CN116886098A - Asymmetric pulse sequence FRI sampling and reconstructing method - Google Patents
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Abstract
The application relates to an asymmetric pulse sequence FRI sampling and reconstructing method, which comprises the following steps: s1, converting an asymmetric pulse signal into a linear combination of Gaussian second derivatives; s2, adopting SoS to check asymmetric pulse sequence filtering, and then uniformly sampling and transforming to obtain Fourier series coefficients; s3, estimating a reconstructed time delay parameter and an amplitude parameter by an improved zero-change filter algorithm based on a Gaussian second derivative model; s4, reconstructing the asymmetric pulse signals by using the reconstructed time delay parameters and the reconstructed amplitude parameters. Based on the established Gaussian second derivative model, the method estimates the true value of an unknown parameter set in the original signal by utilizing an improved zero-change filter algorithm, and finally carries out signal reconstruction, thereby solving the difficulties of asymmetric pulse sequence sampling and reconstruction.
Description
Technical Field
The application belongs to the technical field of signal processing, and particularly relates to an asymmetric pulse sequence FRI sampling and reconstruction method.
Background
Asymmetric pulse sequences are widely available in many fields of application, such as remote sensing image restoration, medical imaging, public security, etc. According to the Nyquist sampling theorem, in order to reconstruct an asymmetric pulse sequence without distortion, the sampling system is required to satisfy the condition that the sampling rate is not lower than twice the highest frequency of the signal. However, with the rapid development of information and communication technologies, the bandwidth of the asymmetric pulse sequence is wider and wider, the sampling rate required for sampling the signal is higher and higher, and the ADC (Analog to Digital Converter) also needs to have a wider analog bandwidth and sampling rate. In addition, high sampling rates can create large amounts of sampled data, and can create significant stress on the back-end data storage, transmission, and processing. Thus, research into asymmetric pulse train undershot sampling and reconstruction methods is increasingly becoming a sampling system design bottleneck.
The undershot sampling theory and method of asymmetric pulse sequences is becoming a research hotspot in recent years. Baechler et al propose a specific model of a signal containing varying degrees of asymmetry and width pulses, which shows that the asymmetry and width of the kexi-lorentz pulse sequence can be estimated using annihilation filter techniques, which model is then extended by Nair et al for multichannel Electrocardiogram (ECG) data compression. Vetterli et al propose a FRI (Finite Rate of Innovation) sampling and reconstruction framework for noise-containing band-limited signals that models the shot noise as a Dirac pulse train to convert the noise-containing band-limited signals to FRI signals. Although the method is suitable for the specific field, the sampling efficiency and the reconstruction accuracy are still low.
Therefore, there is a need to develop an efficient FRI sampling and reconstruction method for undershot sampling and reconstruction of asymmetric pulse sequences.
Disclosure of Invention
Based on the above-mentioned drawbacks and deficiencies of the prior art, it is an object of the present application to at least solve the above-mentioned problems of the prior art, in other words to provide an asymmetric pulse sequence FRI sampling and reconstruction method which meets the aforementioned needs.
In order to achieve the aim of the application, the application adopts the following technical scheme:
an asymmetric pulse sequence FRI sampling and reconstruction method comprising:
s1, converting an asymmetric pulse signal into a linear combination of Gaussian second derivatives;
s2, adopting SoS to check asymmetric pulse sequence filtering, and then uniformly sampling and transforming to obtain Fourier series coefficients;
s3, estimating a reconstructed time delay parameter and an amplitude parameter by an improved zero-change filter algorithm based on a Gaussian second derivative model;
s4, reconstructing the asymmetric pulse signals by using the reconstructed time delay parameters and the reconstructed amplitude parameters.
As a preferred embodiment, step S1 includes the steps of:
s11, converting the asymmetric pulse signals into linear combination of Gaussian second derivatives;
s12, acquiring spectrum information of a linear combination basis function of the Gaussian second derivative and a modeling signal, and carrying out Fourier transformation on the asymmetric pulse signal and the basis signal;
s13, discretizing to generate a discrete frequency spectrum.
As a preferred embodiment, step S2 includes the steps of:
s21, adopting SoS to check asymmetric pulse sequence filtering to generate a filtered signal;
s22, setting a sampling rate, and uniformly sampling the filtered signals to obtain a sampling sample;
s23, carrying out Fourier transform on the sampling sample to obtain a Fourier series coefficient.
As a further preferred embodiment, the uniform sampling is FRI sampling.
As a further preferred embodiment, step S3 comprises the steps of:
s31, acquiring a plurality of continuous samples from the sampled samples to construct a zero-change filter, wherein the zero-change filter is used for annihilating the asymmetric pulse sequence;
s32, calculating a time delay parameter of the annihilation filtered signal of the nulling filter;
s33, calculating amplitude parameters of the annihilation filtered signals of the nulling filter.
As a preferred embodiment, step S4 specifically includes:
s41, forming a parameter set by the time delay parameter and the amplitude parameter;
s42, substituting the parameter set into the asymmetric pulse signal to obtain the reconstructed asymmetric pulse signal. .
Compared with the prior art, the application has the beneficial effects that:
the method models an asymmetric pulse signal into a linear combination form of a series of Gaussian second derivatives, and analyzes and converts the frequency spectrum of the asymmetric pulse signal; then, filtering and sampling the input asymmetric pulse signals at a low speed by designing an FRI sampling structure; and finally, based on the established Gaussian second derivative model, estimating the true value of an unknown parameter set in the original signal by utilizing an improved zero-change filter algorithm, and finally, carrying out signal reconstruction, thereby solving the difficulties of asymmetric pulse sequence sampling and reconstruction.
Drawings
FIG. 1 is a flow chart of an asymmetric pulse sequence FRI sampling and reconstruction method of the present application;
FIG. 2 shows the recovery result of the time delay parameters of an asymmetric pulse sequence FRI sampling and reconstruction method of the present application;
FIG. 3 shows the recovery result of the amplitude parameter 1 of an asymmetric pulse sequence FRI sampling and reconstruction method of the present application;
FIG. 4 shows the recovery result of the amplitude parameter 2 of an asymmetric pulse sequence FRI sampling and reconstruction method of the present application;
fig. 5 shows the final fitting result of an asymmetric pulse sequence FRI sampling and reconstruction method of the present application.
Detailed Description
The technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application.
In the following description, various embodiments of the application are provided, and various embodiments may be substituted or combined, so that the application is intended to include all possible combinations of the same and/or different embodiments described. Thus, if one embodiment includes feature A, B, C and another embodiment includes feature B, D, then the present application should also be considered to include embodiments that include one or more of all other possible combinations including A, B, C, D, although such an embodiment may not be explicitly recited in the following.
The following description provides examples and does not limit the scope, applicability, or examples set forth in the claims. Changes may be made in the function and arrangement of elements described without departing from the scope of the application. Various examples may omit, replace, or add various procedures or components as appropriate. For example, the described methods may be performed in a different order than described, and various steps may be added, omitted, or combined. Furthermore, features described with respect to some examples may be combined into other examples.
The application provides an asymmetric pulse sequence FRI sampling and reconstructing method, a flow chart of which is shown in figure 1, comprising the following steps:
s1, converting an asymmetric pulse signal into a linear combination of Gaussian second derivatives;
s2, adopting SoS to check asymmetric pulse sequence filtering, and then uniformly sampling and transforming to obtain Fourier series coefficients;
s3, estimating a reconstructed time delay parameter and an amplitude parameter by an improved zero-change filter algorithm based on a Gaussian second derivative model;
s4, reconstructing the asymmetric pulse signal by using the reconstructed time delay parameter and the reconstructed amplitude parameter.
An embodiment of the present application provides a specific implementation method of the step S1, including the following steps.
S11, modeling an asymmetric pulse signal x (t) as a linear combination of a series of Gaussian second derivatives:
wherein the constant isRepresenting the duration of the signal; the variable t epsilon [0, T), represents a time variable; constant valueFor the number of combined pulses, the variable k=1, 2..k represents the sequence number of the pulse; unknown parameter t k Representing the time delay of the kth pulse, arranged in ascending order, i.e. 0.ltoreq.t 1 ＜t 2 ＜...＜t k < T. Gaussian and second derivative function h thereof k (t-t k ) The definition is as follows:
wherein the unknown parametersAmplitude being gaussian component; unknown parameters->Amplitude of the gaussian second derivative component; h (t) is a Gaussian function, +.>Is its second derivative, which is specifically as follows:
to sum up, the asymmetric pulse sequence x (t) is expressed as:
s12, acquiring the spectrum information of the modeling signal and the basis function of the linear combination of the Gaussian second derivative. Will beRegarding the base signal, the signal x (t) and the base signal h (t) are fourier transformed, respectively, with the following results:
let H (ω) +.0, define:
s13, discretizing the equation of the S12: order the A sequence number representing the frequency of the signal. Substituting formula (8) to obtain a discrete spectrum:
an embodiment of the present application further provides a specific implementation method of the step S2, including the following steps.
S21, filtering the asymmetric pulse sequence x (t) by adopting a SoS check, and generating a filtered signal. The filtered signal is:
p(t)＝x(t)*g(t) (10)
wherein g (t) is set as the unit impulse response.
S22, setting the sampling rate as f s And (2) uniformly sampling the filtered pulse sequence signal p (T) at the rate of more than or equal to 2K/T, and collecting the following samples:
wherein T is s ＝1/f s Represents the sampling interval, n=0, 1, …, N-1,representing the total number of samples taken, +.>Representing a downward rounding operation on the values (.
Specifically, the above-mentioned samples are FRI samples.
S23, performing discrete Fourier transform on the sampling sample P [ n ] in the step S22 to obtain a Fourier series coefficient P [ m ].
According to the FRI sampling theory, after the asymmetric pulse sequence signal is filtered and ADC, the signal is not distorted in the cut-off frequency of the SoS core, so that the method comprises the following steps:
P[m]＝Y[m] (12)
step S3, estimating parameters by an improved zeroing filter algorithm based on the Gaussian second derivative model. Since the conventional nulling filter algorithm is not suitable for the formula (9), this embodiment of the present application further provides a specific implementation method of the above step S3, which includes the following steps:
s31, acquiring a plurality of continuous samples from the sampled samples to construct a nulling filter. First, 2k+1 consecutive samples p [ n ] are obtained from the FRI sampling structure. Constructing a filter:
wherein the method comprises the steps ofRepresenting the zero point of the filter, the coefficient A [ m ]]By Am]Convolution Y [ m ]]
(m=0, 1, … K) to obtain:
thus, the filterAnnihilation signal Y [ m ]]In other words A [ m ]]*Y[m]＝0。
S32, calculating a time delay parameter of annihilating the filtered signal by the nulling filter. Specifically, step S32 solves equation (14) to obtain a delay parameterWriting equation (14) into a matrix-vector form:
from the a 0=1 a priori information, the system of linear equations (15) can be reduced to the following form:
Yx＝b (16)
wherein, the liquid crystal display device comprises a liquid crystal display device,
x＝[A[1],A[2],...,A[2K]] T (18)
b＝[-Y[0],-Y[1],...,-Y[2K-1],-Y[2K]] T (19)
the Matrix Y is a Hankel Matrix (Hankel Matrix), i.e., the elements on any one of the minor diagonals within the Matrix are equal. According to the nature of the Hank matrix, when the measured value Y [ m ] is non-zero, Y is reversible, and equation (16) has a unique solution:
x＝Y -1 b (20)
in a noisy environment, its least squares solution is generally employed:
x＝(Y T Y) -1 Y T b (21)
when the coefficient A m of the zero-change filter is obtained]Thereafter, unknown parametersCan be simply solved byObtained.
Because ofThe delay parameter ∈>
Then step S33 is executed, the amplitude parameter of the annihilation filtered signal of the nulling filter is calculated, and the amplitude parameter is solvedAccording to equation (9), a system of linear equations is constructed as follows:
y＝Br (23)
wherein, the liquid crystal display device comprises a liquid crystal display device,
y＝[Y[1] Y[2] ... Y[M]] T (24)
r＝[c 1 c 2 ... c K d 1 d 2 ... d K ] T (25)
finally, solving an equation set (23) by a least square method to obtain the amplitude parameterIs the value of (1):
step S4 specifically includes the following two steps, firstly, step S41 is executed, and the time delay parameter and the amplitude parameter form a parameter set
S42, substituting the parameter set into the formula (1) to obtain the reconstructed signal
The method for sampling and reconstructing the FRI of the asymmetric pulse sequence mainly comprises sampling and reconstructing the FRI signal, and the method obtains the amplitude parameter of the original signal by using an improved zero-change filter algorithmAnd delay parameter->Finally obtaining the reconstructed signal +.>
The present application also verifies the method of the above embodiment in a noise-free environment. At the time of experiment, the signal parameters were set as follows: the number of combined pulses k=4 and the pulse width σ=1e-3. To recover the parameters of 1 unknown signal, 2k+1 samples, i.e. 9 samples, need to be taken from the sampling channel. The recovery results are shown in fig. 2 to 5, wherein fig. 2 is the recovery result of the time delay parameter of the method of the present application, fig. 3 is the recovery result of the amplitude parameter 1 of the method of the present application, fig. 4 is the recovery result of the amplitude parameter 2 of the method of the present application, and fig. 5 is the final fitting result of the method of the present application.
The foregoing is merely exemplary embodiments of the present disclosure and is not intended to limit the scope of the present disclosure. That is, equivalent changes and modifications are contemplated by the teachings of this disclosure, which fall within the scope of the present disclosure. Other embodiments of the disclosure will be apparent to those skilled in the art from consideration of the specification and practice of the disclosure herein. This application is intended to cover any variations, uses, or adaptations of the disclosure following, in general, the principles of the disclosure and including such departures from the present disclosure as come within known or customary practice within the art to which the disclosure pertains. It is intended that the specification and examples be considered as exemplary only, with a scope and spirit of the disclosure being indicated by the claims.
Claims (6)
1. An asymmetric pulse sequence FRI sampling and reconstruction method, comprising:
s1, converting an asymmetric pulse signal into a linear combination of Gaussian second derivatives;
s2, adopting SoS to check asymmetric pulse sequence filtering, and then uniformly sampling and transforming to obtain Fourier series coefficients;
s3, estimating a reconstructed time delay parameter and an amplitude parameter by an improved zero-change filter algorithm based on a Gaussian second derivative model;
s4, reconstructing the asymmetric pulse signal by using the reconstructed time delay parameter and the reconstructed amplitude parameter.
2. The method for sampling and reconstructing an asymmetric pulse train FRI as set forth in claim 1, wherein said step S1 comprises the steps of:
s11, converting the asymmetric pulse signals into linear combination of Gaussian second derivatives;
s12, acquiring spectrum information of a linear combination basis function of the Gaussian second derivative and a modeling signal, and carrying out Fourier transformation on the asymmetric pulse signal and the basis signal;
s13, discretizing to generate a discrete frequency spectrum.
3. The method for sampling and reconstructing an asymmetric pulse train FRI as set forth in claim 1, wherein said step S2 comprises the steps of:
s21, adopting SoS to check asymmetric pulse sequence filtering to generate a filtered signal;
s22, setting a sampling rate, and uniformly sampling the filtered signals to obtain sampling samples;
s23, carrying out Fourier transform on the sampling sample to obtain a Fourier series coefficient.
4. The asymmetric pulse train FRI sampling and reconstruction method of claim 3 wherein said uniform sampling is FRI sampling.
5. A method for sampling and reconstructing an asymmetric pulse sequence FRI as claimed in claim 3, wherein said step S3 comprises the steps of:
s31, acquiring a plurality of continuous samples from the sampled samples to construct a zero-change filter, wherein the zero-change filter is used for annihilating an asymmetric pulse sequence;
s32, calculating a time delay parameter of annihilating the filtered signal by the nulling filter;
s33, calculating an amplitude parameter of the filtered signal annihilated by the nulling filter.
6. The method for sampling and reconstructing an asymmetric pulse train FRI according to claim 1, wherein said step S4 is specifically:
s41, forming a parameter set by the time delay parameter and the amplitude parameter;
s42, substituting the parameter set into the asymmetric pulse signal to obtain a reconstructed asymmetric pulse signal.
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