A time delay estimation method for super-resolution phase retrieval of pulse train signals

Abstract

The invention discloses a time delay estimation method for super-resolution phase retrieval of pulse train signals, belonging to the field of signal processing technology. The invention proposes a time delay estimation method for super-resolution phase retrieval of pulse train signals, which has lower computational complexity and better noise robustness than the existing method.

Description

Technical Field

[0001] The invention relates to a time delay estimation method for super-resolution phase retrieval of a pulse train signal, and belongs to the technical field of signal processing. Background Art

[0002] Pulse train signals are widely used in radar, communication, biomedicine and other fields. By measuring and estimating the pulse delay and pulse intensity of the pulse train, it is possible to obtain the target distance, scattering intensity and determine the target shape. Therefore, parameter estimation of pulse train signals is an important task in many engineering applications. In addition, in many measurement scenarios, the amplitude is more difficult to measure accurately than the phase of the signal. This is mainly based on the following two reasons: first, the phase of some signals cannot be measured using existing technology; second, in some measurement processes, the phase information is easily destroyed and lost. In response to this problem, many scholars have realized that in certain specific cases, the original signal can be restored using only the amplitude information of linear measurement. This process of restoring the original signal only from amplitude measurement is called phase retrieval (PR).

[0003] At present, most phase retrieval methods are oriented to discrete signal models, but actual signals are often continuous, and such model mismatch usually leads to inevitable approximation errors. When the original signal is a pulse train signal, its phase can be retrieved in the continuous domain. Because such a process has theoretically infinite resolution, it is also called super-resolution phase retrieval.

[0004] The pulse train signal model targeted by the existing super-resolution phase retrieval method is as follows:

[0005]

[0006] Among them, δ(t) is the Dirac pulse, φ(t) represents the scattering function, and K is the number of pulses. It can be seen that such a pulse train signal x(t) can be obtained by time delay and pulse amplitude In addition, it is known that the autocorrelation function a(t) of x(t) has the same structure as x(t).

[0007]

[0008] And a(t) can be completely determined by the Fourier transform intensity of x(t).

[0009]

[0010] Among them, * represents the convolution operation, represents Fourier transform, represents the inverse Fourier transform, and ψ(t) is the autocorrelation function of φ(t). In summary, the following phase-free Fourier measurement results can be used to achieve super-resolution phase retrieval of the original pulse train signal.

[0011]

[0012] Where Φ(nΩ) is The entire solution framework can be divided into the following three stages. First, the Prony method is used to super-resolve the autocorrelation function to estimate the time delay interval. Sum Amplitude Product

[0013]

[0014] Or estimate the positive delay interval and its corresponding amplitude product. Then use the above results to estimate the delay of the pulse train signal. Finally, estimate the amplitude of the pulse train signal. Summary of the invention

[0015] The purpose of the present invention is to propose a delay estimation method for super-resolution phase retrieval of pulse train signals, which uses a non-redundant positive delay interval Estimated pulse delay This is to solve the problem of high computational complexity and high redundancy of the delay interval in the prior art.

[0016] A time delay estimation method for super-resolution phase retrieval of a pulse train signal, the time delay estimation method for super-resolution phase retrieval of a pulse train signal comprising the following steps:

[0017] Step 1: Remove the estimated positive delay interval The largest element in

[0018]

[0019] Among them, max(·) means taking the maximum value of the elements in the set;

[0020] Step 2: Calculation The sum of any two elements in , and the sum results form a set

[0021] Step 3: Select Among K-2 closest The elements corresponding to The K-2 pairs of elements in the set

[0022]

[0023] Where K is the number of pulses in the pulse train signal;

[0024] Step 4: Determine the search range of the delay to be estimated

[0025]

[0026] Step 5: Determine the initial elements of the delay estimation result. The first initial element is the positive delay interval estimation result. The maximum value in The second initial element is a set The maximum value of the elements in To sum up, the initialization result of the delay estimation is determined.

[0027]

[0028] Step 6: Initialize the search range for the estimated delay. In addition to no longer searching the two initialization elements determined in step 5, set In and p max1 Paired p max0 Also excluded, so the search range of the initial estimated delay is

[0029]

[0030] in, Represents from the collection Remove a collection

[0031] Step 7: Iterate The remaining K-3 elements in are estimated, and j=2, ..., K-2 is defined as the iteration index. Then, at the jth iteration, the search set The element p in is selected as the new element in the estimated pulse delay, and the selection basis of the element is: the new pulse delay estimation result Closest The mathematical representation of this process is as follows:

[0032]

[0033] Among them, ||·||2 represents the l2 norm. In addition, the set The element that is paired with element p must not belong to So update the search scope to:

[0034]

[0035] in, or

[0036] Step 8: Finally return the estimated pulse delay

[0037]

[0038] A storage medium having a computer program stored thereon, wherein the computer program implements the method of claim 1 when executed by a processor.

[0039] An electronic device comprises a processor, a communication interface, a memory and a communication bus, wherein the processor, the communication interface and the memory communicate with each other via the communication bus;

[0040] Memory, used to store computer programs;

[0041] The processor is used to implement the method steps described in claim 1 when executing the program stored in the memory.

[0042] Beneficial effects of the present invention:

[0043] The present invention proposes a delay estimation method for super-resolution phase retrieval of pulse train signals, which uses a non-redundant positive delay interval Estimated pulse delay Compared with the existing methods, the present invention has lower computational complexity and better noise robustness. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 For the validity verification in the noise-free case, Figure 1 (a) is the structure function; Figure 1 (b) Scattering function; Figure 1 (c) original signal; Figure 1 (d) Phaseless measurement; Figure 1 (e) is the reconstruction result;

[0045] Figure 2 Comparison between the existing methods and the proposed method in the case of noise. DETAILED DESCRIPTION

[0046] The technical solutions in the embodiments of the present invention will be described clearly and completely below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0047] Before formally introducing the solution steps, some symbols are declared first.

[0048] is the estimated result of the delay, is the time delay interval for autocorrelation function super resolution The estimated results.

[0049] Then the following three points are explained on the problem of estimating delay using positive delay interval.

[0050] Explanation 1: Even if the labels of the elements in the positive delay interval set are known, only the translation or folding of the original delay can be recovered from the positive delay interval. Its mathematical expression is: The result after translation and folding and The corresponding positive delay interval sets are exactly the same. Known as The trivial ambiguous solution of .

[0051] Explanation 2: In addition to the trivial ambiguous solutions in Explanation 1, there are also some non-trivial ambiguous solutions. k are randomly and independently selected from a sufficiently smooth distribution, then the delay recovery result does not contain non-trivial ambiguous solutions.

[0052] Explanation 3: Even if the non-trivial ambiguous solution is eliminated by Explanation 2, the positive delay interval There are still countless trivial ambiguous solutions. Therefore, in order to determine whether the estimation is successful, the range of feasible solutions needs to be further narrowed. The smallest element in is 0, then according to the definition of trivial ambiguous solution, there are only two feasible solutions. Therefore, successfully estimating any of the two ambiguous solutions is considered a successful estimation.

[0053] Reference Figure 1-Figure 2 As shown, the present invention proposes a time delay estimation method for super-resolution phase retrieval of pulse train signals, and the time delay estimation method for super-resolution phase retrieval of pulse train signals comprises the following steps:

[0054] Step 1: Remove the estimated positive delay interval The largest element in

[0055]

[0056] Among them, max(·) means taking the maximum value of the elements in the set;

[0057] Step 2: Calculation The sum of any two elements in , and the sum results form a set

[0058] Step 3: Select Among K-2 closest The elements corresponding to The K-2 pairs of elements in the set

[0059]

[0060] Where K is the number of pulses in the pulse train signal;

[0061] Step 4: Determine the search range of the delay to be estimated

[0062]

[0063] Step 5: Determine the initial elements of the delay estimation result. The first initial element is the positive delay interval estimation result. The maximum value in The second initial element is a set The maximum value of the elements in To sum up, the initialization result of the delay estimation is determined.

[0064]

[0065] Step 6: Initialize the search range for the estimated delay. In addition to no longer searching the two initialization elements determined in step 5, set In and p max1 Paired p max0 Also excluded, so the search range of the initial estimated delay is

[0066]

[0067] in, Represents from the collection Remove a collection

[0068] Step 7: Iterate The remaining K-3 elements in are estimated, and j=2, ..., K-2 is defined as the iteration index. Then, at the jth iteration, the search set The element p in is selected as the new element in the estimated pulse delay, and the selection basis of the element is: the new pulse delay estimation result Closest The mathematical representation of this process is as follows:

[0069]

[0070] Among them, ||·||2 represents the l2 norm. In addition, the set The element that is paired with element p must not belong to So update the search scope to:

[0071]

[0072] in, or

[0073] Step 8: Finally return the estimated pulse delay

[0074]

[0075] A storage medium having a computer program stored thereon, wherein the computer program implements the method of claim 1 when executed by a processor.

[0076] An electronic device comprises a processor, a communication interface, a memory and a communication bus, wherein the processor, the communication interface and the memory communicate with each other via the communication bus;

[0077] Memory, used to store computer programs;

[0078] The processor is used to implement the method steps described in claim 1 when executing the program stored in the memory.

[0079] The following is a specific embodiment of the present invention:

[0080] 1. Noise-free experiment

[0081] First, the effectiveness of the pulse delay estimation method proposed in the present invention is verified in the absence of noise. The entire experimental process and the results of each part are shown in Figure 1 middle.

[0082] Two Dirac pulse trains with pulse numbers K of 3 and 5 as shown in sub-figure (a) are selected as structure functions. Then the original structure function is filtered using the scattering function shown in sub-figure (b). Sub-figure (c) shows the result after filtering, which is the original signal. The discrete sampling value of its intensity is obtained after Fourier transform of the original signal. The blue line in sub-figure (d) represents the intensity of the Fourier transform of the original signal, and the red line represents the discrete sampling result. The positive delay interval and amplitude product of the original structure function are first estimated using the existing method. Then the pulse delay of the original structure function is estimated using the delay estimation method proposed in the present invention. Finally, the amplitude of the original signal is estimated using the existing method. As shown in sub-figure (e), the original structure signal is drawn based on the estimation results of the delay and amplitude. According to the experimental results, it can be seen that in the absence of noise, the correct estimation of the original pulse train signal parameters can be achieved.

[0083] 2. Noise experiment

[0084] We randomly select K values ​​from the range of 0 to 1 to form the pulse delay set to be restored And set K to 3 and 5 respectively. Then get the delay interval corresponding to the above pulse delay set and positive delay interval And add zero mean additive Gaussian white noise with signal-to-noise ratio ranging from -20dB to 30dB. Finally, the delay is estimated using the existing and proposed delay estimation methods. According to the above introduction to the trivial ambiguous solution, we can know that the delay set There are 2K trivial ambiguous solutions containing 0, of which K are obtained by translation and the other K are obtained by folding. Therefore, we define these 2K trivial ambiguous solutions containing 0 as the set The l2 norm error shown below is defined as the quantitative standard of the reconstruction error.

[0085]

[0086] Here sort(·) means to sort the elements in the set in descending order. The experimental results under the above noise conditions are as follows Figure 2 As shown in Figure 1, the reconstruction error is the average value after 10,000 Monk Carlo experiments. In addition, in the figure, “existing” represents the solution using the existing algorithm, and “proposed” represents the solution using the proposed algorithm.

[0087] The experimental results show that the reconstruction error decreases with the increase of signal-to-noise ratio. In addition, under different pulse numbers, the proposed delay recovery method has better noise robustness than the existing methods.

Claims

1. A time delay estimation method for super-resolution phase retrieval of pulse train signals, characterized in that: The time delay estimation method for super-resolution phase retrieval of pulse train signals comprises the following steps: Step 1: Remove the estimated positive delay interval The largest element in Among them, max(·) means taking the maximum value of the elements in the set; Step 2: Calculation The sum of any two elements in , and the sum results form a set Step 3: Select Among K-2 closest The elements corresponding to The K-2 pairs of elements in the set Where K is the number of pulses in the pulse train signal; Step 4: Determine the search range of the delay to be estimated Step 5: Determine the initial elements of the delay estimation result. The first initial element is the positive delay interval estimation result. The maximum value in The second initial element is a set The maximum value of the elements in To sum up, the initialization result of the delay estimation is determined. Step 6: Initialize the search range for the estimated delay. In addition to no longer searching the two initialization elements determined in step 5, set In and p max1 Paired p max0 Also excluded, so the search range of the initial estimated delay is in, Represents from the collection Remove a collection Step 7: Iterate The remaining K-3 elements in are estimated, and j=2, ..., K-2 is defined as the iteration index. Then, at the jth iteration, the search set The element p in is selected as the new element in the estimated pulse delay, and the selection basis of the element is: the new pulse delay estimation result Closest The mathematical representation of this process is as follows: Among them, ||·||2 represents the l2 norm. In addition, the set The element that is paired with element p must not belong to So update the search scope to: in, or Step 8: Finally return the estimated pulse delay 2. A storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the method according to claim 1 is implemented.

3. An electronic device, characterized in that: It includes a processor, a communication interface, a memory and a communication bus, wherein the processor, the communication interface and the memory communicate with each other through the communication bus; Memory, used to store computer programs; The processor is used to implement the method steps described in claim 1 when executing the program stored in the memory.

Images