frequency response of discrete-time lti systems prof. siripong potisuk
TRANSCRIPT
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:
Can be determined by taking the Z-transform of the governing LCCDE and applying the delay property
The system’s impulse response:
N
k
M
kkk knxbknyanyaZ
1 00 ][][][
NN
MM
zazaa
zbzbb
zX
zYzH
110
110
)(
)()(
)(
)(
}][ {
}][ {)(
zX
zY
nxZ
nyZzH
)(][ 1 zHZnh
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
• 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
• Marginally unstable one or more simple poles on
Nipi 1,1||
the unit circle
Ex. Consider a 2nd order discrete-time LTI system with
(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.
]2[6]1[10]2[32.0]1[2.1][ nxnxnynyny
For a discrete-time LTI system, the frequency responseis defined as
)(
)()(
j
jj
eX
eYeH
Frequency Response
In terms of transfer function,
,)()( jez
j zHeH
The frequency response is just the transfer functionevaluated along the unit circle in the complex z-plane.
Re(z)
Im(z)
H(ej) periodic in with period 2
1
22,)()()(
2,)()()(
2
2
2
2
ssez
FTj
ez
fjj
FF
FzHeHFH
fzHeHeH
sFTjs
fj
For H(z) generated by a difference eq. with realcoefficients,
function) (Odd)}(Re{
)}(Im{tan)(
function)(Even |)(|)(2
0),()(
1
FH
FHF
FHFA
FFFHFH s
Ex. Consider a 2nd order discrete-time system with
64.0
1)(
2
z
zzH
Plot the magnitude and phase responses of the system.Determine also the DC and the high-frequency gain.
Effects of Pole & Zero Locations
• A zero at indicates that the filter
will fully reject spectral component of input at• 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.
111jezz
1
Ex. Roughly sketch the magnitude response of the system with
)8.05.0)(8.05.0)(89.0(
)1()(
2
jzjzz
zzzH
Ex. Roughly sketch the magnitude response of the system with
)7957.04461.11)(683.01(
)0166.11)(1(05634.0)(
211
211
zzz
zzzzH
For a given choice of H(ej) as a function of , thefrequency composition of the output can be shaped: - preferential amplification - selective filtering of some frequencies
Ex. Consider a 1st order IIR digital filter with
cz
zczH
)1)(1(5.0
)(
(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.