Pseudo-Fractional Tap-Length Learning Based Applied Soft Computing for Structure Adaptation of LMS in High Noise Environment

Abstract

The structure of an adaptive time varying linear filter largely depends on its tap-length and the delay units connected to it. The no of taps is one of the most important structural parameters of the liner adaptive filter. Determining the system order or length is not a trivial task. Fixing the tap-length at a fixed value sometimes results in unavoidable issues with the adaptive design like insufficient modeling and adaptation noise. On the other hand a dynamic tap-length adaptation algorithm automatically finds the optimum order of the adaptive filter to have a tradeoff between the convergence and steady state error. It is always difficult to get satisfactory performance in high noise environment employing an adaptive filter for any identification problem. High noise decreases the Signal to noise ratio and sometimes creates wandering issues. In this chapter an improved pseudo-fractional tap-length selection algorithm has been proposed and analyzed to find out the optimum tap-length which best balances the complexity and steady state performance specifically in high noise environment. A steady-state performance analysis has been presented to formulate the steady state tap-length in correspondence with the proposed algorithm. Simulations and results are provided to observe the analysis and to make a comparison with the existing tap-length learning methods.

Full text

Pseudo-Fractional Tap-Length Learning
Based Applied Soft Computing
for Structure Adaptation of LMS
in High Noise Environment
Asutosh Kar and Mahesh Chandra
Abstract The structure of an adaptive time varying linear filter largely depends
on its tap-length and the delay units connected to it. The no of taps is one of the
most important structural parameters of the liner adaptive filter. Determining the
system order or length is not a trivial task. Fixing the tap-length at a fixed value
sometimes results in unavoidable issues with the adaptive design like insufficient
modeling and adaptation noise. On the other hand a dynamic tap-length adaptation
algorithm automatically finds the optimum order of the adaptive filter to have a
tradeoff between the convergence and steady state error. It is always difficult to get
satisfactory performance in high noise environment employing an adaptive filter
for any identification problem. High noise decreases the Signal to noise ratio and
sometimes creates wandering issues. In this chapter an improved pseudo-fractional
tap-length selection algorithm has been proposed and analyzed to find out the
optimum tap-length which best balances the complexity and steady state perfor-
mance specifically in high noise environment. A steady-state performance analysis
has been presented to formulate the steady state tap-length in correspondence with
the proposed algorithm. Simulations and results are provided to observe the
analysis and to make a comparison with the existing tap-length learning methods.
Keywords Adaptive filter Normalized Lease Mean Square (NLMS) algorithm 
Tap-length Structure adaptation System identification Mean Square Error
(MSE) Signal to Noise Ratio (SNR) High noise environment
A. Kar (&)
Department of Electronics and Telecommunication Engineering, IIIT,
Bhubaneswar, India
e-mail: asutosh@iiit-bh.ac.in
M. Chandra
Deptartment of Electronics and Communication Engineering, BIT, Mesra, India
e-mail: shrotriya69@rediffmail.com
S. Patnaik and B. Zhong (eds.), Soft Computing Techniques
in Engineering Applications, Studies in Computational Intelligence 543,
DOI: 10.1007/978-3-319-04693-8_8, Springer International Publishing Switzerland 2014
115

1 Introduction
Linear minimum mean square error (MMSE) adaptive filters are largely used in
various applications such as echo cancellation, channel modelling, equalization etc.
[1–3]. The FIR adaptive filter is widely popular than its IIR counterpart because of its
inherent stability and tapped delay line (TDL) feed forward structure [1,2]. The
performance of the TDL structure of the adaptive filter in which the weights/tap-
coefficients are recursively updated by adaptive algorithm such as the least mean
square (LMS), recursive least square (RLS) is highly affected by the filter order or in
other words the tap-length selection. The LMS algorithm has been extensively used
in many applications because of its simplicity and robustness [1]. A too short order
filter results in inefficient model of the system and increases the mean square error
(MSE) [4,5]. In principle MMSE is a monotonic non increasing function of the filter
order but it is not advisable to have a too long order filter as it introduce adaptation
noise and extra complexity due to more taps [6–9]. Therefore to balance the adaptive
filter performance and complexity there should be an optimum order of the filter. In
many applications, for example an adaptive equalizer used to combat the effects of
time varying multipath channel, the optimum order may vary with time [10]. The
first variable filter order algorithm [11] proved that shorter filters has faster con-
vergence than the larger ones and adjusting the filter order can improve the con-
vergence of the LMS algorithm [12,13]. In [12] a variable filter order algorithm is
proposed by comparing the current MSE to the previous estimated MMSE and in
[13] by using the time constant concept where step size is constant and calculated in
advance. In both [11,12] order can only be increased in one direction from lower to
optimum value of tap-length which motivated further research. Further an algorithm
was proposed in [13] which was efficient than the previous ones. All these algorithms
aim more at improving the convergence rather than finding the optimum filter order.
But the step size control has less effect on filter length control [14]. More relevant
work was proposed in [10] where the filter is partitioned into segments and order is
adjusted by one segment either being added or removed from the filter according to
the difference of the output errors from the last two segments. This algorithm suffers
from the drawback of carefully selecting the segments and use of simple errors rather
than MSE. Based on gradient descent approach another algorithm is proposed [15]
which was proved to be more flexible than [10] but it created the wandering problem.
The fractional tap-length algorithm was first proposed in [6–8] relaxing the con-
straint that the filter order must be an integer. This fractional order estimation pro-
cedure retains the advantage of both segmented filter and gradient decent algorithm
and has less complexity than the previously proposed methods. But it suffers from
noise level and parameter variation due to unconstrained and random use of the leaky
factor and step size used for order adaptation [16,17]. An improved variable tap-
length variable step LMS algorithm produces better convergence and steady state
error performance than the FT-LMS algorithm [16,18]. But it depends on a careful
selection of leaky factor which controls the overall tap-length adaptation. This
algorithm in [16,18] is found to be more suitable for the echo cancellation
116 A. Kar and M. Chandra

applications with the parameter guidelines it suggest. In this chapter a new gradient
search method based on the pseudo-fractional order estimation technique is pro-
posed which finds the optimum filter order dynamically with a modified tap-length
learning procedure. The filter order can be increased to and decreased from any value
to achieve the desired tap-length for structure adaptation. There should be a trade-off
between a suitable steady state tap-length and convergence rate. The steady state
performance analysis of the proposed algorithm shows the importance of variable
error width parameter. The proposed algorithm shows better performance both in
convergence as well as MSE in comparison to the famous FT-LMS [6] and VT-
VLMS algorithm [16,19]. It reduces the overall design complexity and hence proves
to be a cost saving design.
The remainder of this chapter is organized as follows. The problem formulation
for tap-length optimization is shown in Sect. 2 with an acoustic echo cancellation
framework. The existing variable dynamic tap-length estimation algorithms are
discussed in Sect. 3.InSect. 4 the proposed algorithm has been analysed while
setting up variable key parameters. The computer simulation setup has been
designed both system identification frameworks in a high noise environment
keeping the overall signal to noise ratio low in Sect. 5. The results are shown in
Sect. 6 and the improvements of the proposed algorithm are addressed over the
existing ones.
2 Problem Formulation
The need of an optimum dynamic tap-length selection for linear adaptive filters is
explained with a simplified system identification model for single channel acoustic
echo cancellation (AEC) as shown in Fig. 1. In the echo cancellation arrangement
when the near end speaker speaks, signal is transmitted over communication
channel to the far end room.
If the communication system has significant delay the direct path from loud-
speaker to microphone will sound as an echo to the near end although this path is
not caused by an acoustic reflection. The tap-length or number of weights of the
adaptive filter impulse response based AEC setup affects the overall performance
of the echo cancellation. In most filter designs for different unknown plants the
tap-length is unfortunately fixed at some compromise value creating the problem
of too short and too long filters. Too few and too long filter weights results in
undermodelling and slow convergence respectively. So length of the adaptive filter
must vary and approach to optimal value in the time varying scenarios to best
balance the echo cancellation performance. Practically in an identification
framework the length of unknown plant might be very large or small. It can be
compared with the room impulse response of a conference hall and inside car
environments respectively. This results in issues like adaptation noise due to
mismatch of extra taps and system undermodelling because of unavailability of
taps [7]. The requirement is to construct a suitable dynamic tap-length selection
Pseudo-Fractional Tap-Length Learning Based Applied Soft Computing 117

algorithm not only to minimize the structural and computational complexity of the
filter design but also to improve the overall system identification performance.
Type-1 (Too short order filter): Too few filter coefficients results in under
modeling. Suppose there is a typical impulse response from an acoustic arrange-
ment as shown in Fig. 1where the intension is to identify the long non-sparse
system. A too short filter will result in degraded echo cancellation performance [9]
and demonstrates the problem of insufficient modeling.
Type-2 (Too large order filter): The obvious drawback of a too long filter is
slow convergence. It is not suitable to have a too long filter as it increase the filter
design complexity and introduce adaptation noise due to extra coefficients. Sup-
pose there are two filters of different length to model a acoustic echo cancellation
arrangement shown in Fig. 1, where the too long filter convergence slower than the
filter which has same number of coefficient as the acoustic system to be identified.
Due to the mismatch of tap-length the error spreads all over the filter and the
adaptive filter itself introduce echo in this case [9].
3 Existing Algorithms for Dynamic Tap-Length Selection
3.1 The Fractional Variable Tap-Length
Learning Algorithm
The pseudo fractional tap-length that automatically does the structure adaptation in
a dynamic time varying situation was first obtained efficiently by following the
adaptation proposed [1,15]
Lnf ðnþ1Þ¼ Lnf ðnÞW

þ½ðeL
LðnÞðnÞÞ2ðeL
LðnÞDLðnÞðnÞÞ2
Wð1Þ
D/A
A
F
A/D
+
−
Transmission
Channel
(High Noise)
()t
ξ
()xn
()st
()dn
()yn
()en
Far End
(Repeated at Near End)
Fig. 1 AEC setup both for far and near end subscribers
118 A. Kar and M. Chandra

Finally the tap-length L(n?1) in the adaptation of filter weights for next
iteration is formulated as follows, [1]
Lðnþ1Þ¼ Lnf ðnÞ

if LðnÞLnf ðnÞ
[W

W
¼LðnÞotherwise ð2Þ
L
nf
(n), is the tap-length which can take fractional values. ðeL
LðnÞðnÞÞ2ðeL
LðnÞDLðnÞ
ðnÞÞ is the MSE difference with an error spacing of DLnðÞ[1]. The actual order of
the adaptive filter L
nf
(n) is rounded to the nearest integer value to get the optimum
tap-length. In (1) the factor 
Wis the leakage factor which prevents the order to be
increased to an unexpectedly large value and 
Wis the step size for tap-length
adaptation. It follows simple LMS algorithm for weight update. An improved FT-
LMS with a novel methodology to decide variable 
Wis presented in [15].
4 The VT-VSLMS Algorithm
Based on the FT-LMS many algorithms are proposed. One of the recently pro-
posed methods is VT-VSLMS. The dynamic tap-length that can take fractional
value is obtained by the proposed adaptation based on constrained selection of
predefined leaky factor l
f
and a variable error spacing DðnÞas defined in [4],
Lnf ðnÞ¼ Lnf ðn1Þ lf
1lf

þ½ðeL
LðnÞðnÞÞ2ðeL
LðnÞDðnÞðnÞÞ21
ðlog10 lfÞ2ð3Þ
The tap-length L
nf
is rounded to the nearest integer value to get the dynamic
structure where
W¼lf
1lf
and 
W¼1
ðlog10 lfÞ2ð4Þ
The error spacing DðnÞis obtained as the adaptation [4],
DðnÞ¼maxðDmin;Dmax SÞð5Þ
where Sis smoothing parameter which changes as per the variation in the error
spacing and the maximum and minimum value of D.
S¼log 10ðDðnÞÞ
ðDmax DminÞð6Þ
If Schanges from (0, 0.5) the error spacing varies from ðDmin ;0:5DmaxÞ. The
VS-LMS algorithm is used for weight updating and this type of tap-length
selection is useful basically for echo cancellation applications [14].
Pseudo-Fractional Tap-Length Learning Based Applied Soft Computing 119

5 Dynamic Tap-Length Optimization
In this section an idea to search the optimum filter order by establishing the cost
function and defining two most important terms for our analysis i.e. the optimum
and pseudo-optimum filter order will be clearly defined.
The most popular among the existing adaptive algorithm i.e. the least mean
square (LMS) algorithm has the weight adaptation [15,16],
Wðnþ1Þ¼WðnÞþlXðnÞeðnÞð7Þ
where W(n) is the tap weight vector, lis the step-size parameter, X(n) is the
random input vector and e(n) is the error and nis the time index. In adaptive
filtering the misadjustment is defined as the deviation between the final steady
state values from the minimum or optimum steady state value of the error function
[6],
ML¼JLð1Þ  JL;optðnÞ
JL;optðnÞð8Þ
where Lis a parameter relating to the filter order, M
L
is the misadjustment, J
L
(?)
is the steady state MSE and JL;opt nðÞis the minimum MSE. Cross multiplying and
solving it can be shown that the steady state MSE is an integral multiple of the
MMSE. When lthe step size is small enough then according to [6–8],
MLl
2tr½Rð9Þ
where Ris the input correlation matrix and tr R½¼lLr2
xfor a the input signal
having variance. For the stability point of view step size lshould be in the range
[6],
0\l\2
3tr½Rð10Þ
So if filter order Lchanges then step size should also change to make LMS
algorithm stable. So in many scenarios it is advocated to use the normalized LMS
(NLMS) algorithm for better convergence behavior and constant level of misad-
justment under different scenarios.
Wðnþ1Þ¼WðnÞþ 
l
XTðnÞXðnÞXðnÞeðnÞð11Þ
here 
lis the step size for NLMS algorithm. NLMS converges to mean square
for condition [4,15], 0\
l\2. In order to accelerate the convergence of the
algorithm, the step size of the coefficients in the proposed algorithm is updated
according to [6–8]
120 A. Kar and M. Chandra

Wðnþ1Þ¼WðnÞþ l0
XTðnÞXðnÞ½2þPðnÞ XðnÞeðnÞð12Þ
where l0is a constant, r2
X¼XTnðÞXnðÞis the variance of input signal. L(n) is the
instantaneous variable adaptive tap-length.
Previous research works are basically based on LMS [1,2] where the stability
condition should be checked every time the order changes. In our proposed
approach NLMS is used which provides inherent stability and robustness again the
modification to it improves the convergence.
A. Optimum filter order: If the difference of the MSE output of any two
consecutive taps of the adaptive filter falls below a very small positive value, when
the order is increased, then it can be concluded that adding extra taps do not reduce
the MSE. Let define DL¼JL1ð1Þ  JL1ðÞas the difference between the con-
verged MSE when the filter order is increased from L-1toL. Now the optimum
order can be defined as 
Lthat satisfies,
DLdfor all L
Lð13Þ
where dis a very small positive number set pertaining to the system requirement
and min minfLjJL1JLdgis the cost function with respect to the filter order
L. In many cases pseudo-optimum filter order is observed.
B. Pseudo-optimum filter order: Let there exist a positive integer L0that
satisfies,
L0\
Land DL\dð14Þ
where L0is called the pseudo-optimum filter order. If the above condition is
satisfied by a group of concatenated integer L0;L0þ1;......L0þC1 then Cþ1
is called the width of the pseudo-optimum filter order. These taps satisfies the
optimality condition but cannot be treated as the optimal filter order as it under
model the system. The proposed algorithm is designed in such a manner that it can
overcome the suboptimum values which is explained in the next section.
6 Proposed Pseudo-Fractional Tap-Length Optimization
In this chapter the design problem is related with optimizing tap-length related
criterion. An AEC model is considered in which both the optimum tap-length
Lopt nðÞand co-efficient of the unknown room impulse pertaining to that WLopt ðnÞ
are to be identified. The weight update equation in the proposed algorithm is
simplified by using segment-exact method that updates the filter coefficients every
ksample with respect to variable tap-length L(n)[18].
Pseudo-Fractional Tap-Length Learning Based Applied Soft Computing 121

WLðnÞðnþkÞ¼WLðnÞðnÞþlLðnÞðnÞX
nþk1
i¼n
eLðnÞðiÞXLðnÞðiÞð15Þ
where nis the time index, l0is a constant and L(n) is the instantaneous variable
adaptive tap-length obtained from the proposed fractional order estimation algo-
rithm. WLnðÞ
;XLðnÞare the weight and input vector pertaining to L(n). eL
LðnÞnðÞis the
segmented error with respect to L(n) defined as,
eL
kðnÞ¼dðnÞWT
LðnÞ;1:kXLðnÞ;1:kðnÞð16Þ
and dðnÞ¼WT
LOpt ðnÞXLopt ðnÞþtðnÞð17Þ
where 1\k\LnðÞand WT
LnðÞ;1:k;XLnðÞ;1:knðÞare the weight and input vectors
respectively consisting of first kco-efficient of variable structure adaptive filter
with L(n) coefficients. Similarly WT
Lopt ðnÞ;XLopt ðnÞare the weight and input vector
with tap-length of Lopt and t(n) is the system noise. The modified normalized LMS
(NLMS) algorithm is sampled by employing tap-length varying step-size,
lLðnÞðnÞ¼ l0
r2
X;nþk1½2þLðnÞ ð18Þ
where r2
X;nþk1¼XT
LðnÞðnþk1ÞXLðnÞðnþk1Þis the variance of input per-
taining to L(n) and nþk1 initial coefficients.
The mean square of the segmented error is represented as,
QðLÞ
kðnÞ¼E½ðeL
kðnÞÞ2ð19Þ
Hence the cost function for tap-length optimization can be defined as the dif-
ference of MSE with an error spacing of DLnðÞ[5],
minfLjQðLÞ
LDLðnÞðnÞQðLÞ
LðnÞngð20Þ
where nis a small positive number whose value is determined by the system
requirements according to the analysis in [3,5]. Assuming a small misadjustment
after the initial convergence the pseudo fractional tap-length L(n) should vary
within Lopt nðÞþDLnðÞ1;Lopt nðÞþDLnðÞfrom which the true optimum tap-
length of the unknown plant Lopt nðÞcan be obtained [18]. The variable error width
parameter DLnðÞdecides the bias between the unknown optimum tap-length
Lopt nðÞand the steady state tap-length in a system identification framework. It
removes the suboptimum values and finds the optimum tap-length. A large value
of DLnðÞproduces large error width and brings heavy computational complexity
whereas a small DLnðÞslow down the convergence and makes it difficult to
overcome the suboptimum values. The steady state tap-length is approximately
equal to Lopt nðÞþDLnðÞ[18].
In order to maintain the trade-off between convergence and steady state error
122 A. Kar and M. Chandra

^
e2¼q^
e2ðn1Þþð1qÞ^
e2ðnÞð21Þ
DLðnÞ¼minðDL;max;t^
e2ðnÞÞ ð22Þ
where qis the smoothing parameter and tis a constant which will be discussed
later in this section. The steady state analysis of the proposed algorithm is given in
this section. It also provides a general guideline for choice of parameter t.
r2
d¼Ed
2nðÞ

;P¼EX nðÞdnðÞ½ ð23Þ
For arbitrary values of the weights, the total output power consists of two
components.
r2
effiE½e2ðnÞ ¼ QðLÞ
min þQðLÞ
ex ðnÞð24Þ
Now DLð1Þ ¼ 2tr2
t
2l0ð25Þ
)t¼ð2l0ÞDLð1Þ
2r2
tð26Þ
Although different DLnðÞare needed for different applications, whereas for a
certain application it can be easily decided in advance according to the noise
conditions.
Now the algorithm for tap-length adaptation in a time varying environment is
defined as,
Lnf ðnþ1Þ¼ Lnf ðnÞWn

þ½ðeL
k;LðnÞðnÞÞ2ðeL
k;LðnÞDLðnÞðnÞÞ2
Wnð27Þ
Finally the tap-length L(n?1) in the adaptation of filter weights for next
iteration can be formulated as follows,
Lðnþ1Þ¼Lnf ðnÞ

if LðnÞLnf ðnÞ
[Wn

Wn
LðnÞotherwise ð28Þ
L
nf
(n), the tap-length can take fractional values. As the actual tap-length of the
adaptive filter cannot be a fractional value so L
nf
(n) is rounded to the nearest
integer value to get the optimum tap-length. In (26) the factor Wnis the leakage
factor which prevents the order to be increased to an unexpectedly large value and

Wnis the step size for filter order adaptation. In [4] the value of ðWn;
WnÞwas
based on setting a random leaky factor which performed well for FIR systems
especially for the issues of acoustic echo cancellation. In this chapter a unique
method for setting these parameters has been defined which can be applicable both
for infinite impulse response and FIR systems [6].
Wn¼minðWn;max;Wnðiþ1ÞÞ ð29Þ
Pseudo-Fractional Tap-Length Learning Based Applied Soft Computing 123

Wnðiþ1Þ¼ ~
e2
k;LðnÞðiþ1Þ
~
e2
k;LðnÞðiþ1ÞþDLss ðiÞð30Þ
where ~
eL
k;LðnÞðiþ1Þ¼f~
eL
k;LðnÞðiÞþð1fÞeL
k;LoptðnÞðiþ1Þð31Þ
DLss defines the variable error spacing parameter at steady state tap-length
L
ss
(?). fis a partial weight factor At the steady state [6,7],
Wn!ð1fÞr2
t
ð1þfÞDLss ð1Þ ð32Þ
Similarly the adaptation step size depends on the bias between MSE values with
aD
L
difference. If the difference is more, then adaptation should be slow and vice
versa.

Wn¼minð
Wn;max;s
Wn;maxÞ0\s1ð33Þ
where ðWn;min;Wn;max Þand ð
Wn;min;
Wn;maxÞcan be fixed as [16,19]. Again in
this section two cases are discussed with the random tap-length initialization more
and less than the optimum order. This is done to show that the proposed algorithm
is independent of initialization.
The random initialization is made as Lnf 0ðÞ¼L0\Lopt nðÞ(desired optimum
tap-length). Lnf nðÞ, is the tap-length which can take fractional values. In a dynamic
time varying environment when Lnf ðnÞLðnÞ
Wn

Wnthen assign QðLÞ
LDLðnÞ¼
QðLÞ
LðnÞand append one zero weight to the current tap-length to move Lnf 0ðÞ
towards Lopt nðÞ. This arrangement along with the adaptation shown in (27) repre-
sents the increase of tap-length from the initial value to achieve the desired value.
Initially Lnf 0ðÞ¼L0[Lopt nðÞ. Here Lnf decreases to achieve the optimal
value, where the actual order is Lnf ¼\Lnf nðÞ[þ1. The time varying scenario
in this case is defined as, when Lnf ðnÞLðnÞ
W

W, then assign QðLÞ
LDLðnÞ¼
QðLÞ
LðnÞand remove the last weight from the current tap-length.
The combined tap-length adaptation for both the defined occasions can be
specified by employing a direction vector /¼sgn Xn
ðÞwhere Xn¼LnðÞ
Ln1ðÞ. Now (27) is modified as,
Lnf ðnþ1Þ¼½Lnf ðnÞWnþ/½QðLÞ
LDLðnÞQðLÞ
LðnÞ
Wn:ð34Þ
7 Simulation Setup
The performance of the proposed dynamic tap-length selection algorithm along
with the fixed tap-length LMS, FT-LMS [6–8], VT-VSLMS [16,19], variable
leakage factor based advanced FT-LMS algorithm [20] is simulated using
124 A. Kar and M. Chandra

MATLAB platform. The simulation is carried out with the following data.
A system identification framework is considered consisting of an adaptive filter
connected in parallel with an unknown plant. The input signal is a zero mean
white-noise Gaussian sequence with variance r2
X¼1. The system noise is also a
white Gaussian sequence with zero mean and variance r
t
2
of 0.01. The unknown
system is a white Gaussian noise sequence with zero mean and variance of 0.01.
The unknown plant H(n) is weighted by an exponentially decaying impulse
response and the tap-length is set to 300 as shown in Fig. 2.
8 Results and Discussion
At high noise condition keeping the SNR at -20 dB the results of MSE variation for
fixed tap-length LMS and the proposed tap-length variation is shown in Fig. 3over
4,000 iterations and 100 independent Monte Carlo runs. It depicts that the unknown
impulse response can be completely identified with a variable tap-length adaptive
filter consisting of 200 taps which is 100 taps less than the ideal fixed length
selection. Again fixing the tap-length at arbitrary values results in degraded system
performance with 50 and 100 tap order respectively. The improvement of the pro-
posed algorithm has also been tested in high noise environment keeping the SNR at
-20 dB and the result is shown in Fig. 4. Results shown in clearly reveal that both in
high noise environments FT-LMS [8] tap-length selection may lead to undermod-
elling and advanced FT-LMS [19] along with VT-VSLMS [16,19] results in extra
taps to increase the complexity in design. The proposed algorithm achieves the
optimum tap-length at 200 taps without under modeling or over modeling issues.
The variation of error spacing parameter D
L
with increased number of itera-
tions, averaged over 200 Monte Carlo runs has been shown in Fig. 5.IfD
L
is being
varied with respect to number of iterations then two transient points are noticed
Fig. 2 Impulse response of the unknown plant H(n)
Pseudo-Fractional Tap-Length Learning Based Applied Soft Computing 125

between 0–10 and 350–400 numbers of iterations. These transients have been
shown in Figs. 6and 7respectively which depicts that after some initial transition
D
L
attains steady state value which shows that the optimum tap-length has been as
the variation between consecutive converged MSE remains at a fixed value which
has been discussed and mathematically analysed in Sect. 4. The variation for the
proposed algorithm in comparison to its counterparts is between the minimum if
we consider the absolute values. In transition point-1 the DLgoes from 0 to 4, 10
and 15 for the proposed, VT-VLMS and FT-LMS respectively in Fig. 6which
shows a steady increase in value before achieving the steady state up to 390–400
taps. Then it again decreases to -2 as shown in Fig. 6and attains that value till
5,000 iterations which indicates that the desired optimum tap-length has been
achieved.
Fig. 3 MSE versus No. of iterations (SNR =-20 dB)
Fig. 4 Tap-length versus No. of iterations (SNR =-20 dB)
126 A. Kar and M. Chandra

The proposed algorithm makes the best use of the variable error spacing
parameter which affects the tap-length adaptation up to a large extent.
The variable leakage factor based FT-LMS algorithm with fixed Dachieves the
optimum tap-length as shown in Fig. 5over 200 Monte Carlo runs and 10,000
iterations but results in undermodelling the system.
On the other hand the proposed algorithm best adjusts the system performance
in comparison to the VT-VSLMS and FT-LMS algorithm. The tracking effect can
be judged by enclosure dislocation after starting the MSE adaptation for some
iteration. The system model of tracking for FT-LMS and proposed are shown in
Fig. 8for high noise conditions. It can be observed that the tracking capability of
LMS is better for FT-LMS as well as proposed method but the convergence
performance of proposed algorithm is better than FT-LMS.
Fig. 5 D
P
Versus Iterations
Fig. 6 D
L
Versus Iterations (transition point-1)
Pseudo-Fractional Tap-Length Learning Based Applied Soft Computing 127

9 Conclusion
An improved pseudo-fractional tap-length selection for automatic structure
adaptation in a dynamic time varying environment has been proposed. The steady
state mathematical analysis is made in support of the proposed algorithm. The
parameters were set according to the structure adaptation to best adjust the system
performance and convergence in an identification framework. The proposed
algorithm is compared with the existing tap-length learning algorithms and the
improvements are addressed. The computer simulation and results are shown to
verify the analysis.
Fig. 7 D
L
Versus Iterations (transition point-2)
Fig. 8 Tracking performance at SNR =0 dB (high noise)
128 A. Kar and M. Chandra

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