1 Fourier Transformation Fourier Transformasjon Fourier Transformasjon f(x) F(u)
Discrete Fourier Transform Prof. Siripong Potisuk.
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Discrete Fourier Transform Prof. Siripong Potisuk
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Transcript of Discrete Fourier Transform Prof. Siripong Potisuk.
Discrete Fourier Transform
Prof. Siripong Potisuk
Summary of Spectral Representations
Signal type
Transform Frequency Domain
CT, Periodic
Continuous-time Fourier Series
(CTFS)
Discrete Spectrum
CT, Aperiodic
Continuous-time Fourier Transform
(CTFT)
Continuous
Spectrum
DT, Aperiodic
Discrete-time Fourier Transform
(DTFT)
Continuous
Spectrum,
periodic
DT, Periodic
Discrete-time Fourier Series (DTFS)
Discrete Fourier Transform (DFT)
Discrete
Spectrum,
periodic
Computation of DTFT
Computer implementation can be accomplished by:
1. Truncate the summation so that it ranges over finite limits x[n] is a finite-length sequence.2. Discretize to k
evaluate DTFT at a finite number of discrete frequencies
For an N-point sequence, only N values of frequency samplesof X(ej) at N distinct frequency points, are sufficient to determine x[n] and X(ej) uniquely.
10, Nkk
Sequence Truncation
Uniform Frequency Sampling
,)()( jez
j zHeH
10,][
)()(
21
0
/2
Nkenx
eXkX
N
nkjN
n
Nkj
k
N
kjzk
2expRe(z)
Im(z)
z0
z1
z2
z3
z4
z5
z6
z7
1
N = 8
Discrete Fourier Transform
Let x[n] be an N-point signal, and WN be the Nth root of unity. The N-point discrete Fourier Transform of x[n],denoted X(k) = DFT{x[n]}, is defined as
N
nkjN
n
N
n
knN
enx
NkWnxkX
21
0
1
0
][
10,][)(
Inverse Discrete Fourier Transform
Let X(k) be an N-point DFT sequence, and WN be theNth root of unity. The N-point inverse discrete FourierTransform of X(k), denoted x[n] = IDFT{X(k)}, isdefined as
N
nkjN
k
N
k
knN
ekXN
NnWkXN
nx
21
0
1
0
)(1
10,)(1
][
Nth Root of UnitykNW
1 )4
)3
1 )2
1)
4/3
2/
4/
NN
NN
NN
NN
W
jW
W
jW
1*
2/2
)2/(
)8
)7
)6
5)
NN
kN
kN
kN
NkN
kN
NkN
WW
WW
WW
WW
N
jWN
2exp
Matrix Formulation
xWX
Nx
x
x
x
WWW
WWW
WWW
NX
X
X
X
NNN
NN
NN
NNNN
NNNN
]1[
]2[
]1[
]0[
1
1
1
1111
)1(
)2(
)1(
)0(
)1)(1()1(21
)1(242
121
Matrix Formulation
xWN
X
xWX
Nx
x
x
x
WWW
WWW
WWW
N
NX
X
X
X
NNN
NN
NN
NNNN
NNNN
*
1
)1)(1()1(2)1(
)1(242
)1(21
1
]1[
]2[
]1[
]0[
1
1
1
1111
1
)1(
)2(
)1(
)0(
Example 4.2Define a sequence x[n] = 1, 2, 3, 4 when n = 0, 1, 2, 3,respectively. Evaluate its DFT, X(k).
Example 4.3Using the result from example 4.2, evaluate the IDFT to obtain the time-domain sequence, x(n).
DFT Computation Using MATLAB
fft(x) - Computes the N-point DFT of a vector x of length N
fft(x, M) - Computes the M-point DFT of a vector x of length N If N < M, x is zero-padded at the end to make it into a vector of length M If N > M, x is truncated to the first M samples
ifft(X) - Computes the N-point IDFT of a vector X of length N
ifft(X, M) - Computes the M-point IDFT of a vector X of length N If N < M, X is zero-padded at the end to make it into a vector of length M If N > M, X is truncated to the first M samples
DFT Interpretation
)( spectrum Phase
|)(|N
1 spectrum Magnitude
kX
kX
DFT sample X(k) specifies the magnitude and phase angle of the kth spectral component of x[n].
The amount of power that x[n] contains at a normalizedfrequency, fk, can be determined from thepower density spectrum defined as
10,|)(|
)( 2
2
NkN
kXkPN
Example 4.3Consider the sequence given below. Compute andsketch the magnitude, phase, and power densityspectra.
