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Chapter 5
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Deepak Kumar Rout.
Synergy
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Chapter 8. The Discrete Fourier Transform
8.1 Laplace, z-, and Fourier Transforms
8.2 Fourier Transform
8.3 Fourier Series
8.4 Discrete Fourier Transform (DFT)
8.5 Properties of DFS/DFT
8.6 DFT and z-Transform
8.7 Linear Convolution vs. Circular Convolution
8.8 Discrete Cosine Transform(DCT)
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1. Laplace, z-, Fourier Transforms
Analog systems
(continuous time)
Digital Systems
(discrete time)
H(s)
H(z)
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Laplace transform
-z-transform
LHP inside u.c
Fouier transforms
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2. Fourier Transform
(1) continuous aperiodic signals
conti aper
aper conti
x(t) 1 t
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(2) Discrete aperiodic signals
conti per
aper discr
x(n) 1 t ω
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3. Fourier Series
(1) continuous periodic signals
discrete aper
per conti
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X(t) 1 k t T
(2) discrete periodic signals (*Discrete Fourier Series)
discrete per
per discre
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x[n] 1
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4. Discrete Fourier Transform (DFT)
-For a numerical evaluation of Fourier transform and its
inversion, (i.e,computer-aided computation), we need discrete
expression of of both the time and the transform domain data.
-For this,take the advantage of discrete Fourier series(DFS, on page
4), in which the data for both domain are discrete and periodic.
discrete periodic
periodic discrete
-Therefore, given a time sequence x[n], which is
aperiodic and discrete, take the following approach.
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Mip
Top
Top
Mip
DFS
DFT
Reminding that, in DFS
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Define DFT as
(eq)
X[k]
x[n] 1 k n N N
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Graphical Development of DFT
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DFS
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DFT
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5. Property of DFS/DFT (8.2 , 8.6)
(1) Linearity
(2) Time shift
(3) Frequency shift
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(4) Periodic/circular convolution in time
(5) Periodic/circular convolution in frequency
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(6) Symmetry
DFS
DFT
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6. DFT and Z-Transform
(1) Evaluation of from
①If length limited in time, (I.e., x[n]=0, n<0, n>=N)
then
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② What if x[n] is not length-limited?
then aliasing unavoidable. … … … … … …
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(2) Recovery of [or ] from
(in the length-limited case)
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7. Linear Convolution vs. Circular Convolution
(1) Definition
① Linear convolution
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24. Rectangular window of length N
② Circular convolution N
Rectangular window of length N
Periodic
convolution N
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(2) Comparison N
H[n] 2N 2N
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Omit chap. 8.7

26. 8. Discrete cosine transform (DCT)
Definition
- Effects of Energy compaction
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Test signal for computing DFT and DCT

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(a) Real part of N-point DFT; (b) Imaginary part of N-point DFT; (c) N-point DCT-2 of the test signal
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Comparison of truncation errors for DFT and DCT-2
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Appendix: Illustration of DFTs for Derived Signals
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