fourier analysis of discrete time signals for a discrete time sequence we define two classes of...
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Fourier Analysis of Discrete Time Signals
For a discrete time sequence we define two classes of Fourier Transforms:
• the DTFT (Discrete Time FT) for sequences having infinite duration,
• the DFT (Discrete FT) for sequences having finite duration.
X DTFT x n x n e
x n IDTFT X X e d
j n
n
j n
( ) ( ) ( )
( ) ( ) ( )
1
2
x n( )
nN 1
X ( )
discrete timecontinuous frequency
The Discrete Time Fourier Transform (DTFT)
Given a sequence x(n) having infinite duration, we define the DTFT as follows:
…..…..
Observations:
• The DTFT is periodic with period ;
• The frequency is the digital frequency and therefore it is limited to the interval
)(X 2
Recall that the digital frequency is a normalized frequency relative to the sampling frequency, defined as
sF
F 2
0
0
2/sF2/sFsF sF F22
one period of )(X
Example:
0 1N
1
][nx
n
2/sin
2/sin
1
1)(
2/)1(
1
0
Ne
e
eeX
Nj
j
NjN
n
nj
DTFT
since
Example:
)cos(][ 0 nAnx )(
)()(
0
0
j
j
eA
eAX
Discrete Fourier Transform (DFT)
Definition (Discrete Fourier Transform): Given a finite sequence
where
X k x n w w eNkn
n
N
Nj N( ) ( ) , /
0
12
x XDFT
)]1(),...,1(),0([ Nxxxx
its Discrete Fourier Transform (DFT) is a finite sequence
)]1(),...,1(),0([)( NXXXxDFTX
X xIDFT
Definition (Inverse Discrete Fourier Transform): Given a sequence
x nN
X k w w eNkn
k
N
Nj N( ) ( ) , /
1
0
12
)]1(),...,1(),0([ NXXXX
its Inverse Discrete Fourier Transform (IDFT) is a finite sequence
)]1(),...,1(),0([)( NxxxXIDFTx
where
Observations:
• The DFT and the IDFT form a transform pair.
x XDFT
xX
IDFTback to the same signal !
• The DFT is a numerical algorithm, and it can be computed by a digital computer.
DFT as a Vector Operation
,
]1[
]1[
]0[
Nx
x
x
x
Let
]1[
]1[
]0[
}{
NX
X
X
xDFTX
Then:
,
1
)1(
NkN
kN
k
w
we
110 ]1[...]1[]0[1
NeNXeXeXN
x
xekX Tk*][
x
kekekX
N][
1
]1[
]1[
]0[
1
1
111
]1[
]1[
]0[
}{
)1)(1(1
1
Nx
x
x
ww
ww
NX
X
X
xDFTX
NW
NNN
NN
NNN
XWN
x
xWX
NW
TN
N
1
*1
Periodicity: From the IDFT expression, notice that the sequence x(n) can be interpreted as one period of a periodic sequence :
x np ( )
n
x n( )
nN 1
N 2N N 2N
original sequence
periodic repetition
x np ( )
x nN
X k wN
X k w wN
X k w x n Np Nkn
k
N
Nkn
k
N
NkN
Nk n N
k
N
p( ) ( ) ( ) ( ) ( )( )
1 1 1
0
1
0
1
0
1
This has a consequence when we define a time shift of the sequence.
For example see what we mean with . Start with the periodic extensionx n( ) 1
x np ( )
nN N
x np ( ) 1
nN N
A
AB
B
C
C
D
D
x np ( )
If we look at just one period we can define the circular shift
AB
C
DA
B
C
D
x n( ) x n N( ) 1
A B C D D A B C DD
n 0 1 2 3 0 1 2 3
Properties of the DFT:
• one to one with no ambiguity;
• time shift
where is a circular shift
x n X k( ) ( )
DFT x n m w X kN Nkm( ) ( )
x n m N( )
x x x N m x N m x N( ) ( ) ( ) ( ) ( )0 1 1 1
x N m x N x x x N m( ) ( ) ( ) ( ) ( ) 1 0 1 1
x N m x N( ) ( ) 1
x n m N( )
x n( )periodic repetition
• real sequences
• circular convolution
X k X N k( ) ( ) | ( )| | ( )|X k X N k
DFT x n x n X k X k k N1 2 1 2 0 1( ) ( ) ( ) ( ), ,...,
y n x n x n
x k x n k N
circular shiftk
N
( ) ( ) ( )
( ) ( )
1 2
1 20
1
where both sequences must have the same length N. Then:
x x1 2,
Extension to General Intervals of Definition
Take the case of a sequence defined on a different interval:
0n 10 Nn
][nx
How do we compute the DFT, without reinventing a new formula?
0n
][nx
First see the periodic extension, which looks like this:
n
Then look at the period 10 Nn
0n 10 Nn
][nx
n1N
10 Nn
Example: determine the DFT of the finite sequence
33 if 8.0][ || nnx n
3 3 n
][nx
3 n
][nx
Then take the DFT of the vector
]1[],...,3[],3[],...,1[],0[ xxxxxx