introduction and review of fourier transforms
TRANSCRIPT
Phys 515
Introduction and Review of Fourier Transforms
See Ron Bracewell, “The Fourier Transform and its Applications”
Definition of Fourier Transform
F(s) = f (x)e− i2π sx dx, f (x) =−∞
+∞
∫ F(s)e+ i2π sx ds−∞
+∞
∫F(s) is the Fourier transform of f (x).
Other conventions: 2π taken out of exponential andappears outside integral in different ways.
§ existence: integral of |f(x)| must exist, discontinuities finite.
§ See Bracewell Ch. 2 for more details
Phys 515
Even and Odd Functions
§ Let f(x)=sum of an odd and even fcn. Then:
§ If we are dealing with real functions:
§ the FT of an even fcn is real and even § the FT of an odd fcn is imaginary and odd
Phys 515
Some symmetries
Phys 515
f (x) ⊃ F(s) means "F(s) is the Fourier Transform of f (x)."
If f (x) is real, F (s ) 2 is even.
Convolution
Phys 515
Convolution of two fcns f (x) and g(x) is:
f ∗ g ≡ f (x)∗ g(x) = f (u)g(x − u)du−∞
+∞
∫Note that: f ∗ g ≡ g ∗ f , f ∗ (g ∗h) ≡ ( f ∗ g)∗h etc f ∗ (g + h) ≡ f ∗ g + f ∗h
Convolution cont’d
Phys 515
Phys 515
Convolution vs. Cross-Correlation
Correlation and convolution of data are two related operations, and in many cases are identical. Both are a "weighted moving average".
Convolution: reverse time signal, shift, multiply Correlation: shift, multiply Same thing if g is symmetric!
Convolution : Cross correlation :
f ∗ g = g ∗ f = f (u)g(x − u)−∞
∞
∫ du f g = f (u − x)g(u)−∞
∞
∫ du
= f (u)g(u + x)−∞
∞
∫ du ≠ g f
Phys 515
Convolution vs Correlation - FT
Convolution is used for digital filtering.The reason convolution is preferred to correlation for filtering has to do with how the frequency spectra of the two signals interact. Convolving two signals is equivalent to multiplying the frequency spectra of the two signals together - which is easily understood, and is what we mean by filtering.
Convolution : Cross correlation :
f ∗ g = g ∗ f = f (u)g(x − u)−∞
∞
∫ du f g = f (u)g(u + x)−∞
∞
∫ du
FT { f ∗ g} = FT { f } ⋅FT {g} ≡ F ⋅G FT { f g} = F ⋅G∗
Correlation is equivalent to multiplying the complex conjugate of the frequency spectrum of one signal by the frequency spectrum of the other. Complex conjugation is not so easily understood and so convolution is used for digital filtering.
Note: by convention, for complex fcns, the f(u) in the integrals above is replaced with its complex conjugate f*(u).
FT convolution theorem
Autocorrelation
Phys 515
f (x) f (x) = f (u) f (u − x)du−∞
+∞
∫max at x=0 (see Bracewell), even for real f(x)
Power spectrum
§ Power spectrum of f(x) is |F(s)|2.
§ Power spectrum destroys all phase information
§ cannot reconstruct f(x) from |F(s)|2, though you can sometimes/often use the power spectrum to infer some properties of f(x)
Phys 515
Power spectrum cont’d
§ Autocorrelation Theorem: § The Fourier Transform of the autocorrelation function of
f(x) is the power spectrum of f(x)
Phys 515
f (x) ⊃ F(s) ⇒
f (x) f (x) ⊃ F(s) 2
Power spectrum cont’d
§ Rayleigh’s Theorem
§ Power Theorem:
Phys 515
f (x)g∗(x)dx−∞
+∞
∫ = F(s)G∗(s)ds−∞
+∞
∫
f (x) 2 dx−∞
+∞
∫ = F(s) 2 ds−∞
+∞
∫
Special Symbols
Phys 515
Transform Pairs
Phys 515
Transform pairs cont’d
Phys 515
Other Theorems:
§ expansion of time axis causes compression of frequency axis …
§ … in such a way that vertical scale grows so that the area beneath the curves is constant
Phys 515
f (x) ⊃ F(s)⇒
f (ax) ⊃ 1aF
sa
⎛⎝⎜
⎞⎠⎟
1. Similarity
Other Theorems:
2. Shift Theorem:
Note: power spec of a shifted function is identical to the power spec of the original function! (prove)
3. Derivative Theorem: 4. Rotation theorem (2D): If a fcn f(x,y) is rotated by
theta clockwise, its F.T. is also rotated by theta clockwise (not generally known!) - circularly symmetric fcns have circularly symmetric transforms
- http:// www.eee.hku.hk/~work8501/WWW2008/ho4.pdf 5. Convolution Theorem: 6. Power Theorem and Rayleigh’s Theorem: 7. Autocorrelation Theorem: [we’ve already seen these]
Phys 515
f (x + Δ) ⊃ ei2π sΔF(s)
f '(x) ⊃ i2π sF(s)
2D Transforms
§ Entirely analogous to 1D transforms
§ Homework: work your way through the following “workbooks”:
§ http://www.cs.unm.edu/~brayer/vision/fourier.html • some nice illustrations, esp. of rotated image (note
explanation of why it's different)
§ http://www.ysbl.york.ac.uk/~cowtan/fourier/fourier.html • Book of Fourier - some great examples!
Phys 515
The Sampling Theorem
§ Obviously the more samples we take, the better we approximate a measured function.
§ And, the more high frequency power there is in the power spectrum, the more samples we are going to need.
§ Is there a limit? Do we always do better by taking more samples?
§ Sampling Theorem: A continuous band-limited signal can be completely represented by a set of equally spaced samples, if and only if the samples occur at more than twice the frequency of the highest frequency component of the function.
Phys 515
Sampling Theorem
Phys 515 Bracewell p. 223
Sampling Theorem in 2D
http://archive.org/details/Lectures_on_Image_Processing
Phys 515
Exercise
Use Matlab for the following: 1. Construct a “small” 2D Gaussian in the center of
an image which is otherwise zero everywhere. 2. Take its FT. Display it – what does it look like? 3. Make the Gaussian “bigger”. How does the FT
change? Why? 4. Construct a picture like the one on the previous
page. Take its FT. Do you see the features like on the previous page? Make sure you understand what you see. You may have to restrict yourself to the magnitude or power spec of the FT .
Phys 515