transform domain representation of discrete time signals
TRANSCRIPT
1.02, 0.56, 2.09,0.69,-0.95
F
Transform Domain Representation of
Discrete Time Signals
The Discrete Fourier Transform
(II)
Yogananda Isukapalli
1
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F
1,......0
1
0
1,......0
1
0
)2(1
)2(
1,......0)2(1
0
1,......0
1
0
)2(
)(1][
][)(
...
..
)(1][
][)(
-=-
-
=
-=
-
=
-
-
-=
-
=
-=
-
=
-
å
å
å
å
=
=
=
=
=
=
Nnkn
N
N
k
Nk
N
n
knN
Nj
N
Nj
N
Nnnk
NjN
k
Nk
N
n
nkN
j
WkXN
nx
WnxkX
TheneW
eWlet
ekXN
nx
enxkX
p
p
p
p
IDFT
DFT
IDFT
DFT
(1)
(2)
(3)
(4)
Review of DFT (N-Point Transform)
2
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F
We Have proved :
and they are periodic in k and in n.
npN
otherwise
N
k
pnkNW
=-
=
- =å ..
..0
1
0
)(
• DFT inherits the periodicity from the periodicity of complex digital exponentials:From Equation (1)
å
å-
=
--
-
=
+-
=
=+
1
0
)2()2(
1
0
)()2(
][
][)(
N
n
nNN
jnkN
j
N
n
NknN
j
eenx
enxNkX
pp
p
Family of Mutually Orthogonal Complex Exponentials
3
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F
From Equation (2)
1
][)(1
)(1
)(1][
)2(
1
0
)2(
1
0
)2()2(
1
0
)()2(
=
==
=
=+
å
å
å
-
=
-
=
-
=
+
kNN
j
N
n
knN
j
N
n
kNN
jknN
j
N
n
NnkN
j
e
nxekXN
eekXN
ekXN
Nnx
p
p
pp
p
Periodicity
X(k+N) = X(k) k = 0, 1, …..N-1
x[n+N] = x[n] n = 0, 1, …..N-1
1
)(][
)2(
1
0
)2(
=
==
-
-
=
-
ånN
Nj
N
n
nkN
j
e
kXenx
p
p
4
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FDFT Properties :
1) DFT is Linear :
DFTN ( ax1[n] + bx2[n] )
=DFTN ( ax1[n] ) + DFTN ( bx2[n] )
=aX1(k) + bX2(k)
2) IDFT is Linear :
IDFTn(aX1(k) + bX2(k))
= IDFTn(aX1(k)) + IDFTn(bX2(k))
= ax1[n] + bx2[n]
3) DFT is periodic
X(k+N) = X(k)
4) IDFT is periodic
x[n] = x[n+N]
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FDFT representations :
The period : X(0)…….X(N-1) standard frequencies span from:
0 - 2p(N-1)/N rad/sample
The period centered around 0 :
{X(-N/2)…..,X(0),……..X(N/2-1)}
{Normalized frequencies - p to p-2p/N rad/sample
Also Note :
…… 1
21
2
22
++-
-
=
=
NN
NN
XX
XX
11 -- = NXX
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F5) DFT: Time Shift
[ ] å-
=
--=-
1
0
)2(
00 ][][N
n
nkN
jennxnnxDFT
p
( )
)(
][
][
0
0
0
)2(
)2(
)()2(
kXe
mxDFTe
emx
knN
j
knN
j
N
nmkN
j
p
p
p
-
-
><
+-
=
=
=å
kmN
jekXkX
)2(
12 )()(p
-=DFT: If
Then:
shiftcircularmnxnx ..][][ 12 =
7
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F
0 1 2 3
32
1
32
1
Circular shift for m=1
0 3
1 2
1 0
2 3
m=1
shiftcircularmnxnx ..][][ 12 =8
0 1 2 3
0 0
1.02, 0.56, 2.09,0.69,-0.95
F Example:
0 1 2
32
1
3
21
Let:
If x2[n] = x1[n+1] m = -12
13
Which Implies : kj
ekXkX)
32(
12 )()(p
=6) Evaluation of IDFT from DFT(x[n])
[ ]**
*)2(1
0
*
)([1
)(1][
kXDFTN
ekXN
nxnk
NjN
k
=
úû
ùêë
é=
--
=å
p
9
3
12
0 1 2
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F
Thus we see that IDFT is the complex conjugate of the DFT of X*(k) multiplied by 1/N7) DFT: Symmetry Properties
x[n] is real
(1) Re(X(k)) = Re(X(N-k)) k = 1, 2,…N/2-1 N even
(2) Im(X(k)) = Im(X(N-k)) k = 1, 2,…N-1/2 N odd
(3) |X(k)| = |X(N-k)| k = 1, 2,…N/2-1 N even
(4) <X(k) = -<X(N-k) k = 1, 2,…N-1/2 N odd
0 1 2 3 4 5 6
|X(k)|
0 1 2 3 4 5 6
<X(k)
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F
3 3Example:
0 1 2 3 4
x[n] 1
2
1
å=
-=
4
0
)52(
][)(n
knjenxkX
p
Verify :
p
pppp
p
pppp
9.0
)516()
512()
58()
54(
7.0
)58()
56()
54()
52(
73.0)2(33211)2(
08.3)1(33211)1(
10)0(
j
jjjj
j
jjjj
eXeeeeX
eXeeeeX
X
=
++++=
=
++++=
=
----
----
Follows :
p
p
7.0*
9.0*
08.3)1()4(73.0)2()3(
j
j
eXXeXX-
-
==
==
11
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F
12
8) Even functions: If x(n) is an even function xe(n), i.e. xe(n)= xe(-n), then
å-
=
W==1
0
)cos()()()]([N
neeeD nTknxkXnxF
9) Odd functions: If x(n) is an odd function xo(n), i.e. xo(n)= -xo(-n), then
å-
=
W-==1
0
)sin()()()]([N
noooD nTknxjkXnxF
10) Parseval’s theorem: The normalized energy in the signal is given by either of the expressions
å å-
=
-
=
=1
0
1
0
22 )(1)(N
n
N
k
kXN
nx
11) Delta function: 1)]([ =nTFD d
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F
12) The linear and circular cross-correlations of two data sequences may be computed using DFTs. For example, the circular correlation of two finite length periodic sequences )( ),( 21 nxnx pp
can be calculated as
)]()([
1,.....,0 ),()(1)(
2*1
1
21
1
021
kXkXF
NjjnxnxN
jr
D
pp
N
nxcx
-
-
=
=
-=+= å
13) DFTs may also be used in the computation of circular and linear convolutions, for example
length equal of sequences
periodic finite are )( and ),(),(and n,convolutiocircular denotes where
)]()([
)()()(
321
211
213
nxnxnx
kXkXF
nxnxnx
ppp
D
ppp
Ä
=
Ä=-
13