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. Digital Filters design and application.
Case Study.

2. BIOMEDICAL SIGNAL ANALYSIS

3. system H(s) Systems Analysis. Block Diagrams. Transfer function Simple
Input Signal
Output Signal
system
H(s)

4. H1(s) H2(s) Systems Analysis. Block Diagrams. Transfer function Series
System consists of two sub-systems
H1(s)
H2(s)
Sub-system 1
Sub-system 2
Block diagram

5. H1(s) H2(s) Systems Analysis. Block Diagrams. Transfer function
Cascade Connection
System consists of two sub-systems
H1(s) +
Sub-system 1
H2(s) -
Sub-system 2
Block diagram

6. Transfer function. Poles and Zeros
A transfer function H(s) can always be written as a rational function of s, that is as a ratio of two polynomials
Polynomials can be factorized as follows
The are called the zeros of H(s)
The are called the poles of H(s)
Poles define the stability of the system; if Re(pi)<0 LTI is stable

7. system Transfer function. Impulse response
Consider a test signal as the delta-function, d(t). Since
Consequently, if x(t)=d(t) then X(s)=1 and H(s)=Y(s)
The output signal y(t)=h(t) corresponds to the impulse response
Dream: the transfer function H(s) of a system can be obtained by applying an impulse, x(t)=d(t), (whilst the system is in its quiescent state) to the system and measuring its response, y(t).
H(s) is then given by the Laplace transform of y(t).
Input Signal
Output Signal
system

8. Transfer function. Frequency Response
Select the test signal in the form of harmonic functions:
Such signals can be realized experimentally
Substitute s=jw for the transfer function, then the frequency response is
That is, if we know H(s) we can obtain H(w) by replacing s by jw
Note: it is assumed that the system is in a stable steady state.

9. Transfer function. Frequency Response
The frequency response H(jw) is complex function of w
Therefore the polar form is used
is the modulus (gain), the ratio of the amplitudes of the output and the input;
is the phase shift between the output and the input.
Thus, the frequency response is fully specified by the gain and phase over the entire range of frequencies
Both gain and phase are experimentally accessible!

10. Systems Response to a harmonic signal. A cosine input
If a signal Acos(w0t) is applied to a system with transfer function, H(s), the response is still a cosine but with
an amplitude
and phase
Note. We don’t need to use inverse Laplace Transform to estimate the response in time domain.
the system response to

11. Systems Response to a harmonic signal. A cosine input
Example.
Input signal
Transfer function
Output signal y(t) - ?
Rewrite the transfer function as the frequency response in the polar form
Gain
Phase
Calculate the gain and phase for w =2p

12. Fourier Analysis. Spectrum
Most signals encountered in engineering can be represented in both the time and the frequency domains.
These representations are uniquely related. So, alternating the signal in one domain will alter its representation in the other domain as well.
Time domain. The plot of a signal, x(t), as a function of time, t.
Frequency domain. Spectrum. Amplitude, A1, and phase, f1, of harmonic signal(s) of a given frequency, f 1or w1. A
f, (radian) 2p A1 f1
f, (Hz)
f, (Hz)

13. Fourier Analysis. Amplitude response and phase response
Example.
Time domain.
The signal is a sum of a number of harmonic functions.
Frequency domain.
Amplitude spectrum
Phase spectrum
phase, f
amplitude, A
w, (rad./s)
angular frequency, w, (rad./s)

14. Fourier Analysis. Oscillations
Example. Carcadian rhythm
Many species, from bacteria to humans, maintain a daily rhythm of life by way of a circadian clock.
Body temperature ,Tb, and oxygen consumption, V02, from
a chronically instrumented pigeon over a 48-hour period .

15. If the signal is periodic then Fourier series are used.
A periodic signal has the property x(t) = x(t+T),
T is the fundamental period,
f0=1/T or w0 = 2p/T is the fundamental frequency.
Trigonometric form of signal
a0, an, bn are the Fourier coefficients
T is the fundamental period
w0 is the fundamental frequency
For spectrum we need to calculate the coefficients

16. Fourier series of periodic signals
Alternative representation of Fourier series
a0, An, fn are the Fourier coefficients
a0 is the DC component, mean value of the signal
Frequency components occurs at frequencies of nw0 and
they are characterized by amplitudes An and phases fn.
The n=1 term is called the fundamental frequency component and the n=2,3,... components are called 2nd, 3rd,... harmonics respectively. It is an odd harmonic if n is odd and an even harmonic if n is even.
Plots An versus w and fn versus w are amplitude and phase spectra of x(t) respectively.

17. Fourier Series. Parseval’s Theorem
The “power”, P, of a periodic signal, x(t), can be written via Fourier coefficients an and bn,
Time domain
Frequency domain
The power (energy) in the time domain and the frequency domain are equal.
Note, T is the duration of the signal here.

18. Fourier Series
Example. Spectrum of periodic pulse train (the square wave). +A
x(t) t
Fourier series contains infinite number of terms. So the spectrum contains infinite number of peaks. T -A

19. Fourier Series
Example. Spectrum of periodic pulse train (the square wave).
Gibbs Phenomenon.
Approximation of square wave by 125 harmonics

20. Fourier Series. Complex representations
Complex form.
cn are complex coefficients connected with the previously used coefficients as
Parseval’s theorem:
Example. The spectrum of the square wave.

21. Fourier series representation
T is the period of pulses
T1 is the pulse duration
Example. Pulse train.
Fourier series representation
Complex Fourier coefficients
The sinc function

22. Complex Fourier coefficients
T is the period of pulses
T1 is the pulse duration
Example. Pulse train.
Complex Fourier coefficients
The amplitude spectrum
As the period T tends to infinity number of frequency component that occur in the frequency interval [0,w0] tends to infinity too.
|cn|
So, if we consider one pulse (impulse) only, the Fourier series are useless .
|cn|

23. Fourier transform of aperiodic signals.
Fourier transform of x(t) is denoted X(w).
It is a continuous function of the frequency.
The inverse Fourier transform is
Fourier pairs
is the magnitude or amplitude spectrum
is the phase spectrum
Frequency response
Laplace transform
Fourier transform:
Parseval’s theorem:
Energy of an aperiodic signal
Energy (Power) spectrum

24. Fourier transform
Example. Single rectangular pulse
The amplitude spectrum
The phase spectrum

25. Fourier transform
Example. Single exponential pulse
The amplitude spectrum
The phase spectrum

26. Fourier transform
Example. The unit impulse function (delta function) d(t)
White spectrum
The amplitude spectrum
The phase spectrum
Example. Harmonic functions

27. Fourier transform
Example. Harmonic functions

28. Fourier transform of periodic and aperiodic signals
Spectrum is a continuous function of the frequency
Periodic signals
Spectrum of the periodic signal
The period of the signal
Spectrum is discrete, specified in (in)finite number of points only.
A general signal can have both aperiodic and periodic components, consequently the spectrum will have both continuous and discrete components:

29. Fourier transform. Effects of a finite-duration of signal. Edge effect
Consider a harmonic signal, y(t), of a finite duration, T.
The product (multiplication) in the time domain corresponds to the convolution in the frequency domain and vice versa.
The discrete spectrum is transformed to a continuous one