kuliah ke-12 matrek ii fungsi khusus
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MATEMATEKA REKAYASAMO 091204
Fungsi-fungsi Khusus: Fourier and FT
mahmud mustain
Fourier andFourier Transform
Why should we learn Fourier Transform?
Sumber: www.csee.wvu.edu/~xinl/courses/ee465/fourier.ppt
Born: 21 March 1768 in Auxerre, Bourgogne, FranceDied: 16 May 1830 in Paris, France
Joseph Fourier
Joseph’s father was a tailor in AuxerreJoseph was the ninth of twelve childrenHis mother died when he was nine andhis father died the following year Fourier demonstrated talent on mathat the age of 14.In 1787 Fourier decided to train for the priesthood - a religious life or a mathematical life?In 1793, Fourier joined the local Revolutionary Committee
Fourier’s “Controversy” Work
• Fourier did his important mathematical work on the theory of heat (highly regarded memoir On the Propagation of Heat in Solid Bodies ) from 1804 to 1807
• This memoir received objection from Fourier’s mentors (Laplace and Lagrange) and not able to be published until 1815
Napoleon awarded him a pension of 6000 francs, payable from 1 July, 1815. Napoleon awarded him a pension of 6000 francs, payable from 1 July, 1815. However Napoleon was defeated on 1 July and Fourier did not receive any moneyHowever Napoleon was defeated on 1 July and Fourier did not receive any money
Expansion of a Function
Example (Taylor Series)
constant
first-orderterm
second-orderterm
…
Fourier Series
Fourier series make use of the orthogonality relationships of the sine and cosine functions
Examples
Fourier Transform
• The Fourier transform is a generalization of the complex Fourier series in the limit
• Fourier analysis = frequency domain analysis – Low frequency: sin(nx),cos(nx) with a small n– High frequency: sin(nx),cos(nx) with a large n
• Note that sine and cosine waves are infinitely long – this is a shortcoming of Fourier analysis, which explains why a more advanced tool, wavelet analysis, is more appropriate for certain signals
Applications of Fourier Transform
• Physics– Solve linear PDEs (heat conduction, Laplace, wave
propagation)• Antenna design
– Seismic arrays, side scan sonar, GPS, SAR• Signal processing
– 1D: speech analysis, enhancement …– 2D: image restoration, enhancement …
Not Just for EE
• Just like Calculus invented by Newton, Fourier analysis is another mathematical tool
• BIOM: fake iris detection• CS: anti-aliasing in computer graphics• CpE: hardware and software systems
FT in Biometrics
natural fake
FT in CS
Anti-aliasing in 3D graphic display
FT in CpE
• Computer Engineering: The creative application of engineering principles and methods to the design and development of hardware and software systems
• If the goal is to build faster computer alone (e.g., Intel), you might not need FT; but as long as applications are involved, there is a place for FT (e.g., Texas Instrument)
14
Frequency-Domain Analysis of Interpolation
• Step-I: Upsampling
• Step-II: Low-pass filtering• Different interpolation schemes correspond to
different low-pass filters
L/nTxL/nxnx ci
15
Frequency Domain Representation of Upsampling
k
Lkje wLXekxwX
16
Frequency Domain Representation of Interpolation
Introduction toFast Fourier Transform (FFT)
Algorithms
Sumber: ecee.colorado.edu/~ecen4002/10_fft_intro.ppt
ECEN4002 Spring 2003 FFT Intro R. C. Maher 18
Discrete Fourier Transform (DFT)
• The DFT provides uniformly spaced samples of the Discrete-Time Fourier Transform (DTFT)
• DFT definition:
• Requires N2 complex multiplies and N(N-1) complex additions
1
0
2
][][N
n
Nnkj
enxkX
1
0
2
][1][N
n
Nnkj
ekXN
nx
ECEN4002 Spring 2003 FFT Intro R. C. Maher 19
Faster DFT computation?• Take advantage of the symmetry and periodicity
of the complex exponential (let WN=e-j2/N)– symmetry: – periodicity:
• Note that two length N/2 DFTs take less computation than one length N DFT: 2(N/2)2<N2
• Algorithms that exploit computational savings are collectively called Fast Fourier Transforms
*][ )( knN
knN
nNkN WWW
nNkN
NnkN
knN WWW ][][
ECEN4002 Spring 2003 FFT Intro R. C. Maher 20
Decimation-in-Time Algorithm
• Consider expressing DFT with even and odd input samples:
1
02/
1
02/
1
0
21
0
2
1
0
22
22
]12[]2[
)](12[)](2[
][][
][][
NN
NN
r
rkN
kN
r
rkN
r
rkN
kN
r
rkN
oddn
nkN
evenn
nkN
N
n
nkN
WrxWWrx
WrxWWrx
WnxWnx
WnxkX
ECEN4002 Spring 2003 FFT Intro R. C. Maher 21
DIT Algorithm (cont.)• Result is the sum of two N/2 length DFTs
• Then repeat decomposition of N/2 to N/4 DFTs, etc.
samples odd ofDFT N/2
sampleseven ofDFT N/2
][][][ kHWkGkX kN
X[0…7]
x[0,2,4,6]
x[1,3,5,7]
N/2
DFT
N/2
DFT7...0
NW
ECEN4002 Spring 2003 FFT Intro R. C. Maher 22
Detail of “Butterfly”
• Cross feed of G[k] and H[k] in flow diagram is called a “butterfly”, due to shape
rNW
)(
)2(
rN
NrN
W
W
rNW -1
or simplify:
ECEN4002 Spring 2003 FFT Intro R. C. Maher 23
8-point DFT Diagram
0NW
0NW
0NW
0NW
0NW
0NW
0NW
2NW
2NW
2NW
1NW
3NW
1
1
1
1
1
1
1
1
1
1
1
1
X[0…7]x[0,4,2,6,1,5,3,7]
ECEN4002 Spring 2003 FFT Intro R. C. Maher 24
Computation on DSP
• Input and Output data– Real data in X memory– Imaginary data in Y memory
• Coefficients (“twiddle” factors)– cos (real) values in X memory– sin (imag) values in Y memory
• Inverse computed with exponent sign change and 1/N scaling
AL-HAMDULILLAH