1. UNIT V Linear Time Invariant Discrete-Time Systems

2. DT Unit-Impulse Response
Consider the DT SISO system:
If the input signal is and the system has no energy at , the output is called the impulse response of the system
System
System

3. Example
Consider the DT system described by
Its impulse response can be found to be

4. Representing Signals in Terms of Shifted and Scaled Impulses
Let x[n] be an arbitrary input signal to a DT LTI system
Suppose that for
This signal can be represented as

5. Exploiting Time-Invariance and Linearity

6. The Convolution Sum
This particular summation is called the convolution sum
Equation is called the convolution representation of the system
Remark: a DT LTI system is completely described by its impulse response h[n]

7. Block Diagram Representation of DT LTI Systems
Since the impulse response h[n] provides the complete description of a DT LTI system, we write

8. The Convolution Sum for Noncausal Signals
Suppose that we have two signals x[n] and v[n] that are not zero for negative times (noncausal signals)
Then, their convolution is expressed by the two-sided series

9. Example: Convolution of Two Rectangular Pulses
Suppose that both x[n] and v[n] are equal to the rectangular pulse p[n] (causal signal) depicted below

10. The Folded Pulse
The signal is equal to the pulse p[i] folded about the vertical axis

11. Sliding over

12. Sliding over Cont’d

13. Plot of

14. Properties of the Convolution Sum
Associativity
Commutativity
Distributivity w.r.t. addition

15. Properties of the Convolution Sum - Cont’d
Shift property: define
Convolution with the unit impulse
Convolution with the shifted unit impulse
then

16. Direct Form II For a discrete-time system described by
the transfer function is of the form

17. Direct Form II
where

18. Direct Form II

19. Direct Form II

20. Direct Form II For the special case of FIR filters.
(The number of delays has been
changed to M - 1 to conform to
conventions in the DSP literature.)
This type of filter has M - 1 finite
zeros and M - 1 poles at z = 0.

21. Direct Form II
One desirable characteristic of an FIR filter is that it can have
linear phase in its passband.

22. Direct Form II

23. Direct Form II

24. Direct Form II

25. Direct Form II

26. Direct Form II
It can be shown that symmetric or anti-symmetric, even or odd
impulse responses yield linear phase shift in the frequency response.

27. Cascade Realization
Direct Form II is no the only form of realization. There are
several other forms. Two other important forms are the
cascade form and the parallel form.
Each individual factor is realized as a small Direct Form II
subsystem and the subsystems are then cascaded.

28. Parallel Realization Each individual term is realized
as a small Direct Form II subsystem
and the subsystems are then
paralleled.

29. Complex Poles and Zeros
In either the cascade or parallel realization, the first-order
subsystems may have complex poles and/or zeros. In such a case
two first-order subsystems should be combined into one second-
order subsystem to avoid the problem of complex coefficients in the
first-order subsystems. Also, for reasons we will soon see, it is
common to do cascade and parallel realizations with second-order
subsystems even when the poles and/or zeros are real.

30. 1st Order vs 2nd Order
In the case of FIR filters the second-order subsystems take
this form.
2 delays
3 multiplications
2 additions
Compare this two two first-order cascaded stages.
2 delays
4 multiplications
2 additions

31. 1st Order vs 2nd Order
If the FIR filter has linear phase, a fourth-order structure reduces
number of multiplications even further compared with cascaded
first-order or cascaded second-order subsystems.