Discrete Fourier Transform (DFT)
[Pages:20]Discrete Fourier Transform (DFT)
Recall the DTFT:
X() =
x(n)e-jn.
n=-
DTFT is not suitable for DSP applications because
? In DSP, we are able to compute the spectrum only at specific discrete values of ,
? Any signal in any DSP application can be measured only in a finite number of points.
A finite signal measured at N points:
0, n < 0, x(n) = y(n), 0 n (N - 1), 0, n N,
where y(n) are the measurements taken at N points.
EE 524, Fall 2004, # 5
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Sample the spectrum X() in frequency so that
2
X(k) = X(k), =
=
N
N -1
X(k) =
x(n)e-j
2
kn N
DFT.
n=0
The inverse DFT is given by:
x(n) =
1
N -1
X
(k)ej2
kn N
.
N
k=0
1 N-1 x(n) =
N
k=0
N -1
x(m)e-j
2
km N
m=0
ej
2
kn N
N -1
= x(m)
m=0
1
N -1
e-j2
k(m-n) N
N
k=0
= x(n).
(m-n)
EE 524, Fall 2004, # 5
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The DFT pair:
N -1
X(k) =
x(n)e-j2
kn N
analysis
n=0
x(n) =
1
N -1
X
(k)ej
2
kn N
synthesis.
N
k=0
Alternative formulation:
N -1
X(k) =
x(n)W kn
-
W
=
e-j
2 N
n=0
x(n) =
1
N -1
X(k)W -kn.
N
k=0
EE 524, Fall 2004, # 5
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EE 524, Fall 2004, # 5
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Periodicity of DFT Spectrum
N -1
X(k + N ) =
x(n)e-j2
(k+N N
)n
n=0
N -1
=
x(n)e-j2
kn N
e-j2n
n=0
= X(k)e-j2n = X(k) =
the DFT spectrum is periodic with period N (which is expected, since the DTFT spectrum is periodic as well, but with period 2).
Example: DFT of a rectangular pulse:
x(n) =
1, 0 n (N - 1), 0, otherwise.
N -1
X(k) =
e-j2
kn N
=
N (k)
=
n=0
the rectangular pulse is "interpreted" by the DFT as a spectral
line at frequency = 0.
EE 524, Fall 2004, # 5
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DFT and DTFT of a rectangular pulse (N=5)
EE 524, Fall 2004, # 5
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Zero Padding
What happens with the DFT of this rectangular pulse if we increase N by zero padding:
{y(n)} = {x(0), . . . , x(M - 1), 0, 0, . . . , 0 }, N-M positions
where x(0) = ? ? ? = x(M - 1) = 1. Hence, DFT is
N -1
M -1
Y (k) =
y
(n)e-j2
kn N
=
y(n)e-j
2
kn N
n=0
n=0
=
sin(
kM N
)
sin(
k N
)
e-j
k(M -1) N
.
EE 524, Fall 2004, # 5
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DFT and DTFT of a Rectangular Pulse with Zero Padding (N = 10, M = 5)
Remarks:
? Zero padding of analyzed sequence results in "approximating" its DTFT better,
? Zero padding cannot improve the resolution of spectral components, because the resolution is "proportional" to 1/M rather than 1/N ,
? Zero padding is very important for fast DFT implementation (FFT).
EE 524, Fall 2004, # 5
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