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Chapter 7: Fourier Analysis. Fourier Analysis = Series + Transform ◎ Fourier Series -- A periodic ( T ) function f ( x ) can be written as the sum of sines and cosines of varying amplitudes and frequencies. ○ Some function is formed by a finite number
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Chapter 7: Fourier Analysis Fourier Analysis = Series + Transform ◎ Fourier Series -- A periodic (T) function f(x) can be written as the sum of sines and cosines of varying amplitudes and frequencies
○ Some function is formed by a finite number of sinuous functions
Some function requires an infinite number of sinuous functions to compose
Spectrum The spectrum of a periodic function is discrete, consisting of components at dc, , and its multiples, e.g., For non-periodic functions, i.e., The spectrum of the function is continuous
○ In complex form: Compare with
Euler’s formula: 7-6
Discrete case: ◎ Fourier Transform • Vector-Matrix form
Let 7-12
Let ○ Inverse DFT
◎ Properties ○ Linearity: Show:
Application: Noise removal f’ = f + n, n: additive noise It may be easier to identify than n.
○ Scaling: Show: Assignment : Show
Compared with: 7-25
◎ Convolution theorem: Convolution: ◎ Correlation theorem Correlation: * : conjugate 7-27
◎ Fast Fourier Transform (FFT) -- Successive doubling method Let Assume Let N = 2M. 7-28
Let --------- (B) Consider 7-30
---- (C) 7-32
○ Analysis : The Fourier sequence F(u), u = 0 , … , N-1 of f(x), x = 0 , … , N-1can be formed from sequences u = 0 , …… , M-1 Recursively divide F(u) and F(u+M), eventually each contains one element F(u), i.e., u = 0, and F(u) = f(x). 7-33
○ Example: Input { f(0), f(1), ……, f(7) } Computing needs { f(0), f(2), f(4), f(6) } Computing needs { f(1), f(3), f(5), f(7) } { f(0), f(4)},{ f(2), f(6) } { f(0), f(2), f(4), f(6) } even odd { f(1), f(5)}, {f(3), f(7) } { f(1), f(3), f(5), f(7) } 7-35
{ f(2)},{ f(6)} { f(0), f(4)} { f(0)},{ f(4)} { f(2), f(6)} even odd even odd { f(3)},{ f(7)} { f(1), f(5)} { f(1)},{ f(5)} { f(3), f(7)} Reorder the input sequence into {f(0), f(4), f(2), f(6), f(1), f(5), f(3), f(7)} *Bit-Reversal Rule 7-36
。 Time complexity : the length of the input sequence FT: FFT: Times of speed increasing: N FT FFT Ratio 4 16 8 2.0 8 84 24 2.67 16 256 64 4.0 32 1024 160 6.4 64 4096 384 10.67 128 16384 896 18.3 256 65536 2048 32.0 512 262144 4608 56.9 1024 1048576 10240 102.4
○ Inverse FFT ← Given ← compute i. Input into FFT. The output is ii. Taking the complex conjugate and multiplying by N , yields the f(x)
◎ 2D Fourier Transform ○ FT: IFT:
◎ Properties ○ Filtering: every F(u,v) is obtained by multiplying every f(x,y) by a fixed value and adding up the results. DFT can be considered as a linear filtering ○ DC coefficient:
F(u,v) = F*(-u,-v) ○ Conjugate Symmetry:
○ Rotation Polor coordinates:
○ Display: effect of log operation
◎ Filtering in Frequency Domain ○ Low pass filtering I FT m IFT