Chapter 10
Discrete-Time Linear Time-Invariant Systems
Sections 10.1-10-3
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:
x[n]d [n  k ]  
x[n] 

 x[k ]d [n  k ]
k  
x [ k ],n k
0,else
• Notations:
x0 [n]  x[n]d [n  0]  x[0]d [n  0]  x[0], n  0
x1[n]  x[n]d [n  1]  x[1]d [n  1]  x[1], n  1
notes
Representation of Discrete-Time Signals - Example
x[n]  x1[n]  x0 [n]  x1[n]
 x[1]d [n  (1)]  x[0]d [n]  x[1]d [n  (1)]
Convolution for Discrete-Time Systems
• LTI system response can be described using: d[n]
• For time-invariant: d[n-k]h[n-k]
System
h[n]
Impulse Response of a System
• For a linear system: x[k]d[n-k]x[k]h[n-k]
• Remember:
x[n] 
 x[k ]d [n  k ]
k  
• Thus, for LTI:
x[n] 



k  
k  
 x[k ]d [n  k ]  y[n]   x[k ]h[n  k ]  x[n] * h[n]
• We call this the convolution sum
y[n] 

 x[k ]h[n  k ]  x[n] * h[n]
k  
y[n] 
• Remember:

 h[k ]x[n  k ]  h[n] * x[n]
k  
h[n] *d [n  n0 ]  h[n  n0 ] *d [n]  h[n  n0 ]
Convolution for Discrete-Important Properties
• By definition
y[n ]  h[n ] * d [n ]  h[n ]
• Remember (due to time-invariance property):
h[n] *d [n  n0 ]  h[n  n0 ] *d [n]  h[n  n0 ]
• Multiplication
d [n]g[n  n0 ]  d [n]g[n0 ]
Properties of Convolution
• Commutative
• Associative
• Distributive
Example
•
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
Figure 10.3
FIR system contains finite
number of nonzero terms
1
y[n]  ( x[n]  x[n  1]  x[n  2])
3
•
Difference equation
•
To find h[n] we assume x[n]=d[n], thus y[n]=h[n]
•
•
1
y
[
n
]

h
[
n
]

(d [n]  d [n  1]  d [n  2])
– Thus: h[0]=h[1]=h[2]=1/3
3
Since h[n] is finite, the system is FIR
In terms of inputs:

y[n] 
 x[k ]h[n  k ]  x[n] * h[n]
k  
y[n] 

 h[k ]x[n  k ]  h[n] * x[n]
k  
 ...  x[n  3]h[3]  x[n  2]h[2]  x[n  1]h[1]  x[n  0]h[0]  x[n  1]h[1]  ....
 x[n  2]h[2]  x[n  1]h[1]  x[n  0]h[0]
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
Figure 10.3
In terms of inputs:
y[n]  ...  x[n  3]h[3]  x[n  2]h[2]  x[n  1]h[1]  x[n  0]h[0]  x[n  1]h[1]  ....
 x[n  2]h[2]  x[n  1]h[1]  x[n  0]h[0]
Try for different values of n
•
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
Example – cont. (Graphical Representation)
X[m]
h[0]=h[1]=h[2]=1/3
X[n-k]
x[1]=3, x[2]=4.5, x[3]=6
y[n]  ...  x[n  3]h[3]  x[n  2]h[2]  x[n  1]h[1]  x[n  0]h[0]  x[n  1]h[1]  ....
 x[n  2]h[2]  x[n  1]h[1]  x[n  0]h[0]
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?
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)
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
y[n] 
 h[k ]x[n  k ]  h[n] * x[n]

n
 0.6 u[n  k ]u[k ]   0.6
k
k  
h[n]
y[n]

k  

x[n]
k 0
k
 2.5[1  0.6 n 1 ]; n  0
y[0]=1
y[1]=1.6
…..
y(100)=2.5
 Steady State Value is 2.5
Remember These Geometric Series:
Properties of Discrete-Time LTI Systems
• Memory:
y[n] 
– A memoryless system is a pure gain
system: iff h[n]=Kd[n];
• K=h[0] = constant & h[n]=0 otherwise

 x[k ]h[n  k ]  x[n] * h[n]
k  
y[n] 

 h[k ]x[n  k ]  h[n] * x[n]
k  
y[n]  ...  x[n  3]h[3]  x[n  2]h[2]  x[n  1]h[1]  x[n  0]h[0]  x[n  1]h[1]  ....
 x[n  2]h[2]  x[n  1]h[1]  x[n  0]h[0]
• Causality
Note that if k<0depending on
future; Thus h[k] should be zero
to remove dependency on the
future.
– 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].
y[n]   h[k ]x[n  k ]  x[n]h[0]  ...  x[0]h[n]  x[1]h[n  1]  ...
k 0
y[n] 
n
 x[k ]k[n  k ]  x[n]h[0]  ...  x[0]h[n]  x[1]h[n  1]  ...
k  
Properties of Discrete-Time LTI Systems
• Stability
– BIBO: |x[n]|< M
– Absolutely summable:

 | h[k ] | 
k  
• 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]
Example 1
• Assume h[n]= u[n] (1/2)^n
x[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:
1
1
| h[k ] |  | | 
2

1  0.5
k  
k 0 2
k
h[n]
y[n]
Example 2
• Assume h[n]= u[n+1] (1/2)^n
x[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:

1 k
1
| h[k ] |  | |  2 
4

1  0.5
k  
k  1 2
h[n]
y[n]
Example 3
• Assume h[n]= u[n] (2)^n
x[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:

k
|
h
[
k
]
|

|
2
|

 
k  
k 0
h[n]
y[n]