CN104734791A - FRI (final random inspection) based sparse multiband signal frequency spectrum locating method - Google Patents

Abstract

The invention discloses an FRI (final random inspection) based sparse multiband signal frequency spectrum locating method, relates to the technical field of information and communication and solves problems that the number of known sub-bands and frequency bandwidth thereof when signals are restored through an existing broadband modulation converter system. The method includes that the broadband modulation converter system is taken as the research background, the limited innovation rate theory is combined to have the signals processed after appropriate transformation for the problems that the number of the known sub-bands and the frequency bandwidth thereof when the signals are restored through the original broadband modulation converter system, the limiting condition is ingeniously avoided, and the sparse multiband signal sub-bands are located. The sparse multiband signal frequency spectrum locating method is applicable to sparse multiband signal frequency spectrum location.

Description

Based on the sparse multi-band signals frequency spectrum localization method of FRI

Technical field

The present invention relates to Information & Communication Technology field, be specifically related to the sparse multi-band signals frequency spectrum perception technology of modulating wide-band transducer system.

Background technology

Analog because of its confidentiality poor, the problems such as antijamming capability is weak, gradually replace by Digital Signal Processing.The digitlization of the simulated and signal transacting instrument of real world, makes signal sampling become contact analog source and the indispensable bridge of digital signal.For many years, based on the classical signal processing mode of nyquist sampling theorem, almost dominate the acquisition of all signals or image etc., process, storage and transmission.It requires that sampling rate must reach more than the twice of signal bandwidth and accurately can recover primary signal.

Developing rapidly and for the increase of the quantity of information requirement, making the signal bandwidth of carry information more and more wider, under the prerequisite meeting nyquist sampling theorem, the difficulty of wideband signal processing is strengthened day by day, be faced with following problem along with communication:

(1) due to the restriction of sampling thheorem, sample frequency is higher, is a test to existing ADC equipment;

(2) sampled value that gets of high-speed sampling, requires higher to memory device;

(3) classical signal processing mode first samples to compress afterwards, wastes more sampling resource.

How to solve the problem that broadband signal gathers, must process in conjunction with the feature of such signal.For some broadband signal, as burst, frequency hopping, faint but constant signal and opportunity signal etc., its signal only occupies the sub-fraction of frequency spectrum at frequency domain, therefore in whole spectral range, has sparse characteristic, and this type of signal is called as multi-band signal.The restriction of Gonna breakthrough nyquist sampling theorem, must process in conjunction with this feature of such signal.

Traditional sampling theorem encounters bottleneck in process broadband signal, in this context, compressive sensing theory (Compressed Sensing, CS) formally proposed in 2004 by people such as Donoho and Candes, be one and make full use of the openness or compressible brand-new signals collecting of signal and encoding and decoding theory, sampling and compression while successfully achieving signal.This theory make use of the openness of signal or compressibility characteristic, changes signal acquisition and the tupe of high-speed sampling recompression in the past, directly carries out compressed sensing to obtain information to signal.By carrying out solving of a convex optimization problem to the low-dimensional measured value obtained, just can the reconstruct of complete pair signals or other digital processings, thus greatly reduce sample frequency, save memory space.For nature and the artificial signal produced, generally have openness or can find a transform domain, signal being shown under this transform domain openness, the proposition of compressive sensing theory is that signal acquisition and process provide new thinking.

Compressive sensing theory, in discrete digital signal process, can obtain good performance with less sampling resource.Although the compressed sensing of discrete domain is own through obtaining very large development, wants real reduction sampling rate, bringing large change to signal sampling, just the compressed sensing of discrete domain must be generalized to analog domain.In the last few years, along with going deep into of research, Xampling theoretical frame was introduced, in order to solve the compressed sensing problem of analog domain.Wherein, for the process of multiband analog signal, main implementation has analog information transducer (Analog to Information Converter, AIC) and modulation wide-band transducer (Modulated Wideband Converter, MWC).

AIC is otherwise known as random demodulation device, is to be proposed by people such as Sami Kirolos for 2006, its research be when useful frequency content contained in signal is very little relative to the bandwidth of signal, but the situation of positional information the unknown of these useful frequency contents.Under the prerequisite of lack sampling, need to adopt non-linear method to recover, but AIC is difficult to realize down-sampled to broadband signal.

MWC is proposed by Mishali M and Eldar Y.C for 2000, has used for reference the spread spectrum in communication, and design under the inspiration of AIC.MWC mainly solves the Sampling of broadband multi-frequency band signal lower than Nyquist rate, and the sampling rate of a MWC order of magnitude less of Nyquist rate.The sub-band position of recovery process sampling CTF (Continuous To Finite) the module determination multi-band signal of this system, but this process needs known signal sub-band number and band bandwidth.Proposition of the present invention is expected to break this restrictive condition just, without the need on the basis of this type of priori conditions, completes the perception of multi-band signal spectrum structure, for the acquisition, monitoring, interception etc. realizing multi-band signal provides technical support.

Summary of the invention

When recovering signal for existing modulation wide-band transducer system, need the problem of known sub-band number and band bandwidth thereof, the present invention proposes a kind of sparse multi-band signals frequency spectrum localization method based on FRI theory.

Based on the sparse multi-band signals frequency spectrum localization method of FRI, it is realized by following steps:

Step one, multi-band signal is carried out the sparse transformation based on FRI; Be specially:

Using FRI signal as parameter signal model, characterized signal by minority or limited parameter within the one-period time, signal form is:

Wherein, for known function collection, t _kfor translational movement, c _k,rfor amplitude; L is positive integer;

Definition counting function C _x(t _a, t _b), for interval [t computing time _a, t _b] number of interior signal parameter; t _afor timing initial time; t _bfor timing finish time;

Definition innovation speed is ρ:

$\mathrm{\ρ}=\underset{\mathrm{\τ}\→\∞}{\lim }\frac{1}{\mathrm{\τ}}{C}_x(-\frac{\mathrm{\τ}}{2},\frac{\mathrm{\τ}}{2})---\left(2\right)$

In formula: τ is timed interval length;

Analogy is carried out, if collection of functions with multi-band signal for known dirac stream, t _kby sub-band marginal position frequency f _kreplace, with seasonal c _k,r=1, rewrite formula (1) and obtain:

$x\left(f\right)=\underset{k=0}{\overset{2N-1}{\mathrm{\Σ}}}\mathrm{\δ}(f-f_k)---\left(3\right)$

Wherein, N is sub-band number; δ () is dirac stream function; F represents frequency;

Multi-band signal is represented again:

First, Time Continuous Fourier transform X (f) of input signal x (t) is obtained;

Then, differentiate process is carried out to signal X (f), build original FRI signal with this;

Multi-band signal form is expressed as:

$x\left(t\right)=\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\sqrt{E_i{B}_i}\sin c\left(B_i(t-{\mathrm{\τ}}_i)\right)\cos \left({2\mathrm{\πf}}_i(t-{\mathrm{\τ}}_i)\right)---\left(4\right)$

Wherein, E _ifor energy coefficient, τ _ifor time migration, B _ithe bandwidth of each sub-band, f _ifor carrier frequency; T represents the time;

Obtain Time Continuous Fourier transform X (f) form as follows:

$\begin{array}{c}X\left(f\right)=\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{C}_i\mathrm{rect}\left(\frac{f_i}{B_i}\right)*[\mathrm{\δ}(f-f_i)+\mathrm{\δ}(f+f_i)]\\ =\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{C}_i[\mathrm{rect}\left(\frac{f-f_i}{B_i}\right)+\mathrm{rect}\left(\frac{f+f_i}{B_i}\right)]\end{array}---\left(5\right)$

Wherein, C _ibe and E _i, τ _iand B _irelevant constant; Rect () represents rectangular function;

For real multi-band signal, its time continuous fourier transform X (f) is conjugation symmetry, has 2N sub-band; Herein, only pay close attention to the N number of frequency band in positive axis, abbreviation formula (5), obtains:

$X^{*}\left(f\right)=\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{C}_i\mathrm{rect}\left(\frac{f-f_i}{B_i}\right)---\left(6\right)$

To signal X ^*f () processes, select differentiate mode to provide original FRI signal form:

$x\left(f\right)=\underset{k=0}{\overset{N-1}{\mathrm{\Σ}}}\mathrm{\δ}(f-f_k)=\underset{\mathrm{\Δf}\→0}{\lim }\frac{X^{*}(f+\mathrm{\Δf})-X^{*}\left(f\right)}{\mathrm{\Δf}}---\left(7\right)$

Then from multi-band signal, obtain original FRI signal x (f);

In formula, △ f is set frequency interval and △ f ≠ 0; In order to obtain dirac stream, threshold value being set for x (f), making △ f get default value;

F _kcomprise following form:

$f_k=\underset{i=1}{\overset{N}{\mathrm{\Σ}}}[\mathrm{\δ}(f_i+\frac{B_i}{2})+\mathrm{\δ}(f_i-\frac{B_i}{2})]---\left(8\right);$

Step 2, the FRI signal form shown in acquisition formula (3); In order to corresponding with FRI theory, using original FRI signal as time-domain signal, utilize moment t _kreplace frequency f _k; Rewrite formula (3), obtain:

$x\left(t\right)=\underset{k=0}{\overset{2N-1}{\mathrm{\Σ}}}\mathrm{\δ}(t-t_k)---\left(9\right)$

Based on FRI theory, signal is processed, is specially:

First, obtained the Fourier coefficient of original FRI signal by sampled value, utilize sampling to check original FRI signal and sample, its process prescription is:

Wherein, core selection bandwidth of sampling is the sinc function of B; In formula: represent sampling kernel function; T is the sampling period;

Relation between sampled value and its Fourier coefficient as shown in the formula:

By sampled value y _n, obtain Fourier coefficient represent that bandwidth is the sampling core sinc function of B; τ is the dirac stream function cycle; N is integer; J is imaginary unit;

Signal x (t) is adopted the linear combination of its Fourier coefficient to represent:

$x\left(t\right)=\underset{k=0}{\overset{K-1}{\mathrm{\Σ}}}\frac{1}{\mathrm{\τ}}\underset{m\∈z}{\mathrm{\Σ}}\hat{p}\left(\frac{2\mathrm{\πm}}{\mathrm{\τ}}\right)e^{j2\mathrm{\πm}\frac{t-t_k}{\mathrm{\τ}}}=\underset{m\∈Z}{\mathrm{\Σ}}{\hat{x}}_m{e}^{j2\mathrm{\πm}\frac{t}{\mathrm{\τ}}}---\left(11\right)$

Wherein, the Dirac function number that comprises for each cycle dirac stream function of K; for Dirac function Fourier transform;

${\hat{x}}_m=\frac{1}{\mathrm{\τ}}\hat{p}\left(\frac{2\mathrm{\πm}}{\mathrm{\τ}}\right)\underset{k=0}{\overset{K-1}{\mathrm{\Σ}}}{e}^{-j2\mathrm{\πm}\frac{t_k}{\mathrm{\τ}}}---\left(12\right)$

It is the Fourier coefficient of original FRI signal;

Calculate:

${\hat{x}}_m{\hat{p}}^{-1}\left(\frac{2\mathrm{\πm}}{\mathrm{\τ}}\right)=\frac{1}{\mathrm{\τ}}\underset{k=0}{\overset{K-1}{\mathrm{\Σ}}}{a}_k{u}_k^m---\left(13\right)$

Wherein: include t _kinformation;

In order to obtain u in formula (13) _kvalue, definition fall into oblivion filter and its z converts root equal u _kvalue, form is as follows:

In formula: for the Fourier coefficient of original FRI signal x (t); a _kfor dirac stream function amplitude; h _ifor falling into oblivion filter coefficient; with for falling into oblivion the root of filter z conversion;

If h ₀=1, formula (14) is rewritten as matrix form:

Utilize matrix computations solution formula (15), obtain and fall into oblivion filter its root defines value u _kset;

So far, by calculating u _kzero pole point problem, obtain positional information t _k, i.e. original multi-band signal sub-band edge frequency point information f _k, the sparse multi-band signals frequency spectrum completed based on FRI is located.

In step, if under noise existence condition, then FRI signaling protein14-3-3 process is:

First, sampled value form is obtained:

Wherein ε _nrepresent noise;

Due to the existence of noise, the exact value of Fourier coefficient therefore can not be obtained but obtain containing noisy form ${\hat{x}}_m^{*};$

Adopt the impact of Ka Zuo (Cadzow) iterative algorithm stress release treatment, that is: utilize by the noiseless matrix A formed is Teoplitz (Toeplitz) matrix with K grade, and in multi-band signal scene, K is corresponding with sub-band number N; Find a K grade Toeplitz matrix A ', make it in minimum not Lip river than in Nice (Frobenius) norm meaning, with noise matrix A ^*approximate, wherein A ^*by form;

That is: following optimization problem is solved:

meeting rank (A')≤K and A' is Toeplitz matrix (17)

Utilize Cadzow iterative algorithm algorithm to upgrade objective matrix B, until its convergence, wherein utilize A ^*initialization is carried out to B; Then utilize the up-to-date B obtained to obtain h _m.

Cadzow iterative algorithm algorithm is utilized to upgrade objective matrix B, until the method for its convergence is specially:

Input: Noise matrix A ^*;

Export: up-to-date matrix B;

Initialization: order matrix B equals original Noise calculation matrix A ^*;

Step 1: matrix B is decomposed into B=USV ^tform, wherein U and V is unit matrix, and S is diagonal matrix;

Step 2: utilize K maximum in a S element as element on the diagonal of new diagonal matrix S', other positional values of this matrix are zero, build new diagonal matrix S' with this;

Step 3: matrix B is updated to more excellent K grade and is similar to B=US'V ^t;

Step 4: by the operation be averaged matrix B diagonal element, upgrading matrix B is that more excellent Toeplitz is similar to;

Step 5: repeat step 2 until the set point of matrix B convergence or satisfied iterations in advance.

In the present invention, in conjunction with limited innovation Rate Theory, when detecting multi-band signal band position, avoid the restriction needing known sub-band number and band bandwidth thereof in modulation wide-band transducer system in original recovery process dexterously, without the need on the basis of this type of priori conditions, realize the location to multi-band signal sub-bands, relax the restriction to the requirement of multi-band signal spectrum structure.

The present invention carries out secondary rarefaction representation to the multi-band signal possessing spectrum sparse structure, the signal form being applicable to limited innovation Rate Theory is obtained in transform domain, namely this signal can by limited Parametric Representation, and the parameter information of signal is and extracts multi-band signal sub-band edge position information.

The present invention utilizes limited innovation Rate Theory not exclusively to sample to original discrete sparse sequence signal sparse in transform domain, realizes lack sampling process, obtains useful information in signal efficiently, effectively reduces the sampling rate of signal.And utilize sampled value that lack sampling obtains, solve primary signal parameter information by numerical computations mode, for sub-band location provides useful information.In addition, when signal is recovered, under noise existence condition, increase Cadzow iterative process, improve limited innovation Rate Theory to the robust performance of noise.

Accompanying drawing explanation

Fig. 1 is the time domain waveform schematic diagram of original multi-band signal;

The high frequency carrier that modulates information produces to 4 by sinc function is at random utilized in figure;

Fig. 2 is the spectrum structure schematic diagram of original multi-band signal;

This signal is made up of 4 subsignals, and be made up of 8 sub-bands (each subsignal has a pair symmetrical frequency band of conjugation), each sub-band bandwidth is 4MHz, and Whole frequency band scope is 4GHz;

Fig. 3 is limited innovation speed (FRI) signal processing results emulation schematic diagram;

It utilizes its recovery algorithms to obtain the restoring signal overlapped with original FRI signal;

Fig. 4 is original multi-band signal spectrum position information emulator schematic diagram;

It utilizes limited innovation Rate Theory to obtain FRI signal parameter information, and finally determines multi-band signal sub-band position;

Fig. 5 is limited innovation Rate Theory recovery algorithms performance evaluation emulation schematic diagram;

By more whether adopting the difference recovering precision during Cadzow iterative algorithm in figure, show the robustness that limited innovation Rate Theory has noise.

Embodiment

Embodiment one, the signal type utilizing limited innovation speed (FRI) theory to process are the signals with limited innovation speed, namely within the one-period time, such signal can such as, by limited parameter characterization, impulse sequence stream, piecewise linear function etc.No matter whether signal belongs to band limit, and FRI theory all can be utilized to process.Sampling rate is relevant to information rate, can far below Nyquist sampling rate.Specific embodiments is divided into two steps: secondary sparse transformation and the process utilizing FRI theory to signal after conversion of multi-band signal.

1, based on the signal sparse transformation of FRI

In summary, the signal that FRI theory can process must have limited this locality innovation speed, and such as, the piece-wise linear signal that can be characterized by node location and this position amplitude just has limited information speed.And multi-band signal also possesses similar feature, its frequency spectrum does not contain whole nyquist frequency scope, only occupies sub-fraction wherein.Therefore, multi-band signal can be summed up as sparse signal type.From this starting point, this particular frequency spectrum structure can be utilized, obtain the result of our expectation in conjunction with different signal processing methods.Contact FRI is theoretical, can find between multi-band signal and FRI signal, there is certain contact.So, utilize FRI theory can solve some problem of multi-band signal.

Object is locator band position, therefore only needs to grasp sub-band edge position information, and amplitude is insignificant, and only use one, frequency position parameter can represent sub-band marginal information, and this and FRI signal form are quite similar.So, need multi-band signal to be transformed into FRI signal to construct sparse signal model.First, observe FRI signal, seen as parameter signal model, can be characterized signal by minority or limited parameter within the one-period time.Signal form is:

Wherein, for known function collection, such as impulse stream, any translation t _kwith amplitude c _k,r.Here a counting function C is proposed _x(t _a, t _b), its effect is interval [t computing time _a, t _b] number of interior signal parameter.Definition innovation speed is ρ:

$\mathrm{\ρ}=\underset{\mathrm{\τ}\→\∞}{\lim }\frac{1}{\mathrm{\τ}}{C}_x(-\frac{\mathrm{\τ}}{2},\frac{\mathrm{\τ}}{2})---\left(2\right)$

Provide the definition of FRI signal, this signal has limited innovation speed, and its parameter characterization form is as shown in (1), and limited ρ is provided by (2).Analogy is carried out, if collection of functions with multi-band signal for known dirac stream, t _kby sub-band marginal position frequency f _kreplace, with seasonal c _k,r=1, rewrite (1) and obtain:

$x\left(f\right)=\underset{k=0}{\overset{2N-1}{\mathrm{\Σ}}}\mathrm{\δ}(f-f_k)---\left(3\right)$

Wherein, N is sub-band number.This signal belongs to FRI signal model.In order to obtain this form, need again to represent multi-band signal.

First, obtain Time Continuous Fourier transform X (f) of input signal x (t), it shows multiband form.

Then, utilize special transformation to extract marginal information, such as, differentiate process is carried out to signal X (f), build original FRI signal with this.

For simplicity, multi-band signal form is expressed as:

$x\left(t\right)=\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\sqrt{E_i{B}_i}\sin c\left(B_i(t-{\mathrm{\τ}}_i)\right)\cos \left({2\mathrm{\πf}}_i(t-{\mathrm{\τ}}_i)\right)---\left(4\right)$

Wherein, E _ifor energy coefficient, τ _ifor time migration, B _ithe bandwidth of each sub-band, and f _ifor carrier frequency.

Then, its time continuous fourier transform X (f) form is obtained as follows:

$\begin{array}{c}X\left(f\right)=\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{C}_i\mathrm{rect}\left(\frac{f_i}{B_i}\right)*[\mathrm{\δ}(f-f_i)+\mathrm{\δ}(f+f_i)]\\ =\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{C}_i[\mathrm{rect}\left(\frac{f-f_i}{B_i}\right)+\mathrm{rect}\left(\frac{f+f_i}{B_i}\right)]\end{array}---\left(5\right)$

Wherein, C _ibe and E _i, τ _iand B _irelevant constant.In addition, rect () represents rectangular function.Notice, for real multi-band signal, its time continuous fourier transform X (f) is conjugation symmetry, has 2N sub-band.

Only pay close attention to the N number of frequency band in positive axis.Abbreviation (5), obtains:

$X^{*}\left(f\right)=\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{C}_i\mathrm{rect}\left(\frac{f-f_i}{B_i}\right)---\left(6\right)$

Finally, obtain original FRI signal, we are to signal X ^*f () processes.Here select differentiate mode, provide original FRI signal form:

$x\left(f\right)=\underset{k=0}{\overset{N-1}{\mathrm{\Σ}}}\mathrm{\δ}(f-f_k)=\underset{\mathrm{\Δf}\→0}{\lim }\frac{X^{*}(f+\mathrm{\Δf})-X^{*}\left(f\right)}{\mathrm{\Δf}}---\left(7\right)$

Certainly, △ f ≠ 0 is only a very little numerical value here.In order to obtain dirac stream, reasonable threshold value being set for x (f), making △ f get suitable numerical value.In fact, f _kcomprise following special shape:

$f_k=\underset{i=1}{\overset{N}{\mathrm{\Σ}}}[\mathrm{\δ}(f_i+\frac{B_i}{2})+\mathrm{\δ}(f_i-\frac{B_i}{2})]---\left(8\right)$

But we do not need to be grasped f here _i, B _iwith the prior information of N.This process is similar to rim detection.So far, from multi-band signal, original FRI signal x (f) is obtained.

2, FRI signal processing

In a first portion, first consider noiseless scene, the theoretical general principle to dirac stream signal transacting of research and utilization FRI.In ensuing Part II, the performance issue of FRI theory signal recovery algorithms under research noise circumstance.

(1) sampling of FRI signal and recovery

According to above-mentioned steps, the FRI signal form of shape as shown in (3) can be obtained.In order to corresponding with FRI theory, see original FRI signal as time-domain signal, utilize moment t _kreplace frequency f _k.Rewrite (3), obtain:

$x\left(t\right)=\underset{k=0}{\overset{2N-1}{\mathrm{\Σ}}}\mathrm{\δ}(t-t_k)---\left(9\right)$

Then, existing theory is utilized directly to process signal.In fact, according to signal parameter number, processing procedure is suitably improved.The core of FRI theory is with the Fourier coefficient of FRI signal as starting point, numerical computations is utilized to obtain parameter information, such as, in matrix computations, z conversion zero pole point calculating etc.First, the Fourier coefficient of original FRI signal is obtained by sampled value.Utilize sampling to check original FRI signal to sample, its process can be described as

Here, core selection bandwidth of sampling is the sinc function of B.Relation between sampled value and its Fourier coefficient as shown in the formula:

By sampled value y _n, obtain Fourier coefficient carry out subsequent treatment, next enter into the numerical procedure of FRI.

Because signal x (t) can be represented by the linear combination of its Fourier coefficient:

$x\left(t\right)=\underset{k=0}{\overset{K-1}{\mathrm{\Σ}}}\frac{1}{\mathrm{\τ}}\underset{m\∈z}{\mathrm{\Σ}}\hat{p}\left(\frac{2\mathrm{\πm}}{\mathrm{\τ}}\right)e^{j2\mathrm{\πm}\frac{t-t_k}{\mathrm{\τ}}}=\underset{m\∈Z}{\mathrm{\Σ}}{\hat{x}}_m{e}^{j2\mathrm{\πm}\frac{t}{\mathrm{\τ}}}---\left(11\right)$

Wherein,

${\hat{x}}_m=\frac{1}{\mathrm{\τ}}\hat{p}\left(\frac{2\mathrm{\πm}}{\mathrm{\τ}}\right)\underset{k=0}{\overset{K-1}{\mathrm{\Σ}}}{e}^{-j2\mathrm{\πm}\frac{t_k}{\mathrm{\τ}}}---\left(12\right)$

It is the Fourier coefficient of original FRI signal.

Further calculating,

${\hat{x}}_m{\hat{p}}^{-1}\left(\frac{2\mathrm{\πm}}{\mathrm{\τ}}\right)=\frac{1}{\mathrm{\τ}}\underset{k=0}{\overset{K-1}{\mathrm{\Σ}}}{a}_k{u}_k^m---\left(13\right)$

Wherein include t _kinformation.In order to obtain u in (13) _kvalue, definition fall into oblivion filter and its z converts root equal u _kvalue, form is as follows:

Suppose h without loss of generality ₀=1, (14) are rewritten as matrix form:

Utilize matrix computations to solve (15), obtain and fall into oblivion filter its root defines value u _kset.So far, by calculating u _kzero pole point problem, obtain positional information t _k, namely original multi-band signal sub-band edge frequency point information f _k.

(2) the theoretical recovery process of FRI under noise existence condition

In real world, signal bar none pollute by noise.Next, the process to this scene under research FRI theoretical frame.First, obtain as down-sampled values form:

Wherein: ε _nrepresent noise.

Still utilize above-mentioned steps to calculate, due to the existence of noise, we can not obtain the exact value of Fourier coefficient but obtain containing noisy form in FRI theory, in order to the impact of stress release treatment, propose a kind of method, be called Cadzow iterative algorithm.Its thought be make use of by the noiseless matrix A formed is the fact of the Toeplitz matrix with K grade, and in multi-band signal scene, K is corresponding with sub-band number N.Our target be a searching K grade Toeplitz matrix A ', make it in minimum Frobenius norm meaning, with noise matrix A ^*approximate, wherein A ^*by form.This means, we will solve following optimization problem:

meeting rank (A')≤K and A' is Toeplitz matrix (17)

In order to solve (17), we utilize algorithm to upgrade objective matrix B, until its convergence, wherein utilize A ^*initialization is carried out to B.Then utilize the up-to-date B obtained to obtain h _m.The object of this process is the impact reducing noise, and detail is as follows:

Input: Noise matrix A ^*

Export: up-to-date matrix B

Initialization: order matrix B equals original Noise calculation matrix A ^*.

Step 1: matrix B is decomposed into B=USV ^tform, wherein U and V is unit matrix, and S is diagonal matrix;

Step 2: utilize K maximum in a S element as element on the diagonal of new diagonal matrix S', other positional values of this matrix are zero, build new diagonal matrix S' with this;

Step 3: matrix B is updated to more excellent K grade and is similar to B=US'V ^t;

Step 4: by the operation be averaged matrix B diagonal element, upgrading matrix B is that more excellent Toeplitz is similar to;

Step 5: repeat step 2 until the set point of matrix B convergence or satisfied iterations in advance.

The invention discloses a kind of based on limited innovation speed (Finite Rate of Innovation, FRI) theoretical sparse multi-band signals frequency spectrum localization method, relate to Information & Communication Technology field, its objective is and utilize the sparse characteristic that in practical application scene, multi-band signal possesses, effectively reduce the sampling rate of signal, and accurately perception goes out signal band position to be studied, for the acquisition, monitoring, interception etc. realizing multi-band signal provides technical support.The method is to modulate wide-band transducer system for research background, when signal being recovered for original system, need the problem of known sub-band number and band bandwidth thereof, in conjunction with limited innovation Rate Theory, after proper transformation is done to signal, it is processed, avoid this restrictive condition dexterously, realize the location of sparse multi-band signals sub-band position.The invention belongs to a type of frequency spectrum perception, study mainly for band position, its main feature is the prior information without the need to multi-band signal sub-band number and band bandwidth, breaches the restriction to multi-band signal spectrum structure.Because multi-band signal is usual and mimo channel, multi-band OFDM radio ultra wide band system, and cognitive radio networks combines, along with the continuous increase of multi-band signal sub-bands bandwidth, effectively drive the extensive use of modulation wide-band transducer system, as a key technology of modulation wide-band transducer system, the present invention has wide application space and the broad scope of application.In the present invention, first secondary rarefaction representation is carried out to the multi-band signal possessing spectrum sparse structure, in transform domain, obtain the signal form being applicable to limited innovation Rate Theory and carrying out processing.Secondly, utilize limited innovation Rate Theory, lack sampling is carried out to sparse signal in transform domain, effectively reduce the sampling rate of signal, and according to sampled value that lack sampling obtains, this sparse signal is recovered.Finally, when research noise exists, limited innovation Rate Theory is for the robust performance of noise.This design, through simulating, verifying, is determined to obtain good signal recuperation effect and systems axiol-ogy performance.

The present invention carries out secondary rarefaction representation to the multi-band signal possessing spectrum sparse structure, the signal form being applicable to limited innovation Rate Theory is obtained in transform domain, concrete grammar is in transform domain, differentiate process is carried out to the discontinuous multi-band signal frequency-domain expression of frequency spectrum, impulse sequence form is obtained at discontinuity point place, form original discrete sparse sequence signal, this signal can by limited Parametric Representation, meet limited innovation Rate Theory to the requirement of handled signal, extract multi-band signal sub-band edge position information with this.

The present invention utilizes limited innovation Rate Theory not exclusively to sample to original discrete sparse sequence signal sparse in transform domain, realizes lack sampling process, obtains useful information in signal efficiently, effectively reduces the sampling rate of signal.

The present invention utilizes sampled value that lack sampling obtains, design includes the annihilation filter system function of primary signal parameter information, coupling system function zero pole point feature, solves primary signal parameter information by numerical computations mode, i.e. multi-band signal sub-band positional information.Can be proved by emulation, the location utilizing limited innovation theory to realize sparse multi-band signals sub-band position is feasible.

When the present invention utilizes limited innovation Rate Theory to process discrete sparse sequence signal, consider the performance of noise existence condition limited innovation Rate Theory recovery algorithms, the mode adopting matrix circular to decompose carries out the iteration of algorithm, reduce the impact of noise for sampled value to greatest extent, namely in original signal recuperation process, increase Cadzow iterative process, show limited innovation Rate Theory to the robust performance of noise with this.Found by emulation, compared with original algorithm performance not adopting Cadzow iterative process, increase the recovery precision that Cadzow iterative process effectively can improve algorithm.

The present invention has following characteristics and marked improvement:

1, when detecting multi-band signal band position, avoid the restriction needing known sub-band number and band bandwidth thereof in modulation wide-band transducer system in original recovery process dexterously, without the need on the basis of this type of priori conditions, realize the location to multi-band signal sub-bands, relax the restriction to the requirement of multi-band signal spectrum structure.

2, secondary rarefaction representation is carried out to the multi-band signal possessing spectrum sparse structure, the signal form being applicable to limited innovation Rate Theory is obtained in transform domain, namely this signal can by limited Parametric Representation, and the parameter information of signal is and extracts multi-band signal sub-band edge position information.

3, utilize limited innovation Rate Theory not exclusively to sample to original discrete sparse sequence signal sparse in transform domain, realize lack sampling process, obtain useful information in signal efficiently, effectively reduce the sampling rate of signal.

4, sampled value that lack sampling obtains is utilized, design includes the annihilation filter system function of primary signal parameter information, coupling system function zero pole point feature, solves primary signal parameter information by numerical computations mode, reconstruct multi-band signal sub-band positional information.

When 5, utilizing limited innovation Rate Theory to process discrete sparse sequence signal, under noise existence condition, increase Cadzow iterative process, improve limited innovation Rate Theory to the robust performance of noise with this.

Claims (3)

1., based on the sparse multi-band signals frequency spectrum localization method of FRI, it is characterized in that: it is realized by following steps:

Step one, multi-band signal is carried out the sparse transformation based on FRI; Be specially:

Using FRI signal as parameter signal model, characterized signal by minority or limited parameter within the one-period time, signal form is:

Wherein, for known function collection, t _kfor translational movement, c _k,rfor amplitude; L is positive integer;

Definition counting function C _x(t _a, t _b), for interval [t computing time _a, t _b] number of interior signal parameter; t _afor timing initial time; t _bfor timing finish time;

Definition innovation speed is ρ:

$\mathrm{\ρ}=\underset{\mathrm{\τ}\→\∞}{\lim }\frac{1}{\mathrm{\τ}}{C}_x(-\frac{\mathrm{\τ}}{2},\frac{\mathrm{\τ}}{2})---\left(2\right)$

In formula: τ is timed interval length;

Analogy is carried out, if collection of functions with multi-band signal for known dirac stream, t _kby sub-band marginal position frequency f _kreplace, with seasonal c _k,r=1, rewrite formula (1) and obtain:

$x\left(f\right)=\underset{k=0}{\overset{2N-1}{\mathrm{\Σ}}}\mathrm{\δ}(f-f_k)---\left(3\right)$

Wherein, N is sub-band number; δ () is dirac stream function; F represents frequency;

Multi-band signal is represented again:

First, Time Continuous Fourier transform X (f) of input signal x (t) is obtained;

Then, differentiate process is carried out to signal X (f), build original FRI signal with this;

Multi-band signal form is expressed as:

$x\left(t\right)=\underset{i=1}{\overset{N}{\mathrm{\Σ}}}\sqrt{E_i{B}_i}\sin c\left(B_i(t-{\mathrm{\τ}}_i)\right)\cos \left(2\mathrm{\π}{f}_i(t-{\mathrm{\τ}}_i)\right)---\left(4\right)$

Wherein, E _ifor energy coefficient, τ _ifor time migration, B _ithe bandwidth of each sub-band, f _ifor carrier frequency; T represents the time;

Obtain Time Continuous Fourier transform X (f) form as follows:

$\begin{array}{c}X\left(f\right)=\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{C}_i\mathrm{rect}\left(\frac{f_i}{B_i}\right)*[\mathrm{\δ}(f-f_i)+\mathrm{\δ}(f+f_i)]\\ =\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{C}_i[\mathrm{rect}\left(\frac{f-f_i}{B_i}\right)+\mathrm{rect}\left(\frac{f+f_i}{B_i}\right)]\end{array}---\left(5\right)$

Wherein, C _ibe and E _i, τ _iand B _irelevant constant; Rect () represents rectangular function;

For real multi-band signal, its time continuous fourier transform X (f) is conjugation symmetry, has 2N sub-band; Herein, only pay close attention to the N number of frequency band in positive axis, abbreviation formula (5), obtains:

$X^{*}\left(f\right)=\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{C}_i\mathrm{rect}\left(\frac{f-f_i}{B_i}\right)---\left(6\right)$

To signal X ^*f () processes, select differentiate mode to provide original FRI signal form:

$x\left(f\right)=\underset{k=0}{\overset{N-1}{\mathrm{\Σ}}}\mathrm{\δ}(f-f_k)=\underset{\mathrm{\Δf}\→0}{\lim }\frac{X^{*}(f+\mathrm{\Δf})-X^{*}\left(f\right)}{\mathrm{\Δf}}---\left(7\right)$

Then from multi-band signal, obtain original FRI signal x (f);

In formula, △ f is set frequency interval and △ f ≠ 0; In order to obtain dirac stream, threshold value being set for x (f), making △ f get default value;

F _kcomprise following form:

$f_k=\underset{i=1}{\overset{N}{\mathrm{\Σ}}}[\mathrm{\δ}(f_i+\frac{B_i}{2})+\mathrm{\δ}(f_i-\frac{B_i}{2})]---\left(8\right);$

Step 2, the FRI signal form shown in acquisition formula (3); In order to corresponding with FRI theory, using original FRI signal as time-domain signal, utilize moment t _kreplace frequency f _k; Rewrite formula (3), obtain:

$x\left(t\right)=\underset{k=0}{\overset{2N-1}{\mathrm{\Σ}}}\mathrm{\δ}(t-t_k)---\left(9\right)$

In formula: t represents the moment;

Based on FRI theory, signal is processed, is specially:

First, obtained the Fourier coefficient of original FRI signal by sampled value, utilize sampling to check original FRI signal and sample, its process prescription is:

Wherein, core selection bandwidth of sampling is the sinc function of B; In formula: represent sampling kernel function; T is the sampling period;

Relation between sampled value and its Fourier coefficient as shown in the formula:

By sampled value y _n, obtain Fourier coefficient represent that bandwidth is the sampling core sinc function of B; τ is the dirac stream function cycle; N is integer; J is imaginary unit;

Signal x (t) is adopted the linear combination of its Fourier coefficient to represent:

$x\left(t\right)=\underset{k=0}{\overset{K-1}{\mathrm{\Σ}}}\frac{1}{\mathrm{\τ}}\underset{m\∈z}{\mathrm{\Σ}}\hat{p}\left(\frac{2\mathrm{\πm}}{\mathrm{\τ}}\right)e^{j2\mathrm{\πm}\frac{t-t_k}{\mathrm{\τ}}}=\underset{m\∈Z}{\mathrm{\Σ}}{\hat{x}}_m{e}^{j2\mathrm{\πm}\frac{t}{\mathrm{\τ}}}---\left(11\right)$

Wherein, the Dirac function number that comprises for each cycle dirac stream function of K; for Dirac function Fourier transform;

${\hat{x}}_m=\frac{1}{\mathrm{\τ}}\hat{p}\left(\frac{2\mathrm{\πm}}{\mathrm{\τ}}\right)\underset{k=0}{\overset{K-1}{\mathrm{\Σ}}}{e}^{-j2\mathrm{\πm}\frac{t_k}{\mathrm{\τ}}}---\left(12\right)$

It is the Fourier coefficient of original FRI signal;

Calculate:

${\hat{x}}_m{\hat{p}}^{-1}\left(\frac{2\mathrm{\πm}}{\mathrm{\τ}}\right)=\frac{1}{\mathrm{\τ}}\underset{k=0}{\overset{K-1}{\mathrm{\Σ}}}{a}_k{u}_k^m---\left(13\right)$

Wherein: include t _kinformation;

In order to obtain u in formula (13) _kvalue, definition fall into oblivion filter and its z converts root equal u _kvalue, form is as follows:

In formula: for the Fourier coefficient of original FRI signal x (t); a _kfor dirac stream function amplitude; h _ifor falling into oblivion filter coefficient; with for falling into oblivion the root of filter z conversion;

If h ₀=1, formula (14) is rewritten as matrix form:

Utilize matrix computations solution formula (15), obtain and fall into oblivion filter its root defines value u _kset;

So far, by calculating u _kzero pole point problem, obtain positional information t _k, i.e. original multi-band signal sub-band edge frequency point information f _k, the sparse multi-band signals frequency spectrum completed based on FRI is located.

2. the sparse multi-band signals frequency spectrum localization method based on FRI according to claim 1, it is characterized in that in step, if under noise existence condition, then FRI signaling protein14-3-3 process is:

First, sampled value form is obtained:

Wherein ε _nrepresent noise;

Due to the existence of noise, the exact value of Fourier coefficient therefore can not be obtained but obtain containing noisy form

Adopt the impact of Ka Zuo (Cadzow) iterative algorithm stress release treatment, that is: utilize by the noiseless matrix A formed is Teoplitz (Toeplitz) matrix with K grade, and in multi-band signal scene, K is corresponding with sub-band number N; Find a K grade Toeplitz matrix A ', make it in minimum not Lip river than in Nice (Frobenius) norm meaning, with noise matrix A ^*approximate, wherein A ^*by form;

That is: following optimization problem is solved:

meeting rank (A')≤K and A' is Toeplitz matrix (17)

Utilize Cadzow iterative algorithm algorithm to upgrade objective matrix B, until its convergence, wherein utilize A ^*initialization is carried out to B; Then utilize the up-to-date B obtained to obtain h _m.

3. the sparse multi-band signals frequency spectrum localization method based on FRI according to claim 1 according to claim 2, is characterized in that utilizing Cadzow iterative algorithm algorithm to upgrade objective matrix B, until the method for its convergence is specially:

Input: Noise matrix A ^*;

Export: up-to-date matrix B;

Initialization: order matrix B equals original Noise calculation matrix A ^*;

Step 1: matrix B is decomposed into B=USV ^tform, wherein U and V is unit matrix, and S is diagonal matrix;

Step 2: utilize K maximum in a S element as element on the diagonal of new diagonal matrix S', other positional values of this matrix are zero, build new diagonal matrix S' with this;

Step 3: matrix B is updated to more excellent K grade and is similar to B=US'V ^t;

Step 4: by the operation be averaged matrix B diagonal element, upgrading matrix B is that more excellent Toeplitz is similar to;

Step 5: repeat step 2 until the set point of matrix B convergence or satisfied iterations in advance.