1. System Function of discrete-time systems
System representation
Where the input signal is {x(n)} of z-transform X(z)
The output is {y(n)} of z-transform Y(z).
{h(n)}
G(z)
{x(n)}
{y(n)}
X(z)
Y(z) 1

2. System Function of discrete-time systems
The system has an impulse response h(n)
Define so that
and hence
Clearly when X(z) = 1 then G(z) = Y(z) i.e. G(z) is the z-transform of the impulse response h(n) 2

3. Frequency Response
Let the input be
Then the output is
Where
and 3

4. System Function of discrete-time systems
However
While the amplitude & phase responses are
And hence 4

5. System Functions-Amplitude response
Evidently
And hence
Thus
ie for real systems amplitude is an even function, and phase an odd function of frequency 5

6. System Functions-Amplitude response
Moreover from
Since at is finite we obtain 6

7. System Functions-Phase response
From
At we have
Thus for real systems the amplitude response must approach zero frequency with zero slope, while the phase rsponse must be zero at the origin 7

8. System Functions-Phase response
For ,
Hence 8

9. System Functions-Group delay
Thus
where 9

10. Suppression of a frequency band
A real rational transfer function H(z) cannot suppress a band of frequencies completely.
i.e cannot be identically zero for in
This may be demonstrated as follows 10

11. System Function of discrete-time systems
To produce a zero at say we must have in the numerator of H(z) a factor of the form
Therefore for one zero in the band the factor is
and since there are an infinite number of points in the band we need factors in the numerator as 11

12. System Function of discrete-time systems
Clearly the result is not a rational function
Hence it cannot be the transfer function of a digital signal processing system. 12

13. Stability Test
For stability a DSP transfer function must have poles inside the unit circle on the z-plane.
We need to have a means of determining whether the denominator of a given transfer function has all its zeros inside the unit circle.
The procedures for doing so are called stability tests. 13

14. Stability Test Let the transfer function to be tested be
where n is the order of the transfer function. Set A = 1.
For stability Dn(z) must have no zeros in the region 14

15. Stability Test Consider the simple case of a quadratic denominator
Rewrite as (ignore the factor )
If the roots are complex, say
then 15

16. Stability Test Thus and For stability and thus For real roots
If choose root with largest absolute value and make less than 1 16

17. Stability Test Thus And since quantities are positive we obtain
Similarly for
Thus jointly we have 17

18. Stability Test These conditions form the Stability Triangle
Stability region inside triangle 18

19. Stability Test
For higher order functions most tests rely on an iterative precedure that involves
reduction of the polynomial degree by unity
a simple test
Jury-Marden Test: We write Dn(z) as
where is a constant chosen to make of degree (n - 1) 19

20. Stability Test Repeat equation Hence And thus Set so that
is of degree (n-1) when 20

21. Stability Test
Rouche’s Theorem: If the polynomials and are such that in the same region
then has the same number of zeros in that region as 21

22. Stability Test
we observe that Dn(z) has as many zeros as either or depending on whether or
Ie or 22

23. Stability Test
Thus if then is unstable as it has as many zeros as which has at most (n - 1) zeros within |z| < 1.
If then can have as many zeros within |z| < 1 as
The zero at z = 0 can be removed and the procedure repeated for the remaining polynomial 23

24. Stability Test An alternative test: Consider So that
For this equation to be a polynomial we require the constant term in the numerator to be zero so as to be able to cancel through a factor z 24

25. Stability Test
Thus or
The rest of the argument is similar to the previous case 25

26. Further Stability Test
Given that and
show that on the unit circle for any real
Construct
Repeat the previous arguments 26

27. Digital Two-Pairs
The LTI discrete-time systems considered so far are single-input, single-output
Often such systems can be efficiently realised by interconnecting two-input, two-output structures, known as two-pairs 27

28. Digital Two-Pairs
Figures below show two commonly used block diagram representations of a two-pair
Here and denote the two outputs, and and denote the two inputs, where the dependencies on the variable z has been omitted for simplicity 28

29. Digital Two-Pairs
The input-output relation of a digital two-pair is given by
In the above relation the matrix t given by
is called the transfer matrix of the two-pair t 29

30. Digital Two-Pairs
An alternate characterisation of the two-pair is in terms of its chain parameters as
where the matrix G given by
is called the chain matrix of the two-pair G - 30

31. Digital Two-Pairs
The transfer and chain parameters are related as 31

32. Two-Pair Interconnections
Cascade Connection - G-cascade
Here - 32

33. Two-Pair Interconnections
But from figure, and
Substituting the above relations in the first equation on the previous slide and combining the two equations we get
Hence, 33

34. Two-Pair Interconnections
Cascade Connection - t-cascade
Here - 34

35. Two-Pair Interconnections
But from figure, and
Substituting the above relations in the first equation on the previous slide and combining the two equations we get
Hence, 35

36. Two-Pair Interconnections
Constrained Two-Pair
It can be shown that
G(z)
H(z) 36