1. Discrete-Time Linear Time-Invariant Systems Sections 10.1-10-3
Chapter 10
Discrete-Time Linear Time-Invariant Systems
Sections

2. Representation of Discrete-Time Signals
We assume Discrete-Time LTI systems
The signal X[n] can be represented using unit sample function or unit impulse function: d[n]
Remember:
Notations:
notes

3. Representation of Discrete-Time Signals - Example

4. Convolution for Discrete-Time Systems
LTI system response can be described using:
For time-invariant: d[n-k]h[n-k]
For a linear system: x[k]d[n-k]x[k]h[n-k]
Remember:
Thus, for LTI:
We call this the convolution sum
System
d[n]
h[n]
Impulse Response of a System

5. Convolution for Discrete-Important Properties
By definition
Remember (due to time-invariance property):
Multiplication

6. Properties of Convolution
Commutative
Associative
Distributive

7. Example Figure 10.3 Figure 10.3 Given the following block diagram
Find the difference equation
Find the impulse response: h[n]; plot h[n]
Is this an FIR (finite impulse response) or IIR system?
Given x[1]=3, x[2]=4.5, x[3]=6, Plot y[n] vs. n
Plot y[n] vs. n using Matlab
Difference equation
To find h[n] we assume x[n]=d[n], thus y[n]=h[n]
Thus: h[0]=h[1]=h[2]=1/3
Since h[n] is finite, the system is FIR
In terms of inputs:
Figure 10.3
Figure 10.3
FIR system contains finite number of nonzero terms

8. Try for different values of n
Example – cont.
Given the following block diagram
Find the difference equation
Find the impulse response: h[n]; plot h[n]
Is this an FIR (finite impulse response) or IIR system?
Given x[1]=3, x[2]=4.5, x[3]=6, Plot y[n] vs. n
Plot y[n] vs. n using Matlab
In terms of inputs:
Calculate for n=0, n=1, n=2, n=3, n=4, n=5, n=6
n=0; y[0]=0
n=1; y[n]=1
n=2; y[2]=2.5
n=3; y[2\3]=4.5
n=4; y[4]=3.5
n=5; y[5]=2
n=6; y[6]=0
Figure 10.3
Try for different values of n

9. Example – cont. (Graphical Representation)
X[m]
X[n-k]
h[0]=h[1]=h[2]=1/3
x[1]=3, x[2]=4.5, x[3]=6

10. Example Consider the following difference equation:y[n]=ay[n-1]+x[n]
Draw the block diagram of this system
Find the impulse response: h[n]
Is it a causal system?
Is this an IIR or FIR system?

11. Example Consider the following difference equation:y[n]=ay[n-1]+x[n];
Draw the block diagram of this system
Find the impulse response: h[n]
Is this an IIR or FIR system?
We assume x[n]=d[n]
y[n]=h[n]=ah[n-1]+d[n];
y[0]=h[0]=1
y[1]=h[1]=a
y[2]=h[2]=a^2
y[3]=h[3]= a^3
h[n]=a^n ; n>=0
It is IIR (unbounded)
Causal system (current and past)

12. Example Assume h[n]=0.6^n*u[n] and x[n]=u[n]
Find the expression for y[n]
Plot y[n]
Plot y[n] using Matlab
h[n]
x[n]
y[n]
y[0]=1
y[1]=1.6
…..
y(100)=2.5
 Steady State Value is 2.5

13. Remember These Geometric Series:

14. Properties of Discrete-Time LTI Systems
Memory:
A memoryless system is a pure gain system: iff h[n]=Kd[n];
K=h[0] = constant & h[n]=0 otherwise
Causality
y[n] has no dependency on future values of x[n]; thus h[n]=0 for n<0 (note h[n] is non-zero only for d[n=0].
Note that if k<0depending on future; Thus h[k] should be zero to remove dependency on the future.

15. Properties of Discrete-Time LTI Systems
Stability
BIBO: |x[n]|< M
Absolutely summable:
Invertibility:
If the input can be determined from output
It has an inverse impulse response
Invertible if there exists: hi[n]*h[n]=d[n]

16. Example 1 Assume h[n]= u[n] (1/2)^n Memoryless? Casual system? Stable?
Has memory (dynamic): h[n] is not Kd[n] (not pure gain); h[n] is non-zero
Is causal: h[n]=0 for n<0
Stable:
h[n]
x[n]
y[n]

17. Example 2 Assume h[n]= u[n+1] (1/2)^n Memoryless? Casual system?
Stable?
Has memory (dynamic): h[n] is not Kd[n] (not pure gain)
Is NOT causal: h[n] not 0 for n<0; h[-1]=2
Stable:
h[n]
x[n]
y[n]

18. Example 3 Assume h[n]= u[n] (2)^n Memoryless? Casual system? Stable?
Has memory (dynamic): h[n] is not Kd[n] (not pure gain)
Is causal: h[n]=0 for n<0
Not Stable:
h[n]
x[n]
y[n]