Frequency Response of
Discrete-time LTI Systems
Prof. Siripong Potisuk
Transfer Functions

Let x[n] be a nonzero input to an LTI discrete-time system, and
y[n] be the resulting output assuming a zero initial condition.
The transfer function, denoted by H(z), is defined:
Z{ y[n]} Y ( z )
H ( z) 

Z{ x[n]} X ( z )

Can be determined by taking the Z-transform of the governing
LCCDE and applying the delay property
N
M

Z a0 y[n]   ak y[n  k ]   bk x[n  k ]

k 1
k 0


Y ( z ) b0  b1 z 1    bM z  M
H ( z) 

X ( z ) a0  a1 z 1    a N z  N

The system’s impulse response: h[n]  Z 1 H ( z)
BIBO Stability
• BIBO = Bounded-input-bounded-output
• A linear time-invariant (LTI) discrete-time
system with transfer function H(z) is BIBO
stable if and only if the poles of H(z) satisfy
| pi |  1,
1 i  N
• That is, the poles of a stable system, whether
simple or multiple, must all lie strictly within
the unit circle in the complex z-plane
the unit circle
• Marginally unstable  one or more simple
Ex. Consider a 2nd order discrete-time LTI system with
y[n]  1.2 y[n  1]  0.32y[n  2]  10x[n  1]  6 x[n  2]
(a) Determine the transfer function of the system and
comment on the stability of the system.
(b) Determine the zero-state response due to a unit-step
input and the DC gain of the system.
Frequency Response
For a discrete-time LTI system, the frequency response
is defined as
j
Y
(
e
)
j
H (e ) 
X (e j )
In terms of transfer function,
j
H ( e )  H ( z ) z  e j ,      
The frequency response is just the transfer function
evaluated along the unit circle in the complex z-plane.
Im(z)
periodic in 
with period 2
H(ej)

1
Re(z)
H (e j )  H (e j 2f )  H ( z ) z e j 2 f ,    2 f  
H ( F )  H (e
j 2FTs
Fs
Fs
)  H ( z ) z e j 2 FT s ,   F 
2
2
For H(z) generated by a difference eq. with real
coefficients,
Fs
H ( F )  H ( F ), 0  F 
2
A( F )  | H ( F ) | (Evenfunction)

 Im{H ( F )}
 ( F )  tan 
(Odd function)

 Re{H ( F )}
1
Ex. Consider a 2nd order discrete-time system with
z 1
H ( z)  2
z  0.64
Plot the magnitude and phase responses of the system.
Determine also the DC and the high-frequency gain.
Effects of Pole & Zero Locations
j1
• A zero at z  z1  1e indicates that the filter
will fully reject spectral component of input at 1
• Effects of a zero located off the unit circle depends
on its distance from the unit circle.
• A zero at origin has no effect.
• A pole on the unit circle means infinite gain at that
frequency.
• The closer the poles to the unit circle, the higher the
magnitude response.
Ex. Roughly sketch the magnitude response of the
system with
z 2 ( z  1)
H ( z) 
( z  0.89)(z  0.5  j 0.8)(z  0.5  j 0.8)
Ex. Roughly sketch the magnitude response of the
system with
0.05634(1  z 1 )(1  1.0166z 1  z 2 )
H ( z) 
(1  0.683z 1 )(1  1.4461z 1  0.7957z 2 )
For a given choice of H(ej) as a function of , the
frequency composition of the output can be shaped:
- preferential amplification
- selective filtering of some frequencies
Ex. Consider a 1st order IIR digital filter with
0.5(1  c)( z  1)
H ( z) 
z c
(a) Determine c such that the system is BIBO stable.
(b) Without plotting the magnitude response of the
system, determine the type of this filter.
(c) Verify the answer in (b) using MATLAB.