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. The Z-transform
Consider the continuous time signal x(t); sample it with the rate 1/T
the discrete-time signal
(time-series)
Take Laplace transform
xn is not a function of t, so it can be considered as a constant.
Set ; note that s and z are complex
The Z-transform
The inverse Z-transform

3. The Z-transform
The Z-transform for discrete-time linear systems plays same role as the Laplace transform for continuous-time linear systems.
A general form of a discrete-time linear system corresponds to the linear difference equation
It is similar to ODEs, but derivatives are replaced by finite-differences
Derivatives are replaced by the time-shift operator z-1 (in Z-space)
ai and bi are coefficients
xn is the input
yn is the output

4. The Z-transform. Properties
Linearity
Shift to the right (delay)
x[n]=xn
x[n-3]=xn-3

5. The Z-transform. Properties
Shift to the left
x[n]=xn
x[n+3]=xn+3
Remind the Laplace transform of a derivative

6. The Z-transform. Properties
Convolution in the discrete-time domain
Summation in the Discrete−Time Domain (similar to integration)
Initial value theorem
Final value theorem

7. The Z-transform of common discrete-time function.
Delta-function
Step-function (it specifies the causality)
Here we use
Note that the transform exists for |z|<1 only

8. The Z-transform of common discrete-time function.
Geometric sequence (assume the action of u0)
we use
Discrete-time exponent (assume the action of u0)
we use

9. The Z-transform of common discrete-time function.
Discrete-time cosine and sine functions (assume the action of u0)
We can use
Consider as exercises
Note z=esT

10. Transformation Between s and z Domains
Discrete-time Laplace transform
T is the sampling period
Z-transform So
Recall that s is complex and it specifies the system properties.
wS is the angular sampling frequency

11. Transformation Between s and z Domains
The polar form relation is
The second term
defines the unity circle;
the unit circle represents frequencies from zero to the sampling frequency and the frequency response is the discrete−time transfer function evaluated on the unit circle.
Cases:

12. Calculation of the Z-transform
The connection the Laplace and Z- transforms via
the contour integral
a contour C enclosing all singularities (poles) of the Laplace transform.
The integral can be expressed via Cauchy’s Residue Theorem
pk is k-th pole of the Laplace transform
Comment. As a rule tables are used.

13. The inverse Z-transform
The inverse transform is
Can be found via
a. Partial Fraction Expansion
b. The Inversion Integral
c. Long Division of polynomials
The partial Fraction Expansion method is very similar to the partial fraction expansion method that we used in finding the Inverse Laplace transform, that is, we expand into a summation of terms whose inverse is known. The terms are
here k is a constant, r and p represent the residues and poles respectively; these can be real or complex.

14. The inverse Z-transform
Example. Partial Fraction Expansion (via residues)
Multiply by
Next form
Calculate residues

15. The inverse Z-transform
Example. Partial Fraction Expansion (via residues)
Multiply by
Next form
Calculate residues
Finally,

16. The inverse Z-transform
Example. Partial Fraction Expansion (without residues)
Multiply by
Next form
Need to solve
Matlab use
syms n z
xn=2*(0.5)^n−9*(0.75)^n+8; % This is the answer
Xz=ztrans(xn,n,z); simple(Xz) % Verify answer by first taking Z transform of x[n]
ans =
8*z^3/(2*z-1)/(4*z-3)/(z-1)
iztrans(Xz) % Now, verify that Inverse of F(z) gives back f[n]
2*(1/2)^n-9*(3/4)^n+8

17. The Z-transform
The Z-transform for discrete-time linear systems plays same role as the Laplace transform for continuous-time linear systems.
A general form of a discrete-time linear system corresponds to the linear difference equation
It is similar to ODEs, but derivatives are replaced by finite-differences
Derivatives are replaced by the time-shift operator z-1 (in Z-space)
ai and bi are coefficients
xn is the input
yn is the output

18. The Transfer Function of Discrete−Time Systems
A general form of a discrete-time linear system corresponds to the linear difference equation
ai and bi are coefficients
xn is the input
yn is the output
Assuming that all initial conditions are zero, taking the Z transform of both sides
Express Y(z) as a function of X(z)
The transfer function H(z)

19. The Transfer Function of Discrete−Time Systems
Block diagrams.

20. Response to unit pulse (impulse) d [n]
Consider the input x[n]=d[n]. Since X(z)=1 and the transfer function
The discrete−time impulse response h[n] can be found by taking the inverse Z transform of the discrete transfer function H(z)
Consider the structure of H(z)
Here pi are poles; ri are residues.

21. Response to unit pulse (impulse) d [n]
Consider the structure of H(z)
Here pi are poles; ri are residues .
Using the Z-transform of the function an
Write down the inverse Z-transform, that is
the discrete−time impulse response h[n]
The unit circle is the boundary between impulse responses that decay with time (when all poles |pi|<1) and those that increase in an unbounded fashion (if at least one |pi|>1).
The stable system must have its poles inside (or possibly on) the unit circle.

22. Response to unit pulse (impulse) d [n] versus poles

23. Response to unit pulse (impulse) d [n] versus poles

24. complex-valued exponential signal
Frequency response
complex-valued exponential signal
LTI system
LTI system output
impulse response
Use the convolution theorem
Frequency response:
By replacing assume the sampling rate is 1, T=1

25. LTI system output:
Frequency response:

26. LTI system Amplitude (magnitude) response: Phase response:
Sinusoid signal
LTI system output

27. constant coefficient linear difference equation
Time – Domain:
constant coefficient linear difference equation
h(n)
Z – Domain:
transfer function
Frequency – Domain:
frequency response Z
Z-1
FT-1 FT

28. Example The difference equation describing the input−output relationship of a discrete−time system with zero initial conditions, is
Compute:
The transfer function H(z)
b. The frequency response H(jw)
c. Find the poles and zeros and plot them in the complex Z-plane. Comment on the stability of the system.

29. Compute:
The transfer function H(z)
Take the Z-transform

30. Compute:
b. The frequency response H(jw)
Substitute
z=e jw
Long calculations (!)
So we skip and consider a simpler example

31. c. Find the poles and zeros and plot them in the complex Z-plane
c. Find the poles and zeros and plot them in the complex Z-plane. Comment on the stability of the system.
Zeros
Poles
The system is stable.

32. Calculating the frequency response. Simpler example.

33. Example. The impulse response (unit-pulse response) of the system is
Determine its transfer function and frequency response