2D 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
i= j=0
⎧1
⎩0
δ [i , j ] = ⎨
i , j ≠ 0; i ≠ j
⎧1
⎩0
δ [i , j − J ] = ⎨
i = 0; j = J
otherwise
Example 2: Gaussian
Continuous function
f ( x, y ) =
1
e
σ 2π
x2 + y2
2σ 2
Discrete version
1
f [i , j ] =
e
σ 2π
i2 + j2
2σ 2
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
DTFT: Discrete Time FT
CTFS: CT Fourier Series (summation synthesis)
DTFS: DT Fourier Series (summation synthesis)
P: periodical signals
T: sampling period
ωs: sampling frequency (ωs=2π/T)
For DTFT: T=1 → ωs=2π
1D FT: basics
Fourier Transform: 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 sines and cosines
(complex exponentials).
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:
r e jα = r ( cos α + j sin α )
Overview
Transform
Time
Frequency
(Continuous Time)
Fourier Transform
(CTFT)
C
C
(Continuous Time)
Fourier Series (CTFS)
C
Analysis/Synthesis
Duality
F (ω ) = ∫ f (t )e − jωt dt
Self-dual
t
1
jω t
(
)
f (t ) =
F
ω
e
dω
2π ω∫
D
P
T /2
1
F [k ] =
f (t )e − j 2π kt / T dt
∫
T −T / 2
Dual with
DTFT
f (t ) = ∑ F [k ]e j 2π kt / T
k
Discrete Time Fourier
Transform (DTFT)
D
C
P
F (e
jω t
D
D
P
P
Dual with
CTFS
n
f [ n] =
Discrete Time Fourier
Series (DTFS)
) = ∑ f [n]e
− j 2πω n / ωs
ωs / 2
1
ωs
∫
F ( e jωt ) e j 2πω n / ωs d ω
− ωs / 2
1 N −1
F [k ] = ∑ f [n]e − j 2π kn / T
N n=0
N −1
f [n] = ∑ F [k ]e j 2π kn / T
n =0
Self dual
Dualities
SIGNAL DOMAIN
FOURIER DOMAIN
Sampling
DTFT
Periodicity
Periodicity
CTFS
Sampling
Sampling+Periodicity
DTFS/DFT
Sampling +Periodicity
Discrete time signals
•
Sequences of samples
•
Sampled signals
•
f[k]: sample values
•
f(kTs): sample values
•
Assumes a unitary spacing among
samples (Ts=1)
•
The sampling interval (or period) is Ts
•
Non normalized frequency ω
•
Transform
•
Normalized frequency Ω
•
Transform
–
–
–
•
–
–
–
–
DTFT for NON periodic sequences
CTFS for periodic sequences
DFT for periodized sequences
All transforms are 2π periodic
•
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
Ω = ωTs
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
Fourier integral
F (ω ) = ∫ f (t )e − jωt dt
analysis
1
jω t
f (t ) =
F
e
dω
ω
(
)
∫
2π ω
synthesis
t
CTFT: change of notations
■ Fourier Transform of a 1D continuous signal
F (ω ) =
∞
∫
f ( x)e − jω x dx
−∞
“Euler’s formula”
e − jω x = cos (ω x ) − j sin (ω x )
■ Inverse Fourier Transform
1
f ( x) =
2π
Change of notations:
∞
∫
F (ω )e jω x d ω
−∞
ω → 2π u
⎧ω x → 2π u
⎨
⎩ω y → 2π v
Then CTFT becomes
■ Fourier Transform of a 1D continuous signal
∞
F (u ) =
∫
f ( x)e − j 2π ux dx
−∞
“Euler’s formula”
e − j 2π ux = cos ( 2π ux ) − j sin ( 2π ux )
■ Inverse Fourier Transform
∞
f ( x) =
∫
−∞
F (u )e j 2π ux du
CTFS
•
Continuous Time Fourier Series
•
Continuous time periodic signals
– The signal is periodic with period T0
– The transform is “sampled” (it is a series)
our notations
T /2
1 0
− jnω0 t
(
)
Dn =
f
t
e
dt
T
∫
o
T0 −T0 / 2
fT0 (t ) = ∑ Dn e jnω0t
2π
ω0 =
T0
n
table notations
T /2
1
− j 2π kt / T
F [k ] =
f
t
e
dt
(
)
∫
T −T / 2
f (t ) = ∑ F [k ]e j 2π kt / T
k
fundamental frequency
T0↔T
Dn ↔F[k]
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
•
Fourier series are periodic with period equal to the fundamental in the set
(2π/T0)
•
Properties
– even symmetry → only cosinusoidal components
– odd symmetry → only sinusoidal components
CTFS: example 1
CTFS: example 2
From sequences to discrete time signals
•
Looking at the sequence as to a set of samples obtained by sampling a real
signal with 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)
normalized frequency
(sample series)
spectral periodicity in Ω
Ω = ωTs
2π → ωs =
frequency (sampled signal)
2π
Ts
spectral periodicity in ω
– Enclosing the sampling interval Ts in the value of the sequence samples (DFT)
f k = Ts f ( kTs )
DTFT
•
Discrete Time Fourier Transform
•
Discrete time a-periodic signal
•
The transform is periodic and continuous with period
our notations
F (Ω) =
table notations
+∞
∑
F ( e jω ) = ∑ f [n]e − j 2πω n / ωs
f [k ]e − jΩk
k =−∞
1
f [k ] =
2π
normalized
frequency
Ω0 = 2π
∫π F ( Ω )e
2
n
j Ωk
dΩ
f [ n] =
1
ωs
ωs / 2
−
∫ω
⎛Ω⎞
F ( Ω ) = Fc ⎜ ⎟
T
Ω = ωTs ⎝ s ⎠
Ts = 2π / ωs Ts = 2π / ωs
s
F ( e jωt ) e j 2πω n / ωs d ω
/2
non normalized
frequency
Discrete Time Fourier Transform (DTFT)
_
•
F(Ω) can be obtained _from Fc(ω) by replacing ω with Ω/Ts. Thus F(Ω) is
identical to Fc(ω) frequency scaled by a factor 1/Ts
– Ts is the sampling interval in time domain
•
Notations
⎛Ω⎞
F ( Ω ) = Fc ⎜ ⎟
⎝ Ts ⎠
2π
2π
ωs =
→ Ts =
ωs
Ts
ω=
Ω
→ Ω = ωTs
Ts
periodicity of the spectrum
normalized frequency (the spectrum is 2π-periodic)
F (Ω) → F (ωTs ) = F ( 2πω / ωs )
F (Ω) =
+∞
∑
k =−∞
f [k ]e
− j Ωk
→ F (ωTs ) = F (ω ) =
+∞
∑
k =−∞
f [k ]e − j 2 kπω / ωs
DTFT: unitary frequency
(ω = 2π f )
Ω = 2π u
F (Ω) =
∞
∑
f [k ]e
− j Ωk
→ F (u ) =
k =−∞
1
f [k ] =
2π
∫π
2
∞
∑
f [k ]e − j 2π ku
k =−∞
F (Ω)e jΩk d Ω → f [ k ] = ∫ F (u )e j 2π ku du =
∞
⎧
− j 2π ku
=
F
u
f
k
e
(
)
[
]
∑
⎪
k =−∞
⎪⎪
1
⎨
2
⎪ f [k ] = F (u )e j 2π ku du
∫1
⎪
−
⎪⎩
2
1
1
2
∫
−
F (u )e j 2π ku du
1
2
NOTE: when Ts=1, Ω=ω and the spectrum is
2π-periodic. The unitary frequency u=2π/ Ω
corresponds to the signal frequency f=2π/ω.
This could give a better intuition of the
transform properties.
Connection DTFT-CTFT
periodizationFc(ω)
sampling
f(t)
t
0
ω
_
Fc(ω)
f(kTs)
0 Ts
4Ts
t
0
f[k]
0 1
4
k
0
2π/Ts
ω
F(Ω)
2π
Ω
Differences DTFT-CTFT
•
The DTFT is periodic with period Ωs=2π (or ωs=2π/Ts)
•
The discrete-time exponential ejΩk 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
Ω 0 = 2π / N 0
our notations
1
Dr =
N0
f [k ] =
table notations
N 0 −1
∑
f [k ]e
− jr Ω0 k
k =0
N −1
N 0 −1
∑De
r =0
1 N −1
F [ k ] = ∑ f [n]e − j 2π kn / T
N n =0
r
jr Ω0 k
f [k ] = ∑ F [ k ]e j 2π kn / T
n =0
Discrete Fourier Transform (DFT)
Fr =
N 0 −1
∑
k =0
1
fk =
N0
Ω0 =
f k e − jrΩ0 k =
N 0 −1
jr Ω0 k
F
e
∑ r
k =0
N 0 −1
∑
k =0
−j
fk e
1
=
N0
2π
rk
N0
N 0 −1
∑Fe
k =0
jr
2π
k
N0
r
2π
N0
•
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-N0
discrete-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
DFT
0
k
N0
zero padding
0
2π/N0
F(Ω)
2π
4π
r
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, 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
Xk =
N 0 −1
∑
n=0
2
xn =
N0
•
⎡π ⎛
1⎞ ⎤
xn cos ⎢ ⎜ n + ⎟ k ⎥
2⎠ ⎦
⎣ N0 ⎝
k = 0,...., N 0 − 1
N 0 −1
⎧1
⎡π k ⎛
1 ⎞⎤ ⎫
⎨ X 0 + ∑ X k cos ⎢ ⎜ k + ⎟ ⎥ ⎬
2 ⎠⎦ ⎭
k =0
⎣ N0 ⎝
⎩2
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.
Sinusoids
•
Frequency domain characterization of signals
F (ω ) =
+∞
∫
f (t )e − jωt dt
−∞
Signal domain
Frequency domain
Gaussian
rect
sinc function
Images vs Signals
1D
2D
•
Signals
•
Images
•
Frequency
•
Frequency
–
–
Temporal
Spatial
•
Time (space) frequency
characterization of signals
•
Reference space for
–
–
–
–
Filtering
Changing the sampling rate
Signal analysis
….
–
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
y
ωx=1
ωy=0
ωx=0
ωy=1
x
2D Frequency domain
ωy
Large vertical
frequencies correspond
to horizontal 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
ωx
Large horizontal
frequencies correspond
to vertical lines
Vertical grating
ωy
0
ωx
Double grating
ωy
0
ωx
Smooth rings
ωy
ωx
2D box
2D 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)
1D
•
F (ω ) =
+∞
∫
f (t )e − jωt dt , f (t ) =
−∞
∫
F (ω )e jωt dt
−∞
2D Fourier Transform of discrete space signals (2D-DTFT)
1D
F (Ω) =
∞
∑
f [k ]e
− j Ωk
k =−∞
•
+∞
1
, f [k ] =
2π
∫π
F (Ω)e jΩk dt
2
2D Discrete Fourier Transform (2D-DFT)
1D
Fr =
N0 −1
∑ f [k ]e
k =0
− jrΩ0 k
1 N0 −1 jrΩ0k
2π
, f N0 [k ] =
,
F
e
Ω
=
∑ r
0
N0 r =0
N0
2D Continuous Fourier Transform
•
Continuous case (x and y are real) – 2D-CTFT (notation 1)
fˆ (ω x , ω y ) =
+∞
∫
f ( x, y ) e
− j (ω x x + ω y y )
dxdy
−∞
f ( x, y ) =
∫∫
1
4π 2
+∞
∫
j (ω x + ω y )
fˆ (ω x , ω y ) e x y d ω x d ω y
−∞
f ( x, y ) g * ( x, y ) dxdy =
f = g → ∫∫
1
fˆ (ω x , ω y )gˆ * (ω x , ω y ) d ω x d ω y Parseval formula
4π 2 ∫∫
2
2
1
f ( x, y ) dxdy = 2 ∫∫ fˆ (ω x , ω y ) d ω x d ω y
4π
Plancherel equality
2D Continuous Fourier Transform
•
Continuous case (x and y are real) – 2D-CTFT
ω x = 2π u
ω y = 2π v
fˆ ( u , v ) =
+∞
∫
f ( x, y ) e − j 2π ( ux + vy ) dxdy
−∞
f ( x, y ) =
=
1
4π 2
1
4π 2
+∞
∫
2
fˆ ( u , v ) e j 2π ( ux + vy ) ( 2π ) dudv =
−∞
+∞
∫
−∞
ˆf ( u , v ) e j 2π ( ux + vy ) ( 2π )2 dudv
2D Continuous Fourier Transform
•
2D Continuous Fourier Transform (notation 2)
fˆ ( u , v ) =
+∞
∫
f ( x, y ) e − j 2π ( ux + vy ) dxdy
−∞
f ( x, y ) =
+∞
∫
fˆ ( u , v ) e j 2π ( ux + vy ) dudv =
−∞
∞ ∞
∫∫
−∞ −∞
∞ ∞
f ( x, y ) dxdy =
2
∫∫
−∞ −∞
2
fˆ (u , v) dudv
Plancherel’s equality
2D Discrete Fourier Transform
The independent variable (t,x,y) is discrete
Fr =
N 0 −1
∑
f [ k ]e − jrΩ0 k
F [u , v] =
2π
Ω0 =
N0
∑∑
f [i, k ]e − jΩ0 ( ui + vk )
i =0 k =0
k =0
1
f N0 [k ] =
N0
N 0 −1 N 0 −1
N 0 −1
∑
r =0
Fr e jrΩ0 k
1
f N0 [i, k ] = 2
N0
2π
Ω0 =
N0
N 0 −1 N 0 −1
∑∑
u =0 v =0
F [u , v]e jΩ0 ( ui + vk )
Delta
•
Sampling property of the 2D-delta function (Dirac’s delta)
∞
∫ δ ( x − x , y − y ) f ( x, y)dxdy = f ( x , y )
0
0
0
0
−∞
•
Transform of the delta function
F {δ ( x, y )} =
∞ ∞
∫
− j 2π ( ux + vy )
δ
(
x
,
y
)
e
dxdy = 1
∫
−∞ −∞
F {δ ( x − x0 , y − y0 )} =
∞ ∞
∫
− j 2π ( ux + vy )
− j 2π ( ux + vy )
δ
(
x
−
x
,
y
−
y
)
e
dxdy
=
e
0
0
∫
−∞ −∞
0
0
shifting
property
Constant functions
•
Inverse transform of the impulse function
∞ ∞
F −1 {δ (u , v)} =
∫
j 2π ( ux + vy )
j 2π (0 x + v 0)
δ
(
u
,
v
)
e
dudv
=
e
=1
∫
−∞ −∞
•
Fourier Transform of the constant (=1 for all x and y)
k ( x, y ) = 1
F {k } =
∀x, y
∞ ∞
∫
− j 2π ( ux + vy )
e
dxdy = δ (u , v)
∫
−∞ −∞
Trigonometric functions
•
Cosine function oscillating along the x axis
– Constant along the y axis
s( x, y ) = cos(2π fx)
F {cos(2π fx)} =
∞ ∞
∫
− j 2π ( ux + vy )
π
fx
e
dxdy =
cos(2
)
∫
−∞ −∞
∞ ∞
⎡ e j 2π ( fx ) + e − j 2π ( fx ) ⎤ − j 2π (ux + vy )
= ∫ ∫⎢
dxdy
⎥e
2
⎦
−∞ −∞ ⎣
∞ ∞
1
= ∫ ∫ ⎡⎣e− j 2π (u − f ) x + e− j 2π (u + f ) x ⎤⎦ e− j 2π vy dxdy =
2 −∞ −∞
∞
∞
∞
1
1
= ∫ e− j 2π vy dy ∫ ⎣⎡e− j 2π (u − f ) x + e− j 2π (u + f ) x ⎦⎤ dx = 1 ∫ ⎣⎡e− j 2π (u − f ) x + e− j 2π ( u + f ) x ⎦⎤ dx =
2 −∞
2 −∞
−∞
1
[δ (u − f ) + δ (u + f )]
2
Vertical grating
ωy
-2πf
0
2πf
ωx
Ex. 1
Ex. 2
Ex. 3
Magnitudes
Examples
Properties
■ Linearity
af ( x, y ) + bg ( x, y ) ⇔ aF (u , v) + bG (u , v)
− j 2π ( ux0 + vy0 )
■ Shifting
f ( x − x0 , y − x0 ) ⇔ e
■ Modulation
e j 2π ( u0 x + v0 y ) f ( x, y ) ⇔ F (u − u0 , v − v0 )
■ Convolution
f ( x, y ) * g ( x, y ) ⇔ F (u , v)G (u , v)
■ Multiplication
f ( x, y ) g ( x, y ) ⇔ F (u , v) * G (u , v)
■ Separability
f ( x, y ) = f ( x) f ( y ) ⇔ F (u, v) = F (u ) F (v)
F (u, v)
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
∞ ∞
F (u , v) =
∫∫
f ( x, y )e − j 2π (ux + vy ) dxdy =
−∞ −∞
∞ ∞
∫∫
f ( x, y ) e
− j 2π ux − j 2π vy
e
∞
dxdy =
−∞ −∞
∞
=
∫ F (u, y)e
∫e
−∞
− j 2π vy
− j 2π vy
∞
dy ∫ f ( x, y )e− j 2π ux dx =
−∞
dy = F ( u, v )
−∞
2D DFT
1D DFT along
the rows
1D DFT along
the cols
Separability
•
Separable functions can be written as f ( x, y ) = f ( x ) g ( y )
2. The FT of a separable function is the product of the FTs of the two functions
∞ ∞
F (u , v) =
∫∫
f ( x, y )e− j 2π ( ux + vy ) dxdy =
−∞ −∞
∞ ∞
∫∫
−∞ −∞
h( x ) g ( y ) e
= H (u ) G (v )
− j 2π ux − j 2π vy
e
∞
dxdy =
∫
−∞
g ( y)e
− j 2π vy
∞
dy ∫ h( x)e− j 2π ux dx =
f ( x, y ) = h ( x ) g ( y ) ⇒ F ( u , v ) = H ( u ) G ( v )
−∞
2D Fourier Transform of a Discrete function
•
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)
1D
F (Ω ) =
∞
∑
f [k ]e − jΩk , f [ k ] =
k =−∞
2D
F (Ω x , Ω y ) =
f [k ] =
1
4π 2
+∞
k1 =−∞ k2 =−∞
∫π ∫π
Ωx,Ωy: normalized frequency
2π
f [k1 , k2 ]e
F (Ω x , Ω y )e
2 2
∫
F (Ω)e jΩk dt
− j ( k1Ω x + k2 Ω y )
+∞
∑ ∑
1
2π
j ( k1Ω x + k2 Ω y )
d ΩxΩ y
Unitary frequency notations
⎧Ω x = 2π u
⎨
⎩Ω y = 2π v
F (u, v) =
+∞
+∞
∑ ∑
k1 =−∞ k2 =−∞
1/ 2 1/ 2
f [k1 , k2 ] =
∫ ∫
f [k1 , k2 ]e − j 2π ( k1u + k2v )
F (u , v)e − j 2π ( k1u + k2v ) dudv
−1/ 2 −1/ 2
•
The integration interval for the inverse transform has width=1 instead of 2π
–
It is quite common to choose
−1
1
≤ u, v <
2
2
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)
F (u + k , v + l ) =
∞
∞
∑∑
f [m, n]e
− j 2π ( ( u + k ) m + ( v + l ) n )
1
m =−∞ n =−∞
Arbitrary
integers
=
∞
∞
∑∑
f [m, n]e
− j 2π ( um + vn ) − j 2π km − j 2π ln
e
m =−∞ n =−∞
=
∞
∞
∑∑
m =−∞ n =−∞
= F (u , v)
f [m, n]e − j 2π (um + vn )
e
1
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
af [m, n] + bg[m, n] ⇔ aF (u , v) + bG (u , v)
■ Shifting
f [m − m0 , n − n0 ] ⇔ e − j 2π (um0 + vn0 ) F (u , v)
■ Modulation
e
j 2π ( u0 m + v0 n )
f [m, n] ⇔ F (u − u0 , v − v0 )
■ Convolution
f [m, n]* g[m, n] ⇔ F (u , v)G (u , v)
■ Multiplication
f [m, n]g[m, n] ⇔ F (u , v) * G (u , v)
■ Separable functions
f [m, n] = f [m] f [n] ⇔ F (u , v) = F (u ) F (v)
∞
■ Energy conservation
∞
∑∑
m =−∞ n =−∞
1/ 2 1/ 2
f [m, n] =
2
∫ ∫
−1/ 2 −1/ 2
2
F (u, v) dudv
Impulse Train
■ Define a comb function (impulse train) as follows
combM , N [m, n] =
∞
∞
∑ ∑ δ [m − kM , n − lN ]
k =−∞ l =−∞
where M and N are integers
1
comb2 [n]
n
Appendix
2D-DTFT: delta
■ Define Kronecker delta function
⎧1, for m = 0 and n = 0 ⎫
δ [m, n] = ⎨
⎬
⎩0, otherwise
⎭
■ DT Fourier Transform of the Kronecker delta function
F (u , v) =
∞
∞
− j 2π ( um + vn )
− j 2π ( u 0 + v 0 )
⎡
⎤
=
=1
δ
[
m
,
n
]
e
e
∑
∑
⎣
⎦
m =−∞ n =−∞
2D DT Fourier Transform: constant
■ Fourier Transform of 1
f [k , l ] = 1, ∀k , l
F [u , v] =
=
∞
∞
∞
∞
− j 2π ( uk + vl )
⎡
⎤=
1
e
∑
∑
⎣
⎦
k =−∞ l =−∞
∑
∑ δ (u − k , v − l )
k =−∞ l =−∞
periodic with period 1
along u and v
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.