Copyright ? 2001,S,K,Mitra 1
LTI Discrete-Time Systems in
the Transform Domain
? An LTI discrete-time system is completely
characterized in the time-domain by its
impulse response {h[n]}
? We consider now the use of the DTFT and
the z-transform in developing the transform-
domain representations of an LTI system
Copyright ? 2001,S,K,Mitra 2
Finite-Dimensional LTI
Discrete-Time Systems
? We consider LTI discrete-time systems
characterized by linear constant-coefficient
difference equations of the form,
??
??
???
M
k
k
N
k
k knxpknyd
00
][][
Copyright ? 2001,S,K,Mitra 3
Finite-Dimensional LTI
Discrete-Time Systems
? Applying the DTFT to the difference
equation and making use of the linearity and
the time-invariance properties of Table 3.2
we arrive at the input-output relation in the
transform-domain as
where and are the DTFTs of
y[n] and x[n],respectively
)()(
00
?
?
???
?
?? ??? jM
k
kj
k
jN
k
kj
k eXepeYed
)( ?jeY )( ?jeX
Copyright ? 2001,S,K,Mitra 4
Finite-Dimensional LTI
Discrete-Time Systems
? In developing the transform-domain
representation of the difference equation,it
has been tacitly assumed that and
exist
? The previous equation can be alternately
written as
)( ?jeY
)( ?jeX
)()(
00
?
?
???
?
?? ?
?
??
?
? ???
?
??
?
? ? jM
k
kj
k
jN
k
kj
k eXepeYed
Copyright ? 2001,S,K,Mitra 5
Finite-Dimensional LTI
Discrete-Time Systems
? Applying the z-transform to both sides of
the difference equation and making use of
the linearity and the time-invariance
properties of Table 3.9 we arrive at
where Y(z) and X(z) denote the z-transforms
of y[n] and x[n] with associated ROCs,
respectively
)()( zXzpzYzd
M
k
k
k
N
k
k
k ??
?
?
?
? ?
00
Copyright ? 2001,S,K,Mitra 6
Finite-Dimensional LTI
Discrete-Time Systems
? 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
Copyright ? 2001,S,K,Mitra 7
The Frequency Response
? Most discrete-time signals encountered in
practice can be represented as a linear
combination of a very large,possibly
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
Copyright ? 2001,S,K,Mitra 8
The Frequency Response
? An important property of an LTI system is
that for certain types of input signals,called
eigen functions,the output signal is the
input signal multiplied by a complex
constant
? We consider here one such eigen function
as the input
Copyright ? 2001,S,K,Mitra 9
? Consider the LTI discrete-time system with
an impulse response {h[n]} shown below
? Its input-output relationship in the time-
domain is given by the convolution sum
The Frequency Response
x[n] h[n] y[n]
?
?
???
??
k
knxkhny ][][][
Copyright ? 2001,S,K,Mitra 10
The Frequency Response
? If the input is of the form
then it follows that the output is given by
? Let
?????? ? nenx nj,][
nj
k
kj
k
knj eekhekhny ??
???
???
???
?? ?
?
??
?
? ???? ][][][ )(
??
?
???
???
k
kjj ekheH ][)(
Copyright ? 2001,S,K,Mitra 11
The Frequency Response
? Then we can write
? Thus for a complex exponential input signal
,the output of an LTI discrete-time
system is also a complex exponential signal
of the same frequency multiplied by a
complex constant
? Thus is an eigen function of the system
njj eeHny ??? )(][
)( ?jeH
nje ?
nje ?
Copyright ? 2001,S,K,Mitra 12
The Frequency Response
? The quantity is called the frequency
response of the LTI discrete-time system
? provides a frequency-domain
description of the system
? is precisely the DTFT of the impulse
response {h[n]} of the system
)( ?jeH
)( ?jeH
)( ?jeH
Copyright ? 2001,S,K,Mitra 13
The Frequency Response
?,in general,is a complex function
of ? with a period 2p
? It can be expressed in terms of its real and
imaginary parts
or,in terms of its magnitude and phase,
where
)( ?jeH
)()()( ??? ?? jimjrej eHjeHeH
)()()( ???? ? jjj eeHeH
)(a r g)( ???? jeH
Copyright ? 2001,S,K,Mitra 14
The Frequency Response
? The function 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
)( ?jeH
)(??
Copyright ? 2001,S,K,Mitra 15
The Frequency Response
? In some cases,the magnitude function is
specified in decibels as
where G(?) is called the gain function
? The negative of the gain function
is called the attenuation or loss function
dBeH j )(l o g20)( 10 ???G
)()( ???? G A
Copyright ? 2001,S,K,Mitra 16
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 it
follows from Table 3.4 that the magnitude
function is an even function of ?,
and the phase function is an odd function of
?,
)()( ??? ? jj eHeH
)()( ???????
Copyright ? 2001,S,K,Mitra 17
The Frequency Response
? Likewise,for a real impulse response h[n],
is even and is odd
? Example - Consider the M-point moving
average filter with an impulse response
given by
? Its frequency response is then given by
)( ?jre eH )( ?jim eH
o th e r w ise0
101
,
,/ ??? MnM
??
??][ nh
??
?
?
??? 1
0
1)( M
n
nj
M
j eeH
Copyright ? 2001,S,K,Mitra 18
The Frequency Response
? Or,??
??
?
? ???? ?
?
???
?
???
Mn
nj
n
nj
M
j eeeH
0
1)(
? ? ?? ?????
?
??
?
?????
?
??
?
? ??
j
jM
M
jM
n
nj
M e
eee
1
11 1
0
1
2/)1(
)2/s in (
)2/s in (1 ???
?
??? MjeM
M
Copyright ? 2001,S,K,Mitra 19
The Frequency Response
? Thus,the magnitude response of the M-point
moving average filter is given by
and the phase response is given by
)2/s in (
)2/s in ()( 1
?
???? MeH
M
j
? ?p???????
? 02
)1()(
k
M
M
kp?? 2? ?
M/2
Copyright ? 2001,S,K,Mitra 20
Frequency Response
Computation Using MATLAB
? The function H=freqz(h,w) can be used to
determine the values of the frequency response
vector H corresponding to the impulse
response h at a set of given frequency points w
? From H,its real and imaginary parts can be
computed using the functions real and imag,
and its magnitude and phase using the
functions abs and angle
Copyright ? 2001,S,K,Mitra 21
Frequency Response
Computation Using MATLAB
? Example - Program 4_1 can be used to
generate the magnitude and gain responses
of an M-point moving average filter as
shown below
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
M
a
gni
t
ude
? / p
M =5
M =1 4
0 0.2 0.4 0.6 0.8 1
- 200
- 150
- 100
- 50
0
50
100
P
ha
s
e
,de
gr
e
e
s
? / p
M =5
M =1 4
Copyright ? 2001,S,K,Mitra 22
Frequency Response
Computation Using MATLAB
? The phase response of a discrete-time
system when determined numerically 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
Copyright ? 2001,S,K,Mitra 23
Frequency Response
Computation Using MATLAB
? To this end the function unwrap can be
used,provided that 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
Copyright ? 2001,S,K,Mitra 24
Steady-State Response
? Note that the frequency response also
determines the steady-state response of an
LTI discrete-time system to a sinusoidal
input
? Example - Determine the steady-state
output y[n] of a real coefficient LTI
discrete-time system with a frequency
response for an input
????????? nnAnx o ),c o s (][
)( ?jeH
Copyright ? 2001,S,K,Mitra 25
Steady-State Response
? We can express the input x[n] as
where
? Now the output of the system for an input
is simply
][*][][ ngngnx ??
njj oeAeng ???
21][
nj oe ?
njj oo eeH ?? )(
Copyright ? 2001,S,K,Mitra 26
Steady-State Response
? Because of linearity,the response v[n] to an
input g[n] is given by
? Likewise,the output v*[n] to the input g*[n]
is
njjj oo eeHAenv ???? )(][
2
1
njjj oo eeHAenv ??????? )(][*
2
1
Copyright ? 2001,S,K,Mitra 27
Steady-State Response
? Combining the last two equations we get
njjjnjjj oooo eeHAeeeHAe ????????? ?? )()(
2
1
2
1 ][*][][ nvnvny ??
? ?nojjojnojjojoj eeeeeeeHA ???????????? ?? )()()(21
1 c o s ( )
2 ( ) ( )o
j o
oHeAn
? ? ?? ????
Copyright ? 2001,S,K,Mitra 28
Steady-State Response
? Thus,the output y[n] has the same sinusoidal
waveform as the input with two differences,
(1) the amplitude is multiplied by,
the value of the magnitude function at
(2) the output has a phase lag relative to the
input by an amount,the value of the
phase function at
| ( ) |ojHe ?
)( o??
o???
o???
Copyright ? 2001,S,K,Mitra 29
Response to a Causal
Exponential Sequence
? The expression for the steady-state response
developed earlier assumes that the system is
initially relaxed before the application of
the input x[n]
? In practice,excitation x[n] to a discrete-time
system is usually a right-sided sequence
applied at some sample index
? We develop the expression for the output
for such an input
onn ?
Copyright ? 2001,S,K,Mitra 30
Response to a Causal
Exponential Sequence
? Without any loss of generality,assume
for n < 0
? From the input-output relation
we observe that for an input
the output is given by
? ? ??? ?? k knxkhny ][][][
][][][
0
)( nekhny
n
k
knj ?
???
?
???
?? ?
?
??
][][ nenx nj ?? ?
0][ ?nx
Copyright ? 2001,S,K,Mitra 31
Response to a Causal
Exponential Sequence
? Or,
? The output for n < 0 is y[n] = 0
? The output for is given by
][][][
0
neekhny nj
n
k
kj ?
???
?
???
?? ?
?
???
nj
nk
kjnj
k
kj eekheekh ?
?
??
???
?
?
??
???
?
???
??
???
?
???
?? ??
10
][][
0?n
nj
n
k
kj eekhny ?
?
??
???
?
???
?? ?
0
][][
Copyright ? 2001,S,K,Mitra 32
Response to a Causal
Exponential Sequence
? Or,
? The first term on the RHS is the same as
that obtained when the input is applied at
n = 0 to an initially relaxed system and is
the steady-state response,
nj
nk
kjnjj eekheeHny ?
?
??
????
???
?
???
??? ?
1
][)(][
njjsr eeHny ??? )(][
Copyright ? 2001,S,K,Mitra 33
Response to a Causal
Exponential Sequence
? The second term on the RHS is called the
transient response,
? To determine the effect of the above term
on the total output response,we observe
nj
nk
kj
tr eekhny
?
?
??
??
???
?
???
??? ?
1
][][
???
?
?
?
??
?
??
??? ???
011
)( ][][][][
knknk
nkj
tr khkhekhny
Copyright ? 2001,S,K,Mitra 34
Response to a Causal
Exponential Sequence
? For a causal,stable LTI IIR discrete-time
system,h[n] is absolutely summable
? As a result,the transient response is a
bounded sequence
? Moreover,as,
and hence,the transient response decays to
zero as n gets very large
][nytr
??n
0][1 ?? ? ?? nk kh
Copyright ? 2001,S,K,Mitra 35
Response to a Causal
Exponential Sequence
? For a causal FIR LTI discrete-time system
with an impulse response h[n] of length
N + 1,h[n] = 0 for n > N
? Hence,for
? Here the output reaches the steady-state
value at n = N
0][ ?ny tr 1?? Nn
njjsr eeHny ??? )(][
Copyright ? 2001,S,K,Mitra 36
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
Copyright ? 2001,S,K,Mitra 37
The Concept of Filtering
? The key to the filtering process is the IDFT
(or synthesis) equation
? 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
? ??
p
p?
??
p deeXnx
njj )(][
2
1
Copyright ? 2001,S,K,Mitra 38
The Concept of Filtering
? Thus,by appropriately choosing the values
of the magnitude function 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
)( ?jeH
Copyright ? 2001,S,K,Mitra 39
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
()jHe?
c? p0
1
p? c?? ?
Copyright ? 2001,S,K,Mitra 40
The Concept of Filtering
? We apply an input
to this system
? Because of linearity,the output of this
system is of the form
p??????????? 2121 0,c o sc o s][ cnBnAnx
? ?)(c o s)(][ 111 ????? ? neHAny j
? ?)(c o s)( 222 ????? ? neHB j
Copyright ? 2001,S,K,Mitra 41
The Concept of Filtering
? As
the output reduces to
? Thus,the system acts like a lowpass filter
? In the following example,we consider the
design of a very simple digital filter
0)(,1)( 21 ?? ?? jj eHeH
? ?)(c o s)(][ 111 ????? ? neHAny j
Copyright ? 2001,S,K,Mitra 42
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
Copyright ? 2001,S,K,Mitra 43
The Concept of Filtering
? The convolution sum description of this
filter is then given by
? 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
]2[]2[]1[]1[][]0[][ ????? nxhnxhnxhny ]2[]1[][ ?a??b?a? nxnxnx
Copyright ? 2001,S,K,Mitra 44
The Concept of Filtering
? Now,the frequency response of the FIR
filter is given by
????? ??? 2]2[]1[]0[)( jjj ehehheH
???? b??a? jj ee )1( 2 ??????? b?
???
?
???
? ?a? jjjj eeee
2
2
??b??a? je)c o s2(
Copyright ? 2001,S,K,Mitra 45
The Concept of Filtering
? The magnitude and phase functions are
? 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
b??a?? c o s2)( jeH
????? )(
Copyright ? 2001,S,K,Mitra 46
The Concept of Filtering
? Thus,the two conditions that must be
satisfied are
? Solving the above two equations we get
0)1.0c o s (2)( 1.0 ?b?a?jeH
1)4.0c o s (2)( 4.0 ?b?a?jeH
76195.6??a
4 5 6 3 3 5.13?b
Copyright ? 2001,S,K,Mitra 47
The Concept of Filtering
? Thus the output-input relation of the FIR
filter is given by
where the input is
? Program 4_2 can be used to verify the
filtering action of the above system
? ? ]1[456335.13]2[][76195.6][ ?????? nxnxnxny
][)}4.0c o s ()1.0{ c o s (][ nnnnx ???
Copyright ? 2001,S,K,Mitra 48
The Concept of Filtering
? Figure below shows the plots generated by
running this program
0 20 40 60 80 100
-1
0
1
2
3
4
A
m
pl
i
t
ude
T i m e i nde x n
y[ n]
x
2
[ n]
x
1
[ n]
Copyright ? 2001,S,K,Mitra 49
The Concept of Filtering
? The first seven samples of the output are
shown below
Copyright ? 2001,S,K,Mitra 50
The Concept of Filtering
? From this table,it can be seen that,
neglecting the least significant digit,
? 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 and
2f o r))1(4.0c o s (][ ??? nnny
1??n 2??n
Copyright ? 2001,S,K,Mitra 51
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