CN112039498B - Self-adaptive signal processing method and medium based on polymorphic variable step-length least mean square - Google Patents
Self-adaptive signal processing method and medium based on polymorphic variable step-length least mean square Download PDFInfo
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Abstract
The application discloses a self-adaptive signal processing method based on polymorphic variable step-length least mean square, which relates to the technical field of digital signal processing, and the technical scheme is as follows: in the first stage, a steady state MSD between the filter coefficients W (n) and the optimal filter coefficients H (n) is calculated, and a larger step size mu is selected 1 To obtain a faster convergence speed according to the step factor mu 1 Calculating to obtain initial state steady state MSD in the first stage 1 The method comprises the steps of carrying out a first treatment on the surface of the Third stage, calculate the minimum final steady state MSD min Value and according to final steady state MSD min The value calculated the best mu opt The method comprises the steps of carrying out a first treatment on the surface of the Second stage, in initial state MSD 1 And final state MSD min Adding multiple transients in between according to step size factor mu 1 And the optimal step factor mu opt Is to adjust the transient step size factor by a multiple factor of (2)And according to the transient step-size factorCalculating to obtain transient stateUnder abrupt channel, the minimum MSD value under the channel can be reached, the convergence can be carried out at a faster convergence speed, and good performance can be maintained under the condition of abrupt signal change of a stable channel.
Description
Technical Field
The application relates to the technical field of digital signal processing, in particular to a multi-state variable step length least mean square-based adaptive signal processing method and medium.
Background
Adaptive filtering is an optimal filtering method developed in recent years. The method is an optimal filtering method developed on the basis of linear filtering such as wiener filtering and Kalman filtering. Because it has stronger adaptability and better filtering performance. Thus, the method has wide application in engineering practice, especially in information processing technology, such as: signal processing, communication processing, image processing, and the like.
In 1960, widrow and Hoff proposed typical algorithms for adaptive filtering: a least mean square algorithm (Least Mean Square, LMS) based on a minimum mean square error criterion. The algorithm has been widely developed in the last decades and has been applied in the fields of communication, control, radar signal processing, system identification, echo cancellation, etc. However, research finds that the convergence speed and the steady-state mean square error (Mean Least Deviation, MSD) of the LMS algorithm in the signal noise reduction process are contradictory, and the steady-state MSD with a large (small) convergence speed is obtained. To solve the above-mentioned problems of LMS, KWONG proposes a variable step LMS (Variable Step Size Least Mean Square, VSS-LMS) algorithm. The step size is adjusted according to the error size, and although better performance is obtained, four parameters are required to be reasonably adjusted, and the method is difficult to realize. Then, on the basis of the VSS-LMS algorithm, a plurality of improved VSS-NLMS algorithms are provided, and certain convergence rate is improved.
The research context herein is system identification, as shown in fig. 1. At present, the Prob-LMS algorithm has considerable performance. The research finds that the complexity of the algorithm applied to the signal noise reduction process is too high, and the convergence speed in the abrupt channel is slow.
Disclosure of Invention
In the steady-state MSD analysis of the LMS algorithm, the MSD reaching the minimum of the LMS algorithm is determined by the input signal power of the system, the power of the noise signal, the power of the random disturbance signal and the length of the filter, and when the LMS reaches the minimum MSD value, the convergence speed is slow. To solve the above problems of the LMS algorithm, it is an object of the present application to provide an adaptive signal processing method and medium based on Multi-state variable step-size least mean square (Multi-state Variable Step Size Mean Least Square, MVSS-LMS). The MVSS-LMS algorithm is less complex than the Prob-LMS algorithm and overcomes the problem of slow convergence in abrupt channels.
The technical aim of the application is realized by the following technical scheme:
in a first aspect, there is provided a multi-state variable step-size least mean square based adaptive signal processing method, applied to a fast convergence adjustment process of an input signal, comprising:
in the first stage, a steady-state mean square error MSD between the filter coefficient W (n) and the optimal filter coefficient H (n) is calculated, and a larger step size mu is selected 1 To obtain a faster convergence speed according to the step factor mu 1 Calculating to obtain initial state steady state MSD in the first stage 1 ；
Third stage, calculate the minimum final steady state MSD min Value and according to final steady state MSD min The value calculated the best mu opt ；
Second stage, in initial state MSD 1 And final state MSD min Adding multiple transients in between according to step size factor mu 1 And the optimal step factor mu opt Is to adjust the transient step size factor by a multiple factor of (2)And is dependent on the transient step factor->Calculating to obtain transient state
Preferably, the step size factor μ 1 The method comprises the following steps:
where L is the length of the filter, gamma is the kurtosis,is the power of the input signal.
Preferably, the steady-state mean square error MSD 1 The method comprises the following steps:
in the method, in the process of the application,is the power of the noise signal.
Preferably, the optimal step factor μ opt The method comprises the following steps:
preferably, the steady-state mean square error MSD min The method comprises the following steps:
preferably, the multiple factor beta is a step factor mu 1 And the optimal step factor mu opt Is a ratio of (2).
Preferably, the transient step size factorThe regulation is specifically as follows:
dividing the multiple factor beta into n beta i Multiplying by each beta i Corresponds to a transient stateAt the same time correspond to +.>The method comprises the following steps:
wherein beta is 1 ＝β 2 ＝β 3 ＝…＝β n 。
Preferably, the transient step size factorThe method comprises the following steps:
according to beta i (i=1, 2,3.., n.), calculated as:
preferably, the transientThe method comprises the following steps:
in a second aspect, a computer readable medium is provided, on which a computer program is stored, the computer program being executable by a processor to implement the polymorphic variable step-size least mean square based adaptive signal processing method and medium according to any of the first aspects.
Compared with the prior art, the application has the following beneficial effects:
1. the complexity difference of MVSS-LMS depends on the number of transients; if fewer transients are being taken, the complexity is lower; assuming m transients exist, the MVSS-LMS multiplier is 3L+1, the MVSS-NLMS algorithm is 2+1 comparators, 4 adders are fewer, 3 multipliers are fewer, 2 dividers are fewer, but the gain is brought, the convergence speed is improved, the system reaches stability faster, and the gain is considerable.
2. MVSS-NLMS has a higher convergence rate than Prob-LMS and maintains the minimum steady-state mean square error achieved in the time-varying system when the smoothness DNS is less than 0.02 and the kurtosis γ=3 is maintained, and has an excellent convergence rate and steady-state mean square error in the abrupt channel.
Drawings
The accompanying drawings, which are included to provide a further understanding of embodiments of the application and are incorporated in and constitute a part of this specification, illustrate embodiments of the application and together with the description serve to explain the principles of the application. In the drawings:
FIG. 1 is a schematic diagram of a prior art system identification model;
FIG. 2 is a flowchart illustrating operations according to an embodiment of the present application;
FIG. 3 is a graph showing the convergence effect under abrupt signal change of a fixed channel according to an embodiment of the present application;
fig. 4 is a view showing the convergence effect of MVSS-LMS in the embodiment of the present application.
Detailed Description
For the purpose of making the objects, technical solutions and advantages of the present application more apparent, the present application will be further described in detail with reference to the following examples and fig. 1 to 4, which are illustrative embodiments of the present application and the description thereof are only for explaining the present application and are not limiting the present application.
Examples: the adaptive signal processing method based on the multi-state variable step-length least mean square is applied to the rapid convergence adjustment process of an input signal and is defined as MVSS-LMS. Taking the system identification application direction as an example.
As shown in fig. 1 and 2, the system identification model may be divided into a stationary (time-invariant) channel and a non-stationary (time-variant) channel. The filter coefficients of its stationary channel are fixed H (n). The filter coefficients of the non-stationary channel, the vector H (n), follow a random floating model, specifically:
H(n+1)＝H(n)+q(n) (1)
where q (n) is Gaussian white noise independent of x (n), n (n).
A uniform criterion is required for the performance of each algorithm. Steady state MSD between estimated filter coefficient W (n) and optimal filter coefficient H (n), wherein the smoothness (Degree of Nonstationa-city, DNS) of this channel is:the MSD is specifically:
MSD(n)＝Tr(E{(W(n)-H(n))(W(n)-H(n))T}) (2)
where Tr (·) is the trace of the matrix and E (·) is the mathematical expectation of the random variable, MSD is represented in the figure as MSD db It is defined as:
MSD db (n)＝10log 10 (MSD(n)) (3)
the error signal is specifically:
e(n)＝d(n)-y(n) (4)
wherein d (n) =x T (n)H(n)+n(n),y(n)＝X T (n)W(n)。
The estimated filter weights are updated as:
W(n+1)＝W(n)+μX(n)e(n) (5)
the weight error vector is assumed to be:
V(n)＝H(n)-W(n) (6)
the weight error vector of the LMS can be deduced from formulas (1), (5) and (6) as:
V(n+1)＝V(n)-μX(n)(X T (N)V(n)+n(n))+q(n) (7)
from (7), it is desirable that:
wherein:
tr (·) represents the trace of the matrix,expressed as kurtosis of X (n).
From equations (8), (9) it can be deduced that:
as in the formula (11)MSD and step length under the stable channel are:
the LMS algorithm has the following requirements that:
namely:
when (when)The minimum MSD is:
in theory, a large (small) step size results in a fast (slow) convergence speed. From equation (13), the step size is determined by the length L, peak of the filterDegree gamma and power of input signalAnd (3) determining. When the length L and kurtosis gamma of the filter are smaller, the step mu is larger to obtain a faster convergence speed, but the MSD value is not ideal; when step size->And a faster convergence speed cannot be obtained.
The MVSS-LMS provided herein includes a first stage, a second stage, and a third stage. In the first stage, calculating the steady state MSD between the filter coefficient W (n) and the optimal filter coefficient H (n), and selecting a larger mu 1 To obtain a faster convergence speed according to the step factor mu 1 Calculating to obtain initial state steady state MSD in the first stage 1 . A third stage for obtaining the minimum MSD value and selecting the optimal step size factor mu opt And according to the optimal step factor mu opt Calculating to obtain the final state steady state MSD in the third stage min . Second stage, in initial state MSD 1 And final state MSD min Adding multiple transients according to step factor μ1 and optimal step factor μ opt Is to adjust the transient step size factor by a multiple factor of (2)And is dependent on the transient step factor->Calculating to obtain transient->
In the first stage, from equation (14), a step size value can be derived:
wherein mu 1 Correspondingly achievedMSD is:
in the third stage, from equation (14), the optimal step size value can be derived:
wherein mu opt The corresponding MSD is:
in the second stage, mu is calculated first 1 And mu opt The multiple factor of (2) is:
from the beta values, deducing:
wherein beta is 1 ＝β 2 ＝β 3 ＝…＝β n 。
According to beta i (i=1, 2,3.., n.), deriving:
the corresponding MSD is +.>
To sum up, the step size is:
the complexity difference of the MVSS-LMS algorithm depends on the number of transients. If fewer transients are taken, the complexity is lower, m transients are assumed to exist, the MVSS-LMS multiplier is 3L+1, the MVSS-NLMS algorithm has 2m+1 comparators, 4 adders and 3 multipliers and 2 dividers compared with the Prob-LMS algorithm. As shown in table 1:
table 1 complexity comparison
Verification and analysis
As shown in fig. 3 (a) and 3 (b): l=8, snr=20 dB, step u 2 2.4866 ×10 respectively -4 、2.5081×10 -4 The abrupt change test of the signal is performed when the number of iterations is t=500 (algorithm does not reach final steady state MSD), t=20000 (algorithm has reached final steady state MSD), respectively. As can be seen from fig. 3, MVSS-LMS exhibits excellent convergence speed in the signal abrupt change channel and has a small steady-state mean square error.
As shown in fig. 4 (a), l=8, snr=0 dB,10 -5 ，10 -6 DNS corresponding to each of 0.0199, 0.0085 and 0.0028 are faster in mvss-LMS convergence than Prob-LMS.
As shown in fig. 4 (b), l=8, snr=10 dB,10 -5 、10 -6 、10 -7 the corresponding DNS is 0.0638, 0.0277, 0.0088 and 0.0028, and the MVSS-LMS convergence speed is faster than that of the Prob-LMS.
As shown in fig. 4 (c), l=8, snr=20 dB,10 -6 、10 -7 、10 -8 DNS corresponding to each of 0.0859, 0.0282, 0.0089 and 0.0028 are faster in mvss-LMS convergence than Prob-LMS.
As shown in fig. 4 (d), l=8, snr=30 dB,10 -7 、10 -8 、10 -9 the corresponding DNS is 0.0839, 0.0283, 0.0090 and 0.0028, and the MVSS-LMS convergence speed is faster than that of the Prob-LMS.
In summary, compared with the Prob-LMS algorithm, the MVSS-NLMS algorithm has lower complexity, and when the smoothness DNS is less than 0.02 and the kurtosis γ=3 is maintained, the MVSS-LMS algorithm has better convergence rate, maintains the minimum steady-state mean square error achieved under the time varying system, and has excellent convergence rate and steady-state mean square error in abrupt channels.
The foregoing description of the embodiments has been provided for the purpose of illustrating the general principles of the application, and is not meant to limit the scope of the application, but to limit the application to the particular embodiments, and any modifications, equivalents, improvements, etc. that fall within the spirit and principles of the application are intended to be included within the scope of the application.
Claims (5)
1. The adaptive signal processing method based on the multi-state variable step-length least mean square is characterized by being applied to a rapid convergence adjusting process of an input signal and comprising the following steps of:
in the first stage, a steady state MSD between the filter coefficients W (n) and the optimal filter coefficients H (n) is calculated, and a larger step factor mu is selected 1 To obtain a comparison ofFast convergence speed and according to step factor mu 1 Calculating to obtain initial steady state MSD in the first stage 1 ；
Third stage, calculate the minimum final steady state MSD min Value and according to final steady state MSD min Calculating optimal step factor mu opt ；
Second stage, in initial steady state MSD 1 And final steady state MSD min Adding multiple transients in between according to step size factor mu 1 And the optimal step factor mu opt Is to adjust the transient step size factor by a multiple factor of (2)And is dependent on the transient step factor->Calculating to obtain transient->
The optimal step factor mu opt The method comprises the following steps: representing the power of the input signal; />Representing a stationary channel; />Representing the power of the noise signal;
the final steady state MSD min The method comprises the following steps:l is the length of the filter;
the multiple factor beta is a step factor mu 1 And the optimal step factor mu opt Is a ratio of (2);
the transient step size factorThe regulation is specifically as follows:
dividing the multiple factor beta into n beta i Multiplying by each beta i Corresponds to a transient stateAt the same time correspond to +.>The method comprises the following steps:
wherein beta is 1 ＝β 2 ＝β 3 ＝…＝β n ；
The transient step size factorThe method comprises the following steps:
according to beta i (i=1, 2,3.., n.), calculated as:
2. the adaptive signal processing method based on polymorphic variable step-length least mean squares of claim 1, wherein the step-length factor μ 1 The method comprises the following steps:
wherein γ is kurtosis.
3. The adaptive signal processing method based on polymorphic variable step-length least mean squares of claim 2, wherein the initial steady state MSD 1 The method comprises the following steps:
4. the adaptive signal processing method based on polymorphic variable step-length least mean squares of claim 1, wherein the transient stateThe method comprises the following steps:
5. a computer readable medium having stored thereon a computer program for execution by a processor to implement the polymorphic variable step least mean square based adaptive signal processing method of any of claims 1-4.
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