Copyright ? 2001,S,K,Mitra 1
Phase and Group Delays
? The output y[n] of a frequency-selective
LTI discrete-time system with a frequency
response exhibits some delay
relative to the input x[n] caused by the
nonzero phase response
of the system
? For an input
)( ?jeH
)}(a r g {)( ???? jeH
????????? nnAnx o ),c o s (][
Copyright ? 2001,S,K,Mitra 2
Phase and Group Delays
the output is
? Thus,the output lags in phase by
radians
? Rewriting the above equation we get
))(c o s ()(][ ??????? ? ooj neHAny o
)( o??
???
?
???
? ???
?
??
?
?
?
????? ?
o
o
o
j neHAny o )(c o s)(][
Copyright ? 2001,S,K,Mitra 3
Phase and Group Delays
? This expression indicates a time delay,
known as phase delay,at given by
? Now consider the case when the input
signal contains many sinusoidal
components with different frequencies that
are not harmonically related
o???
o
oop
?
?????? )()(
Copyright ? 2001,S,K,Mitra 4
Phase and Group Delays
? In this case,each component of the input
will go through different phase delays when
processed by a frequency-selective LTI
discrete-time system
? Then,the output signal,in general,will not
look like the input signal
? The signal delay now is defined using a
different parameter
Copyright ? 2001,S,K,Mitra 5
Phase and Group Delays
? To develop the necessary expression,
consider a discrete-time signal x[n] obtained
by a double-sideband suppressed carrier
(DSB-SC) modulation with a carrier
frequency of a low-frequency sinusoidal
signal of frequency, o?
c?
)c o s ()c o s (][ nnAnx co ???
Copyright ? 2001,S,K,Mitra 6
Phase and Group Delays
? The input can be rewritten as
where and
? Let the above input be processed by an LTI
discrete-time system with a frequency
response satisfying the condition
)c o s ()c o s (][ 22 nnnx uAA ???? ?
oc ????? ? ocu ?????
)( ?jeH
ujeH ??????? ?f o r1)(
Copyright ? 2001,S,K,Mitra 7
Phase and Group Delays
? The output y[n] is then given by
? Note,The output is also in the form of a
modulated carrier signal with the same
carrier frequency and the same
modulation frequency as the input
? ? ))(c o s ()(c o s][ 22 uuAA nnny ?????????? ??
?????? ????????????? ???????? 2 )()(c o s2 )()(c o s ?? uouc nnA
c?
o?
Copyright ? 2001,S,K,Mitra 8
Phase and Group Delays
? However,the two components have
different phase lags relative to their
corresponding components in the input
? Now consider the case when the modulated
input is a narrowband signal with the
frequencies and very close to the
carrier frequency,i.e,is very small
?? u?
c? o?
Copyright ? 2001,S,K,Mitra 9
Phase and Group Delays
? In the neighborhood of we can express
the unwrapped phase response as
by making a Taylor’s series expansion and
keeping only the first two terms
? Using the above formula we now evaluate
the time delays of the carrier and the
modulating components
c?
)(??c
)()()()( ccccc
cd
d ????
?
????????
???
Copyright ? 2001,S,K,Mitra 10
Phase and Group Delays
? In the case of the carrier signal we have
which is seen to be the same as the phase
delay if only the carrier signal is passed
through the system
c
cc
c
cuc
?
????
?
?????? )(
2
)()( ?
Copyright ? 2001,S,K,Mitra 11
Phase and Group Delays
? In the case of the modulating component
we have
? The parameter
is called the group delay or envelope delay
caused by the system at
cd
d c
u
cuc
????
????
???
?????? )()()(
?
?
cd
d c
cg
????
?????? )()(
c???
Copyright ? 2001,S,K,Mitra 12
Phase and Group Delays
? The group delay is a measure of the
linearity of the phase function as a function
of the frequency
? It is the time delay between the waveforms
of underlying continuous-time signals
whose sampled versions,sampled at t = nT,
are precisely the input and the output
discrete-time signals
Copyright ? 2001,S,K,Mitra 13
Phase and Group Delays
? If the phase function and the angular
frequency ? are in radians per second,then
the group delay is in seconds
? Figure below illustrates the evaluation of
the phase delay and the group delay
Copyright ? 2001,S,K,Mitra 14
Phase and Group Delays
? Figure below shows the waveform of an
amplitude-modulated input and the output
generated by an LTI system
Copyright ? 2001,S,K,Mitra 15
Phase and Group Delays
? Note,The carrier component at the output is
delayed by the phase delay and the envelope
of the output is delayed by the group delay
relative to the waveform of the underlying
continuous-time input signal
? The waveform of the underlying continuous-
time output shows distortion when the group
delay of the LTI system is not constant over
the bandwidth of the modulated signal
Copyright ? 2001,S,K,Mitra 16
Phase and Group Delays
? If the distortion is unacceptable,a delay
equalizer is usually cascaded with the LTI
system so that the overall group delay of the
cascade is approximately linear over the band
of interest
? To keep the magnitude response of the parent
LTI system unchanged the equalizer must
have a constant magnitude response at all
frequencies
Copyright ? 2001,S,K,Mitra 17
Phase and Group Delays
? Example - The phase function of the FIR
filter
is
? Hence its group delay is given by
verifying the result obtained earlier by
simulation
][][][][ 21 ????? nxnxnxny ???
????? )(
1)( ??? g
Copyright ? 2001,S,K,Mitra 18
Phase and Group Delays
? Example - For the M-point moving-average
filter
the phase function is
? Hence its group delay is
??
??][nh
o th e r w ise0
101
,
,/ ??? MnM
? ?????? ????????????
? 0
2
2
)1()(
k M
kMM/2
2 1)( ???? Mg
Copyright ? 2001,S,K,Mitra 19
Frequency Response of the
LTI Discrete-Time System
? The convolution sum description of the LTI
discrete-time system is given by
? Taking the DTFT of both sides we obtain
][][][ knxkhny
k
?? ?
?
???
nj
n
j enyeY ???
???
? ?? ][)(
nj
n k
eknxkh ??
?
???
?
???
? ?
?
??
?
? ? ?? ][][
Copyright ? 2001,S,K,Mitra 20
Frequency Response of the
LTI Discrete-Time System
? Or,
? ?
?
??
?
? ?? ?
???
?
???
????
k
kjj exkheY
?
?? )(][][)(
kj
k
j eexkh ???
???
?
???
??? ?
?
??
?
? ??
?
?? ][][
)( ?jeX
Copyright ? 2001,S,K,Mitra 21
Frequency Response of the
LTI Discrete-Time System
? Hence,we can write
? In the above is the frequency
response of the LTI system
? The above equation relates the input and the
output of an LTI system in the frequency
domain
)()()(][)( ???
?
???
??? ??
?
??
?
? ?? jjj
k
kjj eXeHeXekheY
)( ?jeH
Copyright ? 2001,S,K,Mitra 22
Frequency Response of the
LTI Discrete-Time System
? It follows from the previous equation
? For an LTI system described by a linear
constant coefficient difference equation of
the form we have
)(/)()( ??? ? jjj eXeYeH
?
??
?
??
?
??
?
N
k
kj
k
M
k
kj
kj
ed
epeH
0
0)(
Copyright ? 2001,S,K,Mitra 23
The Transfer Function
? A generalization of the frequency response
function
? The convolution sum description of an LTI
discrete-time system with an impulse
response h[n] is given by
??
???
??
k
knxkhny ][][][
Copyright ? 2001,S,K,Mitra 24
The Transfer Function
? Taking the z-transforms of both sides we get
n
n kn
n zknxkhznyzY ??
???
?
???
?
???
? ? ??
???
?
???
? ??? ][][][)(
? ?
?
???
?
???
?
???
?
???
? ??
k n
nzknxkh ][][
? ?
?
???
?
???
??
???
?
???
??
k
kzxkh
?
?? )(][][
Copyright ? 2001,S,K,Mitra 25
The Transfer Function
? Or,
? Therefore,
? Thus,Y(z) = H(z)X(z)
k
k
zzxkhzY ?
?
???
?
???
?? ?
???
?
???
??
?
?? ][][)(
)(zX )(][)( zXzkhzY
k
k
???
?
???
?? ??
???
?
)(zH
Copyright ? 2001,S,K,Mitra 26
The Transfer Function
? Hence,
? The function H(z),which is the z-transform of
the impulse response h[n] of the LTI system,
is called the transfer function or the system
function
? The inverse z-transform of the transfer
function H(z) yields the impulse response h[n]
)(/)()( zXzYzH ?
Copyright ? 2001,S,K,Mitra 27
The Transfer Function
? Consider an LTI discrete-time system
characterized by a difference equation
? Its transfer function is obtained by taking
the z-transform of both sides of the above
equation
? Thus
?? ?? ??? Mk kNk k knxpknyd 00 ][][
?
?
?
?
?
?
? N
k
k
k
M
k
k
k
zd
zp
zH
0
0)(
Copyright ? 2001,S,K,Mitra 28
The Transfer Function
? Or,equivalently as
? An alternate form of the transfer function is
given by
?
?
?
?
?
?
??
N
k
kN
k
M
k
kM
kMN
zd
zp
zzH
0
0)()(
?
?
?
?
?
?
?
?
?? N
k k
M
k k
z
z
d
pzH
1
1
1
1
0
0
1
1
)(
)(
)(
?
?
Copyright ? 2001,S,K,Mitra 29
The Transfer Function
? Or,equivalently as
? are the finite zeros,and
are the finite poles of H(z)
? If N > M,there are additional zeros
at z = 0
? If N < M,there are additional poles
at z = 0
M???,...,,21
N???,...,,21
)( MN ?
)( NM ?
?
?
?
??
?
?
? N
k k
M
k kMN
z
z
z
d
pzH
1
1
0
0
)(
)(
)( )(
?
?
Copyright ? 2001,S,K,Mitra 30
The Transfer Function
? For a causal IIR digital filter,the impulse
response is a causal sequence
? The ROC of the causal transfer function
is thus exterior to a circle going through the
pole farthest from the origin
? Thus the ROC is given by
?
?
?
??
?
?
? N
k k
M
k kMN
z
z
z
d
pzH
1
1
0
0
)(
)(
)( )(
?
?
kkz ?? m a x
Copyright ? 2001,S,K,Mitra 31
The Transfer Function
? Example - Consider the M-point moving-
average FIR filter with an impulse response
? Its transfer function is then given by
??
??][nh
o th e r w ise,0
10,/1 ??? MnM
?
?
?
??
1
0
1)( M
n
nz
M
zH 1111
( 1 ) [ ( 1 ) ]
MM
M
zz
M z M z z
?
??
????
??
Copyright ? 2001,S,K,Mitra 32
The Transfer Function
? The transfer function has M zeros on the
unit circle at,
? There are poles at z = 0 and a single
pole at z = 1
? The pole at z = 1
exactly cancels the
zero at z = 1
? The ROC is the entire
z-plane except z = 0
Mkjez /2 ?? 10 ??? Mk
-1 - 0.5 0 0.5 1
-1
- 0.5
0
0.5
1
R e a l P a r t
I
m
a
gi
na
r
y P
a
r
t
7
M = 8
1?M
Copyright ? 2001,S,K,Mitra 33
The Transfer Function
? Example - A causal LTI IIR digital filter is
described by a constant coefficient
difference equation given by
? Its transfer function is therefore given by
]1[3.1]3[]2[2.1]1[][ ???????? nynxnxnxny
]3[222.0]2[04.1 ???? nyny
321
321
222.004.13.11
2.1)(
???
???
???
???
zzz
zzzzH
Copyright ? 2001,S,K,Mitra 34
The Transfer Function
? Alternate forms,
? Note,Poles farthest from
z = 0 have a magnitude
? ROC,
2 2 2.004.13.1
12.1)(
23
2
???
???
zzz
zzzH
)7.05.0)(7.05.0)(3.0(
)8.06.0)(8.06.0(
jzjzz
jzjz
?????
?????
-1 - 0.5 0 0.5 1
-1
- 0.5
0
0.5
1
R e a l P a r t
I
m
a
gi
na
r
y P
a
r
t
74.0?z
74.0
Copyright ? 2001,S,K,Mitra 35
Frequency Response from
Transfer Function
? If the ROC of the transfer function H(z)
includes the unit circle,then the frequency
response of the LTI digital filter can
be obtained simply as follows,
? For a real coefficient transfer function H(z)
it can be shown that
??? ? jezj zHeH )()(
)( ?jeH
)(*)()( 2 ??? ? jjj eHeHeH
?????? ?? jezjj zHzHeHeH )()()()( 1
Copyright ? 2001,S,K,Mitra 36
Frequency Response from
Transfer Function
? For a stable rational transfer function in the
form
the factored form of the frequency response
is given by
?
?
?
??
?
?
? N
k k
M
k kMN
z
z
z
d
pzH
1
1
0
0
)(
)(
)( )(
?
?
?
?
?
?
?
?
???
??
??
? N
k k
j
M
k k
j
MNjj
e
e
e
d
peH
1
1)(
0
0
)(
)(
)(
Copyright ? 2001,S,K,Mitra 37
Frequency Response from
Transfer Function
? It is convenient to visualize the contributions
of the zero factor and the pole factor
from the factored form of the
frequency response
? The magnitude function is given by
)( kz ??
)( kz ??
?
?
?
?
?
?
???
??
??
? N
k k
j
M
k k
j
MNjj
e
e
e
d
p
eH
1
1)(
0
0)(
Copyright ? 2001,S,K,Mitra 38
Frequency Response from
Transfer Function
which reduces to
? The phase response for a rational transfer
function is of the form
?
?
?
?
?
?
?
??
??
? N
k k
j
M
k k
j
j
e
e
d
p
eH
1
1
0
0)(
)()/a r g ()(a r g 00 MNdpeH j ?????
??
?
?
?
? ??????
N
k
k
j
M
k
k
j ee
11
)a r g ()a r g (
Copyright ? 2001,S,K,Mitra 39
Frequency Response from
Transfer Function
? The magnitude-squared function of a real-
coefficient transfer function can be
computed using
?
?
?
???
?
???
?
????
????
? N
k k
j
k
j
M
k k
j
k
j
j
ee
ee
d
peH
1
*
1
*2
0
02
))((
))((
)(
Copyright ? 2001,S,K,Mitra 40
Geometric Interpretation of
Frequency Response Computation
? The factored form of the frequency response
is convenient to develop a geometric
interpretation of the frequency response
computation from the pole-zero plot as ?
varies from 0 to 2? on the unit circle
?
?
?
?
?
?
???
??
??
? N
k k
j
M
k k
j
MNjj
e
e
e
d
peH
1
1)(
0
0
)(
)(
)(
Copyright ? 2001,S,K,Mitra 41
Geometric Interpretation of
Frequency Response Computation
? The geometric interpretation can be used to
obtain a sketch of the response as a function
of the frequency
? A typical factor in the factored form of the
frequency response is given by
where is a zero if it is zero factor or is
a pole if it is a pole factor
)( ?? ?? jj ee
?? je
Copyright ? 2001,S,K,Mitra 42
Geometric Interpretation of
Frequency Response Computation
? As shown below in the z-plane the factor
represents a vector starting at
the point and ending on the unit
circle at
??? jez
?? jez
)( ?? ?? jj ee
Copyright ? 2001,S,K,Mitra 43
Geometric Interpretation of
Frequency Response Computation
? As ? is varied from 0 to 2?,the tip of the
vector moves counterclockise from the
point z = 1 tracing the unit circle and back
to the point z = 1
Copyright ? 2001,S,K,Mitra 44
Geometric Interpretation of
Frequency Response Computation
? As indicated by
the magnitude response at a
specific value of ? is given by the product
of the magnitudes of all zero vectors
divided by the product of the magnitudes of
all pole vectors
?
?
?
?
?
?
?
??
??
? N
k k
j
M
k k
j
j
e
e
d
p
eH
1
1
0
0)(
|)(| ?jeH
Copyright ? 2001,S,K,Mitra 45
Geometric Interpretation of
Frequency Response Computation
? Likewise,from
we observe that the phase response
at a specific value of ? is obtained by
adding the phase of the term and the
linear-phase term to the sum of
the angles of the zero vectors minus the
angles of the pole vectors
00 dp /
)( MN ??
)()/a r g ()(a r g 00 MNdpeH j ?????
? ???? ??? ? ?? ? Nk kjMk kj ee 11 )a r g ()a r g (
Copyright ? 2001,S,K,Mitra 46
Geometric Interpretation of
Frequency Response Computation
? Thus,an approximate plot of the magnitude
and phase responses of the transfer function
of an LTI digital filter can be developed by
examining the pole and zero locations
? Now,a zero (pole) vector has the smallest
magnitude when ? = ?
Copyright ? 2001,S,K,Mitra 47
Geometric Interpretation of
Frequency Response Computation
? To highly attenuate signal components in a
specified frequency range,we need to place
zeros very close to or on the unit circle in
this range
? Likewise,to highly emphasize signal
components in a specified frequency range,
we need to place poles very close to or on
the unit circle in this range
Phase and Group Delays
? The output y[n] of a frequency-selective
LTI discrete-time system with a frequency
response exhibits some delay
relative to the input x[n] caused by the
nonzero phase response
of the system
? For an input
)( ?jeH
)}(a r g {)( ???? jeH
????????? nnAnx o ),c o s (][
Copyright ? 2001,S,K,Mitra 2
Phase and Group Delays
the output is
? Thus,the output lags in phase by
radians
? Rewriting the above equation we get
))(c o s ()(][ ??????? ? ooj neHAny o
)( o??
???
?
???
? ???
?
??
?
?
?
????? ?
o
o
o
j neHAny o )(c o s)(][
Copyright ? 2001,S,K,Mitra 3
Phase and Group Delays
? This expression indicates a time delay,
known as phase delay,at given by
? Now consider the case when the input
signal contains many sinusoidal
components with different frequencies that
are not harmonically related
o???
o
oop
?
?????? )()(
Copyright ? 2001,S,K,Mitra 4
Phase and Group Delays
? In this case,each component of the input
will go through different phase delays when
processed by a frequency-selective LTI
discrete-time system
? Then,the output signal,in general,will not
look like the input signal
? The signal delay now is defined using a
different parameter
Copyright ? 2001,S,K,Mitra 5
Phase and Group Delays
? To develop the necessary expression,
consider a discrete-time signal x[n] obtained
by a double-sideband suppressed carrier
(DSB-SC) modulation with a carrier
frequency of a low-frequency sinusoidal
signal of frequency, o?
c?
)c o s ()c o s (][ nnAnx co ???
Copyright ? 2001,S,K,Mitra 6
Phase and Group Delays
? The input can be rewritten as
where and
? Let the above input be processed by an LTI
discrete-time system with a frequency
response satisfying the condition
)c o s ()c o s (][ 22 nnnx uAA ???? ?
oc ????? ? ocu ?????
)( ?jeH
ujeH ??????? ?f o r1)(
Copyright ? 2001,S,K,Mitra 7
Phase and Group Delays
? The output y[n] is then given by
? Note,The output is also in the form of a
modulated carrier signal with the same
carrier frequency and the same
modulation frequency as the input
? ? ))(c o s ()(c o s][ 22 uuAA nnny ?????????? ??
?????? ????????????? ???????? 2 )()(c o s2 )()(c o s ?? uouc nnA
c?
o?
Copyright ? 2001,S,K,Mitra 8
Phase and Group Delays
? However,the two components have
different phase lags relative to their
corresponding components in the input
? Now consider the case when the modulated
input is a narrowband signal with the
frequencies and very close to the
carrier frequency,i.e,is very small
?? u?
c? o?
Copyright ? 2001,S,K,Mitra 9
Phase and Group Delays
? In the neighborhood of we can express
the unwrapped phase response as
by making a Taylor’s series expansion and
keeping only the first two terms
? Using the above formula we now evaluate
the time delays of the carrier and the
modulating components
c?
)(??c
)()()()( ccccc
cd
d ????
?
????????
???
Copyright ? 2001,S,K,Mitra 10
Phase and Group Delays
? In the case of the carrier signal we have
which is seen to be the same as the phase
delay if only the carrier signal is passed
through the system
c
cc
c
cuc
?
????
?
?????? )(
2
)()( ?
Copyright ? 2001,S,K,Mitra 11
Phase and Group Delays
? In the case of the modulating component
we have
? The parameter
is called the group delay or envelope delay
caused by the system at
cd
d c
u
cuc
????
????
???
?????? )()()(
?
?
cd
d c
cg
????
?????? )()(
c???
Copyright ? 2001,S,K,Mitra 12
Phase and Group Delays
? The group delay is a measure of the
linearity of the phase function as a function
of the frequency
? It is the time delay between the waveforms
of underlying continuous-time signals
whose sampled versions,sampled at t = nT,
are precisely the input and the output
discrete-time signals
Copyright ? 2001,S,K,Mitra 13
Phase and Group Delays
? If the phase function and the angular
frequency ? are in radians per second,then
the group delay is in seconds
? Figure below illustrates the evaluation of
the phase delay and the group delay
Copyright ? 2001,S,K,Mitra 14
Phase and Group Delays
? Figure below shows the waveform of an
amplitude-modulated input and the output
generated by an LTI system
Copyright ? 2001,S,K,Mitra 15
Phase and Group Delays
? Note,The carrier component at the output is
delayed by the phase delay and the envelope
of the output is delayed by the group delay
relative to the waveform of the underlying
continuous-time input signal
? The waveform of the underlying continuous-
time output shows distortion when the group
delay of the LTI system is not constant over
the bandwidth of the modulated signal
Copyright ? 2001,S,K,Mitra 16
Phase and Group Delays
? If the distortion is unacceptable,a delay
equalizer is usually cascaded with the LTI
system so that the overall group delay of the
cascade is approximately linear over the band
of interest
? To keep the magnitude response of the parent
LTI system unchanged the equalizer must
have a constant magnitude response at all
frequencies
Copyright ? 2001,S,K,Mitra 17
Phase and Group Delays
? Example - The phase function of the FIR
filter
is
? Hence its group delay is given by
verifying the result obtained earlier by
simulation
][][][][ 21 ????? nxnxnxny ???
????? )(
1)( ??? g
Copyright ? 2001,S,K,Mitra 18
Phase and Group Delays
? Example - For the M-point moving-average
filter
the phase function is
? Hence its group delay is
??
??][nh
o th e r w ise0
101
,
,/ ??? MnM
? ?????? ????????????
? 0
2
2
)1()(
k M
kMM/2
2 1)( ???? Mg
Copyright ? 2001,S,K,Mitra 19
Frequency Response of the
LTI Discrete-Time System
? The convolution sum description of the LTI
discrete-time system is given by
? Taking the DTFT of both sides we obtain
][][][ knxkhny
k
?? ?
?
???
nj
n
j enyeY ???
???
? ?? ][)(
nj
n k
eknxkh ??
?
???
?
???
? ?
?
??
?
? ? ?? ][][
Copyright ? 2001,S,K,Mitra 20
Frequency Response of the
LTI Discrete-Time System
? Or,
? ?
?
??
?
? ?? ?
???
?
???
????
k
kjj exkheY
?
?? )(][][)(
kj
k
j eexkh ???
???
?
???
??? ?
?
??
?
? ??
?
?? ][][
)( ?jeX
Copyright ? 2001,S,K,Mitra 21
Frequency Response of the
LTI Discrete-Time System
? Hence,we can write
? In the above is the frequency
response of the LTI system
? The above equation relates the input and the
output of an LTI system in the frequency
domain
)()()(][)( ???
?
???
??? ??
?
??
?
? ?? jjj
k
kjj eXeHeXekheY
)( ?jeH
Copyright ? 2001,S,K,Mitra 22
Frequency Response of the
LTI Discrete-Time System
? It follows from the previous equation
? For an LTI system described by a linear
constant coefficient difference equation of
the form we have
)(/)()( ??? ? jjj eXeYeH
?
??
?
??
?
??
?
N
k
kj
k
M
k
kj
kj
ed
epeH
0
0)(
Copyright ? 2001,S,K,Mitra 23
The Transfer Function
? A generalization of the frequency response
function
? The convolution sum description of an LTI
discrete-time system with an impulse
response h[n] is given by
??
???
??
k
knxkhny ][][][
Copyright ? 2001,S,K,Mitra 24
The Transfer Function
? Taking the z-transforms of both sides we get
n
n kn
n zknxkhznyzY ??
???
?
???
?
???
? ? ??
???
?
???
? ??? ][][][)(
? ?
?
???
?
???
?
???
?
???
? ??
k n
nzknxkh ][][
? ?
?
???
?
???
??
???
?
???
??
k
kzxkh
?
?? )(][][
Copyright ? 2001,S,K,Mitra 25
The Transfer Function
? Or,
? Therefore,
? Thus,Y(z) = H(z)X(z)
k
k
zzxkhzY ?
?
???
?
???
?? ?
???
?
???
??
?
?? ][][)(
)(zX )(][)( zXzkhzY
k
k
???
?
???
?? ??
???
?
)(zH
Copyright ? 2001,S,K,Mitra 26
The Transfer Function
? Hence,
? The function H(z),which is the z-transform of
the impulse response h[n] of the LTI system,
is called the transfer function or the system
function
? The inverse z-transform of the transfer
function H(z) yields the impulse response h[n]
)(/)()( zXzYzH ?
Copyright ? 2001,S,K,Mitra 27
The Transfer Function
? Consider an LTI discrete-time system
characterized by a difference equation
? Its transfer function is obtained by taking
the z-transform of both sides of the above
equation
? Thus
?? ?? ??? Mk kNk k knxpknyd 00 ][][
?
?
?
?
?
?
? N
k
k
k
M
k
k
k
zd
zp
zH
0
0)(
Copyright ? 2001,S,K,Mitra 28
The Transfer Function
? Or,equivalently as
? An alternate form of the transfer function is
given by
?
?
?
?
?
?
??
N
k
kN
k
M
k
kM
kMN
zd
zp
zzH
0
0)()(
?
?
?
?
?
?
?
?
?? N
k k
M
k k
z
z
d
pzH
1
1
1
1
0
0
1
1
)(
)(
)(
?
?
Copyright ? 2001,S,K,Mitra 29
The Transfer Function
? Or,equivalently as
? are the finite zeros,and
are the finite poles of H(z)
? If N > M,there are additional zeros
at z = 0
? If N < M,there are additional poles
at z = 0
M???,...,,21
N???,...,,21
)( MN ?
)( NM ?
?
?
?
??
?
?
? N
k k
M
k kMN
z
z
z
d
pzH
1
1
0
0
)(
)(
)( )(
?
?
Copyright ? 2001,S,K,Mitra 30
The Transfer Function
? For a causal IIR digital filter,the impulse
response is a causal sequence
? The ROC of the causal transfer function
is thus exterior to a circle going through the
pole farthest from the origin
? Thus the ROC is given by
?
?
?
??
?
?
? N
k k
M
k kMN
z
z
z
d
pzH
1
1
0
0
)(
)(
)( )(
?
?
kkz ?? m a x
Copyright ? 2001,S,K,Mitra 31
The Transfer Function
? Example - Consider the M-point moving-
average FIR filter with an impulse response
? Its transfer function is then given by
??
??][nh
o th e r w ise,0
10,/1 ??? MnM
?
?
?
??
1
0
1)( M
n
nz
M
zH 1111
( 1 ) [ ( 1 ) ]
MM
M
zz
M z M z z
?
??
????
??
Copyright ? 2001,S,K,Mitra 32
The Transfer Function
? The transfer function has M zeros on the
unit circle at,
? There are poles at z = 0 and a single
pole at z = 1
? The pole at z = 1
exactly cancels the
zero at z = 1
? The ROC is the entire
z-plane except z = 0
Mkjez /2 ?? 10 ??? Mk
-1 - 0.5 0 0.5 1
-1
- 0.5
0
0.5
1
R e a l P a r t
I
m
a
gi
na
r
y P
a
r
t
7
M = 8
1?M
Copyright ? 2001,S,K,Mitra 33
The Transfer Function
? Example - A causal LTI IIR digital filter is
described by a constant coefficient
difference equation given by
? Its transfer function is therefore given by
]1[3.1]3[]2[2.1]1[][ ???????? nynxnxnxny
]3[222.0]2[04.1 ???? nyny
321
321
222.004.13.11
2.1)(
???
???
???
???
zzz
zzzzH
Copyright ? 2001,S,K,Mitra 34
The Transfer Function
? Alternate forms,
? Note,Poles farthest from
z = 0 have a magnitude
? ROC,
2 2 2.004.13.1
12.1)(
23
2
???
???
zzz
zzzH
)7.05.0)(7.05.0)(3.0(
)8.06.0)(8.06.0(
jzjzz
jzjz
?????
?????
-1 - 0.5 0 0.5 1
-1
- 0.5
0
0.5
1
R e a l P a r t
I
m
a
gi
na
r
y P
a
r
t
74.0?z
74.0
Copyright ? 2001,S,K,Mitra 35
Frequency Response from
Transfer Function
? If the ROC of the transfer function H(z)
includes the unit circle,then the frequency
response of the LTI digital filter can
be obtained simply as follows,
? For a real coefficient transfer function H(z)
it can be shown that
??? ? jezj zHeH )()(
)( ?jeH
)(*)()( 2 ??? ? jjj eHeHeH
?????? ?? jezjj zHzHeHeH )()()()( 1
Copyright ? 2001,S,K,Mitra 36
Frequency Response from
Transfer Function
? For a stable rational transfer function in the
form
the factored form of the frequency response
is given by
?
?
?
??
?
?
? N
k k
M
k kMN
z
z
z
d
pzH
1
1
0
0
)(
)(
)( )(
?
?
?
?
?
?
?
?
???
??
??
? N
k k
j
M
k k
j
MNjj
e
e
e
d
peH
1
1)(
0
0
)(
)(
)(
Copyright ? 2001,S,K,Mitra 37
Frequency Response from
Transfer Function
? It is convenient to visualize the contributions
of the zero factor and the pole factor
from the factored form of the
frequency response
? The magnitude function is given by
)( kz ??
)( kz ??
?
?
?
?
?
?
???
??
??
? N
k k
j
M
k k
j
MNjj
e
e
e
d
p
eH
1
1)(
0
0)(
Copyright ? 2001,S,K,Mitra 38
Frequency Response from
Transfer Function
which reduces to
? The phase response for a rational transfer
function is of the form
?
?
?
?
?
?
?
??
??
? N
k k
j
M
k k
j
j
e
e
d
p
eH
1
1
0
0)(
)()/a r g ()(a r g 00 MNdpeH j ?????
??
?
?
?
? ??????
N
k
k
j
M
k
k
j ee
11
)a r g ()a r g (
Copyright ? 2001,S,K,Mitra 39
Frequency Response from
Transfer Function
? The magnitude-squared function of a real-
coefficient transfer function can be
computed using
?
?
?
???
?
???
?
????
????
? N
k k
j
k
j
M
k k
j
k
j
j
ee
ee
d
peH
1
*
1
*2
0
02
))((
))((
)(
Copyright ? 2001,S,K,Mitra 40
Geometric Interpretation of
Frequency Response Computation
? The factored form of the frequency response
is convenient to develop a geometric
interpretation of the frequency response
computation from the pole-zero plot as ?
varies from 0 to 2? on the unit circle
?
?
?
?
?
?
???
??
??
? N
k k
j
M
k k
j
MNjj
e
e
e
d
peH
1
1)(
0
0
)(
)(
)(
Copyright ? 2001,S,K,Mitra 41
Geometric Interpretation of
Frequency Response Computation
? The geometric interpretation can be used to
obtain a sketch of the response as a function
of the frequency
? A typical factor in the factored form of the
frequency response is given by
where is a zero if it is zero factor or is
a pole if it is a pole factor
)( ?? ?? jj ee
?? je
Copyright ? 2001,S,K,Mitra 42
Geometric Interpretation of
Frequency Response Computation
? As shown below in the z-plane the factor
represents a vector starting at
the point and ending on the unit
circle at
??? jez
?? jez
)( ?? ?? jj ee
Copyright ? 2001,S,K,Mitra 43
Geometric Interpretation of
Frequency Response Computation
? As ? is varied from 0 to 2?,the tip of the
vector moves counterclockise from the
point z = 1 tracing the unit circle and back
to the point z = 1
Copyright ? 2001,S,K,Mitra 44
Geometric Interpretation of
Frequency Response Computation
? As indicated by
the magnitude response at a
specific value of ? is given by the product
of the magnitudes of all zero vectors
divided by the product of the magnitudes of
all pole vectors
?
?
?
?
?
?
?
??
??
? N
k k
j
M
k k
j
j
e
e
d
p
eH
1
1
0
0)(
|)(| ?jeH
Copyright ? 2001,S,K,Mitra 45
Geometric Interpretation of
Frequency Response Computation
? Likewise,from
we observe that the phase response
at a specific value of ? is obtained by
adding the phase of the term and the
linear-phase term to the sum of
the angles of the zero vectors minus the
angles of the pole vectors
00 dp /
)( MN ??
)()/a r g ()(a r g 00 MNdpeH j ?????
? ???? ??? ? ?? ? Nk kjMk kj ee 11 )a r g ()a r g (
Copyright ? 2001,S,K,Mitra 46
Geometric Interpretation of
Frequency Response Computation
? Thus,an approximate plot of the magnitude
and phase responses of the transfer function
of an LTI digital filter can be developed by
examining the pole and zero locations
? Now,a zero (pole) vector has the smallest
magnitude when ? = ?
Copyright ? 2001,S,K,Mitra 47
Geometric Interpretation of
Frequency Response Computation
? To highly attenuate signal components in a
specified frequency range,we need to place
zeros very close to or on the unit circle in
this range
? Likewise,to highly emphasize signal
components in a specified frequency range,
we need to place poles very close to or on
the unit circle in this range
