Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in...

35
Discrete Fourier Transform Discrete Fourier Transform Nuno Vasconcelos UCSD

Transcript of Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in...

Page 1: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Discrete Fourier TransformDiscrete Fourier Transform

Nuno VasconcelosUCSD

Page 2: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

The Discrete-Space Fourier Transform• as in 1D, an important concept in linear system analysis

is that of the Fourier transform• the Discrete-Space Fourier Transform is the 2D

extension of the Discrete-Time Fourier Transform2211][)( ωω ωω eennxX njnj∑∑ −−

2121

2211

2211

1 2

)(1][

],[),(

ωωωω

ωω

ωω ddeeXnnx

eennxX

njnj

jj

n n

∫∫

∑∑

=

=

• note that this is a continuous function of frequency

2121221 ),()2(

],[ ωωωωπ

ddeeXnnx ∫∫=

q y– inconvenient to evaluate numerically in DSP hardware– we need a discrete version

this is the 2D Discrete Fourier Transform (2D DFT)

2

– this is the 2D Discrete Fourier Transform (2D-DFT)

• before that we consider the sampling problem

Page 3: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Sampling in 2D

• consider an analog signal xc(t1,t2) and let its analog Fourier transform be X (Ω1 Ω2)Fourier transform be Xc(Ω1,Ω2)– we use capital Ω to emphasize that this is analog frequency

• sample with period (T1,T2) to obtain a discrete-space signal

222111 ;2121 ),(],[TntTntc ttxnnx

===

3

Page 4: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Sampling in 2D

• relationship between the Discrete-Space FT of x[n1,n2]and the FT of x (t1 t2) is simple extension of 1D resultand the FT of xc(t1,t2) is simple extension of 1D result

∑ ∑∞ ∞

⎟⎠

⎞⎜⎜⎝

⎛ −−= 2211

212,21),( c T

rT

rXTT

X πωπωωω

DSFT of x[n1,n2] FT of xc(ω1,ω2)

−∞= −∞= ⎠⎝1 2 2121 r r TTTT

“discrete spectrum” “analog spectrum”

• Discrete Space spectrum is sum of replicas of analog• Discrete Space spectrum is sum of replicas of analog spectrum– in the “base replica” the analog frequency Ω1 (Ω2) is mapped

i t th di it l f T ( T )4

into the digital frequency Ω1T1 (Ω2T2)– discrete spectrum has periodicity (2π,2π)

Page 5: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

For exampleΩ2

Ω’

Ω”ω2

Ω’

Ω1

Ω”T2

ω1

2π−Ω”T2

2

1

''''''

TT

Ω=→ΩΩ=→Ω

βα

ω1

Ω’T1

2π2π

• no aliasing if

2π−Ω’T1

⎧ Ω≤Ω '2' TT π

ω1

2π⇔⎩⎨⎧

Ω−≤ΩΩ−≤Ω

22

11

'2''2

TTTT

ππ

⎨⎧ Ω≤

⇔'/1 πT

5⎩⎨ Ω≤

⇔''/2 πT

Page 6: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Aliasing• the frequency (Ω’/π,Ω’’/π)

is the critical sampling frequency

• below it we havealiasing

ω2

aliasing

ω1

• this is just like the 1D case, but now there are morenow there are more possibilities for overlap

6

Page 7: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Reconstruction• if there is no aliasing we can recover the signal in a way

similar to the 1D case

( ) ( )∑ ∑∞ ∞

−−=

2222

1111

2121

sinsin],[),(c

TntT

TntTnnxtty π

π

π

π

t i 2D th ibiliti th i 1D

( ) ( )∑ ∑−∞= −∞= −−1 2

2222

1111

n n TntT

TntT

ππ

• note: in 2D there are many more possibilities than in 1D– e.g. the sampling grid does not have to be rectangular, e.g.

hexagonal sampling when T2 = T1/sqrt(3) and

⎩⎨⎧

= ==

otherwise0odd or even both21;21

21,),(

],[ 222111nnttx

nnx TntTntc

7– in practice, however, one usually adopts the rectangular grid

Page 8: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

• a sequence of images obtained by down-sampling without any filteringg

• aliasing: the low-frequency parts are

li t d th h t th8

replicated throughout the low-res image

Page 9: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

The role of smoothing

none

some

l ta lot

9• too little leads to aliasing• too much leads to loss of information

Page 10: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Aliasing in video• video frames are the result of temporal sampling

– fast moving objects are above the critical frequency– above a certain speed they are aliased and appear to move

backwards– this was common in old western movies and become known as

the “wagon wheel” effect– here is an example: super-resolution increases the frame rate

and eliminates aliasing

from “Space-TimeResolution in Video” by yE. Shechtman, Y. Caspi and M. Irani

(PAMI 2005).

10

Page 11: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

2D-DFT• the 2D-DFT is obtained by sampling the DSFT at regular

frequency intervals

22

211

12,22121 ),(],[

kN

kN

XkkX πωπωωω

===

• this turns out to make the 2D-DFT somewhat harder to work with than the DSFT

it i th i 1D– it is the same as in 1D– you might remember that the inverse transform of the product of

two DFTs is not the convolution of the associated signals– but, instead, the “circular convolution”– where does this come from?

• it is better understood by first considering the 2D

11

it is better understood by first considering the 2D Discrete Fourier Series (2D-DFS)

Page 12: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

2D-DFS• it is the natural representation for a periodic sequence• a sequence x[n1,n2] is periodic of period N1xN2 if

[ ] [ ][ ]

21121 ,,nnNnnx

nNnxnnx∀+=

+=

• note that

[ ] 21221 , ,, nnNnnx ∀+=

∑ ∑∞

−∞=

−∞=

−−=1 2

21212121 ],[),(

n n

jnjn rrnnxrrX

• makes no sense for a periodic signal– the sum will be infinite for any pair r1,r2

ith th 2D DSFT th Z t f ill k h

12

– neither the 2D DSFT or the Z-transform will work here

Page 13: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

2D-DFS• the 2D-DFS solves this problem• it is based on the observation that

– any periodic sequence can be represented as a weighted sum of complex exponentials of the form

222211 nkjnkj ππ

( )10 ,10 ,,

22

1121

222

111

−≤≤−≤≤××

NkNkeekkX

nkN

jnkN

j

– this is a simple consequence of the fact that

10 22 ≤≤ Nk

22 kk ππ

is an orthonormal basis of the space of periodic sequences

10 ,10 , 2211

2222

211

1 −≤≤−≤≤× NkNkeenk

Njnk

Nj ππ

13

– is an orthonormal basis of the space of periodic sequences

Page 14: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

2D-DFS• the 2D-DFS relates x[n1,n2] and X[k1,k2]

( ) [ ] 22111 2221 1 nkjnkjN N ππ

−−− −

( ) [ ] 222

111

1 2

221 1

0 02121 ,,

N N

Nj

Nj

n neennxkkX

ππ

∑∑= =

=

[ ] [ ] 222

111

1

1

2

2

221

0

1

021

2121 ,1,

nkN

jnkN

jN

k

N

keekkX

NNnnx

ππ

∑∑−

=

=

=

• note that X[k1,k2] is also periodic outside

1010 ≤≤≤≤ NkNk• like the DSFT,

properties of the 2D DFS are identical to those of the 1D DFS

10 ,10 2211 −≤≤−≤≤ NkNk

14

– properties of the 2D-DFS are identical to those of the 1D-DFS– with the straightforward extension of separability

Page 15: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Periodic convolution• like the Fourier transform,

– the inverse transform of multiplication is convolution

[ ] [ ] ( ) ( )2121

2121 ,,,, kkYkkXnnxnnxDFS

×↔∗

– however, we have to be careful about how we define convolution– since the sequences have no end, the standard definition

],[],[],[ 221121211 2

knknhkkxnnyk k

−−= ∑ ∑∞

−∞=

−∞=

makes no sense– e.g. if x and h are both positive sequences, this will allways be

infinite

15

infinite

Page 16: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Periodic convolution• to deal with this, we introduce the idea of periodic

convolution• instead of the regular definition

],[],[* 221121 knknhkkxyx −−= ∑ ∑∞ ∞

• which, from now on, we refer to as linear convolution

],[],[ 2211211 2

yk k∑ ∑

−∞= −∞=

• periodic convolution only considers one period of our sequences

1 11 2N N− −

h l diff i i h i li i

],[ ],[ 2211

1

0

1

021

1

1

2

2

knknhkkxhxN

k

N

k

−−= ∑ ∑= =

o

16

• the only difference is in the summation limits

Page 17: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Periodic convolution• this is simple, but produces a convolution which is

substantially different• let’s go back to our example, now assuming that the

sequences have period (N1=3,N2=2)n2 n2

(1) (2)*(3) (4)

(1) (2)

(3) (4)

n1 n1

(1) (2)*(1) (2)

(3) (4)

(1) (2)

(1) (2)

(3) (4)

• as before we need four steps],[ 21 nnx ],[ 21 nnh

17

• as before, we need four steps

Page 18: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Periodic convolution• step 1): express sequences in terms of (k1, k2), and

consider one period only

],[ ],[ 2211

1

0

1

021

1

1

2

2

knknhkkxhxN

k

N

k−−= ∑ ∑

=

=

o

k2 k2

(3) (4)

k1 k1(1) (2)

we next proceed exactly as before

],[ 21 kkx ],[ 21 kkh

18

we next proceed exactly as before

Page 19: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Periodic convolution• step 2): invert h(k1, k2)

1 11 2N N− −

2

],[ ],[ 22110 0

21

1

1

2

2

knknhkkxhxk k

−−= ∑ ∑= =

o

k2

(1)(2)

k2

(3) (4)

k2

(1) (2)

k1

( )( )

(3)(4)

k1(1) (2) k1

(1) (2)

(3) (4)1

],[],[ 2121 kkhkkg −−=],[ 21 kkh ],[ 21 kkh −

19

Page 20: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Periodic convolution• step 3): shift g(k1, k2) by (n1, n2)

1 11 2N N− −

],[ ],[ 22110 0

21

1

1

2

2

knknhkkxhxk k

−−= ∑ ∑= =

o

k2

(1)(2)

k2

(1)(2)

(3)(4)

n2

this sendswhatever is at (0,0) to (n1,n2)

k1

( )( )

(3)(4)

k1n1

],[],[ 2121 kkhkkg −−=][

],[ 2211

knknhnknkg =−−

20

],[ 2211 knknh −−

Page 21: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Periodic convolution• e.g. for (n1,n2) = (1,0)

k2k2

k

(1)(2)n2

k k1

(3)(4)

n1

k1

th t t ld b t i t i lti l th t

],[ 21 kkx],[

],[

2211

2211

knknhnknkg−−

=−−

the next step would be to point-wise multiply the two signals and sum

][][1 11 2

kkhkkhN N

∑ ∑− −

21

],[],[ 22110 0

211 2

knknhkkxhxk k

−−= ∑ ∑= =

o

Page 22: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Periodic convolution• this is where we depart from linear convolution• remember that the sequences are periodic, and we really

only care about what happens in the fundamental periodonly care about what happens in the fundamental period• we use the periodicity to fill the values missing in the

flipped sequencek2

(4)(3)

k2

n

k1(1) (2)

(4)( )

k1

(1)(2)

(3)(4)

n2

=

],[ 2211 knknh −−

n1

],[ 2211 knknh −−

22

Page 23: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Periodic convolution• step 4): we can finally point-wise multiply the two signals

and sum

],[ ],[ 2211

1

0

1

021

1

1

2

2

knknhkkxhxN

k

N

k−−= ∑ ∑

=

=

o

– e.g. for (n1,n2) = (1,0)k2 k2k2

3x1+1x1=4

k1

x =k1k1

(1) (2)

(4)(3)3x1+1x1=4

][ kkx ][ nny],[ 2211 knknh −−

23

],[ 21 kkx ],[ 21 nny],[ 2211

Page 24: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Periodic convolution• note: the sequence that results from the convolution is

also periodic• it is important to keep in mind what we have done• it is important to keep in mind what we have done

– we work with a single period (the fundamental period) to make things manageableb t b th t h i di– but remember that we have periodic sequences

– it is like if we were peeking through a window– if we shift, or flip the sequence we need to remember that – the sequence does not simply move out of the window, but the

next period walks in!!!– note, that this

k2 k2

,can make thefundamentalperiod change k1 k1shift

24

considerably by(1,1)

Page 25: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Discrete Fourier Transform• all of this is interesting,

– but why do I care about periodic sequences?– all images are finite, I could never have such a sequence

• while this is true– the DFS is the easiest route to learn about the discrete Fourierthe DFS is the easiest route to learn about the discrete Fourier

transform (DFT)

• recall that the DFT is obtained by sampling the DSFT

22

211

12,22121 ),(],[

kN

kN

XkkX πωπωωω

===

– we know that when we sample in space we have aliasing in frequency

– well, the same happens when we sample in frequency: we get

25

pp p q y galiasing in time

Page 26: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Discrete Fourier Transform• this means that

– even if we have a finite sequence– when we compute the DFT we are effectively working with a

periodic sequence

• you may recall from 1D signal processing thaty y g p g– when you multiply two DFTs, you do not get convolution in time– but instead something called circular convolution

this is strange: when I flip a signal (e g convolution) it wraps– this is strange: when I flip a signal (e.g. convolution) it wraps around the sequence borders

– well, in 2D it gets much stranger

th l I k h t d t d thi i t• the only way I know how to understand this is to– think about the underlying periodic sequence– establish a connection between the DFT and the DFS of that

26

sequence– use what we have seen for the DFS to help me out with the DFT

Page 27: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Discrete Fourier Transform• let’s start by the relation between DFT and DFS• the DFT is defined as

22

211

12,22121 ),(],[

kN

kN

XkkX πωπωωω

===

(here X(ω1,ω2) is the DSFT) which can be written as

[ ]⎪⎧ <≤∑∑

−−− − NknkN

jnkN

jN N 0 11221 1

22111 2ππ

[ ] [ ]⎪⎩

⎪⎨

<≤<≤

= ∑∑= =

otherwiseNkNk

eennxkkXN

jN

j

n n

000

,,,22

11

0 021

21

222

111

1 2

[ ] [ ]⎪

⎪⎨

<≤<≤

= ∑∑−

=

= NnNn

eekkXNNnnx

nkN

jnkN

jN

k

N

k 00

,1,

22

11221

0

1

021

2121

222

111

1

1

2

2

ππ

27

⎪⎩ otherwise0

Page 28: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Discrete Fourier Transform• comparing this

[ ]⎪⎧ <≤∑∑

−−− − NknkN

jnkN

jN N 0 11221 1

22111 2ππ

[ ] [ ]⎪⎩

⎪⎨ <≤

<≤= ∑∑

= =

otherwiseNkNk

eennxkkXNN

n n

000

,,,22

11

0 021

21

21

1 2

[ ] [ ]⎪⎩

⎪⎨

<≤<≤

= ∑∑−

=

=

hNnNn

eekkXNNnnx

nkN

jnkN

jN

k

N

k 00

,1,

22

11221

0

1

021

2121

222

111

1

1

2

2

ππ

with the DFS

⎪⎩ otherwise0

[ ] [ ] 222

111

1 2221 1 nkN

jnkN

jN N

kkXππ

∑∑−−− −

[ ] [ ]

[ ] [ ] 222

111

1 2

21

1 2

221 1

0 02121

1

,,

nkN

jnkN

jN N

NN

n n

eekkXnnx

eennxkkX

ππ

∑∑

∑∑− −

= =

=

=

28

[ ] [ ]1 20 0

2121

21 ,,k k

eekkXNN

nnx ∑∑= =

=

Page 29: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Discrete Fourier Transform• we see that inside the boxes

110 Nk <≤ 110 Nn <≤

the two transforms are exactly the same22

11

0 Nk <≤ 22

11

0 Nn <≤

• if we define the indicator function of the box

⎪⎧ <≤ Nn01 11

[ ]⎪⎩

⎪⎨ <≤=×

otherwiseNnnnR NN

00

,1,222121

• we can write

[ ] [ ] [ ]212121 ,,, nnRnnxnnx NN ×= [ ] [ ] [ ]212121 ,,, kkRkkXkkX NN ×=

29

[ ] [ ] [ ]212121 ,,,21 NN × [ ] [ ] [ ]212121 ,,,

21 NN ×

Page 30: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Discrete Fourier Transform• note from

[ ] [ ] [ ]212121 ,,,21

nnRnnxnnx NN ×= [ ] [ ] [ ]212121 ,,,21

kkRkkXkkX NN ×=

that working in the DFT domain is equivalent to– working in the DFS domain

[ ] [ ] [ ]212121 21 NN [ ] [ ] [ ]212121 21 NN

– extracting the fundamental period at the end

• we can summarize this as

[ ] [ ] [ ] [ ]2121

2121 ,

,,

, kkXkkXnnxnnx

truncate

eperiodiciz

DFS

eperiodiciz

t t←→

↔←→

• in this way, I can work with the DFT without having to worry about aliasing

eperiodiciztruncate

30

worry about aliasing

Page 31: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Discrete Fourier Transform

[ ] [ ] [ ] [ ]2121

2121 ,

,,

, kkXkkXnnxnnx

truncate

DFS

eperiodiciz→

↔→

• this trick can be used to derive all the DFT properties

[ ] [ ] [ ] [ ]21212121 ,,,, kkXkkXnnxnnxeperiodiciztruncate

←↔←

• e.g. what is the inverse transform of a phase shift?– let’s follow the steps

22 kk ππ

– 1) periodicize: this causes the same phase shift in the DFS

[ ] [ ] 222

111

22

2121 ,,mk

Njmk

Nj

eekkXkkYππ

−−

=

[ ] [ ] 222

111

22

2121 ,,mk

Njmk

Nj

eekkXkkYππ

−−

=

31

Page 32: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Discrete Fourier Transform

[ ] [ ] [ ] [ ]2121

2121 ,

,,

, kkXkkXnnxnnx

truncate

DFS

eperiodiciz

←→

↔←→

– 2) compute the inverse DFS: it follows from the properties of the DFS (page 142 on Lim) that we get a shift in space

eperiodiciztruncate←←

DFS (page 142 on Lim) that we get a shift in space

[ ] [ ]221121 ,, mnmnxnny −−=

– 3) truncate: the inverse DFT is equal to one period of the shifted periodic extension of the sequence

[ ] [ ] [ ]

– in summary, the new sequence is obtained by making the

[ ] [ ] [ ]21221121 ,,,21

nnRmnmnxnny NN ×−−=

32

y q y goriginal periodic, shifting, and taking the fundamental period

Page 33: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Examplek2 k2

periodicize

k1 k1

shift by (1,1)

k2k2

2

k

truncate

k1k1

33

• note that what leaves on one end, enters on the other

Page 34: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

Example• for this reason it is called a circular shift

k2

i l hift

k2

k1

circular shift by (1,1)

k1

• note that this is way more complicated than in 1D• to get it right we really have to think in terms of the

i di t i f thperiodic extension of the sequence• we will see that this shows up in the properties of the

DFT,

34

,• namely that convolution becomes circular convolution

Page 35: Discrete Fourier TransformDiscrete Fourier Transform · – inconvenient to evaluate numerically in DSP hardware –we need a discrete version this is the 2D Discrete Fourier Transform

35