Copyright ? 2001,S,K,Mitra 1
Transform-Domain Representation
of Discrete-Time Signals
? Three useful representations of discrete-time
sequences in the transform domain,
?Discrete-time Fourier Transform (DTFT)
?Discrete Fourier Transform (DFT)
?z-Transform
Copyright ? 2001,S,K,Mitra 2
Discrete-Time Fourier
Transform
? Definition - The discrete-time Fourier
transform (DTFT) of a sequence
x[n] is given by
? In general,is a complex function
of the real variable w and can be written as
)( wjeX
)( wjeX
?
?
???
??
n
njj enxeX ww ][)(
)()()( www ?? jimjrej eXjeXeX
Copyright ? 2001,S,K,Mitra 3
Discrete-Time Fourier
Transform (DTFT)
? and are,respectively,
the real and imaginary parts of,and
are real functions of w
? can alternately be expressed as
where
)( wjeX
)( wjre eX )( wjim eX
)( wjeX
)()()( w?ww ? jjj eeXeX
)}(a r g {)( w?w? jeX
Copyright ? 2001,S,K,Mitra 4
Discrete-Time Fourier
Transform
? is called the magnitude function
? is called the phase function
? Both quantities are again real functions of w
? In many applications,the DTFT is called
the Fourier spectrum
? Likewise,and are called the
magnitude and phase spectra
)( wjeX
)(w?
)( wjeX )(w?
Copyright ? 2001,S,K,Mitra 5
Discrete-Time Fourier
Transform
? For a real sequence x[n],and
are even functions of w,whereas
and are odd functions of w
? Note,
for any integer k
? The phase function ?(w) cannot be uniquely
specified for any DTFT
| ( ) |jXe w
)(w?
)( wjre eX
)( wjim eX
( 2 )( ) | ( ) |j j j kX e X e ew w ? w ???
()| ( ) |jjX e ew ? w?
Copyright ? 2001,S,K,Mitra 6
Discrete-Time Fourier
Transform
? We will assume that the phase function
?(w) is restricted to the following range of
values,
called the principal value
??w???? )(
Copyright ? 2001,S,K,Mitra 7
Discrete-Time Fourier
Transform
? The DTFTs of some sequences exhibit
discontinuities of 2? in their phase
responses
? An alternate type of phase function that is a
continuous function of w is often used
? It is derived from the original phase
function by removing the discontinuities of
2?
Copyright ? 2001,S,K,Mitra 8
Discrete-Time Fourier
Transform
? The process of removing the discontinuities
is called,unwrapping”
? The continuous phase function generated by
unwrapping is denoted as
? In some cases,discontinuities of ? may be
present after unwrapping
)(w?c
Copyright ? 2001,S,K,Mitra 9
Discrete-Time Fourier
Transform
? Example - The DTFT of the unit sample
sequence d[n] is given by
? Example - Consider the causal sequence
1]0[][)( ?d?? d?? w?
?
???
w nj
n
j ene
1],[][ ????? nnx n
Copyright ? 2001,S,K,Mitra 10
Discrete-Time Fourier
Transform
? Its DTFT is given by
as
? ??? ???
?
?
w??
???
w?w
0
][)(
n
njn
n
njnj eeneX
w???
?
?
w? ?? ??
je
n
nje
1
1
0
)(
1???? w? je
Copyright ? 2001,S,K,Mitra 11
Discrete-Time Fourier
Transform
? The magnitude and phase of the DTFT
are shown below )5.01/(1)( w?w ?? jj eeX
-3 -2 -1 0 1 2 3
0.5
1
1.5
2
w / ?
M
a
gni
t
ude
-3 -2 -1 0 1 2 3
- 0.4
- 0.2
0
0.2
0.4
0.6
w / ?
P
ha
s
e
i
n r
a
di
a
ns
Copyright ? 2001,S,K,Mitra 12
Discrete-Time Fourier
Transform
? The DTFT of a sequence x[n] is a
continuous function of w
? It is also a periodic function of w with a
period 2?,
)( wjeX
( 2 ) ( 2 )( ) [ ]j k j k n
n
X e x n ew ? w ?
?
? ? ?
? ? ?
? ?
2[] j n j k n
n
x n e ew?
?
??
? ??
? ? [ ] ( )j n j
n
x n e X eww
?
?
? ? ?
???
Copyright ? 2001,S,K,Mitra 13
Discrete-Time Fourier
Transform
? Therefore
represents the Fourier series representation
of the periodic function
? As a result,the Fourier coefficients x[n] can
be computed from using the
Fourier integral
??
?
???
w?w
n
njj enxeX ][)(
)( wjeX
? w??
?
??
ww deeXnx njj )(
2
1][
)( wjeX
Copyright ? 2001,S,K,Mitra 14
Discrete-Time Fourier
Transform
? Inverse discrete-time Fourier transform,
? Proof,
? w??
?
??
ww deeXnx njj )(
2
1][
? w?????? ???
?
??
w?
???
w? deexnx njj
?
?? ][
2
1][
Copyright ? 2001,S,K,Mitra 15
Discrete-Time Fourier
Transform
? The order of integration and summation can
be interchanged if the summation inside the
brackets converges uniformly,i.e.,if
exists
? Then ? w?????? ??
?
??
w?
???
w? deex njj
?
?? ][
2
1
)( wjeX
)(
)(s in][
2
1][ )(
?
???
?
?
? ??
?????
?
??
?
?
????
?
???
?
??
?w?
??? n
nxex nj
Copyright ? 2001,S,K,Mitra 16
Discrete-Time Fourier
Transform
? Now
? Hence
1,si n ( )
si nc ( )
0,()
nn
nl
nn
?
?
???
? ? ? ?
?? ?
ll
ll
][ ??d? n
][][][)( )(si n][ nxnxn nx ?? ?d??? ???
?
???
?
??? ??
??? ??
Copyright ? 2001,S,K,Mitra 17
Discrete-Time Fourier Transform
()jXe w[]xn
DTFT,analysis equation
IDTFT,synthesis equation
time
domain
frequency
domain
?
?
???
??
n
njj enxeX ww ][)(
? w??
?
??
ww deeXnx njj )(
2
1][
Copyright ? 2001,S,K,Mitra 18
Discrete-Time Fourier
Transform
? Convergence Condition - An infinite series
of the form
may or may not converge
? Let
??
?
???
w?w
n
njj enxeX ][)(
??
??
w?w K
Kn
njj
K enxeX ][)(
Copyright ? 2001,S,K,Mitra 19
Discrete-Time Fourier
Transform
? Then for uniform convergence of,
? Now,if x[n] is an absolutely summable
sequence,i.e.,if
)( wjeX
0)()(lim ?? ww?? jKjK eXeX
???
?
???n
nx ][
w?
Copyright ? 2001,S,K,Mitra 20
Discrete-Time Fourier
Transform
? Then
for all values of w
? Thus,the absolute summability of x[n] is a
sufficient condition for the existence of the
DTFT
??????
?
???
?
???
w?w
nn
njj nxenxeX ][][)(
)( wjeX
Copyright ? 2001,S,K,Mitra 21
Discrete-Time Fourier
Transform
? Example - The sequence for
is absolutely summable as
and its DTFT therefore converges
to uniformly
][][ nnx n ???
1??
?????? ??? ??
?
?
?
??? 1
1][
0n
n
n
n n
)( wjeX
)1/(1 w??? je
Copyright ? 2001,S,K,Mitra 22
Discrete-Time Fourier
Transform
? Since
an absolutely summable sequence has
always a finite energy
? However,a finite-energy sequence is not
necessarily absolutely summable
,][][
2
2 ?
?
??
?
? ??? ?
???
?
??? nn
nxnx
Copyright ? 2001,S,K,Mitra 23
Discrete-Time Fourier
Transform
? Example - The sequence
has a finite energy equal to
? But,x[n] is not absolutely summable
??
??][ nx
00
11
?
?
n
nn
,
,/
6
1 2
1
2 ?
?? ???????
?
?n
x nE
Copyright ? 2001,S,K,Mitra 24
Discrete-Time Fourier
Transform
? To represent a finite energy sequence x[n]
that is not absolutely summable by a DTFT
,it is necessary to consider a mean-
square convergence of,
where
)( wjeX
)( wjeX
0)()(lim
2
?w? ?
?
??
ww
??
deXeX jKj
K
??
??
w?w K
Kn
njj
K enxeX ][)(
Copyright ? 2001,S,K,Mitra 25
Discrete-Time Fourier
Transform
? Here,the total energy of the error
must approach zero at each value of w as K
goes to
? In such a case,the absolute value of the
error may not go to
zero as K goes to and the DTFT is no
longer bounded
?
?
)()( ww ? jKj eXeX
| ( ) ( ) |jj KX e X eww?
Copyright ? 2001,S,K,Mitra 26
Discrete-Time Fourier
Transform
? Example - Consider the DTFT
shown below
?
?
?
??w?w
w?w?
?w
c
cj
LP eH,0
0,1
)(
)( wjLP eH
cw ?0
1
?? cw? w
Copyright ? 2001,S,K,Mitra 27
Discrete-Time Fourier Transform
? The inverse DTFT of is given by
? The energy of is given by
? Therefore,is a finite-energy
sequence,but it is not absolutely summable
)( wjLP eH
][nhLP
sin sin c,c c cnn
n
w w w
? ? ?
????
????
11
[]
22
c cc
c
j n j n
jn
LP
ee
h n e d
j n j n
w ww
w
w
w
??
?
?
??
? ? ? ???
??
?
????? n
?w /c
][nhLP
Copyright ? 2001,S,K,Mitra 28
Discrete-Time Fourier
Transform
? As a result
does not uniformly converge to
for all values of w,but converges to
in the mean-square sense
)( wjLP eH
)( wjLP eH
[ ] s in c
KK
j n j ncc
LP
n K n K
nh n e eww ww
??
??
? ? ? ?
???
??????
Copyright ? 2001,S,K,Mitra 29
Discrete-Time Fourier
Transform
? The mean-square convergence property of
the sequence can be further
illustrated by examining the plot of the
function
for various values of K as shown next
,( ) s in c
K
j j ncc
LP K
nK
nH e eww ww
??
?
??
???
?????
][nhLP
Copyright ? 2001,S,K,Mitra 30
Discrete-Time Fourier Transform
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
w / ?
A
m
pl
i
t
ude
N = 20
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
w / ?
A
m
pl
i
t
ude
N = 30
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
w / ?
A
m
pl
i
t
ude
N = 40
Copyright ? 2001,S,K,Mitra 31
Discrete-Time Fourier
Transform
? As can be seen from these plots,independent
of the value of K there are ripples in the plot
of around both sides of the
point
? The number of ripples increases as K
increases with the height of the largest ripple
remaining the same for all values of K
)(,wjKLP eH
cw?w
Copyright ? 2001,S,K,Mitra 32
Discrete-Time Fourier
Transform
? As K goes to infinity,the condition
holds indicating the convergence of
to
? The oscillatory behavior of
approximating in the mean-
square sense at a point of discontinuity is
known as the Gibbs phenomenon
)(,wjKLP eH
)( wjLP eH
0)()(lim
2
,?w? ?
?
??
ww
??
deHeH jKLPjLP
K
)(,wjKLP eH
)( wjLP eH
Copyright ? 2001,S,K,Mitra 33
Discrete-Time Fourier
Transform
? The DTFT can also be defined for a certain
class of sequences which are neither
absolutely summable nor square summable
? Examples of such sequences are the unit
step sequence ?[n],the sinusoidal sequence
and the exponential sequence
? For this type of sequences,a DTFT
representation is possible using the Dirac
delta function d(w)
)c o s ( ??w no nA?
Copyright ? 2001,S,K,Mitra 34
Discrete-Time Fourier
Transform
? A Dirac delta function d(w) is a function of
w with infinite height,zero width,and unit
area
? It is the limiting form of a unit area pulse
function as ? goes to zero satisfying )(w?p
0( ) l im ( )pd w w????
w
2?? 2?0
?1
)(w?p
Copyright ? 2001,S,K,Mitra 35
Discrete-Time Fourier
Transform
? Example - Consider the complex exponential
sequence
? Its DTFT is given by
where is an impulse function of w and
nj oenx w?][
? ??w?w?d?
?
???
w
k
o
j keX )2(2)(
)(wd
??w??? o
Copyright ? 2001,S,K,Mitra 36
Discrete-Time Fourier
Transform
? The function
is a periodic function of w with a period 2?
and is called a periodic impulse train
? To verify that given above is
indeed the DTFT of we
compute the inverse DTFT of
? ??w?w?d?
?
???
w
k
o
j keX )2(2)(
)( wjeX
nj onx w?][
)( wjeX
Copyright ? 2001,S,K,Mitra 37
Discrete-Time Fourier
Transform
? Thus
where we have used the sampling (or
sifting) property of the impulse function )(wd
? ? w??w?w?d??
?
??
?
???
w
k
nj
o deknx )2(22
1][
njnj
o oede
w?
??
w ?w? w?wd? )(
Copyright ? 2001,S,K,Mitra 38
Commonly Used DTFT Pairs
Sequence DTFT 1][ ?d n
? ??w?d?
?
???k
k )2(21
? ??w?w?d?
?
???
w
k
o
nj ke o )2(2
? ??wd?????
?
???w? kj
k
e
n )2(
1
1][
w??????? jen 1
1)1(],[
Copyright ? 2001,S,K,Mitra 39
DTFT Properties
? There are a number of important properties
of the DTFT that are useful in signal
processing applications
? We will illustrate the applications of some
of the DTFT properties
Copyright ? 2001,S,K,Mitra 40
Table 3.2:General Properties of DTFT
Copyright ? 2001,S,K,Mitra 41
Table 3.3,DTFT Properties,
Symmetry Relations
x[n],A complex sequence
Copyright ? 2001,S,K,Mitra 42
Table 3.4,DTFT Properties,
Symmetry Relations
x[n],A real sequence