k 0 Corresponding to R 0 (k), M represents the number of function values ​​around the correlation peak, Δ represents the fractional delay around the peak, when Δ is equal to the real fractional delay, the corresponding R(Δ) function reaches the maximum value, and the corresponding derivative function reaches the minimum value;

[0021] Taking the derivative of Δ, we get:

[0022]

[0023] When the derivative function is 0 within the value range, the value of the corresponding independent variable Δ is the corresponding fractional-order delay.

[0024] The cross-correlation function method is the most common algorithm for signal time difference estimation, but due to the limitation of sampling rate, the cross-correlation function can only estimate the signal time difference to integer multiple sampling intervals. Fractional order time difference estimation can be performed after using sinc interpolation.

[0025] The beneficial effect of the present invention is that, under certain conditions, the method of the present invention can perform more accurate time difference estimation. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] Figure 1 It is a processing scheme diagram of the present invention;

[0027] Figure 2 It is the principle block diagram of the autocorrelation function algorithm;

[0028] Figure 3 It is the correlation function diagram when the linear frequency modulation delay is 403.2 microseconds;

[0029] Figure 4 This is the derivative function diagram of the sinc interpolation function when the linear frequency modulation delay is 403.2 microseconds (M=6, no noise);

[0030] Figure 5 This is the derivative function diagram of the sinc interpolation function when the linear frequency modulation delay is 403.2 microseconds (M=4, no noise);

[0031] Figure 6 The derivative function diagram of the sinc interpolation function when the linear frequency modulation delay is 403.2 microseconds (M = 4, signal-to-noise ratio = 0dB);

[0032] Figure 7 This is the graph showing the variation of the root mean square error of the linear frequency modulation signal delay estimation with the signal-to-noise ratio;

[0033] Figure 8 This is the correlation function diagram when the BPSK delay is 507.8 microseconds;

[0034] Fig. 9 The derivative function diagram of the sinc interpolation function when the BPSK delay is 507.8 microseconds (M=6, no noise);

[0035] Fig.10 The derivative function diagram of the sinc interpolation function when the BPSK delay is 507.8 microseconds (M=8, no noise);

[0036] Fig.11 The derivative function diagram of the sinc interpolation function when the BPSK delay is 507.8 microseconds (M = 8, signal-to-noise ratio = 0dB);

[0037] Fig.12 This is the graph of the BPSK signal delay root mean square error changing with the signal-to-noise ratio (the signal-to-noise ratio ranges from -10dB at intervals of 2dB to 4dB). Fig.13 This is the graph of the BPSK signal delay root mean square error changing with the signal-to-noise ratio (the signal-to-noise ratio ranges from -10dB at intervals of 0.5dB to -6dB). DETAILED DESCRIPTION

[0038] The present invention will be further described below in conjunction with the accompanying drawings.

[0039] like Figure 1 As shown in FIG. 1 , the processing flow of the method of the present invention from signal transmission to finding the accurate fractional-order time difference is shown. After receiving the signal, the overall processing method is divided into the following two steps:

[0040] S1. Cross-correlation function method

[0041] The principle of the cross-correlation function method is as follows Figure 2 As shown, the cross-correlation function between the two signals is calculated. The cross-correlation function measures the similarity between the two signals and is achieved by calculating the expected value of the product of the two signals. By analyzing the cross-correlation function, its maximum value, that is, the delay p corresponding to the peak-to-peak value of the correlation, is found. This delay is the integer-order time difference between the two signals. The algorithm is simple to calculate and relatively easy to implement.

[0042] The two receivers receive two signals of the signal x(t) after spatial propagation, and the discrete signals obtained after sampling are:

[0043]

[0044] Among them, the source signal s[n] and the zero-mean Gaussian white noise q 1 [n], q 2 [n] are not related to each other. 1 [n] is the reference signal observation, x 2 [n] is the delayed signal observation. n=1,2,…,N, N is the number of signal samples. D is the time difference between two discrete time signals that is an integer multiple of the sampling interval.

[0045] Calculate the signal x 1 [n] and x 2 The cross-correlation function of [n] is

[0046]

[0047] Where, E{·} means finding the mean value, and * means taking the conjugate. Substituting the two signals into the above formula, we can further obtain

[0048]

[0049] Since the signal and noise, as well as the noise and noise, are uncorrelated, the above equation can be simplified to obtain

[0050]

[0051] According to the related function properties r ss [m]≤r ss [0] It can be seen that when m = D, the correlation function Get the maximum value. In practice, the correlation function of a stationary random process can be estimated by using time averaging, that is,

[0052]

[0053] Among them, 2M+1 is the order of cross-correlation, M≤N-1. By performing a peak search, the time difference between the two signals can be obtained.

[0054] S2. After finding the correlation peak, perform fractional time difference estimation based on sinc interpolation near the correlation peak

[0055] When there is a fractional time difference, the sinc function can be interpolated according to the correlation function value calculated near the correlation peak, and the fractional time difference can be searched within a range of 0.5 around the correlation peak. Once the search peak is found, the fractional time difference estimate can be obtained.

[0056] The following explains the basis and formula results of this method

[0057] It is known that the bandwidth of a signal is 2W, that is

[0058]

[0059] Then the signal can be expressed as the sampled signal:

[0060]

[0061] For x(t) and its delayed signal x(tD), after sampling and obtaining their correlation functions, the function values ​​near the correlation peak can be used for sinc interpolation. The formula is:

[0062]

[0063] Taking the derivative of Δ, we get the formula:

[0064]

[0065] When equation (9) is 0 within the range of values, the value of the corresponding independent variable Δ is the corresponding fractional-order delay. 0 (n) is the correlation function obtained in S1, k 0 Corresponding to R 0 (n), M represents the number of function values ​​around the correlation peak, Δ represents the fractional-order delay around the peak, and when Δ is equal to the true fractional-order delay, the corresponding R(Δ) function achieves the maximum value, and its derivative function achieves the minimum value.

[0066] The present invention is described in detail with reference to examples:

[0067] 1. The signal is a linear frequency modulation signal, and the parameters are set to bandwidth of 50000, sampling rate of 100000, sampling time of 0.1 seconds, and carrier frequency of the signal of 0. This parameter sets the sampling interval to 10 microseconds, sets the time difference of the signal to 403.2 microseconds, does not set the noise, and interpolates the function values ​​of the 6 points on the left and right of the correlation peak.

[0068] 2. The signal is a linear frequency modulation signal, and the parameters are set to bandwidth 50000, sampling rate 100000, sampling time 0.1 seconds, and the carrier frequency of the signal is 0. This parameter sets the sampling interval to 10 microseconds, sets the time difference of the signal to 403.2 microseconds, does not set noise, and takes the function value interpolation of the 4 points on the left and right of the correlation peak

[0069] 3. The signal is a linear frequency modulation signal, and the parameters are set to bandwidth 50000, sampling rate 100000, sampling time 0.1 seconds, and the carrier frequency of the signal is 0. This parameter sets the sampling interval to 10 microseconds, sets the signal time difference to 403.2 microseconds, sets the noise to 0dB, and interpolates the function values ​​of the four points on the left and right of the correlation peak.

[0070] 4. The signal is a linear frequency modulation signal, and the parameters are set to bandwidth 50000, sampling rate 100000, sampling time 0.1 seconds, and the carrier frequency of the signal is 0. This parameter sets the sampling interval to 10 microseconds, sets the signal time difference to 403.2 microseconds, sets the noise, and the signal-to-noise ratio from -10dB interval 2dB to 4dB, and interpolates the function values ​​of 2, 4, 8, 16, and 32 points around the correlation peak. Set the number of Monte Carlo to 1000 times,

[0071] 5. The signal is BPSK signal, the parameters are set to bandwidth 50000, sampling rate 100000, sampling time 10 milliseconds, and signal carrier frequency 100000. This parameter sets the sampling interval to 10 microseconds, sets the signal time difference to 507.8 microseconds, does not set noise, and interpolates the function values ​​of the 6 points around the correlation peak.

[0072] 6. The signal is BPSK signal, the parameters are set to bandwidth 50000, sampling rate 100000, sampling time 10 milliseconds, and signal carrier frequency 100000. This parameter sets the sampling interval to 10 microseconds, sets the signal time difference to 507.8 microseconds, does not set noise, and interpolates the function values ​​of 8 points on the left and right of the correlation peak.

[0073] 7. The signal is BPSK signal, the parameters are set to bandwidth 50000, sampling rate 100000, sampling time 10 milliseconds, and signal carrier frequency 100000. This parameter sets the sampling interval to 10 microseconds, sets the signal time difference to 507.8 microseconds, sets the noise to 0dB, and interpolates the function values ​​of 8 points around the correlation peak.

[0074] 8. The signal is BPSK signal, the parameters are set to bandwidth 50000, sampling rate 100000, sampling time 10 milliseconds, and signal carrier frequency 100000. This parameter sets the sampling interval to 10 microseconds, sets the signal time difference to 507.8 microseconds, sets the noise, and the signal-to-noise ratio from -10dB interval 2dB to 4dB, and interpolates the function values ​​of 2, 4, 8, 16, and 32 points around the correlation peak. Set the number of Monte Carlo to 1000 times.

[0075] 9. The signal is BPSK signal, the parameters are set to bandwidth 50000, sampling rate 100000, sampling time 10 milliseconds, and signal carrier frequency 100000. This parameter sets the sampling interval to 10 microseconds, sets the signal time difference to 507.8 microseconds, sets the noise, and the signal-to-noise ratio from -10dB interval 0.5dB to -6dB, and interpolates the function values ​​of 2, 4, 8, 16, and 32 points around the correlation peak. Set the number of Monte Carlo to 1000 times,

[0076] Time difference estimation effect:

[0077] In Example 1, Figure 3 It can be seen that the correlation peak is at 9960 points, the center point is 10000, and the integer point delay is 40 points. At this time, the 6 points around the correlation peak are interpolated, and the formula solves the fractional delay as 40.314828818493942857565402173274. Figure 4 It is a traversal image within [-0.5, 0.5]. It can be seen that the peak is at 0.315, that is, the fractional-order delay obtained by the traversal search is 0.315, which adds up to 40.315 points of delay, which is close to the actual 40.32 points of delay. At the same time, the formula settlement result is basically the same as the traversal search result.

[0078] Figure 5 , Figure 6 Represents the result after changing the number of interpolation points and adding noise. Figure 5 Under the corresponding conditions, the result of the formula solution is 40.320425216457615465973107875154. Figure 6Under the corresponding conditions, the result of the formula solution is 40.317260643854045909003600539862. It can be seen that under different conditions, the formula solution results are basically the same as the search results. At the same time, changing the number of interpolation points and the presence of noise will affect the accuracy of the estimation.

[0079] Figure 7 The influence of the number of interpolation points in the algorithm and the estimation performance of the algorithm under different signal-to-noise ratios are shown. The vertical axis is the logarithmic coordinate with the logarithm of 10. Note that under the condition of 1000 Monte Carlo times, the number of interpolation points around the peak M = 2, 4, 8, 16, and 32 correspond to running times of 82.129964 seconds, 124.271081 seconds, 207.014066 seconds, 368.776851 seconds, and 690.603317 seconds, respectively. It can be seen from the figure that M = 16 and M = 32 have similar estimation accuracy performance, but M = 16 is better in terms of running time.

[0080] Figure 8 It represents the correlation function in Example 5 and the function value near the correlation peak. At this time, the correlation peak position is 1949, the center position is 2000, and the estimated integer point delay is 51. Then the fractional order estimation is performed. Fig. 9 Under the conditions, the formula solution result is 50.779852715040106538598614405543, the traversal search result fractional delay is -0.22, and the final delay is 50.78, which is the same as the actual delay. Under the BPSK signal conditions, the formula settlement result is basically the same as the traversal search result.

[0081] Likewise, Fig.10 , Fig.11 Represents the result after changing the number of interpolation points and adding noise. Fig.10 Under the corresponding conditions, the result of the formula solution is 50.780108456548195933492609642892. Fig.11 Under the corresponding conditions, the result of the formula solution is 50.799200206372089713186291249064. It can be seen that under different conditions, the formula solution results are basically the same as the search results. At the same time, changing the number of interpolation points and the presence of noise will affect the accuracy of the estimation.

[0082] Fig.12 The influence of the number of interpolation points in the algorithm and the estimation performance of the algorithm under different signal-to-noise ratios are shown, and the vertical axis is a logarithmic coordinate with a logarithm of 10. Unlike linear frequency modulation signals, the estimation accuracy of BPSK signals does not differ much when M takes different values ​​after the signal-to-noise ratio is greater than -8, and there is a significant difference when the signal-to-noise ratio is -8.

[0083] Fig.13 The influence of the selection of the number of interpolation points of the algorithm and the estimation performance of the algorithm under different signal-to-noise ratios are shown. The vertical axis is the logarithmic coordinate with the logarithm of 10. Note that under the condition of 1000 Monte Carlo times, the number of interpolation points around the peak M = 2, 4, 8, 16, and 32 correspond to the running time of 76.243198 seconds. It took 120.478735 seconds. It took 211.100362 seconds. It took 391.131343 seconds. It took 753.142683 seconds. From the figure, it can be seen that the different number of points has little effect on the speed of error reduction.

[0084] In summary, the non-cooperative signal fractional-order time difference estimation method based on sinc interpolation proposed in the present invention is effective.

Claims

1. A non-cooperative signal fractional-order time difference estimation method based on band-limited signal reconstruction, characterized in that: The following steps are involved: S1. Receive the source signal s[n] through two receivers to obtain received signals x1[n] and x2[n] respectively: x1[n]=s[n]+q1[n] x2[n]=s[nD]+q2[n] Where q1[n] and q2[n] are zero-mean Gaussian white noises and are uncorrelated with the source signal s[n], x1[n] is the reference signal observation, x2[n] is the delayed signal observation, n=1,2,…,N, N is the number of signal samples, and D is the time difference between x1[n] and x2[n] that is an integer multiple of the sampling interval; Calculate the cross-correlation function of signals x1[n] and x2[n]: Where E{·} represents averaging, * represents conjugation, and m represents time offset. Substitute the expressions of x1[n] and x2[n] into the cross-correlation function and simplify to: It can be obtained that when m = D, the correlation function Get the maximum value and estimate it using time average: Among them, 2M+1 is the order of cross-correlation, M≤N-1; By performing a peak search, we can obtain the integer-order time difference between the two signals; S2. After obtaining the integer-order time difference, perform sinc function interpolation based on the correlation function value calculated near the correlation peak, and search for fractional-order time difference within the range of 0.5 around the correlation peak. Once the search peak is found, the fractional-order time difference estimation can be obtained. Specifically: Perform sinc interpolation near the correlation peak, expressed as: Where 2W is the bandwidth of the signal, R0(k) is the cross-correlation function obtained in S1, k0 corresponds to the correlation peak of R0(k), M represents the number of function values ​​around the correlation peak, Δ represents the fractional-order delay around the peak, and when Δ is equal to the true fractional-order delay, the corresponding R(Δ) function reaches the maximum value, and the corresponding derivative function reaches the minimum value; Taking the derivative of Δ, we get: When the derivative function is 0 within the value range, the value of the corresponding independent variable Δ is the corresponding fractional-order delay.

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