Chapter 4
Frequency-domain
Representation of LTI
Discrete-Time Systems
§ 4.1 LTI Discrete-Time Systems
in the Transform Domain
Such transform-domain representations
provide additional insight into the behavior of
such systems
It is easier to design and implement these
systems in the transform-domain for certain
applications
We consider now the use of the DTFT and the
z-transform in developing the transform-
domain representations of an LTI system
§ 4.1 LTI Discrete-Time Systems
in the Transform Domain
In this course we shall be concerned
with LTI discrete-time systems
characterized by linear constant
coefficient difference equations of the
form:
M
k
k
N
k
k knxpknyd
00
][][
§ 4.1 LTI Discrete-Time Systems
in the Transform Domain
Applying the DTFT to the difference equation
and making use of the linearity and the time-
invariance properties we arrive at the input-
output relation in the transform-domain as
)()(
00
jM
k
kj
k
jN
k
kj
k eXepeYed
where Y(ej?) and X(ej?) are the DTFTs of y[n]
and x[n],respectively
§ 4.1 LTI Discrete-Time Systems
in the Transform Domain
In developing the transform-domain
representation of the difference equation,
it has been tacitly assumed that X(ej?)
and Y(ej?) exist
The previous equation can be alternately
written as
)()(
00
jM
k
kj
k
jN
k
kj
k eXepeYed
§ 4.1 LTI Discrete-Time Systems
in the Transform Domain
Applying the z-transform to both sides of the
difference equation and making use of the
linearity and the time-invariance properties
we arrive at
)()( zXzpzYzd
M
k
k
k
N
k
k
k
00
where Y(z) and X(z) denote the z-transforms of
y[n] and x[n] with associated ROCs,respectively
§ 4.1 LTI Discrete-Time Systems
in the Transform Domain
A more convenient form of the z-domain
representation of the difference equation
is given by
)()( zXzpzYzd
M
k
k
k
N
k
k
k
00
§ 4.2 The Frequency
Response
Most discrete-time signals encountered
in practice can be represented as a
linear combination of a very large,
maybe infinite,number of sinusoidal
discrete-time signals of different angular
frequencies
Thus,knowing the response of the LTI
system to a single sinusoidal signal,we
can determine its response to more
complicated signals by making use of the
superposition property
§ 4.2 The Frequency
Response
The quantity H(ej?) is called the
frequency response of the LTI discrete-
time system
H(ej?) provides a frequency-domain
description of the system
H(ej?) is precisely the DTFT of the
impulse response {h[n]} of the system
§ 4.2 The Frequency
Response
H(ej?),in general,is a complex function of?
with a period 2p
It can be expressed in terms of its real and
imaginary parts
H(ej?)= Hre(ej?) +j Him(ej?)
or,in terms of its magnitude and phase,
H(ej?)=|H(ej?)| e?(?)
where
(?)=argH(ej?)
§ 4.2 The Frequency
Response
The function | H(ej?) | is called the magnitude
response and the function?(?) is called the
phase response of the LTI discrete-time
system
Design specifications for the LTI discrete-time
system,in many applications,are given in
terms of the magnitude response or the phase
response or both
§ 4.2 The Frequency
Response
In some cases,the magnitude function is
specified in decibels as
G(?) = 20log10| H(ej?) | dB
where G(?) is called the gain function
The negative of the gain function
A(?) = - G(?)
is called the attenuation or loss function
§ 4.2 The Frequency
Response
Note,Magnitude and phase functions are real
functions of?,whereas the frequency
response is a complex function of?
If the impulse response h[n] is real then the
magnitude function is an even function of?:
|H(ej?)| = |H(e - j?)|
and the phase function is an odd function of?:
(?) = -?(-?)
§ 4.3 Frequency Response
Computation Using MATLAB
The function freqz(h,w) can be used to
determine the values of the frequency
response vector h at a set of given frequency
points w
From h,the real and imaginary parts can be
computed using the functions real and imag,
and the magnitude and phase functions using
the functions abs and angle
§ 4.3 Frequency Response
Computation Using MATLAB
Example - Program 4_1(p.206) can be
used to generate the magnitude and gain
responses of an M-point moving average
filter as shown below
§ 4.3 Frequency Response
Computation Using MATLAB
The phase response of a discrete-time
system when determined by a computer
may exhibit jumps by an amount 2p
caused by the way the arctangent
function is computed
The phase response can be made a
continuous function of? by unwrapping
the phase response across the jumps
§ 4.3 Frequency Response
Computation Using MATLAB
To this end the function unwrap can be
used,provided the computed phase is in
radians
The jumps by the amount of 2p should
not be confused with the jumps caused
by the zeros of the frequency response as
indicated in the phase response of the
moving average filter
§ 4.4 The Concept of Filtering
One application of an LTI discrete-time
system is to pass certain frequency
components in an input sequence
without any distortion (if possible) and
to block other frequency components
Such systems are called digital filters
and one of the main subjects of
discussion in this course
§ 4.4 The Concept of Filtering
The key to the filtering process is
p
p?
p deeXnx
njj )(][
2
1
It expresses an arbitrary input as a
linear weighted sum of an infinite number
of exponential sequences,or equivalently,
as a linear weighted sum of sinusoidal
sequences
§ 4.4 The Concept of Filtering
Thus,by appropriately choosing the
values of the magnitude function |H(ej?)|
of the LTI digital filter at frequencies
corresponding to the frequencies of the
sinusoidal components of the input,
some of these components can be
selectively heavily attenuated or filtered
with respect to the others
§ 4.4 The Concept of Filtering
To understand the mechanism behind
the design of frequency-selective filters,
consider a real-coefficient LTI discrete-
time system characterized by a
magnitude function
)( jeH
p
c
c
,0
,1
§ 4.4 The Concept of Filtering
We apply an input
x[n]=Acos?1n+Bcos?2n,0<?1<?c<?2<p
to this system
Because of linearity,the output of this
system is of the form
)(c o s)(][ 111 neHAny j
)(c o s)( 222 neHB j
§ 4.4 The Concept of Filtering
As
0)(,1)( 21 jj eHeH
)(c o s)(][ 111 neHAny j
Thus,the system acts like a lowpass
filter
In the following example,we consider
the design of a very simple digital filter
the output reduces to
§ 4.4 The Concept of Filtering
Example - The input consists of a sum of two
sinusoidal sequences of angular frequencies
0.1 rad/sample and 0.4 rad/sample
We need to design a highpass filter that will
pass the high-frequency component of the
input but block the low-frequency component
For simplicity,assume the filter to be an FIR
filter of length 3 with an impulse response,
h[0] = h[2] = a,h[1] = b
§ 4.4 The Concept of Filtering
The convolution sum description of this filter
is then given by
y[n]=h[0]x[n]+h[1]x[n-1]+h[2]x[n-2]
=ax[n] +bx[n-1]+ ax[n-2]
y[n] and x[n] are,respectively,the output and
the input sequences
Design Objective,Choose suitable values of a
and b so that the output is a sinusoidal
sequence with a frequency 0.4 rad/sample
§ 4.4 The Concept of Filtering
Now,the frequency response of the FIR
filter is given by
2]2[]1[]0[)( jjj ehehheH
ba? jj ee )1( 2
b
a? jjjj eeee
2
2
ba? je)c o s2(
§ 4.4 The Concept of Filtering
The magnitude and phase functions are
|H(ej?)|=2acos?+b
(?) = -?
In order to block the low-frequency
component,the magnitude function at
= 0.1 should be equal to zero
Likewise,to pass the high-frequency
component,the magnitude function at? = 0.4
should be equal to one
§ 4.4 The Concept of Filtering
Thus,the two conditions that must be
satisfied are
|H(ej0.1)|=2acos(0.1)+b = 0
|H(ej0.4)|=2acos(0.4)+b = 1
Solving the above two equations we get
a = -6.76195
b = 13.456335
§ 4.4 The Concept of Filtering
Thus the output-input relation of the
FIR filter is given by
y[n] = - 6.76195(x[n]+x[n-2])+13.456335x[n-2]
where the input is
x[n] = {cos(0.1n) + cos(0.4n)}?[n]
Program 4_2 can be used to verify the
filtering action of the above system
§ 4.4 The Concept of Filtering
Figure below shows the plots generated
by running this program
§ 4.4 The Concept of Filtering
The first seven samples of the output are
shown below
§ 4.4 The Concept of Filtering
From this table,it can be seen that,neglecting
the least significant digit,
y[n] = cos(0.4(n-1)) for n? 2
Computation of the present value of the
output requires the knowledge of the present
and two previous input samples
Hence,the first two output samples,y[0] and
y[1],are the result of assumed zero input
sample values at n = -1 and n = -2
§ 4.4 The Concept of Filtering
Therefore,first two output samples
constitute the transient part of the
output
Since the impulse response is of length 3,
the steady-state is reached at n = N = 2
Note also that the output is delayed
version of the high-frequency
component cos(0.4n) of the input,and
the delay is one sample period
§ 4.5 Phase and Group Delays
The output y[n] of a frequency-selective
LTI discrete-time system with a
frequency response H(ej?) exhibits
some delay relative to the input x[n]
caused by the nonzero phase response
(?)=arg{H(ej?)} of the system
For an input
nnAnx o ),c o s (][
§ 4.5 Phase and Group Delays
Thus,the output lags in phase by?(?0)
radians
Rewriting the above equation we get
))(c o s ()(][ ooj neHAny o
o
o
o
j neHAny o )(c o s)(][
the output is
§ 4.5 Phase and Group Delays
This expression indicates a time delay,
known as phase delay,at?=?0 given by
o
oop
)()(
Now consider the case when the input
signal contains many sinusoidal
components with different frequencies that
are not harmonically related
§ 4.5 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
§ 4.5 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?c
of a low-frequency sinusoidal signal of
frequency?0,
)c o s ()c o s (][ nnAnx co
§ 4.5 Phase and Group Delays
where?l =?c -?0 and?u =?c+?0
Let the above input be processed by an
LTI discrete-time system with a
frequency response H(ej?) satisfying the
condition
)c o s ()c o s (][ 22 nnnx uAA
ujeHf o r1)(
The input can be rewritten as
§ 4.5 Phase and Group Delays
Note,The output is also in the form of a
modulated carrier signal with the same
carrier frequency?c and the same
modulation frequency?0 as the input
))(c o s ()(c o s][ 22 uuAA nnny
2 )()(c o s2 )()(c o s uouc nnA
The output y[n] is then given by
§ 4.5 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?l and?u very close
to the carrier frequency?c,i.e,?0 is
very small
§ 4.5 Phase and Group Delays
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
)()()()( ccccc
cd
d
In the neighborhood of?c we can express the
unwrapped phase response?c(?) as
§ 4.5 Phase and Group Delays
In the case of the carrier signal we have
c
cc
c
cuc
)(
2
)()(?
which is seen to be the same as the phase
delay if only the carrier signal is passed
through the system
§ 4.5 Phase and Group Delays
In the case of the modulating component we
have
cd
d c
u
cuc
)()()(
cd
d c
cg
)()(
is called the group delay or envelope delay
caused by the system at?=?c
The parameter
§ 4.5 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
§ 4.5 Phase and Group Delays
If the phase function and the angular
frequency? are in radians per second,
then the group delay is in seconds
Left figure
illustrates the
evaluation of the
phase delay and
the group delay
§ 4.5 Phase and Group Delays
Left figure
shows the
waveform of
an amplitude-
modulated
input and the
output
generated by
an LTI system
§ 4.5 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
§ 4.5 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
§ 4.5 Phase and Group Delays
Example - The phase function of the FIR
filter y[n] = ax[n] + bx[n-1] + ax[n-2]
is?(?) = -?
Hence its group delay is given by
g(?)=1 verifying the result obtained
earlier by simulation
§ 4.6 Frequency Response of
the LTI Discrete-Time System
In the above H(e j?) 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
Hence,we can write
§ 4.6 Frequency Response of
the LTI Discrete-Time System
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)(
It follows from the previous equation
§ 4.7 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 ][][][
§ 4.7 The Transfer Function
Taking the z-transforms of both sides we get n
n kn
n zknxkhznyzY
][][][)(
k n
nzknxkh ][][
k
kzxkh
)(][][
§ 4.8 The Transfer Function
Thus,Y(z) = H(z)X(z)
k
k
zzxkhzY?
][][)(
)(zX
)(][)( zXzkhzY
k
k
)(zH
Or,
Therefore,
§ 4.8 The Transfer Function
Hence,
H(z) = Y(Z)/X(z)
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]
§ 4.8 The Transfer Function
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)(
Consider an LTI discrete-time system
characterized by a difference equation
§ 4.8 The Transfer Function
Or,equivalently as
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
)(
)(
)(
An alternate form of the transfer
function is given by
§ 4.8 The Transfer Function
1,?2,…,?M are the finite zeros,and?1,? 2,…,
N are the finite poles of H(z)
If N > M,there are additional (N-M) zeros at
z = 0
If N < M,there are additional (M-N) poles at
z = 0
N
k k
M
k kMN
z
z
z
d
pzH
1
1
0
0
)(
)(
)( )(
Or,equivalently as
§ 4.8 The Transfer Function
For a causal IIR digital filter,the
impulse response is a causal sequence
The ROC of the causal transfer function
N
k k
M
k kMN
z
z
z
d
pzH
1
1
0
0
)(
)(
)( )(
kkz m a xThus the ROC is given by
is thus exterior to a circle going through
the pole furthest from the origin
§ 4.8 The Transfer Function
Example - Consider the M-point moving-
average FIR filter with an impulse response
][nh
o th e r w ise,0
10,/1 MnM
Its transfer function is then given by
1
0
1)( M
n
nz
M
zH
)]1([
1
)1(
1
1?
zzM
z
zM
z
M
MM
§ 4.8 The Transfer Function
The transfer function has M zeros on the unit
circle at z=ej2pk/M,0? k? M-1
There are M-1 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
M = 8
§ 4.8 The Transfer Function
Example - A causal LTI IIR digital filter
is described by a constant coefficient
difference equation given by
y[n]=x[n-1]-1.2x[n-2]+x[n-3]+1.3y[n-1]
-1.04y[n-2]+0.222y[n-3]
Its transfer function is therefore given
by
321
321
222.004.13.11
2.1)(
zzz
zzzzH
§ 4.8 The Transfer Function
Alternate forms:
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
74.0?zROC:
74.0
Note,Poles farthest from
z=0 have a magnitude
§ 4.9 Frequency Response from
Transfer Function
If the ROC of the transfer function H(z)
includes the unit circle,then the frequency
response H(ej?) of the LTI digital filter can be
obtained simply as follows:
jezj zHeH )()(
)(*)()( 2 jjj eHeHeH
jezjj zHzHeHeH )()()()( 1
For a real coefficient transfer function H(z) it
can be shown that
§ 4.9 Frequency Response from
Transfer Function
For a stable rational transfer function in
the form
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
)(
)(
)(
the factored form of the frequency
response is given by
§ 4.9 Frequency Response from
Transfer Function
It is convenient to visualize the contributions
of the zero factor (z-?k) and the pole factor
(z-?k) from the factored form of the frequency
response
The magnitude function is given by
N
k k
j
M
k k
j
MNjj
e
e
e
d
peH
1
1)(
0
0)(
N
k k
j
M
k k
j
e
e
d
p
1
1
0
0
§ 4.10 Types of Transfer Functions
The time-domain classification of an
LTI digital transfer function sequence is
based on the length of its impulse
response:
- Finite impulse response (FIR) transfer
function
- Infinite impulse response (IIR) transfer
function
§ 4.10 Types of Transfer Functions
Several other classifications are also used
In the case of digital transfer functions with
frequency-selective frequency responses,one
classification is based on the shape of the
magnitude function H(ei?) or the form of the
phase function?(?)
Based on this four types of ideal filters are
usually defined
§ 4.10.1 Ideal Filters
A digital filter designed to pass signal
components of certain frequencies
without distortion should have a
frequency response equal to one at these
frequencies,and should have a
frequency response equal to zero at all
other frequencies
§ 4.10.1 Ideal Filters
The range of frequencies where the
frequency response takes the value of
one is called the passband
The range of frequencies where the
frequency response takes the value of
zero is called the stopband
§ 4.10.1 Ideal Filters
Frequency responses of the four popular types
of ideal digital filters with real impulse
response coefficients are shown below:
§ 4.10.1 Ideal Filters
The frequencies?c,?c1,and?c2 are
called the cutoff frequencies
An ideal filter has a magnitude response
equal to one in the passband and zero in
the stopband,and has a zero phase
everywhere
§ 4.10.1 Ideal Filters
Earlier in the course we derived the inverse
DTFT of the frequency response HLP(ej?)
of the ideal lowpass filter:
hLP[n]=sin?cn/np,-? <n<?
We have also shown that the above impulse
response is not absolutely summable,and
hence,the corresponding transfer function is
not BIBO stable
§ 4.10.1 Ideal Filters
Also,hLP[n] is not causal and is of doubly
infinite length
The remaining three ideal filters are also
characterized by doubly infinite,noncausal
impulse responses and are not absolutely
summable
Thus,the ideal filters with the ideal,brick
wall” frequency responses cannot be realized
with finite dimensional LTI filter
§ 4.10.1 Ideal Filters
To develop stable and realizable transfer
functions,the ideal frequency response
specifications are relaxed by including a
transition band between the passband and the
stopband
This permits the magnitude response to decay
slowly from its maximum value in the
passband to the zero value in the stopband
§ 4.10.1 Ideal Filters
Moreover,the magnitude response is
allowed to vary by a small amount both
in the passband and the stopband
Typical magnitude
response
specifications
of a lowpass filter
are shown as:
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
A second classification of a transfer
function is with respect to its phase
characteristics
In many applications,it is necessary that
the digital filter designed does not
distort the phase of the input signal
components with frequencies in the
passband
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
One way to avoid any phase distortion is
to make the frequency response of the
filter real and nonnegative,i.e.,to design
the filter with a zero phase characteristic
However,it is impossible to design a
causal digital filter with a zero phase
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
For non-real-time processing of real-valued
input signals of finite length,zero-phase
filtering can be very simply implemented by
relaxing the causality requirement
One zero-phase filtering scheme is sketched
below
x[n] v[n] u[n] w[n]H(z) H(z)
u[n]=v[-n],y[n]=w[-n]
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
It is easy to verify the above scheme in the
frequency domain
Let X(ej?),V(ej?),U(ej?),W(ej?),and Y(ej?)
denote the DTFTs of x[n],v[n],u[n],w[n],and
y[n],respectively
From the figure shown earlier and making use
of the symmetry relations we arrive at the
relations between various DTFTs as given on
the next slide
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
V(ej?)= H(ej?)X(ej?),W(ej?)=H(ej?)U(ej?)
U(ej?)= V*(ej?),Y(ej?)= W*(ej?)
Combining the above equations we get
Y(ej?) = W*(ej?) = H*(ej?)U*(ej?)
= H*(ej?)V(ej?) = H*(ej?)H(ej?)X(ej?)
= |H(ej?)|2X(ej?)
This is a zero-phase filter with a frequency response
|H(ej?)|2
x[n] v[n] u[n] w[n]H(z) H(z)
u[n]=v[-n],y[n]=w[-n]
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
The function fftfilt implements the above zero-
phase filtering scheme
In the case of a causal transfer function with a
nonzero phase response,the phase distortion
can be avoided by ensuring that the transfer
function has a unity magnitude and a linear-
phase characteristic in the frequency band of
interest
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
The most general type of a filter with a
linear phase has a frequency response
given by
H(ej?)= ej?D
which has a linear phase from? = 0 to
= 2p
Note also |H(ej?)|=1
(?)=D
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
The output y[n] of this filter to an input
x[n]=Aej?n is then given by
y[n]= Ae-j?Dej?n = Aej?(n-D)
If xa(t) and ya(t) represent the
continuous-time signals whose sampled
versions,sampled at t = nT,are x[n] and
y[n] given above,then the delay between
xa(t) and ya(t) is precisely the group
delay of amount D
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
If D is an integer,then y[n] is identical to
x[n],but delayed by D samples
If D is not an integer,y[n],being delayed
by a fractional part,is not identical to
x[n]
In the latter case,the waveform of the
underlying continuous-time output is
identical to the waveform of the
underlying continuous-time input and
delayed D units of time
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
If it is desired to pass input signal
components in a certain frequency
range undistorted in both
magnitude and phase,then the
transfer function should exhibit a
unity magnitude response and a
linear-phase response in the band of
interest
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
Figure right
shows the
frequency
response if a
lowpass filter with
a linear-phase
characteristic in
the passband
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
Since the signal components in the
stopband are blocked,the phase
response in the stopband can be of any
shape
Example - Determine the impulse
response of an ideal lowpass filter with a
linear phase response:
)( j
LP eH p
c
c
nj oe
,0
0,
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
Applying the frequency-shifting
property of the DTFT to the impulse
response of an ideal zero-phase lowpass
filter we arrive at
p nnn nnnh
o
ocLP,
)(
)(si n][
As before,the above filter is noncausal
and of doubly infinite length,and hence,
unrealizable
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
By truncating the impulse response
to a finite number of terms,a
realizable FIR approximation to the
ideal lowpass filter can be
developed
The truncated approximation may
or may not exhibit linear phase,
depending on the value of n0 chosen
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
If we choose n0= N/2 with N a
positive integer,the truncated and
shifted approximation
NnNn Nnnh cLPp 0,)2/( )2/(si n][^
will be a length N+1 causal linear-
phase FIR filter
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
Figure below shows the filter coefficients
obtained using the function sinc for two
different values of N
N=12 N=13
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
Because of the symmetry of the impulse
response coefficients as indicated in the two
figures,the frequency response of the
truncated approximation can be expressed as:
)(][)( 2/
0
LP
Nj
N
n
nj
LP
j
LP HeenheH ^^
~
where,called the zero-phase
response or amplitude response,is a
real function of?
)(?LPH~
§ 4.11 Linear-Phase FIR
Transfer Functions
It is nearly impossible to design a linear-
phase IIR transfer function
It is always possible to design an FIR
transfer function with an exact linear-
phase response
Consider a causal FIR transfer function
H(z) of length N+1,i.e.,of order N:
Nn nznhzH 0 ][)(
§ 4.11 Linear-Phase FIR
Transfer Functions
The above transfer function has a linear
phase,if its impulse response h[n] is
either symmetric,i.e.,
h[n]=h[N-n],0?n?N
or is antisymmetric,i.e.,
h[n]=-h[N-n],0?n?N
§ 4.11 Linear-Phase FIR
Transfer Functions
Since the length of the impulse response
can be either even or odd,we can define
four types of linear-phase FIR transfer
functions
For an antisymmetric FIR filter of odd
length,i.e.,N even
h[N/2] = 0
We examine next the each of the 4 cases
§ 4.11 Linear-Phase FIR
Transfer Functions
Type 1,N = 8 Type 2,N = 7
Type 3,N = 8 Type 4,N = 7
§ 4.11 Linear-Phase FIR
Transfer Functions
Type 1,Symmetric Impulse Response with
Odd Length
In this case,the degree N is even
Assume N = 8 for simplicity
The transfer function H(z) is given by
321 3210 zhzhzhhzH ][][][][)(
87654 87654 zhzhzhzhzh ][][][][][
§ 4.11 Linear-Phase FIR
Transfer Functions
Because of symmetry,we have h[0]=h[8],
h[1] = h[7],h[2] = h[6],and h[3] = h[5]
Thus,we can write
)]([)]([)( 718 110 zzhzhzH 45362
432 zhzzhzzh ][)]([)]([
)]([)]([{ 33444 10 zzhzzhz
]}[)]([)]([ 432 122 hzzhzzh
§ 4.11 Linear-Phase FIR
Transfer Functions
The corresponding frequency response
is then given by
)3c o s (]1[2)4c o s (]0[2{)( 4 hheeH jj
]}4[)c o s (]3[2)2c o s (]2[2 hhh
The quantity inside the braces is a real
function of?,and can assume positive or
negative values in the range 0?|?|?p
§ 4.11 Linear-Phase FIR
Transfer Functions
where b is either 0 or p,and hence,it is a
linear function of? in the generalized sense
The group delay is given by
b 4)(
4)( )(dd
indicating a constant group delay of 4 samples
The phase function here is given by
§ 4.11 Linear-Phase FIR
Transfer Functions
In the general case for Type 1 FIR filters,
the frequency response is of the form
)()( 2/ HeeH jNj
)(?H~
2/
1 22
)c o s (][2][
N
n
NN nnhh
)(?H~where the amplitude response,also
called the zero-phase response,is of the
form
§ 4.11 Linear-Phase FIR
Transfer Functions
which is seen to be a slightly modified
version of a length-7 moving-average
FIR filter
The above transfer function has a
symmetric impulse response and
therefore a linear phase response
][)( 6215432121610 zzzzzzzH
Example - Consider
§ 4.11 Linear-Phase FIR
Transfer Functions
A plot of the magnitude response of
along with that of the 7-point moving-
average filter is shown below
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
/p
Ma
gn
itu
de
modified filter
moving-average
§ 4.11 Linear-Phase FIR
Transfer Functions
Note the improved magnitude response
obtained by simply changing the first and the
last impulse response coefficients of a moving-
average (MA) filter
It can be shown that we can express
)()()( 54321611210 11 zzzzzzzH
which is seen to be a cascade of a 2-point MA
filter with a 6-point MA filter
Thus,H0(z) has a double zero at z=-1,i.e.,
(? = p)
§ 4.12 Allpass Transfer Function
is called an allpass transfer function
An M-th order causal real-coefficient allpass
transfer function is of the form
a l lf o r,1|)(| 2jeA
M
M
M
M
MM
MM
M zdzdzd
zzdzddzA
1
1
1
1
1
1
1
1
1,..
...
)(
Definition,An IIR transfer function A(z) with
unity magnitude response for all frequencies,
i.e.,
§ 4.12 Allpass Transfer Function
If we denote the denominator polynomials of
AM(z) as DM(z),
MMMMM zdzdzdzD 11111,..)(
)(
)()(
zD
zDz
M M M
M
zA
1
Note from the above that if z=rej? is a pole of a
real coefficient allpass transfer function,then it
has a zero at z=(1/r)e-j?
then it follows that AM(z) can be written as:
§ 4.12 Allpass Transfer Function
The numerator of a real-coefficient
allpass transfer function is said to be the
mirror-image polynomial of the
denominator,and vice versa
)()( zDzzD MMM
~
)(zDM~? We shall use the notation to
denote the mirror-image polynomial of a
degree-M polynomial DM(z),i.e.,
§ 4.12 Allpass Transfer Function
)(
)()(
zD
zDz
M M M
MzA 1
321
321
3 2.018.04.01
4.018.02.0)(
zzz
zzzzA? The expression implies that the
poles and zeros
of a real-
coefficient
allpass function
exhibit mirror-
image symmetry
in the z-plane
§ 4.12 Allpass Transfer Function
Therefore
)(
)(1
1)(
zD
zDz
M M M
MzA
)(
)(
)(
)(1
1
1)()(
zD
zDz
zD
zDz
MM M M
M
M
MMzAzA
1)()(|)(| 12 jezMMjM zAzAeA
Hence
1|)(|jM eA? To show that we observe
that
§ 4.12 Allpass Transfer Function
Now,the poles of a causal stable transfer
function must lie inside the unit circle in
the z-plane
Hence,all zeros of a causal stable allpass
transfer function must lie outside the
unit circle in a mirror-image symmetry
with its poles situated inside the unit
circle
§ 4.12 Allpass Transfer Function
Figure below shows the principal value of the
phase of the 3rd-order allpass function
321
321
3 2.018.04.01
4.018.02.0)(
zzz
zzzzA
0 0.2 0.4 0.6 0.8 1-4
-2
0
2
4
/p
Ph
ase
,de
gre
es
Principal value of phase
§ 4.12 Allpass Transfer Function
A Simple Application
A simple but often used application of an
allpass filter is as a delay equalizer
Let G(z) be the transfer function of a digital
filter designed to meet a prescribed magnitude
response
The nonlinear phase response of G(z) can be
corrected by cascading it with an allpass filter
A(z) so that the overall cascade has a constant
group delay in the band of interest
§ 4.12 Allpass Transfer Function
G(z) A(z)
|)(||)()(| jjj eGeAeG
Overall group delay is the given by the
sum of the group delays of G(z) and A(z)
1|)(|jeA? Since,we have
§ 4.13 Minimum-Phase and
Maximum-Phase Transfer Functions
Both transfer functions have a pole inside the
unit circle at the same location z=-a and are
stable
But the zero of H1(z) is inside the unit circle
at z=-b,whereas,the zero of H2(z) is at z=1/b
situated in a mirror-image symmetry
11121 bazHzH azbzaz bz,,)(,)(
Consider the two 1st-order transfer functions:
§ 4.13 Minimum-Phase and
Maximum-Phase Transfer Functions
Figure below shows the pole-zero plots
of the two transfer functions
H1(z) H2(z)
§ 4.13 Minimum-Phase and
Maximum-Phase Transfer Functions
However,both transfer functions have
an identical magnitude function as
1122111 )()()()( zHzHzHzH
c o ss i n1c o ss i n11 t a nt a n)](a r g [ abjeH
c o ss i n1c o s1 s i n12 t a nt a n)](a r g [ abbjeH
The corresponding phase functions are
§ 4.13 Minimum-Phase and
Maximum-Phase Transfer Functions
Figure below shows the unwrapped
phase responses of the two transfer
functions for a = 0.8 and b =-0.5
§ 4.13 Minimum-Phase and
Maximum-Phase Transfer Functions
From this figure it follows that H2(z) has an
excess phase lag with respect to H1(z)
Generalizing the above result,we can show
that a causal stable transfer function with all
zeros outside the unit circle has an excess
phase compared to a causal transfer function
with identical magnitude but having all zeros
inside the unit circle
§ 4.13 Minimum-Phase and
Maximum-Phase Transfer Functions
A causal stable transfer function with all zeros
inside the unit circle is called a minimum-
phase transfer function
A causal stable transfer function with all zeros
outside the unit circle is called a maximum-
phase transfer function
Any nonminimum-phase transfer function can
be expressed as the product of a minimum-
phase transfer function and a stable allpass
transfer function
Homework
Read the textbook from p.203 to 233,
and from 243 to 248
Problems
4.2,4.5,4.7,4.8,4.12,4.19,4.26
M4.1,M4.3,M4.6
Frequency-domain
Representation of LTI
Discrete-Time Systems
§ 4.1 LTI Discrete-Time Systems
in the Transform Domain
Such transform-domain representations
provide additional insight into the behavior of
such systems
It is easier to design and implement these
systems in the transform-domain for certain
applications
We consider now the use of the DTFT and the
z-transform in developing the transform-
domain representations of an LTI system
§ 4.1 LTI Discrete-Time Systems
in the Transform Domain
In this course we shall be concerned
with LTI discrete-time systems
characterized by linear constant
coefficient difference equations of the
form:
M
k
k
N
k
k knxpknyd
00
][][
§ 4.1 LTI Discrete-Time Systems
in the Transform Domain
Applying the DTFT to the difference equation
and making use of the linearity and the time-
invariance properties we arrive at the input-
output relation in the transform-domain as
)()(
00
jM
k
kj
k
jN
k
kj
k eXepeYed
where Y(ej?) and X(ej?) are the DTFTs of y[n]
and x[n],respectively
§ 4.1 LTI Discrete-Time Systems
in the Transform Domain
In developing the transform-domain
representation of the difference equation,
it has been tacitly assumed that X(ej?)
and Y(ej?) exist
The previous equation can be alternately
written as
)()(
00
jM
k
kj
k
jN
k
kj
k eXepeYed
§ 4.1 LTI Discrete-Time Systems
in the Transform Domain
Applying the z-transform to both sides of the
difference equation and making use of the
linearity and the time-invariance properties
we arrive at
)()( zXzpzYzd
M
k
k
k
N
k
k
k
00
where Y(z) and X(z) denote the z-transforms of
y[n] and x[n] with associated ROCs,respectively
§ 4.1 LTI Discrete-Time Systems
in the Transform Domain
A more convenient form of the z-domain
representation of the difference equation
is given by
)()( zXzpzYzd
M
k
k
k
N
k
k
k
00
§ 4.2 The Frequency
Response
Most discrete-time signals encountered
in practice can be represented as a
linear combination of a very large,
maybe infinite,number of sinusoidal
discrete-time signals of different angular
frequencies
Thus,knowing the response of the LTI
system to a single sinusoidal signal,we
can determine its response to more
complicated signals by making use of the
superposition property
§ 4.2 The Frequency
Response
The quantity H(ej?) is called the
frequency response of the LTI discrete-
time system
H(ej?) provides a frequency-domain
description of the system
H(ej?) is precisely the DTFT of the
impulse response {h[n]} of the system
§ 4.2 The Frequency
Response
H(ej?),in general,is a complex function of?
with a period 2p
It can be expressed in terms of its real and
imaginary parts
H(ej?)= Hre(ej?) +j Him(ej?)
or,in terms of its magnitude and phase,
H(ej?)=|H(ej?)| e?(?)
where
(?)=argH(ej?)
§ 4.2 The Frequency
Response
The function | H(ej?) | is called the magnitude
response and the function?(?) is called the
phase response of the LTI discrete-time
system
Design specifications for the LTI discrete-time
system,in many applications,are given in
terms of the magnitude response or the phase
response or both
§ 4.2 The Frequency
Response
In some cases,the magnitude function is
specified in decibels as
G(?) = 20log10| H(ej?) | dB
where G(?) is called the gain function
The negative of the gain function
A(?) = - G(?)
is called the attenuation or loss function
§ 4.2 The Frequency
Response
Note,Magnitude and phase functions are real
functions of?,whereas the frequency
response is a complex function of?
If the impulse response h[n] is real then the
magnitude function is an even function of?:
|H(ej?)| = |H(e - j?)|
and the phase function is an odd function of?:
(?) = -?(-?)
§ 4.3 Frequency Response
Computation Using MATLAB
The function freqz(h,w) can be used to
determine the values of the frequency
response vector h at a set of given frequency
points w
From h,the real and imaginary parts can be
computed using the functions real and imag,
and the magnitude and phase functions using
the functions abs and angle
§ 4.3 Frequency Response
Computation Using MATLAB
Example - Program 4_1(p.206) can be
used to generate the magnitude and gain
responses of an M-point moving average
filter as shown below
§ 4.3 Frequency Response
Computation Using MATLAB
The phase response of a discrete-time
system when determined by a computer
may exhibit jumps by an amount 2p
caused by the way the arctangent
function is computed
The phase response can be made a
continuous function of? by unwrapping
the phase response across the jumps
§ 4.3 Frequency Response
Computation Using MATLAB
To this end the function unwrap can be
used,provided the computed phase is in
radians
The jumps by the amount of 2p should
not be confused with the jumps caused
by the zeros of the frequency response as
indicated in the phase response of the
moving average filter
§ 4.4 The Concept of Filtering
One application of an LTI discrete-time
system is to pass certain frequency
components in an input sequence
without any distortion (if possible) and
to block other frequency components
Such systems are called digital filters
and one of the main subjects of
discussion in this course
§ 4.4 The Concept of Filtering
The key to the filtering process is
p
p?
p deeXnx
njj )(][
2
1
It expresses an arbitrary input as a
linear weighted sum of an infinite number
of exponential sequences,or equivalently,
as a linear weighted sum of sinusoidal
sequences
§ 4.4 The Concept of Filtering
Thus,by appropriately choosing the
values of the magnitude function |H(ej?)|
of the LTI digital filter at frequencies
corresponding to the frequencies of the
sinusoidal components of the input,
some of these components can be
selectively heavily attenuated or filtered
with respect to the others
§ 4.4 The Concept of Filtering
To understand the mechanism behind
the design of frequency-selective filters,
consider a real-coefficient LTI discrete-
time system characterized by a
magnitude function
)( jeH
p
c
c
,0
,1
§ 4.4 The Concept of Filtering
We apply an input
x[n]=Acos?1n+Bcos?2n,0<?1<?c<?2<p
to this system
Because of linearity,the output of this
system is of the form
)(c o s)(][ 111 neHAny j
)(c o s)( 222 neHB j
§ 4.4 The Concept of Filtering
As
0)(,1)( 21 jj eHeH
)(c o s)(][ 111 neHAny j
Thus,the system acts like a lowpass
filter
In the following example,we consider
the design of a very simple digital filter
the output reduces to
§ 4.4 The Concept of Filtering
Example - The input consists of a sum of two
sinusoidal sequences of angular frequencies
0.1 rad/sample and 0.4 rad/sample
We need to design a highpass filter that will
pass the high-frequency component of the
input but block the low-frequency component
For simplicity,assume the filter to be an FIR
filter of length 3 with an impulse response,
h[0] = h[2] = a,h[1] = b
§ 4.4 The Concept of Filtering
The convolution sum description of this filter
is then given by
y[n]=h[0]x[n]+h[1]x[n-1]+h[2]x[n-2]
=ax[n] +bx[n-1]+ ax[n-2]
y[n] and x[n] are,respectively,the output and
the input sequences
Design Objective,Choose suitable values of a
and b so that the output is a sinusoidal
sequence with a frequency 0.4 rad/sample
§ 4.4 The Concept of Filtering
Now,the frequency response of the FIR
filter is given by
2]2[]1[]0[)( jjj ehehheH
ba? jj ee )1( 2
b
a? jjjj eeee
2
2
ba? je)c o s2(
§ 4.4 The Concept of Filtering
The magnitude and phase functions are
|H(ej?)|=2acos?+b
(?) = -?
In order to block the low-frequency
component,the magnitude function at
= 0.1 should be equal to zero
Likewise,to pass the high-frequency
component,the magnitude function at? = 0.4
should be equal to one
§ 4.4 The Concept of Filtering
Thus,the two conditions that must be
satisfied are
|H(ej0.1)|=2acos(0.1)+b = 0
|H(ej0.4)|=2acos(0.4)+b = 1
Solving the above two equations we get
a = -6.76195
b = 13.456335
§ 4.4 The Concept of Filtering
Thus the output-input relation of the
FIR filter is given by
y[n] = - 6.76195(x[n]+x[n-2])+13.456335x[n-2]
where the input is
x[n] = {cos(0.1n) + cos(0.4n)}?[n]
Program 4_2 can be used to verify the
filtering action of the above system
§ 4.4 The Concept of Filtering
Figure below shows the plots generated
by running this program
§ 4.4 The Concept of Filtering
The first seven samples of the output are
shown below
§ 4.4 The Concept of Filtering
From this table,it can be seen that,neglecting
the least significant digit,
y[n] = cos(0.4(n-1)) for n? 2
Computation of the present value of the
output requires the knowledge of the present
and two previous input samples
Hence,the first two output samples,y[0] and
y[1],are the result of assumed zero input
sample values at n = -1 and n = -2
§ 4.4 The Concept of Filtering
Therefore,first two output samples
constitute the transient part of the
output
Since the impulse response is of length 3,
the steady-state is reached at n = N = 2
Note also that the output is delayed
version of the high-frequency
component cos(0.4n) of the input,and
the delay is one sample period
§ 4.5 Phase and Group Delays
The output y[n] of a frequency-selective
LTI discrete-time system with a
frequency response H(ej?) exhibits
some delay relative to the input x[n]
caused by the nonzero phase response
(?)=arg{H(ej?)} of the system
For an input
nnAnx o ),c o s (][
§ 4.5 Phase and Group Delays
Thus,the output lags in phase by?(?0)
radians
Rewriting the above equation we get
))(c o s ()(][ ooj neHAny o
o
o
o
j neHAny o )(c o s)(][
the output is
§ 4.5 Phase and Group Delays
This expression indicates a time delay,
known as phase delay,at?=?0 given by
o
oop
)()(
Now consider the case when the input
signal contains many sinusoidal
components with different frequencies that
are not harmonically related
§ 4.5 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
§ 4.5 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?c
of a low-frequency sinusoidal signal of
frequency?0,
)c o s ()c o s (][ nnAnx co
§ 4.5 Phase and Group Delays
where?l =?c -?0 and?u =?c+?0
Let the above input be processed by an
LTI discrete-time system with a
frequency response H(ej?) satisfying the
condition
)c o s ()c o s (][ 22 nnnx uAA
ujeHf o r1)(
The input can be rewritten as
§ 4.5 Phase and Group Delays
Note,The output is also in the form of a
modulated carrier signal with the same
carrier frequency?c and the same
modulation frequency?0 as the input
))(c o s ()(c o s][ 22 uuAA nnny
2 )()(c o s2 )()(c o s uouc nnA
The output y[n] is then given by
§ 4.5 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?l and?u very close
to the carrier frequency?c,i.e,?0 is
very small
§ 4.5 Phase and Group Delays
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
)()()()( ccccc
cd
d
In the neighborhood of?c we can express the
unwrapped phase response?c(?) as
§ 4.5 Phase and Group Delays
In the case of the carrier signal we have
c
cc
c
cuc
)(
2
)()(?
which is seen to be the same as the phase
delay if only the carrier signal is passed
through the system
§ 4.5 Phase and Group Delays
In the case of the modulating component we
have
cd
d c
u
cuc
)()()(
cd
d c
cg
)()(
is called the group delay or envelope delay
caused by the system at?=?c
The parameter
§ 4.5 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
§ 4.5 Phase and Group Delays
If the phase function and the angular
frequency? are in radians per second,
then the group delay is in seconds
Left figure
illustrates the
evaluation of the
phase delay and
the group delay
§ 4.5 Phase and Group Delays
Left figure
shows the
waveform of
an amplitude-
modulated
input and the
output
generated by
an LTI system
§ 4.5 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
§ 4.5 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
§ 4.5 Phase and Group Delays
Example - The phase function of the FIR
filter y[n] = ax[n] + bx[n-1] + ax[n-2]
is?(?) = -?
Hence its group delay is given by
g(?)=1 verifying the result obtained
earlier by simulation
§ 4.6 Frequency Response of
the LTI Discrete-Time System
In the above H(e j?) 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
Hence,we can write
§ 4.6 Frequency Response of
the LTI Discrete-Time System
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)(
It follows from the previous equation
§ 4.7 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 ][][][
§ 4.7 The Transfer Function
Taking the z-transforms of both sides we get n
n kn
n zknxkhznyzY
][][][)(
k n
nzknxkh ][][
k
kzxkh
)(][][
§ 4.8 The Transfer Function
Thus,Y(z) = H(z)X(z)
k
k
zzxkhzY?
][][)(
)(zX
)(][)( zXzkhzY
k
k
)(zH
Or,
Therefore,
§ 4.8 The Transfer Function
Hence,
H(z) = Y(Z)/X(z)
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]
§ 4.8 The Transfer Function
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)(
Consider an LTI discrete-time system
characterized by a difference equation
§ 4.8 The Transfer Function
Or,equivalently as
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
)(
)(
)(
An alternate form of the transfer
function is given by
§ 4.8 The Transfer Function
1,?2,…,?M are the finite zeros,and?1,? 2,…,
N are the finite poles of H(z)
If N > M,there are additional (N-M) zeros at
z = 0
If N < M,there are additional (M-N) poles at
z = 0
N
k k
M
k kMN
z
z
z
d
pzH
1
1
0
0
)(
)(
)( )(
Or,equivalently as
§ 4.8 The Transfer Function
For a causal IIR digital filter,the
impulse response is a causal sequence
The ROC of the causal transfer function
N
k k
M
k kMN
z
z
z
d
pzH
1
1
0
0
)(
)(
)( )(
kkz m a xThus the ROC is given by
is thus exterior to a circle going through
the pole furthest from the origin
§ 4.8 The Transfer Function
Example - Consider the M-point moving-
average FIR filter with an impulse response
][nh
o th e r w ise,0
10,/1 MnM
Its transfer function is then given by
1
0
1)( M
n
nz
M
zH
)]1([
1
)1(
1
1?
zzM
z
zM
z
M
MM
§ 4.8 The Transfer Function
The transfer function has M zeros on the unit
circle at z=ej2pk/M,0? k? M-1
There are M-1 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
M = 8
§ 4.8 The Transfer Function
Example - A causal LTI IIR digital filter
is described by a constant coefficient
difference equation given by
y[n]=x[n-1]-1.2x[n-2]+x[n-3]+1.3y[n-1]
-1.04y[n-2]+0.222y[n-3]
Its transfer function is therefore given
by
321
321
222.004.13.11
2.1)(
zzz
zzzzH
§ 4.8 The Transfer Function
Alternate forms:
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
74.0?zROC:
74.0
Note,Poles farthest from
z=0 have a magnitude
§ 4.9 Frequency Response from
Transfer Function
If the ROC of the transfer function H(z)
includes the unit circle,then the frequency
response H(ej?) of the LTI digital filter can be
obtained simply as follows:
jezj zHeH )()(
)(*)()( 2 jjj eHeHeH
jezjj zHzHeHeH )()()()( 1
For a real coefficient transfer function H(z) it
can be shown that
§ 4.9 Frequency Response from
Transfer Function
For a stable rational transfer function in
the form
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
)(
)(
)(
the factored form of the frequency
response is given by
§ 4.9 Frequency Response from
Transfer Function
It is convenient to visualize the contributions
of the zero factor (z-?k) and the pole factor
(z-?k) from the factored form of the frequency
response
The magnitude function is given by
N
k k
j
M
k k
j
MNjj
e
e
e
d
peH
1
1)(
0
0)(
N
k k
j
M
k k
j
e
e
d
p
1
1
0
0
§ 4.10 Types of Transfer Functions
The time-domain classification of an
LTI digital transfer function sequence is
based on the length of its impulse
response:
- Finite impulse response (FIR) transfer
function
- Infinite impulse response (IIR) transfer
function
§ 4.10 Types of Transfer Functions
Several other classifications are also used
In the case of digital transfer functions with
frequency-selective frequency responses,one
classification is based on the shape of the
magnitude function H(ei?) or the form of the
phase function?(?)
Based on this four types of ideal filters are
usually defined
§ 4.10.1 Ideal Filters
A digital filter designed to pass signal
components of certain frequencies
without distortion should have a
frequency response equal to one at these
frequencies,and should have a
frequency response equal to zero at all
other frequencies
§ 4.10.1 Ideal Filters
The range of frequencies where the
frequency response takes the value of
one is called the passband
The range of frequencies where the
frequency response takes the value of
zero is called the stopband
§ 4.10.1 Ideal Filters
Frequency responses of the four popular types
of ideal digital filters with real impulse
response coefficients are shown below:
§ 4.10.1 Ideal Filters
The frequencies?c,?c1,and?c2 are
called the cutoff frequencies
An ideal filter has a magnitude response
equal to one in the passband and zero in
the stopband,and has a zero phase
everywhere
§ 4.10.1 Ideal Filters
Earlier in the course we derived the inverse
DTFT of the frequency response HLP(ej?)
of the ideal lowpass filter:
hLP[n]=sin?cn/np,-? <n<?
We have also shown that the above impulse
response is not absolutely summable,and
hence,the corresponding transfer function is
not BIBO stable
§ 4.10.1 Ideal Filters
Also,hLP[n] is not causal and is of doubly
infinite length
The remaining three ideal filters are also
characterized by doubly infinite,noncausal
impulse responses and are not absolutely
summable
Thus,the ideal filters with the ideal,brick
wall” frequency responses cannot be realized
with finite dimensional LTI filter
§ 4.10.1 Ideal Filters
To develop stable and realizable transfer
functions,the ideal frequency response
specifications are relaxed by including a
transition band between the passband and the
stopband
This permits the magnitude response to decay
slowly from its maximum value in the
passband to the zero value in the stopband
§ 4.10.1 Ideal Filters
Moreover,the magnitude response is
allowed to vary by a small amount both
in the passband and the stopband
Typical magnitude
response
specifications
of a lowpass filter
are shown as:
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
A second classification of a transfer
function is with respect to its phase
characteristics
In many applications,it is necessary that
the digital filter designed does not
distort the phase of the input signal
components with frequencies in the
passband
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
One way to avoid any phase distortion is
to make the frequency response of the
filter real and nonnegative,i.e.,to design
the filter with a zero phase characteristic
However,it is impossible to design a
causal digital filter with a zero phase
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
For non-real-time processing of real-valued
input signals of finite length,zero-phase
filtering can be very simply implemented by
relaxing the causality requirement
One zero-phase filtering scheme is sketched
below
x[n] v[n] u[n] w[n]H(z) H(z)
u[n]=v[-n],y[n]=w[-n]
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
It is easy to verify the above scheme in the
frequency domain
Let X(ej?),V(ej?),U(ej?),W(ej?),and Y(ej?)
denote the DTFTs of x[n],v[n],u[n],w[n],and
y[n],respectively
From the figure shown earlier and making use
of the symmetry relations we arrive at the
relations between various DTFTs as given on
the next slide
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
V(ej?)= H(ej?)X(ej?),W(ej?)=H(ej?)U(ej?)
U(ej?)= V*(ej?),Y(ej?)= W*(ej?)
Combining the above equations we get
Y(ej?) = W*(ej?) = H*(ej?)U*(ej?)
= H*(ej?)V(ej?) = H*(ej?)H(ej?)X(ej?)
= |H(ej?)|2X(ej?)
This is a zero-phase filter with a frequency response
|H(ej?)|2
x[n] v[n] u[n] w[n]H(z) H(z)
u[n]=v[-n],y[n]=w[-n]
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
The function fftfilt implements the above zero-
phase filtering scheme
In the case of a causal transfer function with a
nonzero phase response,the phase distortion
can be avoided by ensuring that the transfer
function has a unity magnitude and a linear-
phase characteristic in the frequency band of
interest
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
The most general type of a filter with a
linear phase has a frequency response
given by
H(ej?)= ej?D
which has a linear phase from? = 0 to
= 2p
Note also |H(ej?)|=1
(?)=D
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
The output y[n] of this filter to an input
x[n]=Aej?n is then given by
y[n]= Ae-j?Dej?n = Aej?(n-D)
If xa(t) and ya(t) represent the
continuous-time signals whose sampled
versions,sampled at t = nT,are x[n] and
y[n] given above,then the delay between
xa(t) and ya(t) is precisely the group
delay of amount D
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
If D is an integer,then y[n] is identical to
x[n],but delayed by D samples
If D is not an integer,y[n],being delayed
by a fractional part,is not identical to
x[n]
In the latter case,the waveform of the
underlying continuous-time output is
identical to the waveform of the
underlying continuous-time input and
delayed D units of time
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
If it is desired to pass input signal
components in a certain frequency
range undistorted in both
magnitude and phase,then the
transfer function should exhibit a
unity magnitude response and a
linear-phase response in the band of
interest
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
Figure right
shows the
frequency
response if a
lowpass filter with
a linear-phase
characteristic in
the passband
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
Since the signal components in the
stopband are blocked,the phase
response in the stopband can be of any
shape
Example - Determine the impulse
response of an ideal lowpass filter with a
linear phase response:
)( j
LP eH p
c
c
nj oe
,0
0,
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
Applying the frequency-shifting
property of the DTFT to the impulse
response of an ideal zero-phase lowpass
filter we arrive at
p nnn nnnh
o
ocLP,
)(
)(si n][
As before,the above filter is noncausal
and of doubly infinite length,and hence,
unrealizable
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
By truncating the impulse response
to a finite number of terms,a
realizable FIR approximation to the
ideal lowpass filter can be
developed
The truncated approximation may
or may not exhibit linear phase,
depending on the value of n0 chosen
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
If we choose n0= N/2 with N a
positive integer,the truncated and
shifted approximation
NnNn Nnnh cLPp 0,)2/( )2/(si n][^
will be a length N+1 causal linear-
phase FIR filter
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
Figure below shows the filter coefficients
obtained using the function sinc for two
different values of N
N=12 N=13
§ 4.10.2 Zero-Phase and Linear-
Phase Transfer Functions
Because of the symmetry of the impulse
response coefficients as indicated in the two
figures,the frequency response of the
truncated approximation can be expressed as:
)(][)( 2/
0
LP
Nj
N
n
nj
LP
j
LP HeenheH ^^
~
where,called the zero-phase
response or amplitude response,is a
real function of?
)(?LPH~
§ 4.11 Linear-Phase FIR
Transfer Functions
It is nearly impossible to design a linear-
phase IIR transfer function
It is always possible to design an FIR
transfer function with an exact linear-
phase response
Consider a causal FIR transfer function
H(z) of length N+1,i.e.,of order N:
Nn nznhzH 0 ][)(
§ 4.11 Linear-Phase FIR
Transfer Functions
The above transfer function has a linear
phase,if its impulse response h[n] is
either symmetric,i.e.,
h[n]=h[N-n],0?n?N
or is antisymmetric,i.e.,
h[n]=-h[N-n],0?n?N
§ 4.11 Linear-Phase FIR
Transfer Functions
Since the length of the impulse response
can be either even or odd,we can define
four types of linear-phase FIR transfer
functions
For an antisymmetric FIR filter of odd
length,i.e.,N even
h[N/2] = 0
We examine next the each of the 4 cases
§ 4.11 Linear-Phase FIR
Transfer Functions
Type 1,N = 8 Type 2,N = 7
Type 3,N = 8 Type 4,N = 7
§ 4.11 Linear-Phase FIR
Transfer Functions
Type 1,Symmetric Impulse Response with
Odd Length
In this case,the degree N is even
Assume N = 8 for simplicity
The transfer function H(z) is given by
321 3210 zhzhzhhzH ][][][][)(
87654 87654 zhzhzhzhzh ][][][][][
§ 4.11 Linear-Phase FIR
Transfer Functions
Because of symmetry,we have h[0]=h[8],
h[1] = h[7],h[2] = h[6],and h[3] = h[5]
Thus,we can write
)]([)]([)( 718 110 zzhzhzH 45362
432 zhzzhzzh ][)]([)]([
)]([)]([{ 33444 10 zzhzzhz
]}[)]([)]([ 432 122 hzzhzzh
§ 4.11 Linear-Phase FIR
Transfer Functions
The corresponding frequency response
is then given by
)3c o s (]1[2)4c o s (]0[2{)( 4 hheeH jj
]}4[)c o s (]3[2)2c o s (]2[2 hhh
The quantity inside the braces is a real
function of?,and can assume positive or
negative values in the range 0?|?|?p
§ 4.11 Linear-Phase FIR
Transfer Functions
where b is either 0 or p,and hence,it is a
linear function of? in the generalized sense
The group delay is given by
b 4)(
4)( )(dd
indicating a constant group delay of 4 samples
The phase function here is given by
§ 4.11 Linear-Phase FIR
Transfer Functions
In the general case for Type 1 FIR filters,
the frequency response is of the form
)()( 2/ HeeH jNj
)(?H~
2/
1 22
)c o s (][2][
N
n
NN nnhh
)(?H~where the amplitude response,also
called the zero-phase response,is of the
form
§ 4.11 Linear-Phase FIR
Transfer Functions
which is seen to be a slightly modified
version of a length-7 moving-average
FIR filter
The above transfer function has a
symmetric impulse response and
therefore a linear phase response
][)( 6215432121610 zzzzzzzH
Example - Consider
§ 4.11 Linear-Phase FIR
Transfer Functions
A plot of the magnitude response of
along with that of the 7-point moving-
average filter is shown below
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
/p
Ma
gn
itu
de
modified filter
moving-average
§ 4.11 Linear-Phase FIR
Transfer Functions
Note the improved magnitude response
obtained by simply changing the first and the
last impulse response coefficients of a moving-
average (MA) filter
It can be shown that we can express
)()()( 54321611210 11 zzzzzzzH
which is seen to be a cascade of a 2-point MA
filter with a 6-point MA filter
Thus,H0(z) has a double zero at z=-1,i.e.,
(? = p)
§ 4.12 Allpass Transfer Function
is called an allpass transfer function
An M-th order causal real-coefficient allpass
transfer function is of the form
a l lf o r,1|)(| 2jeA
M
M
M
M
MM
MM
M zdzdzd
zzdzddzA
1
1
1
1
1
1
1
1
1,..
...
)(
Definition,An IIR transfer function A(z) with
unity magnitude response for all frequencies,
i.e.,
§ 4.12 Allpass Transfer Function
If we denote the denominator polynomials of
AM(z) as DM(z),
MMMMM zdzdzdzD 11111,..)(
)(
)()(
zD
zDz
M M M
M
zA
1
Note from the above that if z=rej? is a pole of a
real coefficient allpass transfer function,then it
has a zero at z=(1/r)e-j?
then it follows that AM(z) can be written as:
§ 4.12 Allpass Transfer Function
The numerator of a real-coefficient
allpass transfer function is said to be the
mirror-image polynomial of the
denominator,and vice versa
)()( zDzzD MMM
~
)(zDM~? We shall use the notation to
denote the mirror-image polynomial of a
degree-M polynomial DM(z),i.e.,
§ 4.12 Allpass Transfer Function
)(
)()(
zD
zDz
M M M
MzA 1
321
321
3 2.018.04.01
4.018.02.0)(
zzz
zzzzA? The expression implies that the
poles and zeros
of a real-
coefficient
allpass function
exhibit mirror-
image symmetry
in the z-plane
§ 4.12 Allpass Transfer Function
Therefore
)(
)(1
1)(
zD
zDz
M M M
MzA
)(
)(
)(
)(1
1
1)()(
zD
zDz
zD
zDz
MM M M
M
M
MMzAzA
1)()(|)(| 12 jezMMjM zAzAeA
Hence
1|)(|jM eA? To show that we observe
that
§ 4.12 Allpass Transfer Function
Now,the poles of a causal stable transfer
function must lie inside the unit circle in
the z-plane
Hence,all zeros of a causal stable allpass
transfer function must lie outside the
unit circle in a mirror-image symmetry
with its poles situated inside the unit
circle
§ 4.12 Allpass Transfer Function
Figure below shows the principal value of the
phase of the 3rd-order allpass function
321
321
3 2.018.04.01
4.018.02.0)(
zzz
zzzzA
0 0.2 0.4 0.6 0.8 1-4
-2
0
2
4
/p
Ph
ase
,de
gre
es
Principal value of phase
§ 4.12 Allpass Transfer Function
A Simple Application
A simple but often used application of an
allpass filter is as a delay equalizer
Let G(z) be the transfer function of a digital
filter designed to meet a prescribed magnitude
response
The nonlinear phase response of G(z) can be
corrected by cascading it with an allpass filter
A(z) so that the overall cascade has a constant
group delay in the band of interest
§ 4.12 Allpass Transfer Function
G(z) A(z)
|)(||)()(| jjj eGeAeG
Overall group delay is the given by the
sum of the group delays of G(z) and A(z)
1|)(|jeA? Since,we have
§ 4.13 Minimum-Phase and
Maximum-Phase Transfer Functions
Both transfer functions have a pole inside the
unit circle at the same location z=-a and are
stable
But the zero of H1(z) is inside the unit circle
at z=-b,whereas,the zero of H2(z) is at z=1/b
situated in a mirror-image symmetry
11121 bazHzH azbzaz bz,,)(,)(
Consider the two 1st-order transfer functions:
§ 4.13 Minimum-Phase and
Maximum-Phase Transfer Functions
Figure below shows the pole-zero plots
of the two transfer functions
H1(z) H2(z)
§ 4.13 Minimum-Phase and
Maximum-Phase Transfer Functions
However,both transfer functions have
an identical magnitude function as
1122111 )()()()( zHzHzHzH
c o ss i n1c o ss i n11 t a nt a n)](a r g [ abjeH
c o ss i n1c o s1 s i n12 t a nt a n)](a r g [ abbjeH
The corresponding phase functions are
§ 4.13 Minimum-Phase and
Maximum-Phase Transfer Functions
Figure below shows the unwrapped
phase responses of the two transfer
functions for a = 0.8 and b =-0.5
§ 4.13 Minimum-Phase and
Maximum-Phase Transfer Functions
From this figure it follows that H2(z) has an
excess phase lag with respect to H1(z)
Generalizing the above result,we can show
that a causal stable transfer function with all
zeros outside the unit circle has an excess
phase compared to a causal transfer function
with identical magnitude but having all zeros
inside the unit circle
§ 4.13 Minimum-Phase and
Maximum-Phase Transfer Functions
A causal stable transfer function with all zeros
inside the unit circle is called a minimum-
phase transfer function
A causal stable transfer function with all zeros
outside the unit circle is called a maximum-
phase transfer function
Any nonminimum-phase transfer function can
be expressed as the product of a minimum-
phase transfer function and a stable allpass
transfer function
Homework
Read the textbook from p.203 to 233,
and from 243 to 248
Problems
4.2,4.5,4.7,4.8,4.12,4.19,4.26
M4.1,M4.3,M4.6
