CN104468427A - Signal efficient sampling and reconstruction method based on FRI time-frequency domain comprehensive analysis - Google Patents

Abstract

The invention provides a signal efficient sampling and reconstruction method based on FRI time-frequency domain comprehensive analysis and relates to the technical field of information and communication. The signal efficient sampling and reconstruction method based on FRI time-frequency domain comprehensive analysis aims to lower Nyquist sampling frequency of signals and improve the sampling accuracy of the signals. In a frequency domain, a frequency spectrum line is used for recording information of high-frequency components of the signals, and the frequency is subjected to logarithm taking and normalized, so that the frequency domain is further compressed. In a time domain, a line segment fitting method is proposed, and time-frequency signals with low frequency are compressed. Since the signals are sampled efficiently in the frequency domain and the time domain, the requirement for the signal sampling quantity is substantially lowered; the signals are processed and recovered in the time domain and the frequency domain according to the FRI theory; meanwhile, the types of the signals capable of being processed according to he FRI theory are expanded, so discrete Dirac flow and continuous signals with high frequency can be both processed according to the FRI theory. The signal efficient sampling and reconstruction method based on FRI time-frequency domain comprehensive analysis is applicable to the signal sampling and reconstruction process.

Description

The signal efficient sampling comprehensively analyzed based on FRI time-frequency domain and signal reconfiguring method

Technical field

The present invention relates to Information & Communication Technology field, be specifically related to a kind of method of signal being carried out to efficient sampling.

Background technology

Digital communication system is by information source, source encoder, channel encoder, modulator, channel, demodulator, channel decoder, the forming of source decoder and the stay of two nights.Wherein, source encoder settling signal sampling, quantize with encode process.Therefore, in communication process, sampling is one of necessary step.The amount of calculation of subsequent treatment is carried out in how many direct impacts of sampling number to signal.At present, in most cases using nyquist sampling theorem as main sampling plan.Nyquist sampling law shows, as sample frequency f _sbe greater than highest frequency f in signal _max2 times time, i.e. f _s>=2f _max, then the digital signal after sampling intactly remains the information in primary signal, ensures that sample frequency is 5 ~ 10 times of signal highest frequency in general practical application.And utilize the sparse characteristic of signal, the sample rate to signal can be reduced significantly, and can very accurately restoring signal.

In fact, there is the sampling plan more much higher than Nyquist Sampling Theorem efficiency.We can utilize the method for some particular sample significantly can reduce sample frequency.Such as: signal y (t)=sin (10000*2* π * t), if utilize nyquist sampling theorem, then sample frequency is at least 20000Hz, namely per secondly needs sampling 20,000 points.And from frequency angle analysis signal, then the spectrogram of this signal only has a spectral line.Namely the amount of information that in spectrogram, point comprises is equivalent in the time domain amount of information of adopting 20,000 points and comprising per second.

But and not all signal is all simple than time domain analysis in frequency-domain analysis.Such as, signal y (t)=t is straight line in the oscillogram of time domain, just can determine a straight line at 2, that is: utilize the information of two points just can represent all information of this root straight line.And in a frequency domain, this signal comprises a lot of frequency spectrum, by emulation, the frequency that the amplitude in spectrogram that obtains is greater than 0.01 just has 31, that is: need the information of 31 points could represent this signal comparatively accurately.

As can be seen here, only all can not obtain best sample effect from time domain or frequency domain, namely utilize the least possible point to represent the information that signal comprises.

Summary of the invention

The present invention is the Nyquist sampling frequency in order to reduce signal, and in order to improve the precision of signal sampling, thus provides a kind of signal efficient sampling of comprehensively analyzing based on FRI time-frequency domain and signal reconfiguring method.

The signal efficient sampling comprehensively analyzed based on FRI time-frequency domain and signal reconfiguring method, it is realized by following steps:

Step one, to primary signal x _nbe fast Fourier transform FFT, by this primary signal x _nbe transformed into frequency domain from time domain, obtain y _k, and make spectrogram; And utilize low pass filter frequency-region signal to be divided into the part of high frequency and the part of low frequency, wherein, the cut-off frequency of low pass filter can make corresponding selection according to concrete application scenarios;

Step 2, the frequency f utilized in spectrogram _kwith amplitude a _krecord the information of high-frequency part, and by the part of high frequency by low pass filter filtering from primary signal;

The frequency of the HFS of frequency-region signal is taken the logarithm, and is normalized, make the frequency distribution of spectrogram in [0,1] scope;

Frequency is taken the logarithm, and normalized concrete steps are as follows:

If the original frequency of primary signal is f ₀, after taking the logarithm, obtain f _log, concrete analytic equation is:

f _log＝log ₁₀(f ₀) (1)

And choose f _login maximum f _max, obtain normalized frequency for:

$\stackrel{\‾}{f}=\frac{f_{\log }}{f_{\max }}---\left(2\right)$

Then, the frequency distribution of all spectral lines in spectrogram is in [0,1] scope;

Step 3, utilize the low frequency part of line-fitting algorithm to frequency-region signal of time domain to analyze, judge whether the low frequency part of frequency-region signal is linear; Be specially:

Step 3 one, utilize the line-fitting algorithm of time domain whether to be linear to the low frequency part of frequency-region signal, if judged result is yes, then perform step 3 three; If judged result is no, then perform step 3 two;

Step 3 two, line segment to be halved, judge that whether linear signal in two intervals be divided into one by one, if judged result is yes, then execution step 3 three; If judged result is no, then perform step 3 four;

The moment of starting point and the slope of this interval lower line segment between step 3 three, recording areas, and perform step 3 five;

Step 3 four, return step 3 two;

Step 4, utilize FRI technology to carry out signal to be reconstructed, to be specially:

Step 4 one, signal x (t) is carried out filtering by filter, the expression formula of filtered signal y (t) is:

y(t)＝h(t)*x(t) (3)

Wherein: the time domain impulse that h (t) is filtering responds, x (t) by position is with amplitude be the stream of pulses signal formed;

$x\left(t\right)=\underset{k=0}{\overset{K-1}{\mathrm{\Σ}}}{a}_k\underset{m\∈z}{\mathrm{\Σ}}p(t-t_k-\mathrm{m\τ})=\underset{k=0}{\overset{K-1}{\mathrm{\Σ}}}{a}_k\frac{1}{\mathrm{\τ}}\underset{m\∈z}{\mathrm{\Σ}}\hat{p}\left(\frac{2\mathrm{\πm}}{\mathrm{\τ}}\right)e^{j2\mathrm{\πm}\frac{t-t_k}{\mathrm{\τ}}}=\underset{m\∈Z}{\mathrm{\Σ}}{\hat{x}}_m{e}^{j2\mathrm{\πm}\frac{t}{\mathrm{\τ}}}---\left(4\right)$

Wherein: the analytic equation that p (t) is pulse, τ is the time span of signal; K is the quantity of signal pulse, and K is positive integer; J is positive integer;

Filtered signal y (t) is sampled, obtains discrete signal y _n, described discrete signal y _nexpression formula be:

Wherein: T is the sampling interval, for sampling function; N represents the n-th sampled point;

Above formula is out of shape, obtains:

$\begin{array}{c}{y}_n=<x\left(t\right),{\mathrm{\&phi;}}_B(\mathrm{nT}-t)>\\ =\underset{m\&Element;Z}{\mathrm{\&Sigma;}}{\hat{x}}_m<e^{j2\mathrm{\&pi;m}\frac{t}{\mathrm{\&tau;}}},{\mathrm{\&phi;}}_B(\mathrm{nT}-t)>\\ =\underset{m\&Element;Z}{\mathrm{\&Sigma;}}{\hat{x}}_m{\widehat{\mathrm{\&phi;}}}_B\left(\frac{2\mathrm{\&pi;m}}{\mathrm{\&tau;}}\right)e^{j2\mathrm{\&pi;n}\frac{\mathrm{\&tau;}}{N}\frac{m}{\mathrm{\&tau;}}}\\ =\frac{1}{B}\underset{m\&le;M=\left[\frac{\mathrm{B\&tau;}}{2}\right]}{\mathrm{\&Sigma;}}{e}^{j2\mathrm{\&pi;}\frac{\mathrm{mn}}{N}}\end{array}---\left(6\right)$

In formula (6), x _mdiscrete Fourier transform, B is the bandwidth of filter, and τ is the time span of signal;

That is:

${\hat{x}}_m={B\hat{y}}_m---\left(7\right)$

Wherein: x _mdiscrete Fourier transform, y _mdiscrete Fourier transform, B is the bandwidth of filter;

Calculate the Fourier series coefficient of x (t) obtain for:

${\hat{x}}_m=\frac{1}{\mathrm{\&tau;}}\hat{p}\left(\frac{2\mathrm{\&pi;m}}{\mathrm{\&tau;}}\right)\underset{k=0}{\overset{K-1}{\mathrm{\&Sigma;}}}{a}_k{e}^{-j2\mathrm{\&pi;m}\frac{t_k}{\mathrm{\&tau;}}}---\left(8\right)$

Above formula is converted:

${\hat{x}}_m{\hat{p}}^{-1}\left(\frac{2\mathrm{\&pi;m}}{\mathrm{\&tau;}}\right)=\frac{1}{\mathrm{\&tau;}}\underset{k=0}{\overset{K-1}{\mathrm{\&Sigma;}}}{a}_k{u}_k^m---\left(9\right)$

Wherein:

$u_k=e^{-j2\mathrm{\&pi;}\frac{t_k}{\mathrm{\&tau;}}}---\left(10\right)$

represent the multiplicative inverse of p, the value of p (t) is that priori is known, for Dirac function stream, $\hat{p}\left(\frac{2\mathrm{\&pi;m}}{\mathrm{\&tau;}}\right)=1;$

Adopt z-conversion represent u _k:

$\hat{h}\left(z\right)=\underset{m=0}{\overset{K}{\mathrm{\&Sigma;}}}{h}_m{z}^{-m}=\underset{m=0}{\overset{k-1}{\mathrm{\&Pi;}}}(1-u_k{z}^{-1})---\left(11\right)$

That is, root equal by the u found _kvalue;

H _mmeet:

$h_m*{\hat{x}}_m=\underset{i=0}{\overset{K}{\mathrm{\&Sigma;}}}{h}_i{\hat{x}}_{m-i}=\underset{i=0}{\overset{K}{\mathrm{\&Sigma;}}}\underset{k=0}{\overset{K-1}{\mathrm{\&Sigma;}}}{a}_k{h}_i{u}_k^{m-i}=\underset{k=0}{\overset{K-1}{\mathrm{\&Sigma;}}}{a}_k{u}_k^m\underset{i=0}{\overset{K}{\mathrm{\&Sigma;}}}{h}_i{u}_k^{-i}=0---\left(12\right)$

Due to this filter { h _mbe called as and bury in oblivion filter;

Make h ₀=1, being write formula (12) as matrix form is:

{ h is obtained by Solving Equations above _m;

{ h _mz conversion zero point be u _k,

t _k＝(lnu _k)/(-j)/2/π×τ (14)

Obtain time delay t thus _k;

Pass through equation:

Solve amplitude a _k;

Step 5, the amplitude a obtained according to step _kwith time delay t _kthe frequency domain part of restoring signal and domain portion;

By time delay t _kaccording to formula

f _log＝t _k×f _max(16)

Calculate the frequency values f of renormalization _log,

Again by f _logcalculate the frequency f of HFS _k, computing formula is such as formula (17):

$f_k={10}^{f_{\log }}---\left(17\right)$

In the time domain, by t _kas the moment of line segment starting point, a _kas the slope of straight line, recover the time domain waveform of the low frequency part of signal;

Step 6, the signal of frequency domain part is changed IFFT to time-domain signal by inverse discrete Fourier transform,

$x\left(n\right)=\frac{1}{N}{\left[\underset{k=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{X}^{*}\left(k\right)W_N^{\mathrm{nk}}\right]}^{*}=\frac{1}{N}{\left\{\mathrm{DET}\right[X^{*}\left(k\right)\left]\right\}}^{*}---\left(18\right)$

Wherein: DFT [] does discrete Fourier transform to function; N represents the inverse discrete Fourier transform doing N point;

After the signal of frequency domain part is transformed to time domain, the Signal averaging with time domain medium and low frequency composition, recovers primary signal.

To primary signal x in step one _nbe fast fourier transform FFT to be specially:

$\begin{array}{c}{y}_k=\underset{n=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{W}_N^{\mathrm{kn}}{x}_n\\ ={\mathrm{\&Sigma;}}_{n=2t}{W}_N^{\mathrm{kn}}{x}_n+{\mathrm{\&Sigma;}}_{n=2t+1}{W}_N^{\mathrm{kn}}{x}_n\\ ={\mathrm{\&Sigma;}}_t{W}_{\frac{N}{2}}^{\mathrm{kt}}{x}_{2t}+W_N^k{\mathrm{\&Sigma;}}_{n=2t+1}{W}_{\frac{N}{2}}^{\mathrm{kt}}{x}_{2t+1}\\ =F_{\mathrm{even}}\left(k\right)+W_N^k{F}_{\mathrm{odd}}\left(k\right)\end{array}---\left(19\right)$

Wherein: use W _nrepresent f _evenk () does discrete Fourier transform, F to even item _oddk () does discrete Fourier transform to odd item.

Whether the low frequency part judging frequency-region signal is linear concrete grammar:

For interval [a, b], judge whether interval [a, b] meets:

$f\left(a\right)+f\left(b\right)-2\&times;f\left(\frac{a+b}{2}\right)<\mathrm{\&epsiv;}---\left(20\right)$

Wherein: ε is error permissible value;

If met, then signal frequency f (x) is linear in interval [a, b]; If do not met, then signal frequency f (x) is nonlinear in interval [a, b].

In step 4 one, h (t) chooses Gaussian function, sinc function, one of B-spline and E spline function.

The beneficial effect that the present invention obtains:

1, the sampling plan in the present invention has high efficiency;

2, the present invention proposes line-fitting algorithm in time-domain signal analysis, and straight line can be utilized to describe the wavy curve of low frequency signals more accurately, can extract the effective information of lower frequency composition so efficiently from time domain.

Although 3, the present invention's sampling count considerably less, can very accurately restoring signal.By Figure 15 and Fig. 4, we can compare restoring signal and primary signal, and the two is almost identical, therefore demonstrates in the present invention the accuracy of recovery algorithms of sampling.

4, the present invention utilizes FRI theoretical dexterously, and has expanded the theoretical treatable signal kinds of FRI.The present invention utilizes f _kand a _krecord frequency domain information, utilizes t _kand a _krecord time-domain information.And FRI theoretical treatable be discrete signal, as dirac stream.The information of these signals can use time delay t _kwith amplitude a _krepresent.The f in frequency domain _kregard the t in FRI theory as _k, a in frequency domain _kregard a in FRI theory as _k.Therefore, frequency domain part can utilize FRI theory to process.If the t in time domain _kregard the t in FRI theory as _k, a in time domain _kregard a in FRI theory as _k, the signal of such domain portion also can process by FRI theory.Therefore, FRI theory can be utilized to be further processed the information of time domain and frequency domain record.Meanwhile, also theoretical for FRI treatable signal kinds is expanded to high frequency continuous signal from discrete dirac stream.

Accompanying drawing explanation

Fig. 1 is signal flow graph of the present invention;

Fig. 2 is the flow chart of line-fitting algorithm;

Fig. 3 is the flow chart utilizing FRI theoretical treatment signal;

Fig. 4 is the wave simulation schematic diagram of primary signal;

Fig. 5 carries out to signal the spectrogram that spectrum analysis obtains;

Fig. 6 is the spectrogram of 0 ~ 100Hz;

Fig. 7 is the FRI signal simulation schematic diagram in frequency domain;

Fig. 8 is the signal simulation schematic diagram that FRI frequency-region signal is obtained by Singh's core filter;

Fig. 9 is the frequency domain FRI signal simulation schematic diagram utilizing the recovery algorithms of FRI theory to reconstruct;

Figure 10 is by after the upper frequency composition filtering in signal, the wave simulation schematic diagram of lower low frequency composition signal;

Figure 11 is the FRI signal simulation schematic diagram in time domain;

Figure 12 is the signal simulation schematic diagram that FRI time-domain signal is obtained by Singh's core filter;

Figure 13 is the time domain FRI signal of reconstruct and original FRI signal simulation schematic diagram;

Figure 14 is the signal waveform emulation schematic diagram of the low-frequency component recovered;

Figure 15 is the wave simulation schematic diagram of restoring signal.

Embodiment

Embodiment one, the signal efficient sampling comprehensively analyzed based on FRI time-frequency domain and signal reconfiguring method, realized by following steps:

Step one, to primary signal x _nbe FFT conversion (Fast Fourier Transformation), i.e. fast Fourier transform, the expression of conversion is as follows:

$\begin{array}{c}{y}_k=\underset{n=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{W}_N^{\mathrm{kn}}{x}_n\\ ={\mathrm{\&Sigma;}}_{n=2t}{W}_N^{\mathrm{kn}}{x}_n+{\mathrm{\&Sigma;}}_{n=2t+1}{W}_N^{\mathrm{kn}}{x}_n\\ ={\mathrm{\&Sigma;}}_t{W}_{\frac{N}{2}}^{\mathrm{kt}}{x}_{2t}+W_N^k{\mathrm{\&Sigma;}}_{n=2t+1}{W}_{\frac{N}{2}}^{\mathrm{kt}}{x}_{2t+1}\\ =F_{\mathrm{even}}\left(k\right)+W_N^k{F}_{\mathrm{odd}}\left(k\right)\end{array}---\left(1\right)$

Wherein, W is used _nrepresent

Y can be obtained by (1) formula _k.Time-domain signal is transformed into frequency domain by this process.Make its spectrogram, line frequency analysis of spectrum of going forward side by side.Utilize high pass filter and low pass filter respectively afterwards, signal is divided into higher frequency content and lower frequency components.

Step 2, utilize the frequency f of frequency spectrum _kwith amplitude a _krecord the information of upper frequency composition.And by higher frequency content by filter filtering from original signal.In order to upper frequency composition is compressed further, frequency is taken the logarithm, and be normalized.Frequency is taken the logarithm, and normalized concrete steps are as follows:

If the original frequency of signal is f ₀, the analytic equation of taking the logarithm is:

f _log＝log ₁₀(f ₀) (2)

And choose f _login maximum f _max, normalized frequency is:

$\stackrel{\&OverBar;}{f}=\frac{f_{\log }}{f_{\max }}---\left(3\right)$

Like this, the frequency of all spectral lines all can be distributed in [0,1] scope.

Step 3, the signal of line-fitting algorithm to lower frequency of time domain is utilized to analyze.

Line-fitting algorithm is one of a kind of innovative point of the present invention.Low frequency signal is because its frequency is lower, that is: the change of signal within the unit interval is slower.Therefore, low frequency signal can regard the curve of slowly change as.These curves slowly changed can utilize straightway to carry out approximate representation.Namely the initial time of these line segments and slope can record the information of low frequency curve.Realize the compression of information in time domain.

Specifically, this line-fitting algorithm first judges whether signal is linear, if signal is linear, and the so moment of starting point and slope of straight line between recording areas.If this line segment is not linear, then line segment is halved, then judge whether signal is linear in two intervals be divided into respectively.If be linear, then the moment of starting point and the slope of straight line between recording areas.If be not linear, then continue to halve to interval, until signal is all approximately linear in each interval.The flow process of this algorithm as shown in Figure 2.

Judge whether each interval can be similar to and see that linear method is as follows:

If the interval judged is as [a, b], if meet:

$f\left(a\right)+f\left(b\right)-2\&times;f\left(\frac{a+b}{2}\right)<\mathrm{\&epsiv;}---\left(4\right)$

Then f (x) can be similar to and regard linear as in interval [a, b].In formula (4), ε is error permissible value.

Step 4, utilize FRI theory carry out follow-up signal transacting and the reconstruct of signal.

FRI theory can process and recovery latency is t _k, amplitude is a _kdiscrete signal.In a frequency domain, the frequency values f that spectrum line normalization is corresponding _krepresent, amplitude a _krepresent.In the time domain, the moment that straight line end points is corresponding also can use t _krepresent, corresponding ordinate of orthogonal axes a _krepresent.If the f in frequency domain _kregard the t in FRI theory as _k, a in frequency domain _kregard a in FRI theory as _k, like this, frequency-region signal just can utilize FRI theory to process.If the t in time domain _kregard the t in FRI theory as _k, a in time domain _kregard a in FRI theory as _k, like this, frequency-region signal just can utilize FRI theory to process.

Therefore, FRI theory can be utilized completely to carry out signal transacting and recovery to time domain and frequency domain respectively.

Utilize the step of FRI processing signals as follows:

First, by signal by specific filter, the time domain impulse response of filter is designated as h (t), and so we by the expression formula of signal y (t) obtained after filter are

y(t)＝h(t)*x(t) (5)

Here, h (t) can choose Gaussian function, sinc function, B-spline and E spline function.And y (t) is sampled, obtain discrete y _n, by sample y _n, provided by following formula:

Above formula is out of shape, obtains:

$\begin{array}{c}{y}_n=<x\left(t\right),{\mathrm{\&phi;}}_B(\mathrm{nT}-t)>\\ =\underset{m\&Element;Z}{\mathrm{\&Sigma;}}{\hat{x}}_m<e^{j2\mathrm{\&pi;m}\frac{t}{\mathrm{\&tau;}}},{\mathrm{\&phi;}}_B(\mathrm{nT}-t)>\\ =\underset{m\&Element;Z}{\mathrm{\&Sigma;}}{\hat{x}}_m{\widehat{\mathrm{\&phi;}}}_B\left(\frac{2\mathrm{\&pi;m}}{\mathrm{\&tau;}}\right)e^{j2\mathrm{\&pi;n}\frac{\mathrm{\&tau;}}{N}\frac{m}{\mathrm{\&tau;}}}\\ =\frac{1}{B}\underset{m\&le;M=\left[\frac{\mathrm{B\&tau;}}{2}\right]}{\mathrm{\&Sigma;}}{e}^{j2\mathrm{\&pi;}\frac{\mathrm{mn}}{N}}\end{array}---\left(7\right)$

That is,

${\hat{x}}_m={B\hat{y}}_m---\left(8\right)$

Here, x (t) by position is with amplitude be the stream of pulses signal formed.

$x\left(t\right)=\underset{k=0}{\overset{K-1}{\mathrm{\&Sigma;}}}{a}_k\underset{m\&Element;z}{\mathrm{\&Sigma;}}p(t-t_k-\mathrm{m\&tau;})=\underset{k=0}{\overset{K-1}{\mathrm{\&Sigma;}}}{a}_k\frac{1}{\mathrm{\&tau;}}\underset{m\&Element;z}{\mathrm{\&Sigma;}}\hat{p}\left(\frac{2\mathrm{\&pi;m}}{\mathrm{\&tau;}}\right)e^{j2\mathrm{\&pi;m}\frac{t-t_k}{\mathrm{\&tau;}}}=\underset{m\&Element;Z}{\mathrm{\&Sigma;}}{\hat{x}}_m{e}^{j2\mathrm{\&pi;m}\frac{t}{\mathrm{\&tau;}}}---\left(9\right)$

Calculate the Fourier series coefficient of x (t), obtain for

${\hat{x}}_m=\frac{1}{\mathrm{\&tau;}}\hat{p}\left(\frac{2\mathrm{\&pi;m}}{\mathrm{\&tau;}}\right)\underset{k=0}{\overset{K-1}{\mathrm{\&Sigma;}}}{a}_k{e}^{-j2\mathrm{\&pi;m}\frac{t_k}{\mathrm{\&tau;}}}---\left(10\right)$

Above formula is converted, can obtain

${\hat{x}}_m{\hat{p}}^{-1}\left(\frac{2\mathrm{\&pi;m}}{\mathrm{\&tau;}}\right)=\frac{1}{\mathrm{\&tau;}}\underset{k=0}{\overset{K-1}{\mathrm{\&Sigma;}}}{a}_k{u}_k^m---\left(11\right)$

Here and represent the multiplicative inverse of p, p (t) is that priori is known, for the sake of simplicity, and our conventional letter $\hat{p}\left(\frac{2\mathrm{\&pi;m}}{\mathrm{\&tau;}}\right)=1.$

In order to obtain u _kvalue, use represent that its filter Z-converts:

$\hat{h}\left(z\right)=\underset{m=0}{\overset{K}{\mathrm{\&Sigma;}}}{h}_m{z}^{-m}=\underset{m=0}{\overset{k-1}{\mathrm{\&Pi;}}}(1-u_k{z}^{-1})---\left(12\right)$

That is, root equal by the u found _kvalue.H _mmeet:

$h_m*{\hat{x}}_m=\underset{i=0}{\overset{K}{\mathrm{\&Sigma;}}}{h}_i{\hat{x}}_{m-i}=\underset{i=0}{\overset{K}{\mathrm{\&Sigma;}}}\underset{k=0}{\overset{K-1}{\mathrm{\&Sigma;}}}{a}_k{h}_i{u}_k^{m-i}=\underset{k=0}{\overset{K-1}{\mathrm{\&Sigma;}}}{a}_k{u}_k^m\underset{i=0}{\overset{K}{\mathrm{\&Sigma;}}}{h}_i{u}_k^{-i}=0---\left(13\right)$

Due to this filter { h _mbe called as and bury in oblivion filter, because it makes signal zero.

Here h is made ₀=1, formula (13) can be write as matrix form and be

This shows, we at least need 2K successive value solve said system.{ h can be obtained by equation above _m.{ h _mz conversion zero point be u _k, and obtain position t thus _k.Amplitude a can be solved by equation below _k:

Here, the process utilizing FRI theory to carry out signal transacting is summed up:

(1) Fourier series is obtained this can by using relational expression is passed through with them ${\hat{x}}_m={B\hat{y}}_m$ Come;

(2) coefficient of the filter buried in oblivion is retrieved formula (14) can be passed through solve;

(3) filter obtained root, its root produce value u _k.We can obtain position t thus _kvalue.

(4) amplitude a is asked _k, formula (15) can be passed through and obtain.

Position can be obtained by step above and amplitude utilize the flow chart of FRI theoretical treatment signal as shown in Figure 3.

Step 5, a obtained according to the recovery algorithms of FRI _kwith t _kthe frequency domain part of restoring signal and domain portion.In a frequency domain, the t of FRI recovery _k, can by t _kfirst calculate the frequency values f of renormalization _log,

f _log＝t _k×f _max(16)

Again by f _logcalculate the frequency f of higher frequency components _k, computing formula is such as formula (17):

$f_k={10}^{f_{\log }}---\left(17\right)$

The a recovered in FRI _kbe the amplitude of higher frequency components.In the time domain, the t of FRI recovery _kbe the moment of line segment starting point, a _kbe the slope of straight line.Therefore, by the t in time domain _kwith a _kthe time domain beamformer of the lower frequency part of signal can be recovered.

Step 6, by the signal of frequency domain part by IFFT conversion (inverse discrete Fourier transform changes) to time domain,

$x\left(n\right)=\frac{1}{N}{\left[\underset{k=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{X}^{*}\left(k\right)W_N^{\mathrm{nk}}\right]}^{*}=\frac{1}{N}{\left\{\mathrm{DET}\right[X^{*}\left(k\right)\left]\right\}}^{*}---\left(18\right)$

And with the Signal averaging of time domain medium and low frequency composition, recover the oscillogram of primary signal.

Specific embodiment:

This sampling plan is applicable to arbitrary continuation signal.In order to verify the high efficiency of sampling plan and the accuracy of restoring signal in the present invention, we enumerate the signal of a more complicated as an example.

Here, primary signal is chosen:

y(t)＝t ³-1.6t ²+2.5t+1.4sin(2πt)+2.2sin(2×2πt)

+cos(5×2πt)+1.2sin(10×2πt)+3.2cos(20×2πt)

+sin(50×2πt)+1.8(100×2πt)+cos(1000×2πt) (19)

+3.6cos(2000×2πt)+2.4cos(5000×2πt)

+2.7sin(10000×2πt)

The oscillogram of primary signal can be drawn as shown in Figure 4.

Do FFT conversion to primary signal, time-domain signal is converted into frequency domain, and available spectrogram as shown in Figure 5.By spectrogram, can the peak frequency of signal be 10000Hz, if utilize nyquist sampling law, then sample frequency must be greater than 20000Hz.

In order to more clearly show the frequency spectrum of low frequency, make the spectrogram of 0 ~ 100Hz as shown in Figure 6.In order to describe signal more accurately, we choose the spectral line that amplitude in spectrogram is greater than 0.01, and such spectral line has 71.Therefore, only carry out from frequency domain the information analyzing needs 71 points.

If analyzed in time domain by low-frequency component little for amplitude in spectral line, counting of sampling can be reduced further.Spectral line information can be utilized record higher frequency content, and by component filtering higher for frequency.The spectrum line of upper frequency has 9, and it is 9 that such frequency domain needs the information of record to count.The frequency at spectrum line place is designated as f _k, amplitude is designated as a _k.And by f _kto take the logarithm and normalization, can compress frequency domain further like this, FRI signal in frequency domain can be obtained thus as shown in Figure 7.

The signal that FRI frequency-region signal is obtained by Singh's core filter as shown in Figure 8.

Utilize frequency domain FRI signal that the recovery algorithms of FRI theory reconstructs as shown in Figure 9.In order to show the accuracy of recovery, also initial FRI frequency-region signal is drawn in fig .9.

The information record in a frequency domain of upper frequency, therefore can reject this part signal from primary signal.After the upper frequency composition filtering in signal, obtain compared with low frequency composition signal oscillogram as shown in Figure 10.

Processed comparatively low frequency composition signal by line-fitting algorithm, that determines to need in time domain to sample counts.Utilize line-fitting algorithm, can by the oscillogram of 30 line segment approximate description low-frequency components.The initial time of line segment is designated as t _k, slope is designated as a _k.FRI signal in time domain can be obtained as shown in figure 11.

The signal that FRI time-domain signal is obtained by Singh's core filter as shown in figure 12.

Utilize time domain FRI signal that the recovery algorithms of FRI theory reconstructs as shown in figure 13; In order to show the accuracy of recovery, also initial FRI time-domain signal is drawn in fig. 13.

According to the moment t recovered _k, amplitude a _kdetermine starting point and the slope of line segment, these line segments are joined end to end recovers the low frequency part of signal, thus, obtains the oscillogram of low frequency composition signal as shown in figure 14.

In a frequency domain, the IFFT that is of the FRI frequency-region signal of recovery is converted, HFS is transformed into time domain from frequency domain.And superpose with the FRI time-domain signal recovered, the oscillogram of the signal that is restored is as shown in figure 15.

By Figure 15, the accuracy of sampling plan restoring signal in the present invention can be found out.

The present invention has following characteristics and marked improvement:

1, the sampling plan in the present invention has high efficiency.Sampling plan of the present invention, for the signal in previous example, needs counting and being only 39 points of sampling, and wherein 9 points are at the high-frequency information of frequency domain tracer signal, and 30 points are in the low frequency information of time domain record signal.And only from time domain analysis, utilize nyquist sampling law to need the information of collection 20000 points.Only analyze from frequency domain, need the position of record 71 spectral lines.As can be seen here, the method that time-frequency domain of the present invention is comprehensively analyzed can reduce required sampling number widely, has the high efficiency of sampling.

2, the present invention proposes line-fitting algorithm in time-domain signal analysis, and straight line can be utilized to describe the wavy curve of low frequency signals more accurately, can extract the effective information of lower frequency composition so efficiently from time domain.

Although 3, the present invention's sampling count considerably less, can very accurately restoring signal.By Figure 15 and Fig. 4, can compare restoring signal and primary signal, the two is almost the same, therefore demonstrates in the present invention the accuracy of recovery algorithms of sampling.

4, the present invention utilizes FRI theoretical dexterously, and has expanded the theoretical treatable signal kinds of FRI.The present invention utilizes f _kand a _krecord frequency domain information, utilizes t _kand a _krecord time-domain information.And FRI theoretical treatable be discrete signal, as dirac stream.The information of these signals can use time delay t _kwith amplitude a _krepresent.The f in frequency domain _kregard the t in FRI theory as _k, a in frequency domain _kregard a in FRI theory as _k.Therefore, frequency domain part can utilize FRI theory to process.If the t in time domain _kregard the t in FRI theory as _k, a in time domain _kregard a in FRI theory as _k, the signal of such domain portion also can process by FRI theory.Therefore, FRI theory can be utilized to be further processed the information of time domain and frequency domain record.Meanwhile, also theoretical for FRI treatable signal kinds is expanded to high frequency continuous signal from discrete dirac stream.

Claims (4)

1. the signal efficient sampling comprehensively analyzed based on FRI time-frequency domain and signal reconfiguring method, is characterized in that: it is realized by following steps:

Step one, to primary signal x _nbe fast Fourier transform FFT, by this primary signal x _nbe transformed into frequency domain from time domain, obtain y _k, and make spectrogram; And utilize low pass filter frequency-region signal to be divided into the part of high frequency and the part of low frequency, wherein, the cut-off frequency of low pass filter makes corresponding selection according to concrete application scenarios;

Step 2, the frequency f utilized in spectrogram _kwith amplitude a _krecord the information of high-frequency part, and by the part of high frequency by low pass filter filtering from primary signal;

The frequency of the HFS of frequency-region signal is taken the logarithm, and is normalized, make the frequency distribution of spectrogram in [0,1] scope;

Frequency is taken the logarithm, and normalized concrete steps are as follows:

If the original frequency of primary signal is f ₀, after taking the logarithm, obtain f _log, concrete analytic equation is:

f _log＝log ₁₀(f ₀) (1)

And choose f _login maximum f _max, obtain normalized frequency for:

$\stackrel{\&OverBar;}{f}=\frac{f_{\log }}{f_{\mathrm{amx}}}---\left(2\right)$

Then, the frequency distribution of all spectral lines in spectrogram is in [0,1] scope;

Step 3, utilize the low frequency part of line-fitting algorithm to frequency-region signal of time domain to analyze, judge whether the low frequency part of frequency-region signal is linear; Be specially:

Step 3 one, utilize the line-fitting algorithm of time domain whether to be linear to the low frequency part of frequency-region signal, if judged result is yes, then perform step 3 three; If judged result is no, then perform step 3 two;

Step 3 two, line segment to be halved, judge that whether linear signal in two intervals be divided into one by one, if judged result is yes, then execution step 3 three; If judged result is no, then perform step 3 four;

The moment of starting point and the slope of this interval lower line segment between step 3 three, recording areas, and perform step 3 five;

Step 3 four, return step 3 two;

Step 4, utilize FRI technology to carry out signal to be reconstructed, to be specially:

Step 4 one, signal x (t) is carried out filtering by filter, the expression formula of filtered signal y (t) is:

y(t)＝h(t)*x(t) (3)

Wherein: the time domain impulse that h (t) is filtering responds, x (t) by position is with amplitude be the stream of pulses signal formed;

$x\left(t\right)=\underset{k=0}{\overset{K-1}{\mathrm{\&Sigma;}}}{a}_k\underset{m\&Element;z}{\mathrm{\&Sigma;}}p(t-t_k-\mathrm{m\&tau;})=\underset{k=0}{\overset{K-1}{\mathrm{\&Sigma;}}}{a}_k\frac{1}{\mathrm{\&tau;}}\underset{m\&Element;z}{\mathrm{\&Sigma;}}\hat{p}\left(\frac{2\mathrm{\&pi;m}}{\mathrm{\&tau;}}\right)e^{j2\mathrm{\&pi;m}\frac{{t-t}_k}{\mathrm{\&tau;}}}=\underset{m\&Element;Z}{\mathrm{\&Sigma;}}{\hat{x}}_m{e}^{j2\mathrm{\&pi;m}\frac{t}{\mathrm{\&tau;}}}---\left(4\right)$

Wherein: the analytic equation that p (t) is pulse, τ is the time span of signal; K is the quantity of signal pulse, and K is positive integer; J is positive integer;

Filtered signal y (t) is sampled, obtains discrete signal y _n, described discrete signal y _nexpression formula be:

Wherein: T is the sampling interval, for sampling function; N represents the n-th sampled point;

Above formula is out of shape, obtains:

$\begin{array}{c}{y}_n=<x\left(t\right),{\mathrm{\&phi;}}_B(\mathrm{nT}-t)>\\ =\underset{m\&Element;Z}{\mathrm{\&Sigma;}}{\hat{x}}_m<e^{j2\mathrm{\&pi;m}\frac{t}{\mathrm{\&tau;}}},{\mathrm{\&phi;}}_B(\mathrm{nT}-t)>\\ =\underset{m\&Element;Z}{\mathrm{\&Sigma;}}{\hat{x}}_m{\widehat{\mathrm{\&phi;}}}_B\left(\frac{2\mathrm{\&pi;m}}{\mathrm{\&tau;}}\right)e^{j2\mathrm{\&pi;n}\frac{\mathrm{\&tau;}}{N}\frac{m}{\mathrm{\&tau;}}}\\ =\frac{1}{B}\underset{m\&le;M=\left[\frac{\mathrm{B\&tau;}}{2}\right]}{\mathrm{\&Sigma;}}{\hat{x}}_m{e}^{j2\mathrm{\&pi;}\frac{\mathrm{mn}}{N}}\end{array}---\left(6\right)$

In formula (6), x _mdiscrete Fourier transform, B is the bandwidth of filter, and τ is the time span of signal;

That is:

${\hat{x}}_m=B{\hat{y}}_m---\left(7\right)$

Wherein: x _mdiscrete Fourier transform, y _mdiscrete Fourier transform, B is the bandwidth of filter;

Calculate the Fourier series coefficient of x (t) obtain for:

${\hat{x}}_m=\frac{1}{\mathrm{\&tau;}}\hat{p}\left(\frac{2\mathrm{\&pi;m}}{\mathrm{\&tau;}}\right)\underset{k=0}{\overset{K-1}{\mathrm{\&Sigma;}}}{a}_k{e}^{-j2\mathrm{\&pi;m}\frac{t_k}{\mathrm{\&tau;}}}---\left(8\right)$

Above formula is converted:

${\hat{x}}_m{\hat{p}}^{-1}\left(\frac{2\mathrm{\&pi;m}}{\mathrm{\&tau;}}\right)=\frac{1}{\mathrm{\&tau;}}\underset{k=0}{\overset{K-1}{\mathrm{\&Sigma;}}}{a}_k{u}_k^m---\left(9\right)$

Wherein:

$u_k=e^{-j2\mathrm{\&pi;}\frac{t_k}{\mathrm{\&tau;}}}---\left(10\right)$

represent the multiplicative inverse of p, the value of p (t) is that priori is known, for Dirac function stream, adopt z-conversion represent u _k:

$\hat{h}\left(z\right)=\underset{m=0}{\overset{K}{\mathrm{\&Sigma;}}}{h}_m{z}^{-m}=\underset{m=0}{\overset{k-1}{\mathrm{\&Pi;}}}(1-u_k{z}^{-1})---\left(11\right)$

That is, root equal by the u found _kvalue;

H _mmeet:

$h_m*{\hat{x}}_m=\underset{i=0}{\overset{K}{\mathrm{\&Sigma;}}}{h}_i{\hat{x}}_{m-i}=\underset{i=0}{\overset{K}{\mathrm{\&Sigma;}}}\underset{k=0}{\overset{K-1}{\mathrm{\&Sigma;}}}{a}_k{h}_i{u}_k^{m-i}=\underset{k=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{a}_k{u}_k^m\underset{i=0}{\overset{K}{\mathrm{\&Sigma;}}}{h}_i{u}_k^{-i}=0---\left(12\right)$

Due to this filter { h _mbe called as and bury in oblivion filter;

Make h ₀=1, being write formula (12) as matrix form is:

{ h is obtained by Solving Equations above _m;

{ h _mz conversion zero point be u _k,

t _k＝(lnu _k)/(-j)/2/π×τ (14)

Obtain time delay t thus _k;

Pass through equation:

Solve amplitude a _k;

Step 5, the amplitude a obtained according to step _kwith time delay t _kthe frequency domain part of restoring signal and domain portion;

By time delay t _kaccording to formula

f _log＝t _k×f _max(16)

Calculate the frequency values f of renormalization _log,

Again by f _logcalculate the frequency f of HFS _k, computing formula is such as formula (17):

$f_k={10}^{f_{\log }}---\left(17\right)$

In the time domain, by t _kas the moment of line segment starting point, a _kas the slope of straight line, recover the time domain waveform of the low frequency part of signal;

Step 6, the signal of frequency domain part is changed IFFT to time-domain signal by inverse discrete Fourier transform,

$x\left(n\right)=\frac{1}{N}{\left[\underset{k=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{X}^{*}\left(k\right)W_N^{\mathrm{nk}}\right]}^{*}=\frac{1}{N}{\left\{\mathrm{DFT}\right[X^{*}\left(k\right)\left]\right\}}^{*}---\left(18\right)$

Wherein: DFT [] does discrete Fourier transform to function; N represents the inverse discrete Fourier transform doing N point;

After the signal of frequency domain part is transformed to time domain, the Signal averaging with time domain medium and low frequency composition, recovers primary signal.

2. the signal efficient sampling comprehensively analyzed based on FRI time-frequency domain according to claim 1 and signal reconfiguring method, is characterized in that in step one primary signal x _nbe fast fourier transform FFT to be specially:

$\begin{array}{c}{y}_k=\underset{n=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{W}_N^{\mathrm{kn}}{x}_n\\ ={\mathrm{\&Sigma;}}_{n=2t}^{\mathrm{}}{W}_N^{\mathrm{kn}}{x}_n+{\mathrm{\&Sigma;}}_{n=2t+1}^{\mathrm{}}{W}_N^{\mathrm{kn}}{x}_n\\ ={\mathrm{\&Sigma;}}_t^{\mathrm{}}{W}_{\frac{N}{2}}^{\mathrm{kt}}{x}_{2t}+W_N^k{\mathrm{\&Sigma;}}_{n=2t+1}^{\mathrm{}}{W}_{\frac{N}{2}}^{\mathrm{kt}}{x}_{2t+1}\\ =F_{\mathrm{even}}\left(k\right)+W_N^k{F}_{\mathrm{odd}}\left(k\right)\end{array}---\left(19\right)$

Wherein: use W _nrepresent f _evenk () does discrete Fourier transform, F to even item _oddk () does discrete Fourier transform to odd item.

3. the signal efficient sampling comprehensively analyzed based on FRI time-frequency domain according to claim 1 and signal reconfiguring method, whether the low frequency part that it is characterized in that judging frequency-region signal is linear concrete grammar is:

For interval [a, b], judge whether interval [a, b] meets:

$f\left(a\right)+f\left(b\right)-2\&times;f\left(\frac{a+b}{2}\right)<\mathrm{\&epsiv;}---\left(20\right)$

Wherein: ε is error permissible value;

If met, then signal frequency f (x) is linear in interval [a, b]; If do not met, then signal frequency f (x) is nonlinear in interval [a, b].

4. the signal efficient sampling comprehensively analyzed based on FRI time-frequency domain according to claim 1 and signal reconfiguring method, it is characterized in that in step 4 one, h (t) chooses Gaussian function, sinc function, one of B-spline and E spline function.