476 293

Noise canceling in audio signal with adaptive filter
John Christakis
Logothetis Antonis
Mpalolakis Antonis
Boulgaris Konstantinos
Gogornas Ioannis
Theodore P. Chinis
Electronic Engineering
Technological Educational Institute of Athens
Agiou Spyridonos
12210
Egaleo
GREECE
Abstract: - In this paper, an adaptive noise canceller will be presented and some useful
observations will be done with the canceling process over the audio signal
1 Introduction
All the physical systems when they operate, they produce physical noise, which is a mixing of
an infinitive number of sinusoidal harmonics with different frequencies. So, the initial signal
information is corrupted with this noise. This complex signal, may be very noisy, so much that
the human ear or other system which may follow it, cannot receive correct the initial signal.
So, an algorithm has to be invented which must be able to separate the physical noise from the
information signal and to output the information signal without noise. This is not possible, as
there is not a perfect system. So, this algorithm should have the ability to reduce the noise level
as much as it can.
A good noise-canceling algorithm is the algorithm, which is presented in this paper. This
algorithm is based on the Least Mean Square algorithm. This system, operates like an automatic
control system with feedback where the feedback performs an adaptation to the filter
coefficients, otherwise it adjust the filtering of the noise in each sample.
2 Problem formulation
A simple audio signal is represented by the following mathematical equation
( ) )*/**2sin()(1 nfsfopinx =
Where 0f is the frequency of the audio signal, fs is the sampling frequency and n
is the discrete time base.

Fig.1 The simple audio signal without noise
When white noise is added to this signal then it becomes
( ) ( ) ( ) ( )nnoisenfsfopinc += */**2sin2
So, the problem is to extract the useful signal from this complex signal.
Fig.2 The audio signal with noise
A simple adaptive filter noise canceller is presented in the following diagram.
Audio signal with noise c(n)- uk Output - yk
Noise(n) - d(n)
error(n)- ek
Fig.3 Adaptive filter noise canceller
2
Adaptive Filter
Wk+1
+

2.1 Brief adaptive filter theory
One of the most successful adaptive algorithms is the LMS algorithm developed by Widrow and
his coworkers. Instead of computing optW which is the optimal solution of the wiener filter, the
LMS coefficients are adjusted from sample to sample in such a way as to minimize the MSE
(mean square error).
The LMS is based on the steepest descent algorithm where the weight vector is updated from
sample to sample as in the next equation.
( ) kkk WW ∇−=+ µ13
Where Wk and k∇ are the weight or the impulse response w (n) of the filter and the true gradient
vectors respectively, at the thk sampling instant. µ Controls the stability and rate of
convergence.
The steepest descent algorithm in the previous equation still requires knowledge of R and P ,
since k∇ is obtained by the next equation.
( ) 0224 =+−==∇ RWP
dW
dJ
k
The LMS algorithm is a practical method of obtaining estimations of the filter weights Wk in real
time without the matrix inversion of the next equation PRWopt
1−
= or the direct computation
of the autocorrelation and cross correlation. The Widrow – Hopf LMS algorithm for updating the
weights from sample to sample is given by
( ) kkkk XeWW µ25 1 +=+
Where: k
T
kkk XWye −=
Clearly, the LMS algorithm above does not require prior knowledge of the signal statistics (that is
the correlations R and P ), but instead uses their instantaneous estimations. The weights
obtained by the LMS algorithm are only estimations, but these estimations improve gradually
with time as the weights are adjusted and the filter learns the characteristics of the signal.
Eventually, the weights converge. The condition of convergence is:
( )
max
106
λ
µ ><
Where maxλ is the maximum eigenvalue of the input data covariance matrix In practice, kW
never reaches the theoretical optimum (the Wiener solution), as the algorithm iteration step can
never be zero but it has a discrete value.
Furthermore, the LMS algorithm can be summarized in the next equation for the implementation
of the filter.
( ) ( ) ( ) ( )∑
+
=
−=
1
0
*7
N
i
i ikukwky
( ) )()()()1(8 κκµκκ euhh +=+
In which: ( )kyde −= )()( κκ
In the above equation, T denotes the transpose of the vector invh . The desired response )(κd
can be found from the adaptive filter scheme to be
3

( ) )()()(9 gwhd T
x −= κκκ
The notation )(κu denotes a vector of samples arranged as
( ) [ ]T
Nuuuku )1(...............)1()()(10 +−−= κκκ
Where N is the order of the filter.
3 Problem solution
The problem solution may be based on standard adaptive techniques and especially on the
adaptive filter noise canceller.
This filter gets the physical noise and filters it in such a way that the algebraic addition with the
complex signal produces a new signal with a very low level of noise. The schematic diagram of
the adaptive filter noise canceller was given above and it is presented again in the following
diagram.
Audio signal with noise x(n) Output – y(n)
Noise d(n)
error(n)
Fig.4 The used adaptive noise canceller
As it can be observed, the adaptive filter noise canceller is a system with feedback as a system
with automatic control. For this reason, there is stability problem. The problem arises when the
error signal’s amplitude increases in accordance with time or when no convergence is obtained.
The mathematical algorithm, which is used by the adaptive filter for the filter’s coefficients
update, is called LMS (Least mean square) algorithm. This algorithm tries to minimize the
square of the error signal so; the audio signal has been reconstructed quite well.
The system output is the error signal. The error signal is the algebraic subtraction between the
audio signal with noise and the noise. When the LMS algorithm performs a good convergence
then the only error signal is the audio signal. So, the error signal must be the noise canceller
output.
The LMS algorithm has also an initial step variable μ, which is the step of the convergence in
each iteration. In the matlab code where the adaptive filter noise canceller is built, the NLMS
algorithm used. The only difference is that the step variable μ, changes in each iteration with the
next equation:
( )
( ) 2
)(11
ku+
=
α
β
κµ
where β is a positive constant, usually less than 2 , and α is a small positive constant. The
symbol
2
)(κu is the energy of the signal in a predefined buffer.
4
Adaptive Filter
Wk+1
+

3.1 Implementation of the adaptive filter noise canceller in matlab and waveforms
The adaptive noise canceller was implemented in matlab as it offers the opportunity for a quick
and very professional way to study the whole operation of the system.
The matlab code, which was written, is presented now.
function [x] = noice_canceller
clear all;
%**********************
%**** Time Base *******
%**********************
N = 2000;
n = 1:1:N;
%***********************
%**** System Vars
%***********************
fo = 500;
Fs = 44100;
db = 5;
%***********************
%***** Signal Generation
%***********************
audio = sin(2*pi*fo/Fs*n); %+ 0.7*sin(2*pi*5*fo/Fs*n));
%**********************
%**** Noise Generation
%*********************
%noise_plus_audio = awgn(audio,db);
noise = wgn(1,N,db)/2;
%noise = 0.1*randn(1,N);
for i=1:1:N
noise_plus_audio(i) = (audio(i) + noise(i)/2);
end
%**************************************
%**** Initialization of Noise Canceller
%**************************************
w0 = zeros(1,32); % Initial filter coefficients
%wi = zeros(1,255);
mu = 0.4; % LMS step size
S = initnlms(w0,mu,[],1);
%***************************************
%***** Apply Noise Canceller over signal
%***************************************
for i=1:1:N
[y,canceller_output(i),S] = adaptnlms(noise(i),noise_plus_audio(i),S);
%***************************************
%***** Calculate FFT of Signals
%***************************************
ff_mix = abs(fft(noise_plus_audio,1024));
ff_output = abs(fft(canceller_output,1024));
%***************************************
%***** Calculate Error
%***************************************
error(i) = audio(i) - canceller_output(i);
%***********************
%*** Plots *************
%***********************
subplot(6,1,1);
plot(noise_plus_audio,'r');
axis([0 N -2 2]);
title('Signal and Noise');
subplot(6,1,2);
plot(audio,'g');
5

axis([0 N -2 2]);
title('Audio Signal');
subplot(6,1,3);
plot(canceller_output,'b');
title('Canceller Output');
axis([0 N -2 2]);
subplot(6,1,4);
plot(ff_mix,'r');
title('FFT of Mixed Signal');
subplot(6,1,5);
plot(ff_output,'r');
title(' FFT of Output');
subplot(6,1,6);
plot(error,'g');
axis([0 N -2 2]);
title('Error of FFT');
drawnow;
end
The output waveforms of the system are plotted in the next figure.
Fig.5 Noise canceller output waveforms
4 Conclusion
To sum up, the adaptive noise canceller is a very efficient and useful system in many
applications with sound, video etc. The only disadvantage is that it needs digital signal processor
DSP for its operation.
References:
1. Simon Haykin, Adaptive Filter Theory, Prentice Hall 2002
6

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