hi im doing a project regarding development of an adptive digital notch filter for the
removal of 50hz noise from an ecg signal.i have the code.now my problem is when do the full
report for methodology.first I explained about adaptive filter and followed by adaptive notch
filter..then I explained about LMS algorithm seems the sign regressor algorithm comes under
LMS..the thing is im confused with this method like it comes under IIR or FIR..because there
also LMS related with IIR and FIR..from what I studied adaptive notch filter always related to
FIR and also IIR..LMS notch filter code comes under IIR or FIR?could anyone briefly explain to
me?it will make to understand the flow of this code working better..thank you

clear all
load data
Fs = 500;
N = 900;
i = [0 : N-1]';

x=data(1:N)+5*sin(2*pi*50* i/Fs)+10*sin(2*pi*48* i/Fs);
u = sin(2*pi*50* i/Fs);

%adaptive filter architecture
L = 20;
step_size = 0.005;
w = zeros(1,L);

%run the adaptive filter
e(L) = x(L);
for k = L : N
regressor = flipud(u(k-L+ 1:k)); 
w = w + step_size * regressor' * e(k);
e(k+1) = x(k) - w * regressor;
end

%compute the spectrum of the initial signal and the filtered signal
f = [0 : Fs/N : Fs/2 - Fs/N]';
F = abs(fft(x));
E = abs(fft(e));

%plot
figure(1);
subplot(411) ;plot(x); title('initial signal');
subplot(412) ;plot(e(2:N+1)); title('initial signal after filtering');
subplot(413) ;plot(f,F( 1:length( f)));title( 'spectrum of initial signal');
subplot(414) ;plot(f,E( 1:length( f)));title( 'spectrum of initial signal after filtering');
figure(2);
plot(i/Fs,x,'b-',i/Fs,e(2:N+1),'r-')

Hi,

the key to understanding this code, is first understanding the various signals that appear in
it. First of all x is the initial signal containing a component of 50Hz which you want to
remove by adaptive filtering. This signal will play the role of the desired signal. Since we
want to remove the frequency component of 50Hz, the reference signal u must be a sinusoid of
50Hz. Now, the LMS adaptive filter will try to use the reference signal u and from it
predict/estimate the desired signal x. So, the output of the adaptive filter, namely u*w, will
be as close as possible to the desired signal x. But since u contains only one frequency, then
the output of the adaptive filter will be an estimation of the corresponding frequency
component in the desired signal, if such one exists. Consequently u*w will be an accurate
approximation of the 50Hz component in x. Now, the error signal of the adaptive filter is
simply e=x-u*w, but based on the above analysis e will be the initial signal with the 50Hz
component removed! Now this process is an IIR process. One way to think of that is that the
input to the overall system is x and the output is e. Hence the transfer function E(z)/D(z)
should indicate whether the filtering is IIR or FIR. If you find this transfer function
("Fundamentals of Adaptive Filtering",Ali Sayed, problem  5.14/page 261), then this
is a transfer function of an IIR notch filter! The derivation of the transfer function is
somehow elaborate but you can try it anyway.

Manolis C. Tsakiris

  
Dear
please refer the book " digital signal processing" using matlab by vinay k.kale and
john g. proakis {BookWare Companian Series} 
-- shankar