1. Analysis of Linear Time Invariant (LTI) Systems
بسم الله الرحمن الرحيم
University of Khartoum
Department of Electrical and Electronic Engineering
Fourth Year ( )
Digital Signal processing
معالجة الإشارة الرقمية
Lecture (4)
Analysis of Linear Time Invariant (LTI) Systems
Dr. Iman Abuel Maaly

2. Discrete-time System
Systems with Finite Duration and Infinite Duration Impulse responses
LTI systems characterized difference equations
Recursive and non-recursive systems
LTI systems characterized by constant coefficient difference equations

3. Systems with Finite Duration and Infinite Duration Impulse responses
LTI systems are divided into two classes: 1. those that have a Finite duration Impulse Response (FIR), and 2. those that have an Infinite duration Impulse Response. (IIR) M
h(n)
y(n)
x(n)
LTI
System

4. 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
An IIR Systems has an infinite duration impulse response.
Its practical implementation as implied by convolution sum is impossible because it requires an infinite number of memory location, additions and multiplications.

6. DSP- Lecture 4 Analysis of LTI
Classes of Linear Time-Invariant Systems (LTI)
IIR Systems
FIR Systems
LTI Described by difference equations
University of Khartoum - Dr. Iman AbuelMaaly

7. Discrete-time System
Systems with Finite Duration and Infinite Duration Impulse responses
LTI systems characterized difference equations
Recursive and non-recursive systems
LTI systems characterized by constant coefficient difference equations

8. Discrete-time System
Systems with Finite Duration and Infinite Duration Impulse responses
LTI systems characterized difference equations
Recursive and non-recursive systems
LTI systems characterized by constant coefficient difference equations

9. Recursive and non-recursive systems
Suppose that we wish to compute the cumulative average of a signal x(n) in the interval 0≤ k≤ n, defined as:

10. The computation of y(n) requires the storage of all the input samples x(k). Since n is increasing, memory requirements grow linearly with time.
By simple arrangement of the equation we get: y(n) can be computed as follows:
and hence 10

11. The cumulative average y(n) can be computed recursive.
This is an example of a recursive system.

12. Recursive systems Determine the computation of the recursive system:

13. Recursive systems
To determine the computation of the recursive system suppose , we begin with n=0 and forward:

14. Recursive systems
If one wishes to compute the response of the system to an input signal x(n) applied at n =n0,, we need the value of y(n0 -1) for n ≥n0
The term y(n0-1) is called the initial condition.

15. Recursive systems
The above two examples are simple recursive systems. Both of them are causal and can be expressed in general as: Where F[.] denotes some function of its arrangements.

16. 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.

17. Non-recursive Systems
The convolution summation for a causal FIR system is:

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

19. Non-recursive Systems
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)]

20. Discrete-time System
Systems with Finite Duration and Infinite Duration Impulse responses
LTI systems characterized difference equations
Recursive and non-recursive systems
LTI systems characterized by constant coefficient difference equations

21. LTI Systems Characterized by Constant Coefficient Difference Equations:
Systems described by constant coefficient linear difference equations are subclasses of the recursive systems introduced in the previous section.

22. A simple recursive system described by a first order difference equation:
Where a is a constant. This system is a linear time invariant system. Its block diagram realization can be shown as below:

23. Apply an input x(n) for n = 0 , The initial condition is y(-1)
We need to solve the equation above. We compute successive values of y(n) for n ≥ 0 beginning with y(0)

24. Part I contains the initial condition of the system.
It is called zero input response yzi of the system.
Part II is the response of the system to the input x(n) .
It is called the zero state response yzs of the system.

25. The zero state response yzs(n)
If the system is initially relaxed at time n=0, the output of the delay should be zero, hence, y(-1) =0
This is called zero state response or forced response,
The above equation is a convolution sum for impulse response,

26. The zero input response yzi(n)
If the system is initially non-relaxed
(i.e., y(-1) ≠ 0) and the input x(n) = 0 for all n, then the response of the system is called zero input response or natural response which is the output of the system without being excited. The zero input response is denoted yzi(n)

27. The above example is a simple recursive system described by a first order difference equation.
The general form of the recursive systems described by linear constant coefficient difference equation is as follows:
or equivalently,
N is the order of the equation

28. Next Lecture
Implementation of discrete time systems