CN105761725A - Time-domain sparsity based FRI signal reconstruction method - Google Patents

Abstract

The invention relates to a time-domain sparsity based FRI signal reconstruction method, and belongs to the technical field of signal processing. The reconstruction method comprises the following steps that (1) quantifying and gridding are carried out on a simulated time axis, and the amount of grids is much high than the amount of unknown parameters of FRI signals; (2) a proper coefficient is selected to carry out weighted summation on an FRI sample, and a measurement vector is obtained and used to represent a sparse linear combination in the time domain; and (3) FRI parameter estimation is converted into optimization of solving a minimal L0 norm, the position of a nonzero element in sparse solution is an estimated value of a time delay parameter, and the value of the nonzero element is an estimated value of an amplitude parameter. The time-domain sparsity based FRI signal reconstruction method is high in reconstruction precision, high in anti-interference capability, and suitable for FRI signal reconstruction in the noise environment.

Description

A kind of FRI signal reconfiguring method openness based on time domain

Technical field

The present invention relates to signal processing technology field, be specifically related to a kind of FRI signal reconfiguring method openness based on time domain.

Background technology

In recent years, a kind of brand-new sampling theory-limited new fixed rate of interest (FiniteRateofInnovation, FRI) theoretical, occur in the visual field of people, this theory can break through the contact of sampling rate and signal bandwidth, it is substantially reduced sampling rate, is expected to solve all drawbacks of Nyquist sampling system.First FRI sampling theory is proposed in 2002 by Vetterli and Marziliano et al., this theory is different from Nyquist sampling thheorem, he points out: for some limited parameter signal, such as burst signal, non-homogeneous spline function, segmentation is sinusoidal wave, can by a limited number of free expressed as parameters, the number of parameter free in the unit interval is called the new fixed rate of interest of signal, signal limited for the new fixed rate of interest is called FRI signal, to FRI signal, as long as selecting suitable sampling kernel function to sample with the speed of the new fixed rate of interest greater than or equal to signal, certain algorithm just can be utilized to estimate free parameter, thus rebuilding primary signal.

FRI signal refers to can by limited the well-determined signal of parameter within the unit interval, it is possible to being called " parametrization signal ", wherein signal number of degrees of freedom, within the unit interval is referred to as the new fixed rate of interest of signal.Assuming there is a time-limited Dirac pulse train, its mathematic(al) representation is as follows:

$x\left(t\right)=\underset{l=0}{\overset{L-1}{\Σ}}{a}_l\mathrm{\δ}(t-t_l),t_l\∈\[0,T)---\left(1^{,}\right)$

Wherein, T is the time span of signal, and L is the quantity of pulse in Dirac pulse train.Obviously the free parameter in signal x (t) only has range parameter a_lWith delay parameter t_l.Introduce a counting function C_x(t_a,t_b), it is used for calculating letter x (t) at interval τ=[t_a,t_b] in the quantity of free parameter, then the new fixed rate of interest ρ of signal x (t) is defined as:

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

If ρ is < ∞, then this signal x (t) has the limited new fixed rate of interest, is referred to as FRI signal.By formula 2, it is possible to calculate the new fixed rate of interest of FRI signal x (t):

$\mathrm{\&rho;}=\frac{2L}{T}---\left(3^{,}\right)$

According to FRI sampling theory, after selecting suitable sampling checking signal x (t) to be filtered, it becomes possible to the sampling rate f greater than or equal to the new fixed rate of interest of signal_s>=ρ carries out sampling and perfectly reconstructing.

As it is shown in figure 1, wherein, x (t) is the FRI signal of a continuous time to FRI sampling structure, and h (t) is the unit impulse response of signal receiver, core of samplingIt is the time reversal of h (t), t_s=1/f_s=1/ ρ is the sampling interval of uniform sampling.

If representing FRI signal x (t) filtering by core of sampling with y (t)=x (t) * h (t), then the sample that FRI sampling obtains is:

Wherein, k=1,2 ..., K represents the quantity of sampled value, K=T/t_sRepresent the sample number altogether obtained.Being different from traditional sampling theory, FRI sampling plan provides multiple sampling core.Index reproducing kernel refer to kernel function and time delay thereof linear combination can the complex exponential form of regeneration, this is a kind of more stable sampling core, M rank index reproducing kernelHave the property that

Wherein, c_m,kFor the coefficient of reproducing kernel, α_m=α₀+ m λ and m=0,1 ..., M is adjustable parameter.

After adopting index regeneration checking signal x (t) filtering and sampling, ensuing key issue is from the sampled value y provided_k(k=1,2 ..., K) in reconstruct original signal x (t), i.e. the reconstruction of FRI signal.The initial document about FRI mainly adopts pulverised filtered method as restructing algorithm, but this algorithm operation quantity speed is slow, and noise is very sensitive；In order to reduce effect of noise, it is proposed that carrying out noise reduction pretreatment with Cadzow algorithm, but each iteration of this algorithm is required for carrying out singular value decomposition and reconstruct, operand is considerable；The time delay of signal is estimated based on ESPRIT and the MUSIC method Subspace Rotation invariance of subspace estimation thought, operand is less, but owing to it estimates signal parameter according to the translation invariance of data covariance matrix subspace, it is therefore desirable to higher sample rate.State space rule directly translation invariance according to Fourier's matrix subspace of sampled data estimates parameter information, it is therefore desirable to data volume less, but amount of calculation is somewhat larger.These methods such as the algorithm such as genetic algorithm of statistics class can obtain maximal possibility estimation under the assumed condition of white Gaussian noise, but owing to operand is excessive, is only used for the situation that signal degree of freedom is less, or processes for off-line data.Up to now, under limited new fixed rate of interest theoretical frame, select which kind of signal reconfiguring method to improve the degree of accuracy of reconstruction signal, be still a key issue.

Summary of the invention

For the FRI signal reconstruction problem under noise circumstance, it is proposed to a kind of FRI signal reconfiguring method openness based on time domain.First pass through the quantization to simulated time axle and gridding processes, be the sparse linear combination in time domain by measuring vector representation；Then pass through the optimization problem solved under a L0 norm and estimate time delay and the magnitude parameters of input FRI signal, thus reconstructing original signal.

Specifically comprising the following steps that of a kind of FRI signal reconfiguring method openness based on time domain of the present invention

Step 1, initialize: assume that FRI signal is x (t), and have t ∈ [0, T)；FRI sampling core is the index reproducing kernel on M rankCoefficient is c_m,k, and have m=0, and 1 ..., M；FRI uniform sampling is spaced apart t_s≤1/ρ；FRI sampled value is y_k(k=1,2 ..., K)；

Wherein, $x\left(t\right)=\underset{l=0}{\overset{L-1}{\&Sigma;}}{a}_l\mathrm{\&delta;}(t-t_l),t_l\&Element;\&lsqb;0,T)---\left(1\right)$

Wherein, ρ is the new fixed rate of interest of FRI signal x (t)；T is the time span of signal, and t is time variable；L is the quantity of pulse in Dirac pulse train；a_lFor range parameter；t_lFor delay parameter；K is the quantity of sampled value；M is adjustable parameter；

Step 2, obtains and measures vector: the character according to index reproducing kernel, and M rank reproducing kernel can regenerate M+1 complex exponential, adopts coefficient c_m,kTo FRI sampled value y_k(k=1,2 ..., K) it is weighted and sues for peace, obtain M+1 measured value τ_m(m=0,1 ..., M), corresponding measurement vector is Γ=[τ₀,τ₁,…,τ_M]^T:

Step 3, by simulated time axle quantify and gridding: by simulated time axle interval [0, T) be divided into N equal portions, i.e. quantization unit Δ=T/N so that random time variable t can use set U={0, Δ, 2 Δs, ..., (N-1) Δ } in unit be usually similar to, i.e. t ≈ n Δ, and have n=0,1 ..., N-1；

Build set V={n₀Δ,n₁Δ,…,n_L-1Δ } so that the delay parameter in FRI signal x (t) can be approximately t_l≈n_lΔ；

Step 4, will measure equation discretization: by the delay parameter t in formula (3)_l=n_lΔ is similar to, and being about to measure vector representation is the linear combination of all elements in set V, thus equation discretization will be measured；

${\mathrm{\&tau;}}_m\&ap;\underset{l=0}{\overset{L-1}{\&Sigma;}}{a}_l{e}^{{\mathrm{\&alpha;}}_m(n_l\mathrm{\&Delta;}/t_s)},withm=0,1,\mathrm{...},M-1---\left(4\right)$

Employing matrix form is expressed as:

Step 5, will measure vector rarefaction representation: due to setMeasure vector and can expand to the linear combination of all elements in set U:

Above-mentioned formula can be reduced to:

Γ=AX (7)

Due to L < < N, a lot of elements in vector X are all zero, be a degree of rarefication are the sparse vector of L；

Step 6, seeks sparse solution: solving of sparse vector X can be converted to the optimization problem solved under a minimum L0 norm:

$\hat{X}=\arg min\left|\right|X|{|}_{0}suchthat\mathrm{\&Gamma;}=AX---\left(8\right)$

Step 7, parameter estimation and signal reconstruction: after trying to achieve sparse solution X, the position correspondence delay parameter of nonzero element, namely the delay parameter of original signal can be usedEstimate；The value correspondence magnitude parameters of nonzero element, namely the range parameter of original signal can be usedEstimate；Finally, original signal is reconstructed

$\hat{x}\left(t\right)=\underset{l=0}{\overset{L-1}{\&Sigma;}}{\hat{a}}_l\mathrm{\&delta;}(t-{\hat{t}}_l),{\hat{t}}_l\&Element;\&lsqb;0,T).---\left(9\right)$

Preferably, the index reproducing kernel described in step 1Exponent number M and FRI signal x (t) between relation be M=2L-1, wherein L is the number of Dirac pulse；Described sampling nuclear energy enough regenerates M+1 complex exponential(m=0,1 ... M), wherein α_m=α₀+ jm λ, and parameter plural number α₀Can freely set with real number λ.

Preferably, the coefficient c described in step 2_m,kCalculation be:

Wherein, functionFor index reproducing kernelDual function, for Quasi Orthogonal Function.

Preferably, in step 3 by simulated time axle quantify and gridding, the number of grid much larger than pulse number, i.e. N > > L.

There is advantages that the reconstruction accuracy of the inventive method is high, and anti-noise jamming ability is strong, is suitable for the FRI signal reconstruction problem under noise circumstance.

Accompanying drawing explanation

Fig. 1 is FRI sampling structure block diagram；

Fig. 2 is under different sample rate, the quality reconstruction schematic diagram of the inventive method；

Fig. 3 is under identical sample rate, the quality reconstruction schematic diagram of algorithms of different.

Detailed description of the invention

Below in conjunction with accompanying drawing, the invention will be further described:

The embodiment of the present invention provides specifically comprising the following steps that of a kind of FRI signal reconfiguring method openness based on time domain

Step 1, initialize: assume that FRI signal is x (t), and have t ∈ [0, T)；FRI sampling core is the index reproducing kernel on M rankCoefficient is c_m,k, and have m=0, and 1 ..., M；FRI uniform sampling is spaced apart t_s≤1/ρ；FRI sampled value is y_k(k=1,2 ..., K).

Wherein: $x\left(t\right)=\underset{l=0}{\overset{L-1}{\&Sigma;}}{a}_l\mathrm{\&delta;}(t-t_l),t_l\&Element;\&lsqb;0,T)---\left(1\right)$

Wherein, ρ is the new fixed rate of interest of FRI signal x (t)；T is the time span of signal, and t is time variable；L is the quantity of pulse in Dirac pulse train；a_lFor range parameter；t_lFor delay parameter；K is the quantity of sampled value；M is adjustable parameter；

Step 2, obtains and measures vector: the character according to index reproducing kernel, M rank reproducing kernel can regenerate M+1 complex exponential.Use coefficient c_m,kTo FRI sampled value y_k(k=1,2 ..., K) it is weighted and sues for peace, M+1 measured value τ can be obtained_m(m=0,1 ..., M), corresponding measurement vector is Γ=[τ₀,τ₁,…,τ_M]^T:

The reconstruction of FRI signal is to estimate unknown parameter, i.e. delay parameter t_lWith range parameter a_lAnd l=0,1 ..., L-1.As long as the quantity M+1 >=2L of the measured value therefore provided, then these parameters just can be recovered accurately.

Step 3, by simulated time axle quantify and gridding: by simulated time axle interval [0, T) be divided into N equal portions, i.e. quantization unit Δ=T/N so that random time variable t can use set U={0, Δ, 2 Δs, ..., (N-1) Δ } in unit be usually similar to, i.e. t ≈ n Δ, and have n=0,1 ..., N-1.In like manner can build set V={n₀Δ,n₁Δ,…,n_L-1Δ } so that the delay parameter in FRI signal x (t) can be approximately t_l≈n_lΔ。

Step 4, will measure equation discretization: by the delay parameter t in formula three_l=n_lΔ is similar to, and being about to measure vector representation is the linear combination of all elements in set V, thus equation discretization will be measured.

${\mathrm{\&tau;}}_m\&ap;\underset{l=0}{\overset{L-1}{\&Sigma;}}{a}_l{e}^{{\mathrm{\&alpha;}}_m(n_l\mathrm{\&Delta;}/t_s)},withm=0,1,\mathrm{...},M-1---\left(4\right)$

Being write as matrix form is:

Step 5, will measure vector rarefaction representation: due to setMeasure vector and can expand to the linear combination of all elements in set U:

Can be reduced to:

Γ=AX (7)

Due to L < < N, a lot of elements in vector X are all zero, be a degree of rarefication are the sparse vector of L.Therefore, formula (7) is regarded as the rarefaction representation measuring vector.

Step 6, seeks sparse solution: solving of sparse vector X can be converted to the optimization problem solved under a minimum L0 norm:

$\hat{X}=\arg min\left|\right|X|{|}_{0}suchthat\mathrm{\&Gamma;}=AX---\left(8\right)$

The method solving this optimization problem has a lot, common are orthogonal matching pursuit OMP algorithm and BP algorithm etc. followed the trail of by base.

Step 7, parameter estimation and signal reconstruction: after trying to achieve sparse solution X, the position correspondence delay parameter of nonzero element, namely the delay parameter of original signal can be usedEstimate；The value correspondence magnitude parameters of nonzero element, namely the range parameter of original signal can be usedEstimate.Finally, original signal is reconstructed

$\hat{x}\left(t\right)=\underset{l=0}{\overset{L-1}{\&Sigma;}}{\hat{a}}_l\mathrm{\&delta;}(t-{\hat{t}}_l),{\hat{t}}_l\&Element;\&lsqb;0,T).---\left(9\right)$

The method that the performance of checking the inventive method is described in detail below:

Adopt several emulation experiment under white noise environment to be verified, and compare with existing FRI reconstructing method.Emulation experiment is provided that the FRI signal of input adopts Dirac pulse train, magnitude parameters a_l～U [0,1], interval between pulse and delay parameter t_lInterval [0, T) in randomly choose；Signal duration T=1s；The minimum quantization unit of simulated time axle is Δ=0.001s, and the grid number therefore divided is N=T/ Δ=1000.

In order to from the performance numerically assessing each reconstructing method, adopt mean square error as evaluation index, relatively mean square error is taken the logarithm in order to convenient:

$MSE\&lsqb;dB\&rsqb;=10\&times;{\log }_{10}\left(\frac{1}{L}\underset{l=0}{\overset{L-1}{\&Sigma;}}{(t_l-{\hat{t}}_l)}^2\right)$

Wherein L is the quantity of pulse, t_lIt is real delay parameter,It it is the delay parameter estimated.Owing to the error of magnitude parameters is directly proportional to the error of delay parameter, therefore have only to weigh the performance of each FRI reconstructing method by the mean square error of delay parameter.

Experiment one, as in figure 2 it is shown, under the white Gaussian noise environment of different signal to noise ratios (SNR is increased to 100 by 0), adopt the inventive method quality reconstruction when without sample rate to compare.The FRI signal that input signal is made up of L=2 Dirac pulse.Wherein delay parameter t_l=[0.256,0.38], magnitude parameters a_l=[0.8,1], the new fixed rate of interest is ρ=2L/T=4.Emulation experiment repeats 100 times, and average reconstruction result contrasts as shown in Figure 2.Figure it is seen that the situation that the inventive method exists at white Gaussian noise, sample rate is more than or equal to new fixed rate of interest ρ, and when namely taking 4Hz, 8Hz and 16Hz, FRI signal reconstruction is respond well.And along with the raising of sample rate, reconstruction accuracy also improves therewith, it is seen that the inventive method is effective.

Experiment two, as it is shown on figure 3, under the white Gaussian noise environment of different signal to noise ratios (SNR is increased to 100 by 0), adopt the inventive method and the existing reconstructing method based on B-spline and E-spline to compare.The FRI signal that input signal is made up of L=4 Dirac pulse.Wherein delay parameter t_l=[0.213,0.452,0.664,0.754], magnitude parameters a_l=[1,0.9,0.].The new fixed rate of interest is ρ=2L/T=8, and sample rate takes 3 times i.e. 24Hz of the new fixed rate of interest.Emulation experiment repeats 100 times, and average reconstruction result contrasts as shown in Figure 3.From figure 3, it can be seen that under identical signal to noise ratio snr, the inventive method is higher than the restructing algorithm based on B-spline and the restructing algorithm reconstruction accuracy based on E-spline, and more stable, embody stronger anti-noise jamming ability.

The reconstruction accuracy of the inventive method is high, and anti-noise jamming ability is strong, is suitable for the FRI signal reconstruction problem under noise circumstance.

Claims (4)

1. one kind based on the openness FRI signal reconfiguring method of time domain, it is characterised in that the process of described reconstructing method includes:

Step 1, initialize: assume that FRI signal is x (t), and have t ∈ [0, T)；FRI sampling core is the index reproducing kernel on M rankWherein, index reproducing kernel coefficient is c_m,k, and have m=0, and 1 ..., M；FRI uniform sampling is spaced apart t_s≤1/ρ；FRI sampled value is

y_k(k=1,2 ..., K)；

Wherein,

Wherein, ρ is the new fixed rate of interest of FRI signal x (t)；T is the time span of signal, and t is time variable；L is the quantity of pulse in Dirac pulse train；a_lFor range parameter；t_lFor delay parameter；K is the quantity of sampled value；M is adjustable parameter；

Step 2, obtains and measures vector: the character according to index reproducing kernel, and M rank reproducing kernel can regenerate M+1 complex exponential, adopts index reproducing kernel coefficient c_m,kTo FRI sampled value y_k(k=1,2 ..., K) it is weighted and sues for peace, obtain M+1 measured value τ_m(m=0,1 ..., M), corresponding measurement vector is Γ=[τ₀,τ₁,…,τ_M]^T:

Step 3, by simulated time axle quantify and gridding: by simulated time axle interval [0, T) be divided into N equal portions, obtain quantization unit Δ=T/N so that random time variable t adopts set U={0, △, 2 △, ..., (N-1) △ } in unit be usually similar to, i.e. t ≈ n Δ, and have n=0,1 ..., N-1；

Build set V={n₀Δ,n₁Δ,…,n_L-1Δ } so that the delay parameter in FRI signal x (t) can be approximately t_l≈n_lΔ；

Step 4, will measure equation discretization: by the delay parameter t in formula (3)_l=n_lΔ is similar to, and being about to measure vector representation is the linear combination of all elements in set V, thus equation discretization will be measured；

Employing matrix form is expressed as:

Step 5, will measure vector rarefaction representation: due to setMeasure vector and can expand to the linear combination of all elements in set U:

Above-mentioned formula can be reduced to:

Γ=AX (7)

Due to L < < N, a lot of elements in vector X are all zero, be a degree of rarefication are the sparse vector of L；

Step 6, seeks sparse solution: solving of sparse vector X can be converted to the optimization problem solved under a minimum L0 norm:

Step 7, parameter estimation and signal reconstruction: after trying to achieve sparse solution X, the position correspondence delay parameter of nonzero element, namely the delay parameter of original signal adoptsEstimate；The value correspondence magnitude parameters of nonzero element, namely the range parameter of original signal adoptsEstimate；Finally draw and reconstruct original signal

2. the FRI signal reconfiguring method openness based on time domain according to claim 1, it is characterised in that the index reproducing kernel described in step 1Exponent number M and FRI signal x (t) between relation be M=2L-1, wherein L is the number of Dirac pulse；Described sampling nuclear energy enough regenerates M+1 complex exponentialWherein α_m=α₀+ jm λ, and parameter plural number α₀Can freely set with real number λ.

3. the FRI signal reconfiguring method openness based on time domain according to claim 1, it is characterised in that the coefficient c described in step 2_m,kCalculation be:

Wherein, functionFor index reproducing kernelDual function, for Quasi Orthogonal Function.

4. the FRI signal reconfiguring method openness based on time domain according to claim 1, it is characterised in that in step 3 by simulated time axle quantify and gridding, the number of grid much larger than pulse number, i.e. N > > L.