Convergence Analysis of a Mixed Controlled Adaptive Algorithm

The least mean square (LMS) algorithm [1] is one of the most widely used adaptive schemes. Several works have been presented using the LMS or its variants [2–14], such as signed LMS [8], the least mean fourth (LMF) algorithm and its variants [15], or the mixed LMS-LMF [16–18] all of which are intuitively motivated.

The LMS algorithm is optimum only if the noise statistics are Gaussian. However, if these statistics are different from Gaussian, other criteria, such as -norm ( ), perform better than the LMS algorithm. An alternative to the LMS algorithm which performs well when the noise statistics are not Gaussian is the LMF algorithm. A further improvement is possible when using a mixture of both algorithms, that is, the LMS and the LMF algorithms [16].

In this respect, existing algorithms based on mixed-norm (MN) criteria have been used in system identification behaving robustly in Gaussian and non-Gaussian environments. These algorithms are based on a fixed combination of the LMS and the LMF algorithms or a time varying combination of them. The time variation is used in adapting the mixed control parameter to compensate for nonstationarities and time-varying environments. The combination of error norms governed by a mixture parameter is introduced to yield a better performance than algorithms derived from a single error norm. Very attractive results are found through the use of mixed-norm algorithms [16–18]. These are based on the minimization of a mixed norm cost function in a controlled fashion, that is [16–18],

(1)

where the error is defined as

(2)

is the desired value, is the filter coefficient of the adaptive filter, is the input vector, is the additive noise, and is the mixing parameter between zero and one and set in this range to preserve the unimodal character of the cost function. It is clear from (1) that if the algorithm reduces to the LMS algorithm; if, however, the algorithm is the LMF. A careful choice for in the interval (0,1) will enhance the performance of the algorithm. The algorithm for adjusting the tap coefficients, , is given by the following recursion:

(3)

Adaptive filter algorithms designed through the minimization of equation (1) have a disadvantage when the absolute value of the error is greater than one. This makes the algorithm go unstable unless either a small value of the step size or a large value of the controlling parameter is chosen such that this unwanted instability is eliminated. Unfortunately, a small value of the step size will make the algorithm converge very slowly, and a large value of the controlling parameter will make the LMS algorithm essentially dominant.

The rest of the paper is organized as follows. In Section 2, the description of the proposed algorithm is addressed, while Section 3 deals with the convergence analysis. Section 4 details the derivation of the excess mean-square-error. The simulation results are reported in Section 5, and finally Section 6 concludes the main findings of the paper and outlines possible further work.

To overcome the above-mentioned problem, a modified approach is proposed where both constraints of the step size and the control parameter are eliminated. The proposed criterion consists of the cost function (1) where the -norm is substituted for the -norm. Ultimately, this should eliminate the instability in the -norm and retains the good features of (1), that is, the mixed nature of the criterion if . The proposed scheme is defined as,

(4)

If , the cost function defined by (4) reduces to the LMS algorithm whatever the value of in the range for which the unimodality of the cost function is preserved.

For , the algorithm reduces to the -norm adaptive algorithm, and moreover if results in the familiar signed LMS algorithm [14].

The value range of the lower-order is selected to be because

1. (1)

   for , the cost function may easily become large valued when the magnitude of the output error , leading to a potentially considerable enhancement of noise, and

    
2. (2)

   for , the gradient decreases in a positive direction, resulting in an obviously undesirable attribute for being used as a cost function. Setting the value of within the range provides a situation where the gradient at is very much lower than that for the cases with . This means that the resulting algorithm can be less sensitive to noise.

    

For , gives less weight for larger error and this tends to reduce the influence of aberrant noise, while it gives relatively larger weight to smaller errors and this will improve the tracking capability of the algorithm [19].

2.1. Convex Property of Cost Function

The cost function is a convex function defined on for , where and are the dimensions of and , respectively.

Proof.

(5)

Let be the joint probability density function of and . Taking the expectation value of the above, after multiplying its both sides by , one obtains the following:

(6)

This shows that the cost function is convex.

2.2. Analysis of the Error Surface

Case 1 .

Let the input autocorrelation matrix be , and the cross-correlation vector that describes the cross-correlation between the received signal ( ) and the desired data ( ) . The error function can be more conveniently expressed as follows:

(7)

It is clear from (7) that the mean-square-error (MSE) is precisely a quadratic function of the components of the tap coefficients, and the shape associated with it is hyperparaboloid. The adaptive process continuously adjusts the tap coefficients, seeking the bottom of this hyperparaboloid.

(8)

Case 2 .

It can be shown as well that the error function for the feedback section will have a global minimum since the latter one is a convex function. As in the feedforward section, the adaptive process will continuously seek the bottom of the error function of the feedback section.

2.3. The Updating Scheme

The updating scheme is given by,

(9)

and sufficient condition for convergence in the mean of the proposed algorithm can be shown to be given by:

(10)

where is the trace operation of the autocorrelation matrix .

In general, the step size is chosen small enough to ensure convergence of the iterative procedure and produce less misadjustment error.

In this section, the convergence analysis of the proposed algorithm is detailed. The following assumptions which are quite similar to what is usually assumed in literature and which can also be justified in several practical instances are used during the conver thegence analysis of the mixed controlled algorithm. For example, these are quite similar to what is usually assumed in the literature [14, 15, 20–22], and which can also be justified in several practical instances.

1. (A1)

   The input signal is zero mean and having variance .

    
2. (A2)

   The noise is a zero-mean independent and identically distributed process and is independent of the input signal and having zero odd moments.

    
3. (A3)

   The step-size is small enough for the independence assumption [14] to be valid. As a consequence, the weight-error vector is independent of the input .

    

While assumptions (A1-A2) can be justified in several practical instances, assumption (A3) can only be attained asymptotically. The independence assumption [14] is very common in the literature and is justified in several practical instances [21]. The assumption of small step size is not necessarily true in practice but has been commonly used to simplify the analysis [14].

During the convergence analysis of the proposed algorithm only the case of is considered as it is carried out for the first time. Cases for can be found, for example, in [16–18].

The weight error is defined to be

(11)

3.1. First Moment Behavior of the Weight Error Vector

We start by evaluating the statistical expectation of both sides of (9) which looks after subtracting of both sides to give

(12)

After substituting the error defined by (2) in the above equation and taking the expectation of its both sides, this results in:

(13)

Here at this point, we have to evaluate the expression using Price's theorem [20] in the following way:

(14)

note that in the second step of this equation the error has been substituted.

Now, we are ready to evaluate expression (13), and it is given by,

(15)

It is to show that the mis-alignment vector will converge to the zero vector if the step-size, , is given by

(16)

A more restrictive, but sufficient and simpler, condition for convergence of (12) in the mean is

(17)

where is the largest eigenvalue of the autocorrelation matrix , since in general , and is the minimum MSE.

An inspection of (16) will immediately show that if the convergence does occur, the root mean-squared estimation error at time is such that

(18)

where the mean-square value of the estimation error can be shown to be

(19)

(a) Discussion

It can be seen from (18) that, a sufficient condition for the algorithm to converge in the mean, the following must hold:

(20)

Consequently, when , the convergence for the LMS algorithm is proved.

3.2. Second Moment Behavior of the Weight Error Vector

From (12) we get the following expression for :

(21)

Let define the second moment of the misalignment vector therefore, the above equation becomes, after taking the expectation of both of its sides, the following:

(22)

Before finalizing the above expression, let us evaluate the following quantities taking into account that they are Gaussian and zero mean [20]:

(23)

(24)

(25)

and finally,

(26)

Substituting expressions (23)–(26) in (22) results in the following:

(27)

During the derivation of the above equation, expressions and are evaluated, respectively, as follows:

(28)

and

(29)

Both of these expressions are substituted in (22) to result in its simplified form (27).

Now, denote by and the limiting values of and , respectively; then closed-form expressions for the limiting (steady-state) values of the second moment matrix and error power are derived next.

It is assumed that the autocorrelation matrix, , is positive definite [23] with eigenvalues, ; hence, it can be factorized as;

(30)

where is the diagonal matrix of eigenvalues

(31)

and is the orthonormal matrix whose th column is the eigenvector of associated with the th eigenvalue, that is,

(32)

which results in

(33)

hence (27) can be written as

(34)

We are now ready to decompose the above matrix equation into its scalar form as:

(35)

where

(36)

and is the th scalar element of the matrix .

Two cases can be considered for the step size so that the weight vector converges in the mean square sense.

(1) Case

In this case, (35) consists of the off-diagonal elements of matrix and will look like the following:

(37)

consequently, the range of the step size parameter is dictated by

(38)

As it was in the case of the mean convergence, a sufficient condition for mean square convergence is

(39)

(2) Case

In this case, (35) consists of only the diagonal elements of matrix and will look like the following:

(40)

correspondingly, the range of the step size parameter for convergence in the mean square sense is given by

(41)

(b) Discussion

Note that will result in zero in the denominator of expression (41) and therefore will make take any value in the range of positive numbers, a contradiction with the ranges of values for the step sizes of LMS and LMF algorithms. Moreover, any value for in will make of the step size set by (41) less than zero, also this condition is discarded. This concludes that it is safer to use the more realistic bounds of (39) which will guarantee stability regardless of the value of , and therefore will be considered here.

Once again, it is easy to see that if the convergence in the mean-square occurs, consequently the following occurs

(42)

In this section, the derivation of the EMSE will be performed for the general case of . First, let us define the a priori estimation error

(43)

Second, the following assumption is to be used in the following ensuing analysis:

(A4)The a priori estimation error with zero-mean is independent of .

The updating scheme of the proposed algorithm defined in (9) can be set up into the following recursion:

(44)

where the error function is given by

(45)

where .

In order to find the expression of the EMSE of the algorithm (defined as ), we need to evaluate the following relation:

(46)

Taking the left-hand side of (46), we can write

(47)

At this point, we make use of the Taylor series expansion to expand with respect to around as

(48)

where and are, respectively, the first-order and second-order derivatives of with respect to evaluated around , and denotes the third, and higher-order terms of .

Using (45), we can write

(49)

Similarly, we can obtain

(50)

Substituting (48) in (47) we get

(51)

Using (A4) and ignoring , we obtain

(52)

Using (49), we get

(53)

Using the Price's theorem to evaluate the expectation as

(54)

where . So (53) becomes

(55)

Now taking the right-hand side of (46), we require . So, we write

(56)

Therefore,

(57)

Using (A2) and (A4) and ignoring , we write (57) as

(58)

By using (56), we can evaluate as

(59)

Therefore, using (56) and (59), we can evaluate

(60)

Now letting

(61)

(62)

(63)

we can write (58) as

(64)

and subsequently (46) can be concisely expressed as

(65)

and the EMSE can be evaluated as

(66)

In this section, the performance analysis of the proposed mixed controlled adaptive algorithm is investigated in an unknown system identification problem for different values of and different values of the mixing parameter . The simulations reported here are based on an FIR channel system identification defined by the following channel:

(67)

Three different noise environments have been considered namely, Gaussian, uniform, and Laplacian. The length of the adaptive filter is the same as that of the unknown system. The learning curves are obtained by averaging 600 independent runs. Two scenarios are considered for the case of the value of , that is, and . The performance measure considered here is the excess mean-square-error (EMSE).

Figures 2, 3, and 4 depict the convergence behavior of the proposed algorithm for different values of in a white Gaussian noise, Laplacian noise, and uniform noise, respectively, for the case of . As can be depicted from these figures the best performance is obtained when . More importantly, the best noise statistics for this scenario is when the noise is Laplacian distributed. An enhancement in performance is obtained, and about a 2 dB improvement is achieved for all values of . Also, one can notice that the worst performance is obtained when the noise is uniformly distributed.

Figures 5, 6, 7, 8, 9 and 10 report the performance of the proposed algorithm for an SNR of 0 dB, 10 dB and 20 dB, respectively, for the case of . Figures 5 and 6 are the result of the simulations for and , respectively. A consistency in performance of the proposed algorithm in these scenarios for the uniform noise as far as the lowest EMSE is reached by the proposed algorithm. Similar behaviour is obtained by the proposed algorithm in Figures 7 and 8 where Figures 7 and 8 report the simulations results of the proposed algorithm for and , respectively, for an SNR of 10 dB.

In the case of an SNR of 20 dB, Figures 9 and 10 depict the results. The case of is shown in Figure 9 while that of is shown in Figure 10. One can see that, even though the proposed algorithm is still performing better in the uniform noise environment, as shown in Figure 9, for , however, identical performance is obtained by the different noise environments when as reported in Figure 10. The theoretical findings confirm these results as will be seen later.

From the above results, one can conclude that when the proposed algorithm is biased towards the LMF algorithm, in contrast to the case when , the proposed algorithm is biased towards the LMS algorithm.

Next, to assess further the performance of the proposed algorithm for the same steady-state value, two different cases for are considered, that is, and . Figures 11 and 12 illustrate the learning behavior of the proposed algorithm for and , respectively, both are for . As can be seen from these figures that the best performance is obtained with uniform noise while the worst performance is obtained with Laplacian. The mixing variable had little effect on the speed of convergence of the proposed algorithm when the noise is uniformly and Gaussian distributed. However, as can be seen from Figure 12 in the case of Laplacian noise, has decreased the speed of convergence of the proposed algorithm from 55000 iterations (in the case of ) to almost 2000 iterations. A gain of 3500 iterations in favor of the proposed algorithm when the noise is Laplacian distributed.

Finally, the analytical results for the steady-state EMSE derived for the proposed algorithm given in (66) are compared with the ones obtained from simulation for Gaussian, Laplacian, and uniform noise environments with an SNR of 0 dB, 10 dB, and 20 dB. This comparison is reported in Tables 1-2, and as can be seen from these tables, a close agreement exists between theory and the simulation results as mentioned earlier, for the case of and , that similar performance by the different noise environments is obtained for and SNR of 20 dB as shown in Table 2.

	Gaussian		Laplacian		Uniform
	Theoretical	Simulation	Theoretical	Simulation	Theoretical	Simulation
0 dB	−16.9	−16.85	−9.62	−9.82	−22.81	−22.6
10 dB	−26.02	−26.53	−19.33	−19.99	−31.64	−31.29
20 dB	−44.14	−43.93	−40.34	−40.55	−45.14	−45.43

	Gaussian		Laplacian		Uniform
	Theoretical	Simulation	Theoretical	Simulation	Theoretical	Simulation
0 dB	−22.47	−22.6	−14.26	−16.02	−27.65	−26.59
10 dB	−28.7	−28.64	−26.41	−26.32	−29.15	−29.57
20 dB	−39.28	−39.87	−39.24	−39.58	−39.28	−39.92

A new adaptive scheme for system identification has been introduced, where a controlling parameter in the range is used to control the mixture of the two norms. The derivation of the algorithm is worked out, and the convexity property is proved for this algorithm. Existing algorithms, for example [16–18] can be considered as a special case of the proposed algorithm. Also, the first moment behaviour as well as the second moment behaviour of the weights are studied. Bounds for the step size on the convergence of the proposed algorithm are derived. Finally, the steady-state analysis was carried out; simulation results performed for the purpose of validating theory are found to be in good agreement with the theory developed.

The proposed algorithm has been applied so far to a system identification scenario, for example, echo cancellation. As a future extension, recent work is going on the application of the proposed algorithm to mitigate the effects of intersymbol interference in a communication system.