CN107592115A - A kind of sparse signal restoration methods based on non-homogeneous norm constraint - Google Patents
A kind of sparse signal restoration methods based on non-homogeneous norm constraint Download PDFInfo
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Abstract
The present invention relates to a kind of sparse signal restoration methods based on non-homogeneous norm constraint, L1 norm minimum strategies are improved using NNC norm constraints strategy, so as to realize that innovatory algorithm better adapts to degree of rarefication situation of change.Different from MP algorithms, the present invention need not pre-set the prioris such as the degree of rarefication of sparse signal, only need to set iteration step length, with reference to method of Lagrange multipliers, sparse signal is estimated by adjusting thresholds.So as to obtain the parameters such as the position of sparse signal and amplitude.Have the beneficial effect that：The present invention is based on NNC norm constraints, under method of Lagrange multipliers optimizing strategy, to being improved on the basis of the classical method for minimizing L1 norm constraints, and then obtains sparse solution.Due to the use of NNC norms so that the present invention is calculated Optimized Iterative and sparse signal recovers have very big advantage.
Description
Technical field
The invention belongs to marine acoustics and field of underwater acoustic signal processing, is related to a kind of sparse letter based on non-homogeneous norm constraint
Number restoration methods, estimate it for the sparse signal of time domain or transform domain, suitable for the estimation of ocean underwater acoustic channel, water
The content such as sound data compression and recovery.
Background technology
The problems such as underwater sound data compression and underwater acoustic channel are estimated can all be attributed to be recovered to sparse signal, to random point
The sparse signal position of cloth and amplitude are estimated.Use at present mainly have matching pursuit algorithm (Matching pursuit,
) and the base method for tracing based on L1 norm minimums (Basis pursuit, BP) MP.Wherein, matching pursuit algorithm referring to
《Sparse channel estimation via matching pursuit with application to
equalization》, this article is published in for 2002《IEEE Transactions on Communications》50th phase, rise
The beginning page number is 374.Based on the base method for tracing of L1 norm minimums referring to《Proportionate adaptive filters
from a basis pursuit perspective》This article is published in for 2010《IEEE Signal Processing
Letters》17th phase, first page number 985.
Because the essence for seeking most sparse solution is to seek zero Norm minimum solution, and zero norm minimum can not engineering reality in practice
It is existing, therefore switch to L1 norm minimums and carry out approximation, therefore, although its solution procedure can obtain sparse solution, and algorithm is simple, but
Precision is inadequate, and the match tracing rule based on greedy strategy is based on matching primitives, and its core is also to apply L1 norm constraint items,
However, the intrinsic redundant computation of matching pursuit algorithm and the redundancy to dictionary select characteristic, the algorithm estimated result is caused to improve journey
Spend limited.
The content of the invention
Technical problems to be solved
In order to avoid the shortcomings of the prior art, the present invention proposes a kind of sparse signal based on non-homogeneous norm constraint
Restoration methods, the problems such as overcoming existing sparse signal recovery algorithms estimated accuracy not high.
Technical scheme
A kind of sparse signal restoration methods based on non-homogeneous norm constraint, it is characterised in that step is as follows：
Step 1：If A is sensing matrix, y is measurement signal, and x is sparse signal to be estimated, wherein A dimension be M ×
N, and M ＜ N, measurement process are：
Y=Ax；
Step 2：Constructing object function is：
Wherein,0≤pi≤1.i,pi, n represents element position respectively, element equivalency index at i-th
Variable and vector dimension；
Step 3：Setting iterations is L, threshold value t=σ | | x | |, the setting of threshold value control parameter and signal noise are horizontal
Relevant, span is in 0 ＜ σ ＜ 1；
Step 4：Initial estimate isPrimary iteration number i=0
Step 5：It is iterated according to iterative as follows
I=i+1
G=diag (| x1|,…,|xn|)+δI
X=F-1GATλ=F-1GAT(AF-1GAT)-1y
Wherein diag is that diagonalization is carried out to vector to obtain diagonal matrix, and I is unit matrix, and δ is to prevent algorithm from entering
The small positive number set during ill computing；TTo carry out transposition computing, λ=(AGA to matrixT)-1Y is Lagrange multiplier；
Step 6：Stop iteration as i ＞ L, obtain sparse signal estimate x；Otherwise 5~step 6 of repeat step.
Beneficial effect
A kind of sparse signal restoration methods based on non-homogeneous norm constraint proposed by the present invention are a kind of based on non-homogeneous
(Non-uniform norm constraint, NNC) method of estimation of norm constraint, the invention utilize NNC norm constraint strategies
L1 norm minimum strategies are improved, so as to realize that innovatory algorithm better adapts to degree of rarefication situation of change.With MP algorithms
Difference, the present invention need not pre-set the prioris such as the degree of rarefication of sparse signal, only need to set iteration step length, bright with reference to glug
Day multiplier method, estimates sparse signal by adjusting thresholds.So as to obtain the parameters such as the position of sparse signal and amplitude.
Have the beneficial effect that：The present invention is based on NNC norm constraints, under method of Lagrange multipliers optimizing strategy, to warp
Allusion quotation is improved on the basis of minimizing the method for L1 norm constraints, and then obtains sparse solution.Due to the use of NNC norms so that
The present invention is calculated Optimized Iterative and sparse signal recovers have very big advantage.
Brief description of the drawings
Fig. 1 is that the inventive method estimates performance comparison figure from classical way under different degree of rarefications.
Fig. 2 is estimation performance comparison figure of the inventive method from classical way in different pendulous frequencies.
Fig. 3 is the inventive method and the sparse signal restoration result and primary signal comparison diagram of classical way estimation.
Fig. 4 is estimation performance comparison figure of the inventive method from classical way in the case of different snr of received signal.
Embodiment
In conjunction with embodiment, accompanying drawing, the invention will be further described：
Technical scheme
1. sparse signal estimation problem, is concretely comprised the following steps：
(1) A is set as sensing matrix, and y is measurement signal, and x is sparse signal to be estimated, and wherein A dimension is M × N, M
＜ N, specific measurement process are：
Y=Ax (1)
(2) it was complete system to measure process, there is infinite multiresolution in theory, and most sparse it is an object of the present invention to search out
One group of solution, meet formula (1), therefore, the object function of optimization problem is：
Wherein | | x | |0The number of nonzero element is represented, the direct solution of formula (2) is extremely complex, therefore traditional algorithm converts
For
Wherein | | x | |1Represent the absolute value sum of each element.
(3) limited precision of traditional algorithm is considered, new object function proposed by the present invention is
Wherein0≤pi≤1.
(4) specific method of sparse signal x estimations is：
1. given sensing matrix A and measurement signal y, sets algorithm iteration number L and threshold value σ.
2. output information：Sparse signal estimate x
3. initialize：Initial estimate isThreshold value t=σ | | x | |：Iterations i=0.
4. whether evaluation algorithm end condition meets, i.e., whether i ＞ L, if so, then stopping iteration, if it is not, then according to as follows
Iterative iteration：
I=i+1 (5)
Wherein diag is that diagonalization is carried out to vector to obtain diagonal matrix, and I is unit matrix, and δ is to prevent algorithm computing
There is the small positive number set when morbid state calculates.
G=diag (| x1|,…,|xn|)+δI (7)
X=F-1GATλ=F-1GAT(AF-1GAT)-1y (8)
WhereinTTo carry out transposition computing to matrix.
Specific embodiment：
Reference picture 1, using Gaussian Profile matrix as sensing matrix, it is 100 to set sparse signal length, position and size
Meet the Gauss rule change of zero mean unit variance.The signal to noise ratio OSNR of reception signal is defined as
Wherein v is Complex-valued additive random noise.
Fig. 1 parameter settings are as follows：OSNR=20dB, M=50, N=100, degree of rarefication increase to 30 from 3, and iterations is set
Be set to 50, σ=0.08, it will be seen from figure 1 that when degree of rarefication is more than 6, NNC SNR is more than MP and L1 algorithms, this be because
There is certain adaptability for degree of rarefication increase for NNC algorithms.
In data compression, it is desirable to which compression ratio is bigger, but compression ratio means that greatly recovering difficulty increases.Fig. 2 is used for testing
The performance of each algorithm during different pendulous frequencies.Parameter is arranged to N=100, OSNR=15dB, degree of rarefication 15, pendulous frequency
60 are changed to from 30, change step 5, mode caused by sparse signal is as before, the result such as figure that 100 averagings of emulation obtain
Shown in 2, it can be seen that in this case, NNC algorithms are high compared to the SNR of MP and L1 algorithms, this, which has benefited from NNC norms, suitably to adjust
Whole degree of rarefication match condition, so as to improve recovery precision.
Further to investigate the present invention under different signal to noise ratio to the restorability of sparse signal.Channel impulse response is set
Function such as Fig. 3 (a) so, it is all 100 that involved parameter, which is arranged to pendulous frequency and channel exponent number, degree of rarefication 15, when connecing
Signal-to-Noise is received when be 15dB, the result that three kinds of methods obtain respectively such as Fig. 3 (b) (c) (d) so, as can be seen from the figure
There is false multipath in L1 and MP algorithms, and result and simulated channel that NNC algorithms obtain are close.Further to quantify these calculations
Difference between method, the signal to noise ratio of reception signal is arranged to changing value, from 3 to 30dB, change step 3dB.Other specification
As before, obtained result is as shown in Figure 4.It can be seen that the present invention after snr of received signal is higher than 9dB, recovers ratio of precision
MP and L1 algorithms have very big lifting.
The present invention recovers to achieve obvious implementation result in emulation in sparse signal, and sparse signal is recovered,
And sparse underwater acoustic channel is have estimated, the estimated accuracy of such current algorithm is improved under certain condition.
Claims (1)
1. a kind of sparse signal restoration methods based on non-homogeneous norm constraint, it is characterised in that step is as follows：
Step 1：If A is sensing matrix, y is measurement signal, and x is sparse signal to be estimated, and wherein A dimension is M × N, and M
＜ N, measurement process are：
Y=Ax；
Step 2：Constructing object function is：
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Wherein,0≤pi≤1.i,pi, n represents element position respectively, element equivalency index variable at i-th
And vector dimension；
Step 3：Setting iterations is L, threshold value t=σ | | x | |, the setting of threshold value control parameter is relevant with signal noise level,
Span is in 0 ＜ σ ＜ 1；
Step 4：Initial estimate isPrimary iteration number i=0
Step 5：It is iterated according to iterative as follows
I=i+1
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G=diag (| x1|,…,|xn|)+δI
X=F-1GATλ=F-1GAT(AF-1GAT)-1y
Wherein diag is that diagonalization is carried out to vector to obtain diagonal matrix, and I is unit matrix, and δ is to prevent algorithm from entering morbid state
The small positive number set during computing；T is to carry out transposition computing, λ=(AGA to matrixT)-1Y is Lagrange multiplier；
Step 6：Stop iteration as i ＞ L, obtain sparse signal estimate x；Otherwise 5~step 6 of repeat step.
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Cited By (3)
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| CN108880558A (en) * | 2018-06-08 | 2018-11-23 | 西北工业大学 | A kind of underwater sound signal condensation matrix optimization method based on discrete cosine transform |
| CN110048723A (en) * | 2019-04-10 | 2019-07-23 | 深圳大学 | Robust sparse signal restoration methods |
| CN111711584A (en) * | 2020-05-22 | 2020-09-25 | 西北工业大学 | Sparse underwater acoustic channel estimation method based on proportional normalization minimum mean square error |
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Cited By (4)
| Publication number | Priority date | Publication date | Assignee | Title |
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| CN108880558A (en) * | 2018-06-08 | 2018-11-23 | 西北工业大学 | A kind of underwater sound signal condensation matrix optimization method based on discrete cosine transform |
| CN110048723A (en) * | 2019-04-10 | 2019-07-23 | 深圳大学 | Robust sparse signal restoration methods |
| CN111711584A (en) * | 2020-05-22 | 2020-09-25 | 西北工业大学 | Sparse underwater acoustic channel estimation method based on proportional normalization minimum mean square error |
| CN111711584B (en) * | 2020-05-22 | 2022-06-07 | 西北工业大学 | Sparse underwater acoustic channel estimation method based on proportional normalization minimum mean square error |
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