Multiple Sub-Filter using Variable Step Size and Partial ... - IEEE Xplore

2012 IEEE 8th International Colloquium on Signal Processing and its Applications

Multiple Sub-Filter using Variable Step Size and Partial Update for Acoustic Echo Cancellation Manoj Sharma#, Ravinder Nath Electrical Engineering Department National Institute of Technology Hamirpur-177005 (HP), India [email protected], [email protected] Abstract—In this paper a M-max partial update and variable step size normalized least mean square (M-max VSS NLMS) algorithm for multiple sub-filter (MSF) based Acoustic echo cancellation has been studied. Here computational complexity is controlled by M-max partial update approach where fast convergence is achieved by variable step size NLMS algorithm with MSF parallel structure. Further the condition for convergence analysis has been derived for MSF using M-max VSS NLMS algorithm. Simulation results shows that M-max partial update coefficients with VSS NLMS algorithm for MSF has fast convergence as compared to the single long filter (SLF) and all other combinations. As expected the convergence rate of all the algorithms is fast when the input is a sample function of white random process as compared to that of correlated signal.

time varying which leads to further degradation in performance of the algorithm. There are number of methods available in literature to alleviate the slow convergence limitation [5], [6]. One such method is variable step size approach as suggested in [5]. Another way to mitigate the slowly convergent adaptive filter problem in time domain is to use decomposition to get multiple sub-filter (MSF) parallel structure [6] instead of using single long filter (SLF) [6]. The long length impulse response of acoustic echo channel is realized by lower order parallel sub-filters. The decomposition is done by partitioning the SLF into smaller MSF. The idea of decomposing the input signal vector and the weight vector into sub vectors was presented in [6]. The parallel structure distributes the load of adjusting a long adaptive filter by one adaptive algorithm into lower order MSF updated individually by LMS adaptive algorithm [3]. For MSF structure different adaptive algorithms are constructed depending upon the how the error signal is generated. The error signal used for weight updation of adaptive filter can be obtained at each stage of the sub-filter or it can be a common error obtained at the last stage named different error and common error respectfully. In different error algorithm convergence improves but the steady state error also increases as the number of sub-filter increases. But the common error adaptation algorithm for MSF can be able to overcome the high steady state error problem with little sacrifice in convergence speed due to coupling of each weight update equation. Computational complexity can be reduced by introducing partial updation of filter coefficients. It is also seen that low complexity with fast convergence can be achieved by partial updation with variable step size (VSS) [7].

Keywords-acoustic echo cancellation; variable step size; adaptive algorithm; partial update adaptive filtering; multiple subfilter.

I.

INTRODUCTION

In hands-free applications, the loudspeaker emitted signal in a loudspeaker-enclosure-microphone (LEM) system reaches the microphone not only directly but also by reflections from neighboring objects. The signal received at the microphone is therefore, a superposition of the delayed, attenuated, and filtered versions of the emitted signal. Thus, the received signal contains a direct path and extra paths resulting acoustic echo. However echo is not noticed if the delay associated with the acoustic feedback loops is small as in the LEM [1], [2]. The echo problem is severe in the hands-free telephony and teleconferencing etc. when communicating over long distances and using digital wireless channel where long processing time is needed for signal compression, channel encoding etc. [3]. An acoustic echo canceller [4] can be used to tackle this problem as it generates an estimate of echo signal which is received at the microphone and subtracts from it. Acoustic echo cancellers are usually realized by adaptive FIR filters having thousands of coefficients and least mean square (LMS) as an adaptive algorithm [3]. But LMS algorithm converges poorly when the length of adaptive filter is large and the input signal has large eigen value spread. It has been found that generally the adaptive LMS algorithm with lower filter order has faster convergence speed. Also, the acoustic channel is

This convergence and computational complexity problem further can be reduced by proposing new algorithm containing both MSF structure and partial updation with variable step size. As M-max partial update is well known method for reducing computational complexity. M-max partial update is an example of data dependent approach. Data dependent approaches results in sluggish convergence when the input is a sample function of white random process, but works well for highly correlated. So we studied a low complexity M-max

# Manoj Sharma is pursuing M.Tech in department of electrical engineering, a National institute of technology, Hamirpur-177005, India

978-1-4673-0961-5/12/$31.00 ©2012 IEEE

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2012 IEEE 8th International Colloquium on Signal Processing and its Applications

partial filter coefficients update variable step size normalized least mean square (M-max VSS NLMS) algorithm for MSF structure to achieve fast convergence. The presented simulation results shows that the proposed MSF structure for M-max VSS NLMS algorithm achieves higher rate of convergence compared to normalized least mean square (NLMS) [8], variable step size normalized least mean square (VSS NLMS) [9], and M-max variable step size normalized least mean square (M-max VSS NLMS) [7] with single long filter (SLF) for both uncorrelated and correlated inputs. II.

sub

,

(1)

Then

1 0

is the

1 ,………

0, 1, 2, … ,

0, 1, 2, … ,

(7)

as

;

norm and

0, 1, 2, … ,

1

1

1

(3)

(9)

1

(10)

Taking the squared norm and expectation on both sides of (10), we obtain 1

1

2

Whereas for common error MSF structure, the error signal is given by,

(11)

We can write (11) in the following form

(6)

and is the AWGN and Here, the impulse response of acoustic channel.

(8)

In terms of weight error vector, we write (9) as,

(4) (5)

∑

is the

So the weight updation equation for MSF using M-max NLMS different error algorithm can be written as,

The error signal for different error MSF structure is given by;

1, 2, … ,

1

|

Using the above terms we define the sub-selected tap input vector for each sub-filter,

We initialize each sub filter coefficient vector as, ;

|

is defined as squared Where . regularization parameter.

0, 1, 2, … , 1 and 0, 1, 2, … … …

1

is

∆

(12)

Here ∆ 2

In the M-max NLMS algorithm [9], only those taps corresponding to the M largest magnitude tap inputs are selected for updating at each iteration with 1 .

Here

|

sub-filter input vector

0

1 is given

1

(2)

So, let us define as,

|

Now defining matrix

THE M-MAX VSS NLMS ADAPTIVE ALGORITHM FOR MULTIPLE SUB-FILTER STRUCTURE

Let is the input vector and it is partitioned into vectors each of length .

0, 1, 2, … ,

Where element by

If ∆ is maximized, then the mean square deviation (MSD) iteration to will undergo the largest decrease from iteration. 1

tap selection matrix for each sub filter

The optimum step size will be found by setting the derivative of ∆ with respect to and setting to zero, hence we obtain

is defined as; ,…,

(13)

,

0, 1, 2, … ,

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2012 IEEE 8th International Colloquium on Signal Processing and its Applications

As the error is controlled by both active filter coefficients (which are updated) and inactive filter coefficients (which remains un-updated) for each sub-filter such that, total error becomes

We can approximate to . So we can write (5) as

for

(21) Here, is a positive constant that can be approximated from (18). From (18), we can notice that is inversely proportional to SNR. While calculating the optimum step size from (21), and are not available, as the problem is that and consequently is unknown therefore we need to estimate these quantities.

large enough close (14)

Taking expectation on both sides of (19) and (20),

as process and Assuming the noise sequence statistically independent of the regressor data and neglecting on the past noise samples, we the dependency of simplify the (13); So the numerator part of (13) is given as.

(22) (23) Using (14) in (22) and (23), we get; (24) (25) We are now in the position to estimate these quantities with recursions,

(15)

̂ ̂

And the denominator part of (13) is given by, Here

and

1 1

1 1

are smoothing factors (0

(26) (27) ,

1).

Finally, substituting (26) and (27) in (21), the filter update equation for M-max VSS NLMS different error algorithm becomes.

(16)

1

Substituting (15) and (16) in (13) then the optimum step size is given as,

(28) Here, ̂

(17)

(29)

(18)

and The step size changes with the ̂ quantities and the constant can be approximated from is selected according to equation (18). For stability stability condition

0, 1, 2, … , Here,

Further to simplify our formulation for as, and

, we define

0

2

Here, (19) (20)

by e(n) in the expressions of MFurther, by replacing max VSS NLMS different error algorithm, we can get M-max VSS NLMS common error algorithm as presented in Table 1.

Substituting (19) and (20) in (17), we get

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2012 IEEE 8th International Colloquium on Signal Processing and its Applications

III. SIMULATION RESULTS With the help of simulation we demonstrated the performance of M-max VSS NLMS different error and common error algorithm for both uncorrelated and highly correlated input. We measured and obtained an impulse response, using an ordinary PC loudspeaker and an unidynamic microphone kept at a distance of 1.0 in a room of dimension10 10 8 , which has been sampled at 8 and shown in Fig. 1. The obtained response is in agreement to what has been reported in the existing literature The correlated input signal has been obtained by AR(1) process as 1 . Where 0.9 and is zero mean white Gaussian noise sequence and for uncorrelated input signal is set to zero.

Table 1 M-MAX VSS NLMS ALGORITHM FOR MSF 1.

Data 0.01, 0.001 0 , 1, 0 1 =acoustic channel output =white Gaussian noise = length of each sub filter N=number of sub-filters , 1 , … … … ; input vector ; sub-filter input vector =step size for sub filter 0, 1, 2, … , 1 2.

Initialize 0 0

0, 1, 2, … ,

1

3. Compute for , , ,……… ; Sub-Selected tap input vector 1 Fig. 1 Impulse response of a typical enclosure

; Tap selection matrix ,…, 1 0

|

|

0, 1, 2, … ,

|

The study has been made at signal to noise ratio (SNR) 0.99, 0.01 and = 30 and for 0.99, 0.1 0, 1, 2, … , 1. The simulation results shown are obtained by ensemble averaging over 200 independent trials and the length of the impulse response is 1000. While for NLMS algorithm has been chosen as the maximum value which lead to stability. From Fig. 2 and Fig. 3 It can be observed from the results, for both the cases, that MSF parallel structures for M-max VSS NLMS algorithm has faster convergence with less computational complexity than SLF structure for M-max VSS NLMS algorithm.

|

1

(a). M-max VSS NLMS Different Error Algorithm

̂ ̂

1 1 ̂

1 1

0

NLMS

-5

M-max VSS NLMS, M=3L/5 M-max VSS NLMS, M=4L/5 VSS NLMS M-max VSS NLMS Different Error, M=4L/5

1 -10

M SE [dB ]

(b). M-max VSS NLMS Common Error Algorithm

-15

M-max VSS NLMS Common Error, M=4L/5

-20

̂ ̂

1 1 ̂

1 1

-25 -30 -35

1

0

0.5

1

1.5

Number of iterations, k

2

2.5

3

Fig. 2 Comparison between NLMS, M-max VSS NLMS and MSF structure of M-max VSS NLMS algorithms for uncorrelated input

264

4

x 10

2012 IEEE 8th International Colloquium on Signal Processing and its Applications

REFERENCES

0

[1]

NLMS VSS NLMS

-5

M-max VSS NLMS, M=3L/5

[2]

M-max VSS NLMS, M=4L/5 -10

M-MAX VSS NLMS Different Error, M=4L/5 M-MAX VSS NLMS Common Error, M=4L/5

-15

M S E [d B ]

[3]

-20

-25

[4]

-30

-35

0

0.5

1

1.5

2

2.5

Number of iterations, k

[5]

3 4

x 10

Fig. 3 Comparison between NLMS, M-max VSS NLMS and MSF structure of M-max VSS NLMS algorithms for correlated input

IV.

[6]

CONCLUSION

[7]

In this paper we have presented the convergence analysis for a partial update variable step size multiple sub filter based acoustic echo cancellation. From the simulation results it has been inferred that M-max VSS NLMS common error algorithm achieves fast convergence rate with less computational complexity. Though M-max VSS NLMS different error algorithm provides faster convergence, it has poor misadjustment which is not desirable.

[8]

[9]

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