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Relatives of DFT
Definitions
Formation
Relationship
Mathematic
Relationship
Properties of DFT
Periodical
Convolution Law
Circular Convolution Law
Symmetry
Relatives of DFT
Equations
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Laplace
Transform
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Fourier Transform
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Z Transform
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DTFT
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DFS
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DFT
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(k = 0, 1, ? N-1)
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Formation
Relationship
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LT
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Complex
Exponential
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s→jω
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Imaginary
Exponential
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FT
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Time
Continuous
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ejω→z , t→n
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Time Discrete
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ZT
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Frequency
Continuous
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z→ejΩ
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Frequency
Discrete
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DTFT
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Inf. Time, Cont. Freq.
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(-∞,∞)→(0,N-1)
Ω→2πk/N
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Finite Time, Discr. Freq.
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DFS
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Periodic
Signal
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ck→X[k]
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Nonperiodic Signal
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DFT
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Mathematic
Relationship
Laplace &
Z
Fourier & Z
DFT & Z
DFT & DTFT
Properties of DFT
Periodical Convolution Law
Given
Periodical Sequences x1[n], x2[n],
x3[n]
with period of N,
DFT of them are X1[k], X2[k],
X3[k].
If X3[k] = X1[k]
X2[k] , then
Circular
Convolution Law
Circular Shift ( Non-Periodical x[m] )
xp[n-m]RN[n]
m
Samples Circular Shift of x[n], where
RN[n] is Rectangular
Window of length N.
Circular Convolution:
Circular
Convolution Law
Given Finite Length Sequences x[n], h[n], y[n]
with same Length N, DFT of them are X [k],
H [k],
Y [k]. If Y [k] = X [k]
H [k] , then
y[n] = x[n] # h[n]
 :Convolution can be done by DFT !
 :If length of two sequences are M
& N,
one
of them must be expanded into
a (M+N-1)
periodical sequence.
Symmetry
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Signal
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DFT
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x[n]
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X[k]
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x*[n]
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X* [-k]
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x*
[-n]
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X*[k]
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Re{ x[n] }
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Xe[k]
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j Im{ x[n] }
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Xo[k]
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xe[n]
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Re{ X[k] }
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xo[n]
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j Im{ X[k]
}
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X[n]
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N x[-k]
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Even Symmetry Sequence:
xe[n] = (
x[n] + x* [-n]
) / 2
Xe[k] = ( X[k] + X* [-k] ) / 2
Odd Symmetry Sequence:
xo[n] = ( x[n] - x* [-n] ) / 2
Xo[k] = ( X[k] - X* [-k] ) / 2
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