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Fourier Transform
Overview
• Signals as functions (1D, 2D)– Tools
• 1D Fourier Transform– Summary of definition and properties in the different cases
• CTFT, CTFS, DTFS, DTFT• DFT
• 2D Fourier Transforms– Generalities and intuition– Examples– A bit of theory
• Discrete Fourier Transform (DFT)
• Discrete Cosine Transform (DCT)
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
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]
Example 1: δ function
[ ]⎩⎨⎧
≠≠==
=jiji
jiji
;0,001
,δ
[ ]⎩⎨⎧ ==
=−otherwise
JjiJji
0;01
,δ
Example 2: Gaussian
2
22
2
21),( σ
πσ
yx
eyxf+
=
2
22
2
21],[ σ
πσ
ji
ejif+
=
Continuous function
Discrete version
Example 3: Natural image
Example 3: Natural image
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
Continuous Time Fourier Transform (CTFT)
Time is a real variable (t)
Frequency is a real variable (ω)
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]
Continuous Time Fourier Transform (CTFT)
T=1
Fourier Transform
• Cosine/sine signals are easy to define and interpret.
• However, it turns out that the 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
• A complex exponential signal:
( )e cos sinjr r jα α α= +
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
Then CTFT becomes
Fourier Transform of a 1D continuous signal
2( ) ( ) j uxF u f x e dxπ∞
−
−∞
= ∫
Inverse Fourier Transform
2( ) ( ) j uxf x F u e duπ∞
−∞
= ∫
( ) ( )2 cos 2 sin 2j uxe ux j uxπ π π− = −“Euler’s formula”
CTFT: change of notations
Fourier Transform of a 1D continuous signal
( ) ( ) j xF f x e dxωω∞
−
−∞
= ∫
Inverse Fourier Transform1( ) ( )
2j xf x F e dωω ω
π
∞
−∞
= ∫
( ) ( )cos sinj xe x j xω ω ω− = −“Euler’s formula”
Change of notations:
222
x
y
uuv
ω πω πω π
→
→⎧⎨ →⎩
CTFT
• Replacing the variables
• More compact notations
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
∞−
−∞
∞
−∞
=
=
∫
∫
Sinusoids
• Frequency domain characterization of signals
Frequency domain (spectrum, absolute value of the transform)
Signal domain
( ) ( ) j tF f t e dtωω+∞
−
−∞
= ∫
Gaussian
Frequency domain
Time domain
rect
sinc function
Frequency domain
Time domain
Example
Example
Discrete Fourier Transform (DFT)
The easiest way to get to it
Time is a discrete variable (t=n)
Frequency is a discrete variable (f=k)
DFT
• The DFT can be considered as a generalization of the CTFT to discrete series
• 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] 0• This is also the case for the CTFT
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
−−
=−
=
=
=
∑
∑0,1, , 10,1, , 1
n Nk N= −= −
……
1 12 0 /
0 0
1 1[0] [ ] [ ]N N
j n N
n n
F f n e f n fN N
π− −
−
= =
= = =∑ ∑
Example 1
time
amplitude
0
frequency
|F[k]|F[0] low-pass
characteristic
Example 2
time
amplitude
0
frequency
|F[k]|F[0]=0
0
band-pass (or high-pass) characteristic
-M M
DFT
• About M2 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 n
F k f n e f n j kn N j j kn NN N
π π π− −
−
= =
= = −∑ ∑
frequency component k discrete trigonometric functions
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”
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
−
= +
=
⎧ ⎫= +⎪ ⎪⎪ ⎪⎨ ⎬⎧ ⎫
= −⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭⎩ ⎭
=
magnitude or spectrum
phase or angle
power spectrum
Example
Let’s take a bit more advanced perspective…
Book: Lathi, Signal Processing and Linear Systems
OverviewTransform Time Frequency Analysis/Synthesis Duality
(Continuous Time) Fourier Transform (CTFT)
C C Self-dual
(Continuous Time) Fourier Series (CTFS)
C
P
D Dual with DTFT
Discrete Time Fourier Transform (DTFT)
D C
P
Dual with CTFS
Discrete Time Fourier Series (DTFS)
D
P
D
P
Self dual
( ) ( )
1( ) ( )2
j t
tj t
F f t e dt
f t F e d
ω
ω
ω
ω
ω ωπ
−=
=
∫
∫/ 2
2 /
/ 2
2 /
1[ ] ( )
( ) [ ]
Tj kt T
T
j kt T
k
F k f t e dtT
f t F k e
π
π
−
−
=
=
∫
∑
12 /
01
2 /
0
1[ ] [ ]
[ ] [ ]
Nj kn T
nN
j kn T
n
F k f n eN
f n F k e
π
π
−−
=−
=
=
=
∑
∑
( )2
( ) [ ]
1[ ]2
j k
k
j k
F f k e
f k F e dππ
+∞− Ω
=−∞
Ω
Ω =
= Ω Ω
∑
∫
Linking continuous and discrete domains
• DT signals can be seen as sampled versions of CT 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 (DF) transform (f=k) → CTFS– Discrete time signals have periodic transform → DTFT– DT periodic signals have DF periodic transforms → DTFS, DFT
t
ampl
itude
ampl
itude
n
fk=f(kTs)
Ts
Dualities
FOURIER DOMAINSIGNAL DOMAIN
Sampling
Periodicity
Periodicity
Sampling
DTFT
CTFS
Sampling+Periodicity Sampling +PeriodicityDTFS/DFT
Discrete time signalsSequences of samples
• f[k]: sample values
• Assumes a unitary spacing among samples (Ts=1)
• Normalized frequency Ω
• Transform– DTFT for NON periodic sequences– CTFS for periodic sequences– DFT for periodized sequences
• All transforms are 2π periodic
Sampled signals
• f(kTs): sample values
• The sampling interval (or period) is Ts
• Non normalized frequency ω
• Transform– DTFT– CSTF– DFT– BUT accounting for the fact that the
sequence values have been generated by sampling a real signal → fk=f(kTs)
• All transforms are periodic with period ωs
sTωΩ =2
ssTπω =
2 2s s s ss
T TTπω πΩ = = =
Connection DTFT-CTFT
0 Ts 4Ts
0 1 4 k
t
t
ω
ω
Ω
0
0 2π
F(Ω)
Fc(ω)_0
2π/Ts
Fc(ω)sampling periodization
f(t)
f(kTs)
f[k]
CTFS
• Continuous Time Fourier Series
• Continuous time periodic signals– The signal is periodic with period T– The transform is “sampled” (it is a series)
/ 22 /
/ 2
2 /
1[ ] ( )
( ) [ ]
Tj kt T
T
j kt T
k
F k f t e dtT
f t F k e
π
π
−
−
=
=
∫
∑
t
ampl
itude
T
coefficients of the Fourier series
periodic signal
CTFS
• Representation of a continuous time signal as a sum of orthogonal components in a complete orthogonal signal space
– The exponentials are the basis functions
• Properties– even symmetry → only cosinusoidal components– odd symmetry → only sinusoidal components
DTFT
• Discrete Time Fourier Transform
• Discrete time a-periodic signal
• The transform is periodic and continuous with period
( )
( ) ( )
2 /
/ 2 /2 /
/ 2 /
[ ] [ ]
1 2[ ]
2 2 22
s s
s s
s s
s s
j n j nT
n nT
j n j nTj t j t
s s T
s s ss s
F f n e f n e
f n F e e d F e e dT
T TT
πω ω ω
ω ππω ω ωω ω
ω π
ω
πω ωωπ πω πωω ω
ω π
− −
− −
= =
= =
= → = =
∑ ∑
∫ ∫
non normalized frequency
2 /s sT π ω=
2s
sTπω =
sampling interval in time ↔ periodicity in frequencythe closer the samples, the farther the replicas
n
DTFT with normalized frequency
• Normalized frequency: change of variables
( )2
( ) [ ]
1[ ]2
jn
n
jk
F f k e
f n F e dππ
+∞− Ω
=−∞
Ω
Ω =
= Ω Ω
∑
∫
normalized frequency
( ) 1c
s s
F FT T
⎛ ⎞ΩΩ = ⎜ ⎟
⎝ ⎠
sTωΩ =
2s πΩ =
2 2s s s ss
T TTπω πΩ = = =
periodicity in the normalized frequency domain
Relation to the CTFT
If Ts>1, the DTFT can be seen as a stretched periodizedversion of the CTFT.
DTFT with normalized frequency
• F(Ω) can be obtained from Fc(ω) by replacing ω with Ω /Ts.
• Thus F(Ω) is identical to F(ω) frequency scaled and stratched by a factor 1/Ts, where Ts is the sampling interval in time domain
• Notations
( )
( ) ( )
( ) 2 /
1
2 2
( ) 2 /
( ) [ ] ( ) [ ] s
cs s
s ss s
ss
s s
j nj ns
n k
F FT T
TT
TT
F F T F
F f n e F T F f n e πω ω
π πωω
ω ω
ω ω π ω
ω ω+∞ +∞
−− Ω
=−∞ =−∞
⎛ ⎞ΩΩ = ⎜ ⎟
⎝ ⎠
= → =
Ω= →Ω =
Ω → =
Ω = → = =∑ ∑
_
periodicity of the spectrum
normalized frequency (the spectrum is 2π-periodic)
DTFT CTFT
DTFT with unitary frequency( )
2
12
2 2
12 12
2
12
2
12
2 2
( ) [ ] ( ) [ ]
1[ ] ( ) [ ] ( ) ( )2
( ) [ ]
[ ] ( )
j n j nu
n n
j n j nu j nu
j nu
n
j nu
u f
F f n e F u f n e
f n F e d f n F u e du F u e du
F u f k e
f n F u e du
π
π π
π
π
π
π ω π
π
∞ ∞− Ω −
=−∞ =−∞
Ω
−
∞−
=−∞
−
Ω = =
Ω = → =
= Ω Ω→ = =
⎧ =⎪⎪⎪⎨⎪ =⎪⎪⎩
∑ ∑
∫ ∫ ∫
∑
∫
NOTE: when Ts=1, Ω=ω and the spectrum is 2π-periodic. The unitary frequency u=2π/ Ωcorresponds to the signal frequency f=2π/ω. This could give a better intuition of the transform properties.
Summary
• Sampled signals are sequences of sampels
• Looking at the sequence as to a set of samples obtained by sampling a real signal with sampling frequency ωs we can still use the formulas for calculating the transforms as derived for the sequences by
– Stratching the time axis (and thus squeezing the frequency axis if Ts>1)
– Enclosing the sampling interval Ts in the value of the sequence samples (DFT)
( )k s sf T f kT=
22
s
ss
T
T
ωππ ω
Ω =
→ =
normalized frequency(sample series) frequency (sampled signal)
spectral periodicity in Ω spectral periodicity in ω
Connection DTFT-CTFT
0 Ts 4Ts
0 1 4 k
t
t
ω
ω
Ω
0
0 2π
F(Ω)
Fc(ω)_0
2π/Ts
Fc(ω)sampling periodization
f(t)
f(kTs)
f[k]
Differences DTFT-CTFT
• The DTFT is periodic with period Ωs=2π (or ωs=2π/Ts)
• The discrete-time exponential ejΩn has a unique waveform only for values of Ω in a continuous interval of 2π
• Numerical computations can be conveniently performed with the Discrete Fourier Transform (DFT)
DTFS
• Discrete Time Fourier Series
• Discrete time periodic sequences of period N0– Fundamental frequency
0 02 / NπΩ =
00
00
12 /
001
2 /
0
1[ ] [ ]
[ ] [ ]
Njkn N
nN
jkn N
k
F k f n eN
f n F k e
π
π
−−
=−
=
=
=
∑
∑
DTFS: Example
[Lathi, pag 621]
N0=20Ts=1Ωs=2πΩ0= π/10
Discrete Fourier Transform (DFT)
• The DFT transforms N0 samples of a discrete-time signal to the same number of discrete frequency samples
• The DFT and IDFT are a self-contained, one-to-one transform pair for a length-N0discrete-time signal (that is, the DFT is not merely an approximation to the DTFT as discussed next)
• However, the DFT is very often used as a practical approximation to the DTFT
0 00 0
0 00 0
21 1
0 021 1
0 00 0
00
[ ]
1 1[ ] [ ] [ ]
2
N N j nkjn k N
n nn n
N N jn kjn k N
k k
F k f e f e
f n F k e F k eN N
N
π
π
π
− − −− Ω
= =
− −Ω
= =
= =
= =
Ω =
∑ ∑
∑ ∑
DFT
n
zero padding
0 N0
k0 2π
F(Ω)
4π2π/N0
DFT
n
zero padding
0N0
k0 2π
F(Ω)
4π2π/N0
Increasing the number of zeros augments the “resolution” of the transform since the samples of the DFT gets “closer”
Properties
( ) ( )( ) ( ) ( )
( ) ( ) ( ) ( )'
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 − = − = = −∫ ∫
For real signals f(t)
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
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).
Symmetries
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.
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– ….
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=1ωy=0 ωx=0
ωy=1
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
Vertical grating
ωx
ωy
0
Double grating
ωx
ωy
0
Smooth rings
ωx
ωy
2D box2D sinc
Margherita Hack
log amplitude of the spectrum
Einstein
log amplitude of the spectrum
What we are going to analyze
• 2D Fourier Transform of continuous signals (2D-CTFT)
• 2D Fourier Transform of discrete space signals (2D-DTFT)
• 2D Discrete Fourier Transform (2D-DFT)
( ) ( ) , ( ) ( )j t j tF f t e dt f t F e dtω ωω ω+∞ +∞
−
−∞ −∞
= =∫ ∫
0 00 0
0
1 1
00 00 0
1 2[ ] , [ ] ,N N
jr k jr kr N r
k rF f k e f k F e
N Nπ− −
− Ω Ω
= =
= = Ω =∑ ∑
2
1( ) [ ] , [ ] ( )2
j k j k
kF f k e f k F e dt
ππ
∞− Ω Ω
=−∞
Ω = = Ω∑ ∫
1D
1D
1D
2D Continuous Fourier Transform
• Continuous case (x and y are real) – 2D-CTFT (notation 1)
( ) ( ) ( )
( ) ( ) ( )2
ˆ , ,
1 ˆ, ,4
x y
x y
j x yx y
j x yx y x y
f f x y e dxdy
f x y f e d d
ω ω
ω ω
ω ω
ω ω ω ωπ
+∞− +
−∞
+∞+
−∞
=
=
∫
∫
( ) ( ) ( ) ( )
( ) ( )
* *2
222
1 ˆ, , , ,ˆ4
1 ˆ, ,4
x y x y x y
x y x y
f x y g x y dxdy f g d d
f g f x y dxdy f d d
ω ω ω ω ω ωπ
ω ω ω ωπ
=
= → =
∫∫ ∫∫
∫∫ ∫∫
Parseval formula
Plancherel equality
2D Continuous Fourier Transform
• Continuous case (x and y are real) – 2D-CTFT
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
2
222
222
22
ˆ , ,
1 ˆ, , 24
1 ˆ , 24
x
y
j ux vy
j ux vy
j ux vy
uv
f u v f x y e dxdy
f x y f u v e dudv
f u v e dudv
π
π
π
ω πω π
ππ
ππ
+∞− +
−∞
+∞+
−∞
+∞+
−∞
=
=
=
= =
=
∫
∫
∫
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
2D Discrete Fourier Transform
00
00
0
1
0
1
00
00
[ ]
1[ ]
2
Njr k
rk
Njr k
N rr
F f k e
f k F eN
Nπ
−− Ω
=
−Ω
=
=
=
Ω =
∑
∑
0 00
0 00
0
1 1( )
0 0
1 1( )
20 00
00
[ , ] [ , ]
1[ , ] [ , ]
2
N Nj ui vk
i k
N Nj ui vk
Nu v
F u v f i k e
f i k F u v eN
Nπ
− −− Ω +
= =
− −Ω +
= =
=
=
Ω =
∑ ∑
∑ ∑
The independent variable (t,x,y) is discrete
[Lathi’s notations]
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
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∞ ∞
− + +
−∞ −∞
= = =∫ ∫
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
∞ ∞− − − + −
−∞ −∞
∞ ∞ ∞− − − − + − − − +
−∞ −∞ −∞
⎡ ⎤= + =⎣ ⎦
⎡ ⎤ ⎡ ⎤= + = + =⎣ ⎦ ⎣ ⎦
− + +
∫ ∫
∫ ∫ ∫
Vertical grating
ωx
ωy
0-2πf 2πf
Ex. 1
Ex. 2
Ex. 3
Magnitudes
Examples
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 + ⇔ − −
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 DFT 1D DFT along the rows
1D DFT along the cols
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=
• Fourier Transform of a 2D a-periodic signal defined over a 2D discrete grid– The grid can be thought of as a 2D brush used for sampling the continuous
signal with a given spatial resolution (Tx,Ty)
2D Fourier Transform of a Discrete function
2
1( ) [ ] , [ ] ( )2
j k j k
kF f k e f k F e dt
ππ
∞− Ω Ω
=−∞
Ω = = Ω∑ ∫
( )
( )
1 2
1 2
1 2
1 2
22 2
( , ) [ , ]
1[ ] ( , )4
x y
x y
j k k
x yk k
j k kx y x y
F f k k e
f k F e dπ ππ
− Ω + Ω+∞ +∞
=−∞ =−∞
Ω + Ω
Ω Ω =
= Ω Ω Ω Ω
∑ ∑
∫ ∫
1D
2D
Ωx,Ωy: normalized frequency
Unitary frequency notations
1 2
1 2
1 2
2 ( )1 2
1/ 2 1/ 22 ( )
1 21/ 2 1/ 2
22
( , ) [ , ]
[ , ] ( , )
x
y
j k u k v
k k
j k u k v
uv
F u v f k k e
f k k F u v e dudv
π
π
ππ
+∞ +∞− +
=−∞ =−∞
− +
− −
Ω =⎧⎨Ω =⎩
=
=
∑ ∑
∫ ∫
• The integration interval for the inverse transform has width=1 instead of 2π– It is quite common to choose
1 1,2 2
u v−≤ <
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
Properties
• Linearity
• shifting
• modulation
• convolution
• multiplication
• separability
• energy conservation properties also exist for the 2D Fourier Transform of discrete signals.
• NOTE: in what follows, (k1,k2) is replaced by (m,n)
2D 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= ⇔ =1/ 2 1/ 2
2 2
1/ 2 1/ 2
[ , ] ( , )m n
f m n F u v dudv∞ ∞
=−∞ =−∞ − −
=∑ ∑ ∫ ∫
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
Appendix
2D-DTFT: delta
Define Kronecker delta function
DT 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 n
F u v m n e eπ πδ∞ ∞
− + − +
=−∞ =−∞
⎡ ⎤= = =⎣ ⎦∑ ∑
2D DT Fourier Transform: 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
