ip-l5 -fourier transform...fourier transform • different formulations for the different classes of...
TRANSCRIPT
![Page 1: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/1.jpg)
Fourier Transform
1
![Page 2: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/2.jpg)
Contents
• Signals as functions (1D, 2D) – Tools
• Continuos Time Fourier Transform (CTFT)
• Discrete Time Fourier Transform (DTFT)
• Discrete Fourier Transform (DFT)
• Discrete Cosine Transform (DCT)
• Sampling theorem
2
![Page 3: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/3.jpg)
Signals as functions
1. Continuous functions of real independent variables – 1D: f=f(x) – 2D: f=f(x,y) x,y – Real world signals (audio, ECG, images)
2. Real valued functions of discrete variables – 1D: f=f[k] – 2D: f=f[i,j] – Sampled signals
3. Discrete functions of discrete variables – 1D: y=y[k] – 2D: y=y[i,j] – Sampled and quantized signals – For ease of notations, we will use the same notations for 2 and 3
3
![Page 4: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/4.jpg)
Fourier Transform
• Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous/
discrete time and frequency variables. – Notations:
• CTFT: continuous time FT: t is real and f real (f=ω) (CT, CF) • DTFT: Discrete Time FT: t is discrete (t=n), f is real (f=ω) (DT, CF) • CTFS: CT Fourier Series (summation synthesis): t is real AND the function is periodic, f
is discrete (f=k), (CT, DF) • DTFS: DT Fourier Series (summation synthesis): t=n AND the function is periodic, f
discrete (f=k), (DT, DF) • P: periodical signals • T: sampling period • ωs: sampling frequency (ωs=2π/T) • For DTFT: T=1 → ωs=2π
• This is a hint for those who are interested in a more exhaustive theoretical approach
4
![Page 5: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/5.jpg)
Images as functions
• Gray scale images: 2D functions – Domain of the functions: set of (x,y) values for which f(x,y) is defined : 2D lattice
[i,j] defining the pixel locations – Set of values taken by the function : gray levels
• Digital images can be seen as functions defined over a discrete domain {i,j: 0<i<I, 0<j<J}
– I,J: number of rows (columns) of the matrix corresponding to the image – f=f[i,j]: gray level in position [i,j]
5
![Page 6: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/6.jpg)
Example 1: δ function
[ ]⎩⎨⎧
≠≠
===
jijiji
ji;0,001
,δ
[ ]⎩⎨⎧ ==
=−otherwise
JjiJji
0;01
,δ
6
![Page 7: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/7.jpg)
Example 2: Gaussian
2
22
2
21),( σ
πσ
yx
eyxf+
=
2
22
2
21],[ σ
πσ
ji
ejif+
=
Continuous function
Discrete version
7
![Page 8: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/8.jpg)
Example 3: Natural image
8
![Page 9: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/9.jpg)
Example 3: Natural image
9
![Page 10: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/10.jpg)
Mathematical Background: Complex Numbers
• A complex number x is of the form:
a: real part, b: imaginary part
• Addition
• Multiplication
![Page 11: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/11.jpg)
Mathematical Background: Complex Numbers (cont’d)
• Magnitude-Phase (i.e.,vector) representation
Magnitude:
Phase:
φ Phase – Magnitude notation:
![Page 12: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/12.jpg)
Mathematical Background: Complex Numbers (cont’d)
• Multiplication using magnitude-phase representation
• Complex conjugate
• Properties
![Page 13: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/13.jpg)
Mathematical Background: Complex Numbers (cont’d)
• Euler’s formula
• Properties
j
![Page 14: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/14.jpg)
Mathematical Background: Sine and Cosine Functions
• Periodic functions
• General form of sine and cosine functions:
![Page 15: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/15.jpg)
Mathematical Background: Sine and Cosine Functions
Special case: A=1, b=0, α=1
π
π
![Page 16: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/16.jpg)
Mathematical Background: Sine and Cosine Functions (cont’d)
Note: cosine is a shifted sine function:
• Shifting or translating the sine function by a const b
cos( ) sin( )2
t t π= +
![Page 17: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/17.jpg)
Mathematical Background: Sine and Cosine Functions (cont’d)
• Changing the amplitude A
![Page 18: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/18.jpg)
Mathematical Background: Sine and Cosine Functions (cont’d)
• Changing the period T=2π/|α| consider A=1, b=0: y=cos(αt)
period 2π/4=π/2
shorter period higher frequency (i.e., oscillates faster)
α =4
Frequency is defined as f=1/T
Alternative notation: sin(αt)=sin(2πt/T)=sin(2πft)
![Page 19: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/19.jpg)
Fourier Series Theorem
• Any periodic function can be expressed as a weighted sum (infinite) of sine and cosine functions of varying frequency:
is called the “fundamental frequency”
![Page 20: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/20.jpg)
Fourier Series (cont’d)
α1
α2
α3
![Page 21: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/21.jpg)
Continuous Fourier Transform (FT)
• Transforms a signal (i.e., function) from the spatial domain to the frequency domain.
where
(IFT)
![Page 22: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/22.jpg)
Why is FT Useful?
• Easier to remove undesirable frequencies.
• Faster perform certain operations in the frequency domain than in the spatial domain.
![Page 23: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/23.jpg)
Example: Removing undesirable frequencies
remove high frequencies
reconstructed signal
frequencies noisy signal
To remove certain frequencies, set their corresponding F(u) coefficients to zero!
![Page 24: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/24.jpg)
How do frequencies show up in an image?
• Low frequencies correspond to slowly varying information (e.g., continuous surface).
• High frequencies correspond to quickly varying information (e.g., edges)
Original Image Low-passed
![Page 25: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/25.jpg)
Example of noise reduction using FT
![Page 26: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/26.jpg)
Frequency Filtering Steps
1. Take the FT of f(x):
2. Remove undesired frequencies:
3. Convert back to a signal:
We’ll talk more about this later .....
![Page 27: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/27.jpg)
Definitions
• F(u) is a complex function:
• Magnitude of FT (spectrum):
• Phase of FT:
• Magnitude-Phase representation:
• Energy of f(x): P(u)=|F(u)|2
![Page 28: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/28.jpg)
Continuous Time Fourier Transform (CTFT)
Time is a real variable (t)
Frequency is a real variable (ω)
Signals : 1D
28
![Page 29: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/29.jpg)
The idea A signal can be interpreted as en electromagnetic wave. This consists of lights of different “color”, or frequency, that can be split apart usign an optic prism. Each component is a “monochromatic” light with sinusoidal shape.
Following this analogy, each signal can be decomposed into its “sinusoidal” components which represent its “colors”.
Of course these components in general do not correspond to visible monochromatic light. However, they give an idea of how fast are the changes of the signal.
29
![Page 30: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/30.jpg)
CTFT: Concept
■ A signal can be represented as a weighted sum of sinusoids.
■ Fourier Transform is a change of basis, where the basis functions consist of sins and cosines (complex exponentials).
[Gonzalez Chapter 4]
30
![Page 31: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/31.jpg)
Continuous Time Fourier Transform (CTFT)
T=1
31
![Page 32: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/32.jpg)
Fourier Transform
• Cosine/sine signals are easy to define and interpret.
• Analysis and manipulation of sinusoidal signals is greatly simplified by dealing with related signals called complex exponential signals.
• A complex number has real and imaginary parts: z = x+j y
• The Eulero formula links complex exponential signals and trigonometric functions
( )e cos sinjr r jα α α= +cosα = e
iα + e−iα
2
sinα = eiα − e−iα
2i32
![Page 33: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/33.jpg)
CTFT
• Continuous Time Fourier Transform
• Continuous time a-periodic signal
• Both time (space) and frequency are continuous variables – NON normalized frequency ω is used
• Fourier integral can be regarded as a Fourier series with fundamental frequency approaching zero
• Fourier spectra are continuous – A signal is represented as a sum of sinusoids (or exponentials) of all
frequencies over a continuous frequency interval
( ) ( )
1( ) ( )2
j t
tj t
F f t e dt
f t F e d
ω
ω
ω
ω
ω ωπ
−=
=
∫
∫
analysis
synthesis
Fourier integral
33
![Page 34: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/34.jpg)
CTFT of real signals
• Real signals: each signal sample is a real number
• Property: the CTFT is symmetric
34
( ) ( )
( ) ( ) ( )
( ){ } ( ) ( ) ( )'
ˆ
ˆ ˆ
Pr
ˆ' 'j t j t
f t f
f t f foof
f t f t e dt f t e dt fω ω
ω
ω ω
ω
∗
+∞ +∞−
−∞ −∞
→
− → − =
ℑ − = − = = −∫ ∫
f̂ −ω( ) = f ω( )
ω
![Page 35: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/35.jpg)
Sinusoids
• Frequency domain characterization of signals
Frequency domain (spectrum, absolute value of the transform)
Signal domain
( ) ( ) j tF f t e dtωω+∞
−
−∞
= ∫
35
![Page 36: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/36.jpg)
Gaussian
Frequency domain
Time domain
36
![Page 37: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/37.jpg)
rect
sinc function
Frequency domain
Time domain
37
![Page 38: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/38.jpg)
Example
38
![Page 39: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/39.jpg)
Example
39
![Page 40: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/40.jpg)
Properties
40
![Page 41: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/41.jpg)
CTFT
• Change of variables for simplified notations: ω=2πu
• More compact notations (same as in GW)
2(2 ) ( ) ( ) j uxF u F u f x e dxππ∞
−
−∞
= = =∫( )2 21( ) ( ) 2 ( )
2j ux j uxf x F u e d u F u e duπ ππ
π
∞ ∞
−∞ −∞
= =∫ ∫
( )
2
2
( ) ( )
( )
j ux
j ux
F u f x e dx
f x F u e du
π
π
∞−
−∞
∞
−∞
=
=
∫
∫
41
![Page 42: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/42.jpg)
Images vs Signals
1D
• Signals
• Frequency – Temporal – Spatial
• Time (space) frequency characterization of signals
• Reference space for – Filtering – Changing the sampling rate – Signal analysis – ….
2D
• Images
• Frequency – Spatial
• Space/frequency characterization of 2D signals
• Reference space for – Filtering – Up/Down sampling – Image analysis – Feature extraction – Compression – ….
42
![Page 43: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/43.jpg)
43
![Page 44: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/44.jpg)
44
![Page 45: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/45.jpg)
2D Continuous FT
45
![Page 46: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/46.jpg)
2D Frequency domain
ωx
ωy
Large vertical frequencies correspond to horizontal lines
Large horizontal frequencies correspond to vertical lines
Small horizontal and vertical frequencies correspond smooth grayscale changes in both directions
Large horizontal and vertical frequencies correspond sharp grayscale changes in both directions
46
![Page 47: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/47.jpg)
2D spatial frequencies
• 2D spatial frequencies characterize the image spatial changes in the horizontal (x) and vertical (y) directions
– Smooth variations -> low frequencies – Sharp variations -> high frequencies
x
y
ωx=1 ωy=0 ωx=0
ωy=1
47
![Page 48: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/48.jpg)
2D Continuous Fourier Transform
• 2D Continuous Fourier Transform (notation 2)
( ) ( ) ( )
( ) ( ) ( )
2
2
ˆ , ,
ˆ, ,
j ux vy
j ux vy
f u v f x y e dxdy
f x y f u v e dudv
π
π
+∞− +
−∞
+∞+
−∞
=
= =
∫
∫
22 ˆ( , ) ( , )f x y dxdy f u v dudv∞ ∞ ∞ ∞
−∞ −∞ −∞ −∞
=∫ ∫ ∫ ∫ Plancherel’s equality
48
![Page 49: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/49.jpg)
Delta
• Sampling property of the 2D-delta function (Dirac’s delta)
• Transform of the delta function
0 0 0 0( , ) ( , ) ( , )x x y y f x y dxdy f x yδ∞
−∞
− − =∫
{ } 2 ( )( , ) ( , ) 1j ux vyF x y x y e dxdyπδ δ∞ ∞
− +
−∞ −∞
= =∫ ∫
{ } 0 02 ( )2 ( )0 0 0 0( , ) ( , ) j ux vyj ux vyF x x y y x x y y e dxdy e ππδ δ
∞ ∞− +− +
−∞ −∞
− − = − − =∫ ∫ shifting property
49
![Page 50: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/50.jpg)
Constant functions
• Inverse transform of the impulse function
• Fourier Transform of the constant (=1 for all x and y)
( ) 2 ( )
( , ) 1 ,
, ( , )j ux vy
k x y x y
F u v e dxdy u vπ δ∞ ∞
− +
−∞ −∞
= ∀
= =∫ ∫
{ }1 2 ( ) 2 (0 0)( , ) ( , ) 1j ux vy j x vF u v u v e dudv eπ πδ δ∞ ∞
− + +
−∞ −∞
= = =∫ ∫
50
![Page 51: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/51.jpg)
Trigonometric functions
• Cosine function oscillating along the x axis – Constant along the y axis
{ } 2 ( )
2 ( ) 2 ( )2 ( )
( , ) cos(2 )
cos(2 ) cos(2 )
2
j ux vy
j fx j fxj ux vy
s x y fx
F fx fx e dxdy
e e e dxdy
π
π ππ
π
π π∞ ∞
− +
−∞ −∞
∞ ∞ −− +
−∞ −∞
=
= =
⎡ ⎤+= ⎢ ⎥
⎣ ⎦
∫ ∫
∫ ∫
[ ]
2 ( ) 2 ( ) 2
2 2 ( ) 2 ( ) 2 ( ) 2 ( )
12
1 112 2
1 ( ) ( )2
j u f x j u f x j vy
j vy j u f x j u f x j u f x j u f x
e e e dxdy
e dy e e dx e e dx
u f u f
π π π
π π π π π
δ δ
∞ ∞− − − + −
−∞ −∞
∞ ∞ ∞− − − − + − − − +
−∞ −∞ −∞
⎡ ⎤= + =⎣ ⎦
⎡ ⎤ ⎡ ⎤= + = + =⎣ ⎦ ⎣ ⎦
− + +
∫ ∫
∫ ∫ ∫
51
![Page 52: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/52.jpg)
Vertical grating
ωx
ωy
0
52
![Page 53: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/53.jpg)
Double grating
ωx
ωy
0
53
![Page 54: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/54.jpg)
Smooth rings
ωx
ωy
54
![Page 55: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/55.jpg)
Vertical grating
ωx
ωy
0 -2πf 2πf
55
![Page 56: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/56.jpg)
2D box 2D sinc
56
![Page 57: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/57.jpg)
CTFT properties
■ Linearity
■ Shifting
■ Modulation
■ Convolution
■ Multiplication
■ Separability
( , ) ( , ) ( , ) ( , )af x y bg x y aF u v bG u v+ ⇔ +
( , )* ( , ) ( , ) ( , )f x y g x y F u v G u v⇔
( , ) ( , ) ( , )* ( , )f x y g x y F u v G u v⇔
( , ) ( ) ( ) ( , ) ( ) ( )f x y f x f y F u v F u F v= ⇔ =
0 02 ( )0 0( , ) ( , )j ux vyf x x y x e F u vπ− +− − ⇔
0 02 ( )0 0( , ) ( , )j u x v ye f x y F u u v vπ + ⇔ − −
57
![Page 58: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/58.jpg)
Separability
1. Separability of the 2D Fourier transform – 2D Fourier Transforms can be implemented as a sequence of 1D Fourier
Transform operations performed independently along the two axis
( )
2 ( )
2 2 2 2
2
( , ) ( , )
( , ) ( , )
( , ) ,
j ux vy
j ux j vy j vy j ux
j vy
F u v f x y e dxdy
f x y e e dxdy e dy f x y e dx
F u y e dy F u v
π
π π π π
π
∞ ∞− +
−∞ −∞
∞ ∞ ∞ ∞− − − −
−∞ −∞ −∞ −∞
∞−
−∞
= =
= =
= =
∫ ∫
∫ ∫ ∫ ∫
∫
2D FT 1D FT along the rows
1D FT along the cols
58
![Page 59: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/59.jpg)
Separability
• Separable functions can be written as
2. The FT of a separable function is the product of the FTs of the two functions
( ) ( )
( ) ( )
2 ( )
2 2 2 2
( , ) ( , )
( ) ( )
j ux vy
j ux j vy j vy j ux
F u v f x y e dxdy
h x g y e e dxdy g y e dy h x e dx
H u G v
π
π π π π
∞ ∞− +
−∞ −∞
∞ ∞ ∞ ∞− − − −
−∞ −∞ −∞ −∞
= =
= =
=
∫ ∫
∫ ∫ ∫ ∫
( ) ( ) ( ) ( ) ( ) ( ), ,f x y h x g y F u v H u G v= ⇒ =
( ) ( ) ( ),f x y f x g y=
59
![Page 60: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/60.jpg)
Discrete Time Fourier Transform (DTFT)
Applies to Discrete time (sampled) signals and time series
1D
60
![Page 61: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/61.jpg)
Bahadir K.
Gunturk
61
Fourier Transform: 2D Discrete Signals
■ Fourier Transform of a 2D discrete signal is defined as
where
2 ( )( , ) [ , ] j um vn
m nF u v f m n e π
∞ ∞− +
=−∞ =−∞
= ∑ ∑
1 1,2 2
u v−≤ <
1/ 2 1/ 22 ( )
1/ 2 1/ 2
[ , ] ( , ) j um vnf m n F u v e dudvπ +
− −
= ∫ ∫
■ Inverse Fourier Transform
![Page 62: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/62.jpg)
Properties
• Periodicity: 2D Fourier Transform of a discrete a-periodic signal is periodic – The period is 1 for the unitary frequency notations and 2π for normalized
frequency notations. – Proof (referring to the firsts case)
( )2 ( ) ( )( , ) [ , ] j u k m v l n
m nF u k v l f m n e π
∞ ∞− + + +
=−∞ =−∞
+ + = ∑ ∑
( )2 2 2[ , ] j um vn j km j ln
m nf m n e e eπ π π
∞ ∞− + − −
=−∞ =−∞
= ∑ ∑
2 ( )[ , ] j um vn
m nf m n e π
∞ ∞− +
=−∞ =−∞
= ∑ ∑
1 1
( , )F u v=
Arbitrary integers
62
![Page 63: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/63.jpg)
Bahadir K.
Gunturk
63
Fourier Transform: Properties
■ Periodicity: Fourier Transform of a discrete signal is periodic with period 1.
( )2 ( ) ( )( , ) [ , ] j u k m v l n
m nF u k v l f m n e π
∞ ∞− + + +
=−∞ =−∞
+ + = ∑ ∑
( )2 2 2[ , ] j um vn j km j ln
m nf m n e e eπ π π
∞ ∞− + − −
=−∞ =−∞
= ∑ ∑
2 ( )[ , ] j um vn
m nf m n e π
∞ ∞− +
=−∞ =−∞
= ∑ ∑
1 1
( , )F u v=
Arbitrary integers
![Page 64: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/64.jpg)
Bahadir K.
Gunturk
64
Fourier Transform: Properties
■ Linearity, shifting, modulation, convolution, multiplication, separability, energy conservation properties also exist for the 2D Fourier Transform of discrete signals.
![Page 65: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/65.jpg)
Bahadir K.
Gunturk
65
DTFT Properties
■ Linearity
■ Shifting
■ Modulation
■ Convolution
■ Multiplication
■ Separable functions
■ Energy conservation
[ , ] [ , ] ( , ) ( , )af m n bg m n aF u v bG u v+ ⇔ +
0 02 ( )0 0[ , ] ( , )j um vnf m m n n e F u vπ− +− − ⇔
[ , ] [ , ] ( , )* ( , )f m n g m n F u v G u v⇔
[ , ]* [ , ] ( , ) ( , )f m n g m n F u v G u v⇔
0 02 ( )0 0[ , ] ( , )j u m v ne f m n F u u v vπ + ⇔ − −
[ , ] [ ] [ ] ( , ) ( ) ( )f m n f m f n F u v F u F v= ⇔ =2 2[ , ] ( , )
m nf m n F u v dudv
∞ ∞∞ ∞
=−∞ =−∞ −∞ −∞
=∑ ∑ ∫ ∫
![Page 66: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/66.jpg)
Bahadir K.
Gunturk
66
Fourier Transform: Properties
■ Define Kronecker delta function
■ Fourier Transform of the Kronecker delta function
1, for 0 and 0[ , ]
0, otherwisem n
m nδ= =⎧ ⎫
= ⎨ ⎬⎩ ⎭
( ) ( )2 2 0 0( , ) [ , ] 1j um vn j u v
m nF u v m n e eπ πδ
∞ ∞− + − +
=−∞ =−∞
⎡ ⎤= = =⎣ ⎦∑ ∑
![Page 67: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/67.jpg)
Bahadir K.
Gunturk
67
DTFT Properties
■ Fourier Transform of 1
To prove: Take the inverse Fourier Transform of the Dirac delta function and use the fact that the Fourier Transform has to be periodic with period 1.
( )2( , ) 1 ( , ) 1 ( , )j um vn
m n k lf m n F u v e u k v lπ δ
∞ ∞ ∞ ∞− +
=−∞ =−∞ =−∞ =−∞
⎡ ⎤= ⇔ = = − −⎣ ⎦∑ ∑ ∑ ∑
![Page 68: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/68.jpg)
Bahadir K.
Gunturk
68
Impulse Train
■ Define a comb function (impulse train) as follows
, [ , ] [ , ]M Nk l
comb m n m kM n lNδ∞ ∞
=−∞ =−∞
= − −∑ ∑
where M and N are integers
2[ ]comb n
n
1
![Page 69: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/69.jpg)
Bahadir K.
Gunturk
69
Impulse Train
combM ,N [m,n] δ[m− kM ,n− lN ]l=−∞
∞
∑k=−∞
∞
∑
[ ] 1, ,k l k l
k lm kM n lN u vMN M N
δ δ∞ ∞ ∞ ∞
=−∞ =−∞ =−∞ =−∞
⎛ ⎞− − ⇔ − −⎜ ⎟⎝ ⎠
∑ ∑ ∑ ∑
1 1,( , )
M N
comb u v, [ , ]M Ncomb m n
combM ,N (x, y) δ x − kM , y − lN( )l=−∞
∞
∑k=−∞
∞
∑
• Fourier Transform of an impulse train is also an impulse train:
![Page 70: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/70.jpg)
Bahadir K.
Gunturk
70
Impulse Train
2[ ]comb n
n u
1 12
12
1 ( )2comb u
12
![Page 71: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/71.jpg)
Bahadir K.
Gunturk
71
Impulse Train
( ) 1, ,k l k l
k lx kM y lN u vMN M N
δ δ∞ ∞ ∞ ∞
=−∞ =−∞ =−∞ =−∞
⎛ ⎞− − ⇔ − −⎜ ⎟⎝ ⎠
∑ ∑ ∑ ∑
1 1,( , )
M N
comb u v, ( , )M Ncomb x y
combM ,N (x, y) = δ x − kM , y − lN( )l=−∞
∞
∑k=−∞
∞
∑
• In the case of continuous signals:
![Page 72: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/72.jpg)
Bahadir K.
Gunturk
72
Impulse Train
2( )comb x
x u
1 12
12
1 ( )2comb u
12
2
![Page 73: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/73.jpg)
2D DTFT: constant
■ Fourier Transform of 1
To prove: Take the inverse Fourier Transform of the Dirac delta function and use the fact that the Fourier Transform has to be periodic with period 1.
( )2
[ , ] 1, ,
[ , ] 1
( , )
j uk vl
k l
k l
f k l k l
F u v e
u k v l
π
δ
∞ ∞− +
=−∞ =−∞
∞ ∞
=−∞ =−∞
= ∀
⎡ ⎤= =⎣ ⎦
= − −
∑ ∑
∑ ∑ periodic with period 1 along u and v
73
![Page 74: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/74.jpg)
Consequences
Sampling (Nyquist) theorem
74
![Page 75: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/75.jpg)
Bahadir K.
Gunturk
75
Sampling
x
xM
( )f x
( )Mcomb x
u
( )F u
u
1( )* ( )M
F u comb u
u1M
1 ( )M
comb u
x
( ) ( )Mf x comb x
![Page 76: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/76.jpg)
Bahadir K.
Gunturk
76
Sampling
x
( )f x
u
( )F u
u
1( )* ( )M
F u comb u
x
( ) ( )Mf x comb x
WW−
M
W
1M1 2W
M>Nyquist theorem: No aliasing if
![Page 77: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/77.jpg)
Bahadir K.
Gunturk
77
Sampling
u
1( )* ( )M
F u comb u
x
( ) ( )Mf x comb x
M
W
1M
If there is no aliasing, the original signal can be recovered from its samples by low-pass filtering.
12M
![Page 78: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/78.jpg)
Bahadir K.
Gunturk
78
Sampling
x
( )f x
u
( )F u
u
1( )* ( )M
F u comb u
( ) ( )Mf x comb x
WW−
W
1MAliased
![Page 79: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/79.jpg)
Bahadir K.
Gunturk
79
Sampling
x
( )f x
u
( )F u
u[ ]( )* ( ) ( )Mf x h x comb x
WW−
1M
Anti-aliasing filter
uWW−
( )* ( )f x h x12M
![Page 80: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/80.jpg)
Bahadir K.
Gunturk
80
Sampling
u[ ]( )* ( ) ( )Mf x h x comb x
1M
u( ) ( )Mf x comb x
W
1M
■ Without anti-aliasing filter:
■ With anti-aliasing filter:
![Page 81: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/81.jpg)
Bahadir K.
Gunturk
81
Sampling in 2D (images)
x
y
u
v
uW
vW
x
y
( , )f x y ( , )F u v
M
N
, ( , )M Ncomb x y
u
v
1M
1N
1 1,( , )
M N
comb u v
![Page 82: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/82.jpg)
Bahadir K.
Gunturk
82
Sampling
u
v
uW
vW
,( , ) ( , )M Nf x y comb x y
1M
1N
1 2 uWM>No aliasing if and
1 2 vWN>
![Page 83: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/83.jpg)
Bahadir K.
Gunturk
83
Interpolation (low pass filtering)
u
v
1M
1N
1 1, for and v( , ) 2 2
0, otherwise
MN uH u v M N
⎧ ≤ ≤⎪= ⎨⎪⎩
12N
12M
Ideal reconstruction filter:
![Page 84: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/84.jpg)
Bahadir K.
Gunturk
84
Anti-Aliasing
a=imread(‘barbara.tif’);
![Page 85: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/85.jpg)
Bahadir K.
Gunturk
85
Anti-Aliasing
a=imread(‘barbara.tif’); b=imresize(a,0.25); c=imresize(b,4);
![Page 86: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/86.jpg)
Bahadir K.
Gunturk
86
Anti-Aliasing
a=imread(‘barbara.tif’); b=imresize(a,0.25); c=imresize(b,4); H=zeros(512,512); H(256-64:256+64, 256-64:256+64)=1; Da=fft2(a); Da=fftshift(Da); Dd=Da.*H; Dd=fftshift(Dd); d=real(ifft2(Dd));
![Page 87: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/87.jpg)
Discrete Fourier Transform (DFT)
Applies to finite length discrete time (sampled) signals and time series
The easiest way to get to it
Time is a discrete variable (t=n)
Frequency is a discrete variable (f=k)
87
![Page 88: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/88.jpg)
DFT
• The DFT can be considered as a generalization of the CTFT to discrete series
• It is the FT of a discrete (sampled) function of one variable
– The 1/N factor is put either in the analysis formula or in the synthesis one, or the 1/sqrt(N) is put in front of both.
• Calculating the DFT takes about N2 calculations
12 /
01
2 /
0
1[ ] [ ]
[ ] [ ]
Nj kn N
nN
j kn N
k
F k f n eN
f n F k e
π
π
−−
=−
=
=
=
∑
∑
88
![Page 89: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/89.jpg)
In practice..
• In order to calculate the DFT we start with k=0, calculate F(0) as in the formula below, then we change to u=1 etc
• F[0] is the average value of the function f[n] in k=0 – This is also the case for the CTFT
• The transformed function F[k] has the same number of terms as f[n] and always exists
• The transform is always reversible by construction so that we can always recover f given F
1 12 0 /
0 0
1 1[0] [ ] [ ]N N
j n N
n nF f n e f n f
N Nπ
− −−
= =
= = =∑ ∑
89
![Page 90: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/90.jpg)
Highlights on DFT properties
time
amplitude
0
Frequency (k)
|F[k]| F[0] low-pass
characteristic
90
The DFT of a real signal is symmetric The DFT of a real symmetric signal (even like the cosine) is real and symmetric The DFT is N-periodic Hence The DFT of a real symmetric signal only needs to be specified in [0, N/2]
0 N/2 N
![Page 91: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/91.jpg)
Visualization of the basic repetition
• To show a full period, we need to translate the origin of the transform at u=N/2 (or at (N/2,N/2) in 2D)
|F(u-N/2)|
|F(u)|
f n[ ]e2πu0n → f k −u0[ ]
u0 =N2
f n[ ]e2π N
2n= f n[ ]eπNn = −1( )n f n[ ]→ f k − N
2#
$%&
'(
![Page 92: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/92.jpg)
DFT
• About N2 multiplications are needed to calculate the DFT
• The transform F[k] has the same number of components of f[n], that is N
• The DFT always exists for signals that do not go to infinity at any point
• Using the Eulero’s formula
( ) ( )( )1 1
2 /
0 0
1 1[ ] [ ] [ ] cos 2 / sin 2 /N N
j kn N
n nF k f n e f n j kn N j j kn N
N Nπ π π
− −−
= =
= = −∑ ∑
frequency component k discrete trigonometric functions
92
![Page 93: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/93.jpg)
2D example
no translation after translation
![Page 94: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/94.jpg)
Going back to the intuition
• The FT decomposed the signal over its harmonic components and thus represents it as a sum of linearly independent complex exponential functions
• Thus, it can be interpreted as a “mathematical prism”
94
![Page 95: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/95.jpg)
DFT
• Each term of the DFT, namely each value of F[k], results of the contributions of all the samples in the signal (f[n] for n=1,..,N)
• The samples of f[n] are multiplied by trigonometric functions of different frequencies
• The domain over which F[k] lives is called frequency domain
• Each term of the summation which gives F[k] is called frequency component of harmonic component
95
![Page 96: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/96.jpg)
DFT is a complex number
• F[k] in general are complex numbers
[ ] [ ]{ } [ ]{ }[ ] [ ] [ ]{ }
[ ] [ ]{ } [ ]{ }
[ ][ ]{ }[ ]{ }
[ ] [ ]
2 2
1
2
Re Im
exp
Re Im
Imtan
Re
F k F k j F k
F k F k j F k
F k F k F k
F kF k
F k
P k F k
−
= +
=
⎧ ⎫= +⎪ ⎪⎪ ⎪⎨ ⎬⎧ ⎫
= −⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭⎩ ⎭
=
R
R
magnitude or spectrum
phase or angle
power spectrum
96
![Page 97: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/97.jpg)
Stretching vs shrinking
97
stretched shrinked
![Page 98: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/98.jpg)
Periodization vs discretization
• DT (discrete time) signals can be seen as sampled versions of CT (continuous time) signals
• Both CT and DT signals can be of finite duration or periodic
• There is a duality between periodicity and discretization – Periodic signals have discrete frequency (sampled) transform – Discrete time signals have periodic transform – DT periodic signals have discrete (sampled) periodic transforms
t
ampl
itude
ampl
itude
n
fk=f(kTs)
Ts
98
Linking continuous and discrete domains
![Page 99: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/99.jpg)
Increasing the resolution by Zero Padding
• Consider the analysis formula
• If f[n] consists of n samples than F[k] consists of N samples as well, it is discrete (k is an integer) and it is periodic (because the signal f[n] is discrete time, namely n is an integer)
• The value of each F[k], for all k, is given by a weighted sum of the values of f[n], for n=1,..,N-1
• Key point: if we artificially increase the length of the signal adding M zeros on the right, we get a signal f1[m] for which m=1,…,N+M-1. Since
99
F k[ ] = 1N
f n[ ]e−2π jknN
n=0
N−1
∑
f1 m[ ] =f [m] for 0 ≤m < N
0 for N ≤m < N +M
"
#$
%$
![Page 100: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/100.jpg)
Increasing the resolution through ZP
• Then the value of each F[k] is obtained by a weighted sum of the “real” values of f[n] for 0≤k≤N-1, which are the only ones different from zero, but they happen at different “normalized frequencies” since the frequency axis has been rescaled. In consequence, F[k] is more “densely sampled” and thus features a higher resolution.
100
![Page 101: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/101.jpg)
Increasing the resolution by Zero Padding
n
zero padding
0 N0
k 0 2π
F(Ω) (DTFT) in shade F[k]: “sampled version”
4π 2π/N0
101
![Page 102: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/102.jpg)
Zero padding
n
zero padding
0 N0
k 0 2π
F(Ω)
4π 2π/N0
Increasing the number of zeros augments the “resolution” of the transform since the samples of the DFT get “closer”
102
![Page 103: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/103.jpg)
Summary of dualities
FOURIER DOMAIN SIGNAL DOMAIN
Sampling Periodicity
Sampling Periodicity
DTFT
CTFS
Sampling+Periodicity Sampling +Periodicity DTFS/DFT
103
![Page 104: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/104.jpg)
Discrete Cosine Transform (DCT)
Applies to digital (sampled) finite length signals AND uses only cosines.
The DCT coefficients are all real numbers
104
![Page 105: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/105.jpg)
Discrete Cosine Transform (DCT)
• Operate on finite discrete sequences (as DFT)
• A discrete cosine transform (DCT) expresses a sequence of finitely many data points in terms of a sum of cosine functions oscillating at different frequencies
• DCT is a Fourier-related transform similar to the DFT but using only real numbers
• DCT is equivalent to DFT of roughly twice the length, operating on real data with even symmetry (since the Fourier transform of a real and even function is real and even), where in some variants the input and/or output data are shifted by half a sample
• There are eight standard DCT variants, out of which four are common
• Strong connection with the Karunen-Loeven transform – VERY important for signal compression
105
![Page 106: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/106.jpg)
DCT
• DCT implies different boundary conditions than the DFT or other related transforms
• A DCT, like a cosine transform, implies an even periodic extension of the original function
• Tricky part – First, one has to specify whether the function is even or odd at both the left and
right boundaries of the domain – Second, one has to specify around what point the function is even or odd
• In particular, consider a sequence abcd of four equally spaced data points, and say that we specify an even left boundary. There are two sensible possibilities: either the data is even about the sample a, in which case the even extension is dcbabcd, or the data is even about the point halfway between a and the previous point, in which case the even extension is dcbaabcd (a is repeated).
106
![Page 107: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/107.jpg)
Symmetries
107
![Page 108: IP-L5 -Fourier Transform...Fourier Transform • Different formulations for the different classes of signals – Summary table: Fourier transforms with various combinations of continuous](https://reader035.vdocuments.site/reader035/viewer/2022062609/60f867eed36f1e0f1d64cc54/html5/thumbnails/108.jpg)
DCT
0
0
1
00 0
1
000 0
1cos 0,...., 12
2 1 1cos2 2
N
k nn
N
n kk
X x n k k NN
kx X X kN N
π
π
−
=
−
=
⎡ ⎤⎛ ⎞= + = −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
⎧ ⎫⎡ ⎤⎛ ⎞= + +⎨ ⎬⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎩ ⎭
∑
∑
• Warning: the normalization factor in front of these transform definitions is merely a convention and differs between treatments.
– Some authors multiply the transforms by (2/N0)1/2 so that the inverse does not require any additional multiplicative factor.
• Combined with appropriate factors of √2 (see above), this can be used to make the transform matrix orthogonal.
108