Chapter 3
Discrete Fourier Transform
Review


The DTFT provides the frequency-domain ()
representation for absolutely summable sequences.
The z-transform provides a generalized frequencydomain ( z ) representation for arbitrary sequences.
Features in common
 Defined for infinite-length sequences.
 Functions of continuous variable (  or z ).
 They are not numerically computable transform.
We need a numerically computable transform, that is
Discrete Fourier Transform (DFT)
Copyright © 2005. Shi Ping CUC
Chapter 3
Discrete Fourier Transform
Content
The Family of Fourier Transform
The Discrete Fourier Series (DFS)
The Discrete Fourier Transform (DFT)
The Properties of DFT
The Sampling Theorem in Frequency Domain
Approximating to FT (FS) with DFT (DFS)
Summary
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The Family of Fourier Transform
 Introduction
Fourier analysis is named after Jean Baptiste Joseph
Fourier (1768-1830), a French mathematician and
physicist.
A signal can be either continuous or discrete, and it can
be either periodic or aperiodic. The combination of
these two features generates the four categories of
Fourier Transform.
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The Family of Fourier Transform
 Aperiodic-Continuous－Fourier Transform
X ( j ) 
x( t ) 
1
2




x ( t )e


 j t
dt
X ( j )e
j t
d
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The Family of Fourier Transform
 Periodic-Continuous－Fourier Series
X ( jk  0 ) 
1
T0

T0 2
T0 2
x ( t )e

x( t ) 
 X ( jk
0
)e
 jk 0 t
dt
jk 0 t
k 
 0  2F 
2
T0
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The Family of Fourier Transform
 Aperiodic-Discrete－DTFT
X (e
j

)
 x( n)e
 jn
n  
x ( n) 
1
2

  X (e

j
)e
jn
d
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The Family of Fourier Transform
 Periodic-Discrete－DFS (DFT)
N 1
X (k ) 
 x( n)e
j
2
nk
N
n0
x ( n) 
1
N
N 1
 X ( k )e
j
2
nk
N
k 0
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The Family of Fourier Transform
 Summary
Time function
Frequency function
Continuous and Aperiodic
Aperiodic and Continuous
Continuous and Periodic( T )
0
Discrete ( T ) and Aperiodic
Aperiodic and Discrete(  0 
Periodic(  s 
2
T
2
T0
)
) and Continuous
Periodic(   2 )
s
Discrete ( T ) and Periodic ( T )
0
T
and Discrete(  0 
2
T0
)
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The Discrete Fourier Series (DFS)
 Definition
Periodic time functions can be synthesized as a linear
combination of complex exponentials whose frequencies
are multiples (or harmonics) of the fundamental frequency
Periodic continuous-time function x( t )  x( t  rT )
fundamental
frequency
j
2
t
T
e

x( t ) 
 X (k )e
j
2
kt
T
k 
Periodic discrete-time function x( n)  x( n  rN )
fundamental
frequency
j
e
2
N
n
x ( n) 
1
N
N 1
 X (k )e
k 0
j
2
kn
N
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The Discrete Fourier Series (DFS)
1
N
N 1
N 1
e
j
2
j
rn
N
n0
 x( n)e
j
2
rN
r  mN
1,
1 1 e



2
j
r
N
0,
N
1 e
N
rn
N
n0
1
  X ( k )
k 0
N
N 1
2
1
 
n0  N
N 1
N 1
e
n0
j
2
N
N 1
 X ( k )e
elsewhere
j
2
N
k 0
( k r )n
kn
  j 2 rn
N
e



  X (r )

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The Discrete Fourier Series (DFS)
N 1
X (k ) 
 x( n)e
j
2
kn
N
n0
N 1
Because:
X ( k  mN ) 
 x( n)e
j
2
( k  mN ) n
N
n0
N 1

 x( n)e
j
2
N
kn
 X (k )
n0
The X (k ) is a periodic sequence with fundamental
period equal to N
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The Discrete Fourier Series (DFS)
Let W N  e
 j 2N
~
~
X ( k )  DFS [ x ( n )] 
N 1

nk
~
x ( n )W N
n0
1
~
~
x ( n )  IDFS [ X ( k )] 
N
N 1

~
 nk
X ( k )W N
k 0
Copyright © 2005. Shi Ping CUC
The Discrete Fourier Series (DFS)
 Relation to the z-transform
~
 x ( n ),
x(n)  
 0,
0  n  N 1
elsewhere
N 1
X (z) 

x(n)z
n
,
n0
~
X (k ) 
N 1

x ( n )( e
j 2N k
)
n
n0
~
X (k )  X ( z ) |
ze
j 2 k
N
~
The DFS X ( k ) represents N evenly spaced samples of
the z-transform X (z ) around the unit circle.
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The Discrete Fourier Series (DFS)
 Relation to the DTFT
~
 x ( n ),
x(n)  
 0,
X (e
j
0  n  N 1
elsewhere
N 1
)

x ( n )e
 j n
，
~
X (k ) 
n0
~
X (k )  X (e
N 1

 j
x ( n )e
2
nk
N
n0
j
) |  2  k
N
The DFS is obtained by evenly sampling the DTFT at 2 N
intervals. It is called frequency resolution and represents the
sampling interval in the frequency domain.
Copyright © 2005. Shi Ping CUC
The Discrete Fourier Series (DFS)
~
X (k )  X (e
j
) |  2  k
N
jIm[z]
frequency resolution
N=8
2
N
k 0
Re[z]
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The Discrete Fourier Series (DFS)
 The properties of DFS

Linearity
~
~
~
~
DFS[ax1 ( n)  bx2 ( n)]  aX 1 ( k )  bX 2 ( k )

Shift of a sequence
 mk ~
~
DFS[ x ( n＋m )]  W N X ( k )  e

j
2
N
mk
~
X (k )
Modulation
~
~
DFS[W x ( n)]  X ( k  l )
ln
N
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The Discrete Fourier Series (DFS)

if
then
Periodic convolution
~
~
~
Y (k )  X 1 (k )  X 2 (k )
~
~
y ( n)  IDFS[Y ( k )] 
N 1
~
~
 x1 ( m ) x2 ( n  m )
m 0
N 1

~ (m ) x
~ (n  m )
x
 2
1
m 0
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1
~
~
~
y ( n)  IDFS[ X 1 ( k ) X 2 ( k )] 
N
N 1
N 1

mk
~


  x1 ( m )W N
N k 0  m 0
1
~
~
 nk
 X 1 ( k ) X 2 ( k )W N
k 0
~
 nk
 X 2 ( k )W N

N 1
~
( n m ) k 
 X 2 ( k )W N

k 0

N 1
N 1
1
~
  x1 ( m )
m 0
N

N 1
N 1
~ (m ) x
~ (n  m ) 
x
 1
2
m 0
~ (m ) x
~ (n  m )
x
 2
1
m 0
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The Discrete Fourier Transform (DFT)
 Introduction
The DFS provided us a mechanism for numerically
computing the discrete-time Fourier transform. But most of the
signals in practice are not periodic. They are likely to be of
finite length.

Theoretically, we can take care of this problem by defining a
periodic signal whose primary shape is that of the finite length
signal and then using the DFS on this periodic signal.

Practically, we define a new transform called the Discrete
Fourier Transform, which is the primary period of the DFS.

This DFT is the ultimate numerically computable Fourier
transform for arbitrary finite length sequences.

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The Discrete Fourier Transform (DFT)
 Finite-length sequence & periodic sequence
x (n) Finite-length sequence that has N samples
~ ( n) periodic sequence with the period of N
x
Window operation
~
0  n  N 1
 x ( n),
x ( n)  
0, elsewhere
~ ( n) R ( n)
x ( n)  x
N
Periodic extension
~ ( n) 
x

 x( n  rN )
r 
~ ( n)  x (( n))
x
N
Copyright © 2005. Shi Ping CUC
The Discrete Fourier Transform (DFT)
 The definition of DFT
N 1
X ( k )  DFT[ x ( n)] 
 x( n)W N ,
0  k  N 1
nk
n0
x ( n)  IDFT[ X ( k )] 
1
N
N 1
X (k ) 
 x( n)W
nk
N
N 1
 X ( k )W N
 nk
,
0  n  N 1
n0
~
RN ( k )  X ( k ) RN ( k )
n0
x ( n) 
1
N
N 1
 X ( k )W N
 nk
~ ( n) R ( n)
RN ( n )  x
N
n0
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The Properties of DFT
 Linearity
DFT[ax1 (n)  bx2 (n)]  aX 1 (k )  bX 2 (k )
N3-point DFT, N3=max(N1,N2)
 Circular shift of a sequence
DFT[ x(( n  m )) N RN ( n)]  W
km
N
X (k )
 Circular shift in the frequency domain
DFT[W
 nl
N
x( n)]  X (( k  l )) N RN ( k )
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The Properties of DFT
 The sum of a sequence
N 1
X (k ) k 0 
N 1
 x(n)W N

nk
n 0
k 0
 x ( n)
n 0
 The first sample of sequence
x ( 0) 
1
N
N 1
 X (k )
k 0
 DFT[ x( n)]  X ( k )
DFT[ X ( n)]  Nx(( N  k )) N RN ( k )
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The Properties of DFT
 Circular convolution
x1 ( n)
N
 N 1

x 2 ( n)    x1 ( m ) x 2 (( n  m )) N  RN ( n)
 m 0

 N 1

   x 2 ( m ) x1 (( n  m )) N  RN ( n)  x 2 ( n)
 m 0

DFT[ x1 (n)
N
N
x1 ( n)
x2 (n)]  X 1 (k ) X 2 (k )
 Multiplication
DFT[ x1 ( n)  x 2 ( n)] 
1
N
X 1 (k )
N
X 2 (k )
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The Properties of DFT
 Circular correlation
Linear correlation


n  
n  
rxy ( m ) 
 x ( n) y * ( n  m )   x ( n  m ) y * ( n)
Circular correlation
N 1
rxy ( m ) 
 x( n) y * (( n  m ))
N
RN ( m )
n0
N 1

 x(( n  m ))
N
y * ( n)RN ( m )
n0
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The Properties of DFT
if
R xy ( k )  X ( k )  Y ( k )
then
rxy ( m )  IDFT[ R xy ( k )]
*
N 1

 x( n) y * (( n  m ))
N
RN ( m )
n0
N 1

 x(( n  m ))
N
y * ( n)RN ( m )
n0
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The Properties of DFT
 Parseval’s theorem
N 1
 x ( n) y ( n) 
*
n0
let
1
N
N 1
 X ( k )Y (k )
*
k 0
x ( n)  y( n)
N 1
then
 x ( n) x
*
( n) 
n0
N 1
 x ( n)
n0
2
1
N 1
X (k ) X

N
*
(k )
k 0

1
N
N 1
 X (k )
2
k 0
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The Properties of DFT
 Conjugate symmetry properties of DFT

xep (n) and xop (n)
Let x (n) be a N-point sequence
~( n)  x(( n))
x
N
1 ~
1


~
~
xe ( n)  [ x ( n)  x (  n)]  [ x (( n)) N  x (( N  n)) N ]
2
2
1 ~
1


~
~
xo ( n)  [ x ( n)  x (  n)]  [ x (( n)) N  x (( N  n)) N ]
2
2
*
~
~
It can be proved that
x ( n)  x (  n)
e
e
*
~
~
xo ( n)   xo (  n)
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The Properties of DFT
Circular
conjugate
symmetric
component
~ ( n) R ( n)
xep ( n)  x
e
N

1
2
Circular
conjugate
antisymmetric
component
x(( n))

N

 x (( N  n)) N RN ( n)
~ ( n) R ( n)
xop ( n)  x
o
N

1
2
x(( n))

N

 x (( N  n)) N RN ( n)
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The Properties of DFT
x( n)  xep ( n)  xop ( n)
xep ( n)  x (( N  n)) N RN ( n)
*
ep
xop ( n)   x (( N  n)) N RN ( n)
*
op
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The Properties of DFT

X ep (k ) and X op (k )
X ( k )  X ep ( k )  X op ( k )
X ep ( k )  X (( N  k )) N RN ( k )
*
ep
X op ( k )   X (( N  k )) N RN ( k )
*
op
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The Properties of DFT
Re[X ep ( k )]  Re[X ep (( N  k )) N RN ( k )]
Im[ X ep ( k )]   Im[ X ep (( N  k )) N RN ( k )]
Re[X op ( k )]   Re[X op (( N  k )) N RN ( k )]
Im[ X op ( k )]  Im[ X op (( N  k )) N RN ( k )]
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The Properties of DFT

Circular even sequences
if
x( n)  x(( N  n)) N RN ( n)
then
X ( k )  X (( N  k )) N RN ( k )

Circular odd sequences
if
x( n)   x(( N  n)) N RN ( n)
then
X ( k )   X (( N  k )) N RN ( k )
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The Properties of DFT

Conjugate sequences
DFT[ x ( n)]  X ((  k )) N RN ( k )
*
*
 X (( N  k )) N RN ( k )  X ( N  k )
*
*
DFT[ x ((  n)) N RN ( n)]
*
 DFT[ x (( N  n)) N RN ( n)]  X ( k )
*
*
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The Properties of DFT

Complex-value sequences
DFTRe[x ( n)]  X ep ( k )

1
2
X (( k ))
*
N

 X (( N  k )) N RN ( k )
DFT j Im[ x ( n)]  X op ( k )

1
2
X (( k ))
N

 X (( N  k )) N RN ( k )
*
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The Properties of DFT
DFT[ xep ( n)]  Re[X ( k )]
1

*
 DFT  [ x (( n)) N  x (( N  n)) N ]RN ( n)
2

DFT[ xop ( n)]  j Im[ X ( k )]
1

*
 DFT  [ x (( n)) N  x (( N  n)) N ]RN ( n)
2

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The Properties of DFT

Real-value sequences
if x( n) is real - value sequence
then X ( k )  X (( N  k )) N RN ( k )
*

Imaginary-value sequences
if x( n) only has imaginary part
then X ( k )   X (( N  k )) N RN ( k )
*
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The Properties of DFT

Summary
x ( n)  Re[x ( n)]  j Im[ x ( n)]



X ( k )  X ep ( k )

X op ( k )
x ( n) 

xop ( n)

xep ( n)


X ( k )  Re[X ( k )]  j Im[ X ( k )]
example
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The Properties of DFT
 Linear convolution & circular convolution

x1 ( n)
N1 point sequence, 0≤n≤ N1-1
x2 ( n)
N2 point sequence, 0≤n≤ N2-1
Linear convolution
yl ( n)  x1 ( n)  x 2 ( n)
N 1 1


 x (m ) x
1
m
2
(n  m ) 
 x (m ) x
1
2
(n  m )
m 0
yl (n) L point sequence, L= N1+N2-1
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The Properties of DFT

Circular convolution
We have to make both x1 ( n) and x2 ( n) L-point
sequences by padding an appropriate number of zeros
in order to make L point circular convolution.
 x1 ( n), 0  n  N 1  1
x1 ( n)  
N1  n  L  1
 0,
 x2 ( n), 0  n  N 2  1
x 2 ( n)  
N2  n  L  1
 0,
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The Properties of DFT
yc ( n)  x1 ( n)
L
 L 1

x 2 ( n)    x1 ( m ) x 2 (( n  m )) L  RL ( n)
m 0


 L 1

   x1 ( m )  x 2 ( n  rL  m ) RL ( n)
r 
m 0

  L 1

    x1 ( m )x 2 ( n  rL  m ) RL ( n)
r   m 0

 

   y l ( n  rL ) RL ( n)
r 

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The Properties of DFT



yc ( n)    yl ( n  rL) RL ( n)
r  

if
L  N1  N 2  1
then
that is
yc ( n )  y l ( n )
x1 ( n)
L
x 2 ( n)  x1 ( n)  x 2 ( n)
Copyright © 2005. Shi Ping CUC
return
The Sampling Theorem in Frequency Domain
 Sampling in frequency domain
~
X ( k )  X ( z ) |z W k 
N

 x(m )W N
km
m 
1
~
~
x N ( n)  IDFS[ X ( k )] 
N
N 1
~
 kn
X
(
k
)
W

N
k 0
 
km 
 kn

x
(
m
)
W
W

N 
N

N k 0  m  

1

N 1
1
  x( m )
m
N
N 1
k ( mn) 
WN

k 0


 x( n  rN )
r 
Copyright © 2005. Shi Ping CUC
The Sampling Theorem in Frequency Domain
~
x N ( n) 

 x( n  rN )
r 
Frequency Sampling Theorem
For M point finite duration sequence, if the frequency
sampling number N satisfy:
NM
then
~
x N (n)  x N (n) RN (n)  x(n)
Copyright © 2005. Shi Ping CUC
The Sampling Theorem in Frequency Domain
 Interpolation formula of X (z )
N 1
X (z) 

n0

1
N

1
N
N 1

k 0
N 1

k 0
x(n)z
n
N 1
 1
  
n0  N
N 1

X ( k )W
 nk
N
k 0
 N  1  nk  n 
1
X (k )  W N z  
N
 n0

N 1

k 0
 n
z

 N 1
 k 1
X (k )  W N z
 n0
 1  W N Nk z  N  1  z  N
X ( k )

 k 1 
N
 1WN z


N 1



X (k )
 1W
k 0

n
k
N
z
1
Copyright © 2005. Shi Ping CUC
The Sampling Theorem in Frequency Domain
X (z) 
1 z
N
 k (z) 
1
N
N 1
X (k )
 1W
k 0
1 z
N 1W
N 1
k
N
z
1


X ( k ) k ( z )
k 0
N
k
N
z
1
Interpolation
function
Copyright © 2005. Shi Ping CUC
The Sampling Theorem in Frequency Domain
 Interpolation formula of X (e j )
N 1
X (e
jw
)

X ( k ) K ( e
j
N 1
)
k 0
 ( ) 

k 0
sin  2N

N sin  2 

e
 j
X ( k )  ( 
2
k)
N
 N2 1 
Interpolation
function
Copyright © 2005. Shi Ping CUC
return
Approximating to FT (FS) with DFT (DFS)
 Approximating to FT of continuous-time aperiodic
signal with DFT
CTFT
X ( j ) 
x( t ) 
1
2




x ( t )e


 j t
dt
X ( j )e
j t
d
Copyright © 2005. Shi Ping CUC
Approximating to FT (FS) with DFT (DFS)

Sampling in time domain
t  nT ,

X ( j ) 
x(t ) 
2
x ( nt ) 

x ( t )e

1

1
2
dt  T ,
 j t



dt 

T
n  

dt  T 

x ( nT ) e
 j  nT
n  



X ( j )e
S
0
j t
X ( j )e
d
j  nT
d
 s  2f s 
2
T
Copyright © 2005. Shi Ping CUC
Approximating to FT (FS) with DFT (DFS)

Truncation in time domain
t : (0 ~ T0 ) , T0  NT ,
n : (0 ~ N  1)
N 1
X ( j  )  T   x ( nT ) e
 j  nT
n0
x ( nT ) 
1
2

S
X ( j )e
j  nT
d
0
Copyright © 2005. Shi Ping CUC
Approximating to FT (FS) with DFT (DFS)

Sampling in frequency domain
  k 0 ,
T0 
1

F0
 0T 
d   0 ,
N
 NT ,
fs
2
T 
T0
N 1
N 1
d 
0

0
n0
 0  2F0 
2
T0
2
N
X ( jk  0 )  T   x ( nT ) e
n0

s
 jk  0 nT
N 1
 T   x ( n )e
 j
2
nk
N
n0
 T  DFT [ x ( n )]
Copyright © 2005. Shi Ping CUC
Approximating to FT (FS) with DFT (DFS)
0
x ( nT ) 
2
 F0  N 
 fs 
1
N
1
N
N 1

X ( jk  0 ) e
k 0
N 1

X ( jk  0 ) e
j
2
nk
N
k 0
N 1

jk  0 nT
X ( jk  0 ) e
j
2
nk
N
k 0
 f s  IDFT [ X ( jk  0 )] 
demo
1
T
 IDFT [ X ( jk  0 )]
Copyright © 2005. Shi Ping CUC
Approximating to FT (FS) with DFT (DFS)
 Approximating to FS of continuous-time periodic
signal with DFS
X ( jk  0 ) 
1
T0

T0
x ( t )e
dt
0

x( t ) 
 jk 0 t
 X ( jk
0
)e
jk 0 t
k 
 0  2F0 
2
T0
Copyright © 2005. Shi Ping CUC
Approximating to FT (FS) with DFT (DFS)

Sampling in time domain
t  nT ,
X ( jk  0 ) 

1
dt  T ,
T
T0
N 1

n0
x ( nT ) e

T0  NT
 jk  0 nT

T0
N 1
dt 
0
1
N
T
n0
N 1

j
x ( n )e
2
nk
N
n0
 DFS [ x ( n )]
N
Copyright © 2005. Shi Ping CUC
Approximating to FT (FS) with DFT (DFS)
Truncating in frequency domain

T0  NT ,  f s  NF0 ,

x(t ) 

X ( jk  0 ) e
let k : (0, N  1)
jk  0 t
k  
N 1
x ( nT ) 

jk  0 nT
X ( jk  0 ) e
k 0
 N 
1
N


X ( jk  0 ) e
j
2
nk
N
k 0
N 1

N 1
X ( jk  0 ) e
j
2
N
nk
 N  IDFS [ X ( jk  0 )]
k 0
Copyright © 2005. Shi Ping CUC
Approximating to FT (FS) with DFT (DFS)
 Some problems

Aliasing
Sampling in time domain:
fs  2 fh ,
T 
1

1
fs
2 fh
Otherwise, the aliasing will occur in frequency domain
1
Sampling in frequency domain: T0 
F0
T0 Period in time domain F0 Frequency resolution
fs
F0

T0
T
N
f h and F0
is contradictory
Copyright © 2005. Shi Ping CUC
Approximating to FT (FS) with DFT (DFS)

Spectrum leakage
x1 ( n), infinite - length sequence
x2 ( n)  x1 ( n) RN ( n), finite - length sequence
X 2 (e
j
)  X 1 (e
j
)  W R (e
j
)
Spectrum extension (leakage)
Spectrum aliasing
demo
Copyright © 2005. Shi Ping CUC
Approximating to FT (FS) with DFT (DFS)

Fence effect
0 

2

N
0

2F0
fs
fs
,
F0 
fs
N
Frequency resolution
F0 
fs
N

1
NT

1
T0
demo
Copyright © 2005. Shi Ping CUC
Approximating to FT (FS) with DFT (DFS)

Comments

Zero-padding is an operation in which more zeros are
appended to the original sequence. It can provides closely
spaced samples of the DFT of the original sequence.

The zero-padding gives us a high-density spectrum and
provides a better displayed version for plotting. But it does
not give us a high-resolution spectrum because no new
information is added.

To get a high-resolution spectrum, one has to obtain
more data from the experiment or observation.
demo
example
Copyright © 2005. Shi Ping CUC
return
Summary
z-transform of x(n)
The frequency representations of x(n) Complex
frequency domain
X (z )
DTFT of x(n)
j
z

e
Frequency
domain
ZT
interpolation
2
j
k Time
z  e Nsequence
x (n)
DTFT
X (e
j

)
2
DFT of x(n)
Discrete frequency
domain
k
DFT
N
interpolation
X (k )
Copyright © 2005. Shi Ping CUC
return
Illustration of the four Fourier transforms
Fourier Transform
Signals that are continuous
and aperiodic
Fourier Series
Signals that are continuous
and periodic
DTFT
Signals that are discrete and
aperiodic
Discrete Fourier Series
Signals that are discrete and
periodic
Copyright © 2005. Shi Ping CUC
~ (m)
x
1
n0
n1
n2
n3
n4
n5
n6
0
m
~ (n  m)
x
2
~
y ( n)
m
0
0 1 2 3 4 5 6
n
return
Copyright © 2005. Shi Ping CUC
xep ( n) 
1
2
x(n)  x

(( N  n)) N RN ( n)

x (n)
0
5
0
5
0
5
n
x(( N  n)) N
5
n
xep (n)
n
return
Copyright © 2005. Shi Ping CUC
xep ( n)  x (( N  n)) N RN ( n)
*
ep
xep (n)
0
5
0
5
n
xep (( N  n)) N
n
RN (n)
return
Copyright © 2005. Shi Ping CUC
10  (0.8) R11 ( n)
n
Original sequence
x(n)
10
5
0
0
1
2
3
4
5
6
7
8
Circular conjugate symmetric component
9
10 n
0
1
2
3
4
5
6
7
8
Circular conjugate antisymmetric component
9
10 n
0
1
9
10 n
xep(n)
10
5
0
xop(n)
4
2
0
-2
-4
2
3
4
5
6
7
8
return
Copyright © 2005. Shi Ping CUC
Circular even sequence x(n)
10
5
0
0
1
2
3
4
5
6
The DFT of x(n)
7
8
9
10 n
X (k )
40
20
0
0
1
2
3
4
5
6
7
8
9
10 k
X (( N  k )) N RN (n)
40
20
0
0
1
2
3
4
5
6
7
8
9
10 k
return
Copyright © 2005. Shi Ping CUC
Circular odd sequence x(n)
4
2
0
-2
-4
0
1
2
3
4
5
6
7
The imaginary part of DFT[x(n)]
8
9
10 n
8
9
10 k
8
9
10 k
X (k )
10
0
-10
0
1
2
3
4
5
6
7
 X (( N  k )) N RN (n)
10
0
-10
0
1
2
3
4
5
6
7
return
Copyright © 2005. Shi Ping CUC
X ( k )  X (( N  k )) N RN ( k )
*

X (0)  X (( N  k )) N RN ( k )
*

 X ( 0)
*
k 0
 X (0) is a real number
if N is even
X(
N
2

)  X (( N  k )) N RN ( k )
 X(
*

k
N
2
N
X (
*
N
)
2
) is a real number
2
return
Copyright © 2005. Shi Ping CUC
X ( k )   X (( N  k )) N RN ( k )
*

X (0)   X (( N  k )) N RN ( k )
*

  X ( 0)
*
k 0
 X (0) is an imaginary number
if N is even
X(
N
2

)   X (( N  k )) N RN ( k )
 X(
*

k
N
2
N
 X (
*
N
)
2
) is an imginary number
2
return
Copyright © 2005. Shi Ping CUC
x1 ( n) , x2 ( n) N-point real-value sequences
X 1 (k )  DFT[ x1 (n)],
X 2 (k )  DFT[ x2 (n)]
y( n)  x1 ( n)  jx 2 ( n)
Y ( k )  DFT[ y( n)]  DFT[ x1 ( n)  jx 2 ( n)]
 DFT[ x1 ( n)]  jDFT[ x 2 ( n)]  X 1 ( k )  jX 2 ( k )
X 1 ( k )  DFTRe[ y( n)]  Yep ( k ) 
X 2 ( k )  DFTIm[ y( n)] 
1
j
Yop ( k ) 
1
2
Y (k )  Y
1
2j

Y (k )  Y

(( N  k )) N RN ( k )


(( N  k )) N RN ( k )
return
Copyright © 2005. Shi Ping CUC
Linear convolution
Circular convolution N = 6
12
12
10
10
8
8
6
6
4
4
2
2
0
0
0
1
2
3
4
5
6
7
8
9 n
x (n)  [1,2,2],
1
Circular convolution
N=7
0
12
10
10
8
8
6
6
4
4
2
2
0
0
1
2
3
4
5
6
7
8
9 n
2
3
4
5
6
7
8
9 n
x2 (n)Circular
 [1,2
,3,2],
convolution N = 5
12
0
1
0
1
2
3
4
5
6
7
8
9n
return
Copyright © 2005. Shi Ping CUC
 ( )
Magnitude Response, N = 8
1
0.8
2
4
N
N
0.6
0.4
0.2
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
frequency in pi units
0.4
0.6
0.8
1
0.4
0.6
0.8
1
Phase Response
1
pi
0.5
0
-0.5
-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
frequency in pi units
return
Copyright © 2005. Shi Ping CUC
X(k),N = 8
6
5
4
3
2
1
0
0
1
2
3
X (0)( )
4
4
2
6
X (1) (  X ( 3))4(

)

X ( 2)N( 
) N
N
5
6
7
k
2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
frequency in pi units
0.7
0.8
0.9
1
return
Copyright © 2005. Shi Ping CUC
xa ( t )  10  (0.8)
X a ( j)
t
10
50
8
40
FT
6
30
4
20
2
10
0
0
5
10
t
15
20
25
0
-1
X (e
x (n)
10
50
8
40
6
DTFT30
4
20
2
10
0
0
5
-0.5
10
15
n
20
25
0
-2
j
0
rad
0.5
1
)
-1
0
pi
1
2
Copyright © 2005. Shi Ping CUC
X (e
x(n) R11 ( n)
j
)  R(e
j
)
50
10
40
8
30
DTFT
6
20
4
10
2
0
-10
-5
~ ( n)
x
N
0
5
10
0
n
50
-2
-1
~
X N (k )
0
pi
1
2
10
40
8
DFS 30
6
20
4
10
2
0
-10
0
n
10
0
-10
0
k
10
Copyright © 2005. Shi Ping CUC
x N (n)
50
X N (k )
10
40
8
DFT
6
20
4
10
2
0
30
-10
0
n
10
0
-10
0
k
10
return
Copyright © 2005. Shi Ping CUC
x1 ( n)
0
X 1 (e
n
j

X 2 (e
n
)
0
x2 ( n)
0

R( e
n
)
0
RN (n)
0
j
0
j
)

return
Copyright © 2005. Shi Ping CUC
x( n)  [1,1,1,1]
4
DTFT
DFT
2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
x( n)  [1,1,1,1,0,0,0,0]
4
1.6
1.8
2 pi
1.8
2 pi
1.8
2 pi
DTFT
DFT
2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
x( n)  [1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
4
DTFT
DFT
2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
return
Copyright © 2005. Shi Ping CUC
signal
x(n),
x( n)  cos(
0.48
n0<=n<=19
)  cos(0.52n)
2
1
0
-1
-2
0
2
4
6
8
10
n
12
14
16
18
20
20
X 20 ( k )
15
10
5
0
0
0.1
0.2
0.3
0.4
0.5
pi
0.6
0.7
0.8
0.9
1
Copyright © 2005. Shi Ping CUC
signal x(n), 0<=n<=19+80 zeros
2
1
0
-1
-2
0
10
20
30
40
50
n
60
70
80
90
100
20
X 100 ( k )
15
10
5
0
0
0.1
0.2
0.3
0.4
0.5
pi
0.6
0.7
0.8
0.9
1
Copyright © 2005. Shi Ping CUC
signal x(n), 0<=n<=99
2
1
0
-1
-2
0
10
20
30
40
50
n
60
70
80
90
100
50
X 100 ( k )
40
30
20
10
0
0
0.1
0.2
0.3
0.4
0.5
pi
0.6
0.7
0.8
0.9
1
Copyright © 2005. Shi Ping CUC
signal x(n), 0<=n<=99+300 zeros
2
1
0
-1
-2
0
50
100
150
200
n
250
300
350
400
60
X 400 ( k )
40
20
0
0
0.1
0.2
0.3
0.4
0.5
pi
0.6
0.7
0.8
0.9
1
return
Copyright © 2005. Shi Ping CUC
Suppose
F0  10 Hz,
Determine T0 ,
Solution
T
1
f h  4 kHz
T,
N
T0 
1


F0
1

1
 0.1 s
10
1
fs
2 fh
2  4  10
N 
T0
0.1
T
N 2
m

0.125  10
2
10
3
3
 0.125 ms
 800
 1024
return
Copyright © 2005. Shi Ping CUC