CN105761725B - A kind of FRI signal reconfiguring method based on time domain sparsity - Google Patents
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Abstract
The present invention relates to signal processing technology fields, and in particular to a kind of FRI signal reconfiguring method based on time domain sparsity.The reconstructing method includes the following steps: that (1) carries out quantization to simulated time axis and gridding is handled, and the number of grid is significantly larger than the unknown parameter number of FRI signal；(2) it chooses coefficient appropriate and summation is weighted to FRI sample, to obtain measurement vector, and this measurement vector is expressed as the combination of the sparse linear in time domain；(3), FRI Parameter Estimation Problem is converted to the optimization problem solved under a minimum L0 norm, the position of nonzero element is the estimated value of delay parameter in sparse solution, and the value of nonzero element is the estimated value of magnitude parameters.FRI signal reconfiguring method provided by the invention based on time domain sparsity, reconstruction accuracy is high, and anti-noise jamming ability is strong, the FRI signal reconstruction problem being applicable under noise circumstance.
Description
Technical field
The present invention relates to signal processing technology fields, and in particular to a kind of signal reconstruction side FRI based on time domain sparsity
Method.
Background technique
In recent years, the limited new fixed rate of interest (Finite Rate of Innovation, the FRI) reason of a kind of brand-new sampling theory-
By, appeared in the visual field of people, which can break through the connection of sampling rate and signal bandwidth, substantially reduce sampling speed
Rate is expected to solve various drawbacks of Nyquist sampling system.FRI sampling theory is first by Vetterli and Marziliano etc.
People proposed that the theory is different from Nyquist sampling thheorem, he points out: for certain limited parameter signals, such as pulse in 2002
String signal, non-homogeneous spline function, segmentation sine wave etc., can be by a limited number of free expressed as parameters, will be in the unit time
The number of free parameter is known as the new fixed rate of interest of signal, and the limited signal of the new fixed rate of interest is known as FRI signal, to FRI signal, as long as choosing
It is sampled with suitable sampling kernel function with the rate for being greater than or equal to the new fixed rate of interest of signal, so that it may utilize certain algorithm
Free parameter is estimated, to rebuild original signal.
FRI signal refers to the signal that can be uniquely determined by limited parameter within the unit time, alternatively referred to as " parametrization
Signal ", wherein number of degrees of freedom, of the signal within the unit time is referred to as the new fixed rate of interest of signal.Assuming that have one it is time-limited
Dirac pulse train, mathematic(al) representation are as follows:
Wherein, T is the time span of signal, and L is the quantity of pulse in Dirac pulse train.Obviously in signal x (t)
Free parameter there was only range parameter alWith delay parameter tl.Introduce a counting function Cx(ta,tb), exist for calculating letter x (t)
Time interval τ=[ta,tb] in free parameter quantity, then the new fixed rate of interest ρ of signal x (t) is defined as:
If ρ < ∞, this signal x (t) have the limited new fixed rate of interest, referred to as FRI signal.It, can be with by formula 2
Calculate the new fixed rate of interest of FRI signal x (t):
According to FRI sampling theory, after selecting suitable sampling checking signal x (t) to be filtered, it will be able to height
In or equal to the new fixed rate of interest of signal sampling rate fs>=ρ is sampled and is perfectly reconstructed.
FRI sampling structure is as shown in Figure 1, wherein x (t) is the FRI signal of a continuous time, and h (t) is that signal receives
The unit impulse response of equipment samples coreIt is the time reversal of h (t), ts=1/fs=1/ ρ is between the sampling of uniform sampling
Every.
If indicating that FRI signal x (t) passes through the filtering of sampling core, FRI sampling with y (t)=x (t) * h (t)
Obtained sample are as follows:
Wherein, k=1,2 ..., K indicate the quantity of sampled value, K=T/tsIndicate the sample number being always obtained.Different from passing
The sampling theory of system, FRI sampling plan provide a variety of sampling cores.Index reproducing kernel refers to linear group of kernel function and its time delay
The form that can regenerate complex exponential is closed, this is a kind of more stable sampling core, M rank index reproducing kernelWith following property
Matter:
Wherein, cm,kFor the coefficient of reproducing kernel, αm=α0+ m λ and m=0,1 ..., M is adjustable parameter.
After being filtered and sampled using index regeneration checking signal x (t), next critical issue is from the sampling provided
Value ykOriginal signal x (t), the i.e. reconstruction of FRI signal are reconstructed in (k=1,2 ..., K).It is initially main about the document of FRI
Using pulverised filtered method as restructing algorithm, however the algorithm operation quantity speed is slow and very sensitive to noise；In order to
The influence for reducing noise, proposes and carries out noise reduction pretreatment with Cadzow algorithm, however each iteration of the algorithm requires to carry out
Singular value decomposition and reconstruct, operand are considerable；ESPRIT and MUSIC method subspace based on subspace estimation thought
Rotational invariance estimates the time delay of signal, and operand is smaller, but since it is according to the translation of data covariance matrix subspace
Invariance estimates signal parameter, it is therefore desirable to higher sample rate.State space rule is directly according in Fu of sampled data
The translation invariance of leaf matrix subspace estimates parameter information, it is therefore desirable to data volume it is less, but calculation amount is somewhat larger.
These methods of algorithm such as genetic algorithm available maximal possibility estimation under the assumed condition of white Gaussian noise of class is counted,
But since operand is excessive, it is only used for the lesser situation of signal freedom degree, or handle for off-line data.It is so far
Only, under limited new fixed rate of interest theoretical frame, which kind of signal reconfiguring method is selected to improve the accuracy of reconstruction signal, is still one
Critical issue.
Summary of the invention
For the FRI signal reconstruction problem under noise circumstance, a kind of signal reconstruction side FRI based on time domain sparsity is proposed
Method.First by the quantization and gridding processing to simulated time axis, measurement vector is expressed as the sparse linear group in time domain
It closes；Then by the time delay and magnitude parameters that solve the optimization problem under a L0 norm to estimate input FRI signal, thus
Reconstruct original signal.
Specific step is as follows for a kind of FRI signal reconfiguring method based on time domain sparsity of the present invention:
Step 1, initialize: assuming that FRI signal be x (t), and have t ∈ [0, T)；FRI samples the index that core is M rank and regenerates
CoreCoefficient is cm,k, and have m=0,1 ..., M；T is divided between FRI uniform samplings≤1/ρ；FRI sampled value is yk(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 Dirac arteries and veins
Rush the quantity of pulse in sequence；alFor range parameter；tlFor delay parameter；K is the quantity of sampled value；M is adjustable parameter；
Step 2, obtain measurement vector: according to the property of index reproducing kernel, M rank reproducing kernel can regenerate M+1 and refer to again
Number, using coefficient cm,kTo FRI sampled value yk(k=1,2 ..., K) is weighted and sums, and obtains M+1 measured value τm(m=0,
1 ..., M), corresponding measurement vector is Γ=[τ0,τ1,…,τM]T:
Step 3, simulated time axis is quantified and gridding: by simulated time axis section [0, T) be divided into N equal portions, i.e.,
Quantization unit Δ=T/N, so that any time variable t can use the element in set U={ 0, Δ, 2 Δs ..., (N-1) Δ }
Carry out approximate, i.e. t ≈ n Δ, and has n=0,1 ..., N-1；
Construct set V={ n0Δ,n1Δ,…,nL-1Δ } so that the delay parameter in FRI signal x (t) can be approximately tl
≈nlΔ；
Step 4, equation discretization will be measured: by the delay parameter t in formula (3)l=nlΔ is next approximate, i.e., will measurement
Vector is expressed as the linear combination of all elements in set V, so that equation discretization will be measured；
It is indicated using matrix form are as follows:
Step 5, vector rarefaction representation will be measured: due to setMeasurement vector, which can be extended in set U, to be owned
The linear combination of element:
Above-mentioned formula can simplify are as follows:
Γ=AX (7)
It is the sparse vector that a degree of rarefication is L since many elements in L < < N, vector X are all zero；
Step 6, it seeks sparse solution: the optimization solved under a minimum L0 norm can be converted to the solution of sparse vector X
Problem:
Step 7, parameter Estimation and signal reconstruction: after acquiring sparse solution X, the position of nonzero element corresponds to time delay ginseng
Number, the i.e. delay parameter of original signal are availableTo estimate；The value of nonzero element corresponds to magnitude parameters, the i.e. width of original signal
It is available to spend parameterTo estimate；Finally, reconstructing original signal
Preferably, index reproducing kernel described in step 1Order M and FRI signal x (t) between relationship be M=
2L-1, wherein L is the number of Dirac pulse；The sampling nuclear energy enough regenerates M+1 complex exponential
Wherein αm=α0+ jm λ, and parameter plural number α0It can freely be set with real number λ.
Preferably, coefficient c described in step 2m,kCalculation are as follows:
Wherein, functionFor index reproducing kernelDual function, be Quasi Orthogonal Function.
Preferably, simulated time axis is quantified in step 3 and gridding, the number of grid is much larger than pulse number, i.e. N >
> L.
The invention has the following beneficial effects: the reconstruction accuracy of the method for the present invention height, and anti-noise jamming ability is strong, are applicable in
FRI signal reconstruction problem under noise circumstance.
Detailed description of the invention
Fig. 1 is FRI sampling structure block diagram；
Fig. 2 is the quality reconstruction schematic diagram of the method for the present invention under different sample rates；
Fig. 3 is the quality reconstruction schematic diagram of algorithms of different under identical sample rate.
Specific embodiment
The present invention will be further explained below with reference to the attached drawings:
The embodiment of the present invention provides a kind of FRI signal reconfiguring method based on time domain sparsity, and specific step is as follows:
Step 1, initialize: assuming that FRI signal be x (t), and have t ∈ [0, T)；FRI samples the index that core is M rank and regenerates
CoreCoefficient is cm,k, and have m=0,1 ..., M；T is divided between FRI uniform samplings≤1/ρ；FRI sampled value is yk(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 Dirac arteries and veins
Rush the quantity of pulse in sequence；alFor range parameter；tlFor delay parameter；K is the quantity of sampled value；M is adjustable parameter；
Step 2, obtain measurement vector: according to the property of index reproducing kernel, M rank reproducing kernel can regenerate M+1 and refer to again
Number.With coefficient cm,kTo FRI sampled value yk(k=1,2 ..., K) is weighted and sums, and M+1 measured value τ can be obtainedm(m=
0,1 ..., M), corresponding measurement vector is Γ=[τ0,τ1,…,τM]T:
The reconstruction of FRI signal is the unknown parameter of estimation, i.e. delay parameter tlWith range parameter alAnd l=0,1 ...,
L-1.Therefore quantity M+1 >=2L of only measured value to be offered, then these parameters can accurately be restored.
Step 3, simulated time axis is quantified and gridding: by simulated time axis section [0, T) be divided into N equal portions, i.e.,
Quantization unit Δ=T/N, so that any time variable t can use the element in set U={ 0, Δ, 2 Δs ..., (N-1) Δ }
Carry out approximate, i.e. t ≈ n Δ, and has n=0,1 ..., N-1.Set V={ n can similarly be constructed0Δ,n1Δ,…,nL-1Δ } so that
Delay parameter in FRI signal x (t) can be approximately tl≈nlΔ。
Step 4, equation discretization will be measured: by the delay parameter t in formula threel=nlΔ come it is approximate, i.e., will measure to
Amount is expressed as the linear combination of all elements in set V, so that equation discretization will be measured.
Write as matrix form are as follows:
Step 5, vector rarefaction representation will be measured: due to setMeasurement vector, which can be extended in set U, to be owned
The linear combination of element:
It can simplify are as follows:
Γ=AX (7)
It is the sparse vector that a degree of rarefication is L since many elements in L < < N, vector X are all zero.Therefore, public
Formula (7) is regarded as the rarefaction representation to measurement vector.
Step 6, it seeks sparse solution: the optimization solved under a minimum L0 norm can be converted to the solution of sparse vector X
Problem:
The method for solving this optimization problem has very much, common are orthogonal matching pursuit OMP algorithm and base tracking BP is calculated
Method etc..
Step 7, parameter Estimation and signal reconstruction: after acquiring sparse solution X, the position of nonzero element corresponds to time delay ginseng
Number, the i.e. delay parameter of original signal are availableTo estimate；The value of nonzero element corresponds to magnitude parameters, the i.e. width of original signal
It is available to spend parameterTo estimate.Finally, reconstructing original signal
The method that the performance of verifying the method for the present invention is described in detail below:
It is verified using several emulation experiments under white noise environment, and is compared with existing FRI reconstructing method
Compared with.Emulation experiment is provided that the FRI signal of input using Dirac pulse train, magnitude parameters al~U [0,1], pulse it
Between time interval, that is, delay parameter tlSection [0, T) in random selection；Signal duration T=1s；The minimum of simulated time axis
Quantization unit is Δ=0.001s, therefore the grid number divided is Δ=1000 N=T/.
In order to from the performance for numerically assessing each reconstructing method, using mean square error as evaluation index, for convenience than
Logarithm relatively is taken to mean square error:
Wherein L is the quantity of pulse, tlIt is true delay parameter,It is the delay parameter of estimation.Due to magnitude parameters
Error is directly proportional to the error of delay parameter, therefore only needs to be measured each FRI reconstructing method with the mean square error of delay parameter
Performance.
Experiment one, as shown in Fig. 2, being used under the white Gaussian noise environment of different signal-to-noise ratio (SNR increases to 100 by 0)
Quality reconstruction of the method for the present invention when not having to sample rate is compared.Input signal is made of L=2 Dirac pulse
FRI signal.Wherein delay parameter tl=[0.256,0.38], magnitude parameters al=[0.8,1], the new fixed rate of interest are ρ=2L/T=4.It is imitative
True experiment repeats 100 times, and average reconstruction result comparison is as shown in Figure 2.Figure it is seen that the method for the present invention is in Gauss
White noise there are the case where, sample rate be greater than or equal to new fixed rate of interest ρ, that is, when taking 4Hz, 8Hz and 16Hz, FRI signal reconstruction effect
Well.And with the raising of sample rate, reconstruction accuracy is also increased accordingly, it is seen that the method for the present invention is effective.
Experiment two, as shown in figure 3, being used under the white Gaussian noise environment of different signal-to-noise ratio (SNR increases to 100 by 0)
The method of the present invention is compared with the existing reconstructing method based on B-spline and E-spline.Input signal is by L=4
The FRI signal of Dirac pulse composition.Wherein delay parameter tl=[0.213,0.452,0.664,0.754], magnitude parameters al=
[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
Secondary, average reconstruction result comparison is as shown in Figure 3.From figure 3, it can be seen that the method for the present invention ratio is based under identical Signal to Noise Ratio (SNR)
The restructing algorithm of B-spline and restructing algorithm reconstruction accuracy based on E-spline are higher and more stable, embody relatively strong
Anti-noise jamming ability.
The reconstruction accuracy of the method for the present invention is high, and anti-noise jamming ability is strong, the FRI signal reconstruction being applicable under noise circumstance
Problem.
Claims (4)
1. a kind of FRI signal reconfiguring method based on time domain sparsity, which is characterized in that the process of the reconstructing method includes:
Step 1, initialize: assuming that FRI signal be x (t), and have t ∈ [0, T)；FRI samples the index reproducing kernel that core is M rankWherein, index reproducing kernel coefficient is cm,k, and have m=0,1 ..., M；T is divided between FRI uniform samplings≤1/ρ；FRI is adopted
Sample value is yk, 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 Dirac pulse sequence
The quantity of pulse in column；alFor range parameter；tlFor delay parameter；K is the quantity of sampled value；M is adjustable parameter；
Step 2, obtain measurement vector: according to the property of index reproducing kernel, M rank reproducing kernel can regenerate M+1 complex exponential, adopt
With index reproducing kernel coefficient cm,kTo FRI sampled value yk, k=1,2 ..., K be weighted and sum, and obtains M+1 measured value τm, m
=0,1 ..., M, corresponding measurement vector are Γ=[τ0,τ1,…,τM]T:
Wherein αm=α0+ jm λ, and parameter plural number α0It can freely be set with real number λ,
Step 3, simulated time axis is quantified and gridding: by simulated time axis section [0, T) be divided into N equal portions, the amount of obtaining
Change unit △=T/N, so that any time variable t is usually close using the member in set U={ 0, △, 2 △ ..., (N-1) △ }
Seemingly, i.e. t ≈ n △, and have n=0,1 ..., N-1；
Construct set V={ n0△,n1△,…,nL-1△ } so that the delay parameter in FRI signal x (t) can be approximately tl≈nl
△；
Step 4, equation discretization will be measured: by the delay parameter t in formula (3)l=nl△ is next approximate, i.e., will measure vector
It is expressed as the linear combination of all elements in set V, so that equation discretization will be measured；
It is indicated using matrix form are as follows:
Step 5, vector rarefaction representation will be measured: due to setMeasurement vector can be extended to all elements in set U
Linear combination:
Above-mentioned formula can simplify are as follows:
Γ=AX (7)
Due to L < < N, many elements in vector X are all zero, are the sparse vectors that a degree of rarefication is L；
Step 6, it seeks sparse solution: the optimization problem solved under a minimum L0 norm can be converted to the solution of sparse vector X:
Step 7, parameter Estimation and signal reconstruction: after acquiring sparse solution X, the position of nonzero element corresponds to delay parameter, i.e.,
The delay parameter of original signal usesTo estimate；The value of nonzero element corresponds to magnitude parameters, the i.e. range parameter of original signal
UsingTo estimate；It finally obtains and reconstructs original signal
2. the FRI signal reconfiguring method according to claim 1 based on time domain sparsity, which is characterized in that institute in step 1
The index reproducing kernel statedOrder M and FRI signal x (t) between relationship be M=2L-1, wherein L is Dirac pulse
Number；The sampling nuclear energy enough regenerates M+1 complex exponentialM=0,1 ... M, wherein αm=α0+ jm λ, and parameter plural number
α0It can freely be set with real number λ.
3. the FRI signal reconfiguring method according to claim 1 based on time domain sparsity, which is characterized in that institute in step 2
The coefficient c statedm,kCalculation are as follows:
Wherein, functionFor index reproducing kernelDual function, be Quasi Orthogonal Function.
4. the FRI signal reconfiguring method according to claim 1 based on time domain sparsity, which is characterized in that will in step 3
The number of the quantization of simulated time axis and gridding, grid is much larger than pulse number, i.e. N > > L.
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