1. Digital Signal Processing Lecture 9 Review of LTI systems
بسم الله الرحمن الرحيم
University of Khartoum
Department of Electrical and Electronic Engineering
Diploma/M. Sc. Program in Telecommunication and Information Systems
Digital Signal Processing
Lecture 9
Review of LTI systems
Dr. Iman AbuelMaaly

2. System design and implementation
In practice, system design and implementation are usually treated jointly rather than separately.
Often, the system design is driven by the method of implementations and implementation constraints such as cost, hardware limitations, size limitations and power requirements.

3. Types of Systems Finite Impulse Response (FIR)
Infinite Impulse Response (IIR)
Recursive System
Non-recursive

4. Finite Impulse Response (FIR) Systems
Digital Filters
Finite Impulse Response (FIR) Systems
FIR systems has a finite memory:
The convolution formula reduces to
The system acts as a window that views only the most recent input signal samples for forming the ouptut.

5. Infinite Impulse Response (IIR) Systems
Digital Filters
Infinite Impulse Response (IIR) Systems
An IIR Systems has an infinite duration impulse response.

6. Recursive systems
Recursive systems can be expressed in general as: Where F[.] denotes some function of its arrangements. In recursive systems the response y(n) depends on past output and the present and past inputs.

7. Recursive systems
A simple recursive system described by a first order difference equation:
Where a is a constant. This system is a LTI system. Its block diagram realization can be shown as below:

8. An example of a Recursive Filter
Recursive systems
An example of a Recursive Filter

9. Non-recursive Systems
In contrast, if y(n) depends only on the present and past inputs, then,
Such a system is called non-recursive system.
Finite Impulse Response systems (FIR) have the above form.

10. An example of nonrecursive system described by a difference equation:
Non-recursive Systems
An example of nonrecursive system described by a difference equation:

11. An example of nonrecursive system described by a difference equation:
Non-recursive Systems
An example of nonrecursive system described by a difference equation:

12. The basic form of:
a) Causal non recursive system:
b) Causal recursive system

13. Note
The feed back which contains a delay elements is crucial in recursive systems.
For recursive systems: to compute the output which is excited with an input at time n, you need to compute all the previous values (i.e., y(0), y(1), ..y(n-1))
Whereas for non-recursive systems the output can be calculated in any order [i.e., y(20), y(11),..y(3)]

14. Structure for the Realization of LTI Systems
Consider the first-order system
Which is realized below

15. Structure for realization of LTI systems
This realization uses separate delays for both the input and the output, and is called the direct form I structure.

16. Structure for realization of LTI systems
This system can be viewed as two LTI systems, the first is a non-recursive, system described by
The second is a recursive system described by
If we can interchange the order of the cascaded LTI systems, the response of the system remain the same 16
2009

17. v(n) -

18. The resulting figure will be
And we get

19. The two delay elements contain the same input w(n) and the same output w(n-1) and thus can be merged into one delay as shown below
The new design requires only one delay and hence is more efficient in terms of memory requirements.

20. It is called direct form II structure and it is used extensively in practical applications.
These structures can be generalized for the general LTI recursive system described by the difference equation

21. The Direct Form I Structure
v(n)

22. The Direct Form I structure
This structure requires M+N delays and N+M+1 multiplications.
It is a cascade of non-recursive system
And a recursive system

23. The Direct Form II Structure for N>M

24. x(n) y(n) w(n) b0 -a1 b1 w(n-1) -a2 b2 -a3 b3 -aN-2 bM (M=N-2) -aN-1
w(n-N)

25. The Direct Form II structure
This structure is the cascade of the recursive system
Followed by the non-recursive system

26. The Direct Form II structure
If N>M, the number of delays is equal to N
If M>N, the number of delays is equal to M
(i.e, the number of delays is the max of {M,N})
In both cases the required number of multiplications is N+M+1
Direct form II is called the canonic form.

27. The general LTI recursive system is described by the difference equation
Special cases:
ak =0, for k=0, 1,2, ..
M=0

28. Then the above equation becomes
ak =0, for k=0, 1,2, ..
Then the above equation becomes
This is a non- recursive system. It is an FIR system with an impulse response

29. Which is a “purely recursive” system
(b) M=0, then
Which is a “purely recursive” system

30. LTI systems described by a second order difference equation
The second order systems are usually used as basic building blocks for realizing higher order systems
The most general second-order system is described as follows:
If N=M=2

31. The direct form II structure is as follows

32. Structure realization of this system is as follows:
Special cases
(a) If we set a1= a2=0, then
Which is the FIR system.
Structure realization of this system is as follows:

33. (b) If we set b1= b2=0, we obtain the purely recursive second-order system described by

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