1. ES97H Biomedical Signal Processing
Lecturer: Dr Igor Khovanov
Office: D207
Syllabus:
Biomedical Signal Processing. Examples of signals.
Linear System Analysis. Laplace Transform. Transfer Function.  
Frequency Response. Fourier Transform.
Discrete Signal Analysis. Digital (discrete-time) systems. Z-transform.
Filtering. Analogue Filters design and application.
Case Study.

2. Envelope Extraction
Problem: Estimate EMG activity to airflow during inspiration.
The intricate variations in the EMG signal in the signal are often not of interest. A measure of the total or integrated electrical activity, ideally reflecting the global activity in the pool of active motor units of the muscle, would serve the purposes of most analyses. The smoothed envelope of the EMG signal is commonly used under these circumstances.

3. Envelope Extraction
Approach 1. The filters commonly used for calculation of the envelope.
Consider the digital signal xi
Step 1. Calculation of the rectified signal |xi|
Step 2. Selection of an appropriate low-pass filter to severely attenuate high frequencies, and filter the rectified signal

4. Envelope Extraction via low-pass filtering
Matlab Script: Plot the rectified signal |xi|
close all;clear all;
% load the signal
fs=10000; % Sampling frequency;...
Ts=1/fs; % Sampling period
fn=fs/2; % Nyquist frequency
dinput=load('emg2.dat');
signal=abs(dinput(:,1));
Ns=length(signal);
time=[0:1:(Ns-1)]*Ts;
figure(1)
% plot the signal
hold on % to draw next plot in the same figure
plot(time,signal,'-b','linewidth',1)
grid on;
box on;
hold on;
xlabel(' \it t, [sec] ','fontname','times','fontsize',24);
ylabel('\it EMG, [\mu V] ','fontname','times','fontsize',24);

5. Envelope Extraction via low-pass filtering
Matlab Script: Plot the rectified signal |xi|

6. Envelope Extraction via low-pass filtering
Matlab Script: Estimate the spectrum of the rectified signal |xi|
% Calculate FFT
xfft=fft(signal); % complex coefficient of Fourier transform
p=abs(xfft(1:Ns/2))/Ns*2; % amplitude for positive frequencies only
freq=[0:Ns/2-1]/(Ns/2)*fn; % frequencies, for which FFT was applied
%%%%% Plot the amplitude spectrum
f2=figure(2);
movegui(f2, 'northwest');
%semilogy(freq,p,'-m')
loglog(freq,p,'-b')
hold on
grid on
box on
xlabel ('frequency, f, Hz'); % x label
ylabel ('Amplitude spectrum'); % y label

7. Envelope Extraction via low-pass filtering
Matlab Script: Estimate the spectrum of the rectified signal |xi|

8. Envelope Extraction via low-pass filtering
Matlab Script: Design the filter (Butterworth)
% Design filter
wcutoff=5;
wn=wcutoff/(fs/2);
filtorder=5;
[b1,a1]=butter(filtorder,wn);
%[b1,a1]=ellip(filtorder,3,60,wn);
% plot the frequency response of the filter (on the same plot as signal sppectrum)
[G1z,wT]=freqz(b1,a1,16384);
freqf=wT/(2*pi*Ts);
loglog(freqf,abs(G1z)*pmax,'--r');

9. Envelope Extraction via low-pass filtering
Matlab Script: Design the filter (Butterworth)

10. Envelope Extraction via low-pass filtering
Matlab Script: Apply the filter
% perform filtering by FILTFILT
s2=filtfilt(b1,a1,signal);
%%%% Calculate and plot FFT
xfft=fft(s2); % complex coefficient of Fourier transform
p=abs(xfft(1:Ns/2))/Ns*2; % amplitude for positive frequencies only
loglog(freq,p,'-k')
figure(1)
plot(time,s2,'-k','linewidth',2)

11. Envelope Extraction via low-pass filtering
Matlab Script: Apply the filter (Butterworth)

12. Envelope Extraction via low-pass filtering
Matlab Script: Apply the filter (Elliptic)

13. Envelope Extraction or Matlab script Matlab script
Approach 2. Moving average technique can be used for calculation of the envelope.
Consider the digital signal xi
Step 1. Calculation of the rectified signal |xi|
Step 2. Selection of an appropriate window length, M, and application of the window or
Matlab script
% Moving Average Window with the centre
% at a given time moment
for k=1:Ns, % consider each time moment
k1=k-iwin;
if k1<1, k1=1; end
k2=k+iwin;
if k2>Ns,k2=Ns;end
xx=signal(k1:k2);
mx1(k)=mean(xx);
end
plot(time,mx1,'-k','linewidth',2);
Matlab script
% Moving Average Window
for k=1:Ns, % consider each time moment,
% form a window and average
k2=k+iwin;
if k2>Ns,
k2=Ns;
end
xx=signal(k:k2);
mx(k)=mean(xx);
plot(time,mx,'-r','linewidth',2);

14. Envelope Extraction or
Approach 2. Moving average technique can be used for calculation of the envelope.
Consider the digital signal xi
Step 1. Calculation of the rectified signal |xi|
Step 2. Selection of an appropriate window length, M, and application of the window or

15. In Matlab there is a function y=hilbert(x) xH=Imag(y);
Envelope Extraction
Approach 3. The envelogram via Hilbert transform
Consider the digital signal xi
Step 1. Calculate Hilbert transform xH (below we use notations for continuous signals)
To form an analytical signal
Then the signal x(t) can be presented as
Where the amplitude and phase are coming from the analytical signal
Step 2. Calculate the amplitude of the analytical signal a(t) (i.e. |y(t)|), which is the estimated envelope.
In Matlab there is a function y=hilbert(x)
xH=Imag(y);

16. Examples of Hilbert transform
Amplitude modulation.

17. Examples of Hilbert transform
Frequency (phase) modulation.

18. Envelope Extraction via Hilbert Transform
Matlab script: …
%Approach Envelope extraction by Hilbert transform
asignal=hilbert(signal);
signalH=imag(asignal);
ampHilbert = sqrt((signal.^2)+(signalH.^2));
plot(time,ampHilbert,'-k')
title(' Envelope extraction via Hilbert transform','fontname','Times','fontsize',12);