Low complexity algorithm for updating the coefficients of adaptive 2

International Journal of Electronics and Communication Engineering & Technology (IJECET),
ISSN 0976 – 6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 3, May – June (2013), © IAEME
63
LOW COMPLEXITY ALGORITHM FOR UPDATING THE
COEFFICIENTS OF ADAPTIVE FILTER
G.Prasannakumar1
, K.Indirapriyadarsini2
1
(ECE/Vishnu Institute of Tech/ Jntuk,Bhimavaram, India)
2
(ECE/ Swarnandhra institute of engg/Jntuk, Narsapuram, India)
ABSTRACT
This paper presents a novel algorithm which can dynamically change the update rate
of the coefficients of adaptive filter by analyzing the actual application environments. This
low complexity algorithm changes the update rate of the coefficients of the adaptive filter
dynamically by analyzing the actual application environment. This algorithm builds a
nonlinear relationship between the update rate and the minimum error. Change in update rate
is based on time partition method, which updates the coefficients for every ‘m’ samples,
where m is down sampling rate. If the coefficients are updated for every two samples, it
results reduction in computations by half, further increase of down sampling rate reduces
more number of computations, but the convergence time increases. To minimize the
convergence time, the update rate can be adjusted dynamically by using the relation between
down sampling factor and error, in this method. Acoustic echo cancellation (AEC)
experiments indicate that the scheme proposed in this paper performs significantly better than
traditional algorithms
Keywords: AEC, AGC, FIR, LMS, MSE
1. INTRODUCTION
The last decades have witnessed a rapid increase in the use of Internet voice
communications systems which require the use of an acoustic echo cancellation (AEC) to
eliminate acoustic feedback from the loudspeaker to the microphone. The AEC is an adaptive
filter that estimates the acoustic transfer function and utilizes this estimation to remove the
acoustic echo from the microphone signal. Adaptation algorithm used in AEC is required to
possess good convergence and tracking properties, without posing excessive computational
requirements[1]. On the other hand, in order to be effective, AEC algorithm typically requires
INTERNATIONAL JOURNAL OF ELECTRONICS AND
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ISSN 0976 – 6464(Print)
ISSN 0976 – 6472(Online)
Volume 4, Issue 3, May – June, 2013, pp. 63-69
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64
filters with thousands of coefficients. When applying classical adaptive filter to the AEC
problem, the resulting computational complexity might be prohibitively high. The most
widely used adaptation algorithm is the least mean square error (LMS) method [2], [3].
Although LMS is easy to realize, the giant computational works will cause a long output
delay which is intolerable, as its everely prevents a nature, full-duplex speech conversation. A
significant reduction in the computational burden can be achieved by using frequency domain
adaptive filtering [4]. The time domain linear convolution is efficiently implemented in the
frequency domain, but data gathering might introduce an inherent delay [5], [6].This paper
proposes a low complexity algorithm to update the weights of adaptive filter. Based on lots of
practical experiments, the characteristic is found that the most coefficients of systems change
slowly or keep constant during the conversation. So we need not to update the coefficients of
adaptive filter for each sample. A nonlinear function relationship between the update-rate and
the minimum error is introduced in the new method. The remaining sections of this paper are
organized as follows. Section 2 gives a brief introduction of the adaptive and mean square
error surface. In Section 3, we derive the new updating method based on the deducing of
steepest descent method. In Section 4, we present experimental results, and finally in Section
5, we draw some conclusions.
2. ADAPTIVE FILTER
Adaptive filter is composed of filtering sub-system and adaptation algorithm [7]. The
former can be categorized as system identification, channel equalization and echo
cancellation based on how the structures are chosen. And the latter can apply various
criterions to adjust the parameters of filtering sub system. The factors that influence the
choice of the adaptation algorithm are the speed of convergence to optimal operating
condition, the minimum error at convergence and computational complexity. Lots of research
works are focused on quickening the convergence speed and reducing the minimum error
[8],[9], and there is a few to pay attentions to decreasing the computational complexity.
2.1. Mean square error surface
Consider the adaptive filter demonstrated in Fig. 1, we need to use the adaptation
algorithm to adjust the coefficients of the filter, which decreases the difference between the
input signal vector x and the reference signal d. At time n ,x(n) is defined as:
X(n)=[ x(n),x(n-1),x(n-2),------x(n-N+1)]T
(1)
The weight vector w(n) is:
W (n) = [w0 (n) w1(n) w2(n) ----wN-1(n)]T
(2)
Then the filter output can be expressed as:
Y (n) = WT
(n)X(n) (3)
The errore(n)is defined as:
e (n)=d(n)-wT
(n)x(n). (4)

65
error desired
Fig 1: Typical structure of the adaptive filter using input and error signals to update its tap
weights
The adaptive processing is based on the minimization of the mean square error criterion,
defined as:
ξ(n)=E(e2
(n)) (5)
By (5) we can deduce that the mean square error ξ (n) is a quadratic function of the filter
coefficient vector w(n), and it is a super parabolic in L +1 dimension with a single minimum
point. Usually we can use the steepest descent method to search the minimum point along the
parabola surface [10]. And the gradient vector ߘ is defined as:
డక
డ௪
ൌ ߘ ൌ
డక
డ௪బ
డక
డ௪భ
െ െ
డక
డ௪೙
(6)
Adaptation algorithm is initialized with a random state, then it moves along the negative of
gradient direction at each step, and reaches the minimum point finally. The coefficients can
be updated by:
W(n+1)=w(n)+µ(-∇(n)) (7)
Where µ is the step length, usually it is given by experience.
3. A NEW METHOD UPDATING THE COEFFICIENTS VECTOR
In practical application, the input data should be processed promptly in a short time.
The computational complexity is a very important factor that influences the choice of the
adaptation algorithm. The traditional adaptation algorithms such as LMS update the
coefficients for each sample [3], which cannot satisfy the requirement of online processing.
Many stationary signals can be processed with low tracking speed in the real environment.
desired
Error desired signal
OutputInput Transversal
Filter
Adaptive
weight control
mechanism

66
And a conclusion can be deduced from the mean square error surface that we need not to
adjust the coefficients vector since the output have reached the minimum error, if the system
is invariable, because that ∇ = 0 .And the low complexity method of updating the coefficients
is viable. That is mean the updating rate can be decreased when the adaptation algorithm
searches around the minimum point. It is a tradeoff between computational complexity and
convergence speed. To change the updating rate, we utilize a time partition method which
updating the coefficients only once in every m samples. Because the most computation of
adaptation algorithm is consumed to update the coefficients vector, the complexity can be
reduced almost a half when the coefficients of the adaptive filter is updated only once in
every 2 samples, which is called factor-of-2 down-sampling. And according to the increasing
of down-sampling factor m, the computational complexity will decrease. We can conclude
from equation (7) that changing the updating rate of w(n) will only decelerates the speed of
searching the minimum point, the adaptation algorithm will approach convergence at last.
And it cost double time for factor-of-2 down-sampling method to get the optimal coefficients.
According to the increasing of factor m, the convergence time will increases, and it is
illustrated in Fig. 2.
Fig 2: Convergence speed of down-updating
Along with the reducing of complexity, the convergence speed is slow-down at the
same time. To solve this problem, we proposed a method to adjust the update rate
dynamically. It builds a function between the down-sampling factor m and the minimum
errore(n), as:
m=β
ଵ
ଵାୣ୶୮ ሺఈ௘ሺ௡ሻೖሻ
(8)
Where α controls the shape of the function, β controls the value range, and k controls the
smoothness degree. If m = 0 , filter weights are updated at each sample. If m =1, the weights
are updated in every two samples, so on and so forth

67
Fig 3: Relationship of e(n) and m with various k
Fig. 3 illustrates the curves of equation (8) with k =1, 2,3, 4,5,8, α=50, β=20.
For k = 3, the update rate is relatively fast at the beginning of convergence stage, and with the
decreasing of e(n) , the update rate is slow down accordingly. For k =1, 2 , when e(n) nearly
reach the minimum point, the update rate is still so fast that complexity cannot be reduced in
the stable stage. For k = 4,8 , the update rate is too slow when e(n) is faraway from the
minimum point, and it will take a longtime for the filter to converge to the optimal state. So
the median value of k should be selected to ensure the smoothness of the curve for the
tradeoff of convergence speed and computation.
4. EXPERIMENT AND RESULTS
We use the number of multiplications per input sample as a measure of the
computational complexity of the various algorithms. The complexity of each algorithm is
normalized by the complexity of k =1 (i.e., the complexity of k =1 is 100%).
The filter is chosen to be 20 taps. This number of taps was chosen for two reasons. It
was small enough to limit processing time; however it was large enough to show good
convergence in the MATLAB simulations. The adaptive filter taps at first are initialized to
the null vector.
In low complexity algorithm the filter weights of adaptive filter are updated according
to the equation 4 and equation 8 in this the step size parameter µ is selected a small value of
0.01, the scaling factor α is selected in the range of 3000 to 15000, the range of down
sampling rate β is selected in the range of 10 to 20.

68
Fig 4: Echo cancellation using low complexity algorithm
Table 1: comparison of number of computations and convergence time in LMS and low
complexity algorithm
5. CONCLUSION
We have proposed a new method to change the update rate of adaptation algorithm by
establishing a function between update rate and error signal. The value of k is fixed on the
range of 1.4-1.6 that will ensure the better performance of the adaptive filter. This method
keeps a fast update rate at the convergence stage and has a low computation complex at the
stable stage. The results of echo cancellation indicate that the new method has the fast
convergence speed, the low computation complexity, and the same minimum error as the
traditional method. However, the parameters of the algorithm should be taken into account
strictly depend on the application environment, which implies that a complex tuning activity
might be required to find the optimal settings.
6. REFERENCES
[1] Y. Bendel, D. Burshtein, O. Shalvi, “Delay less Frequency Domain Acoustic Echo
Cancellation”, IEEE Transactions on Speech and Audio Processing, 2001, vol. 9, no. 5, pp.
589-597.
[2] T. J. Shan, T. Kailath, “Adaptive Algorithm with An Automatic Gain Control Feature”,
IEEE Transactions on Circuits and Systems, 1988, vol.35, no.1, pp. 122-127.
[3] S. C. Chan, Y. Zhou, “Improved Generalized proportionate Step-size LMS Algorithms
and Performance Analysis”, In: Proceedings of International Symposium on Circuits and
Systems, 2006, pp. 2325-2328.
Number of
computations
Convergence time
LMS
algorithm
64,821
0.05
(approximately)
Low
complexity
algorithm
29,110 0.05

69
[4] J. J. Shynk, “Frequency-domain and Multirate Adaptive Filtering”, IEEE Signal
Processing Mag., 1992, vol. 9, pp.14-37.
[5] G. Clark, S. Mitra, S. Parker, “Block Implementation of Adaptive Digital Filters”, IEEE
Transactions on Circuits and Systems, 1981, vol. CAS-28, pp. 584-592.
[6] G. A. Clark, S. R. Parker, S. K. Mitra, “A Unified Approach to Time-and frequence-
domain Realization of FIR Adaptive Digital Filters”, IEEE Transactions on Speech,
Audio and Signal Processing, 1983, vol. ASSP-31, pp. 1073-1083.
[7] B. Widrow, Adaptive Filters, In Aspects of Network and System Theory. New York:
Hoit, Rinehart and Winson, 1970.
[8] M. Nekuii, M. Atarodi, “A Fast Converging Algorithm for Network Echo Cancellation”,
Signal Processing Letters, 2004, vol. 11, no. 4, pp. 427- 430.
[9] S. Ohta, Y. Kajikawa, Y. Nomura, “Acoustic Echo Cancellation Using Sub-adaptive
Filter”, International Conference on Acoustics, Speech and Signal Processing,
2007, vol. 1, pp. 85-88.
[10] Yao Tian-Ren, Sun Hong. Advanced Digital Signal Processing. Wuhan: HUST Press,
1999.
[11] Prabira Kumar Sethy and Professor Dr. Subrata Bhattacharya, “Noise Cancellation in
Adaptive Filtering Through RLS Algorithm using TMS320C6713DSK”, International
Journal of Electronics and Communication Engineering &Technology (IJECET), Volume 3,
Issue 1, 2012, pp. 154 - 159, ISSN Print: 0976- 6464, ISSN Online: 0976 –6472.
[12] Er. Ravi Garg and Er. Abhijeet Kumar, “Comparasion of SNR and MSE for Various
Noises using Bayesian Framework”, International Journal of Electronics and Communication
Engineering & Technology (IJECET), Volume 3, Issue 1, 2012, pp. 76 - 82, ISSN Print:
0976- 6464, ISSN Online: 0976 –6472.

Low complexity algorithm for updating the coefficients of adaptive 2

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