presentation-8 discrete fourier transformehm.kocaeli.edu.tr/upload/duyurular/121118081844827aa.pdf · d z ( ( } ( Á ] v } Á ] v p w [>q@ ')7 &rhiilflhqwv [a>q@ :lqgrzhg 6ljqdo

MEH329DIGITAL SIGNAL PROCESSING

-8-Discrete Fourier Transform

Discrete Fourier TransformIntroduction

MEH329 Digital Signal Processing 2

• Frequency analysis should be performed inreal applications.

• X(ejΩ) is not computationally convenientrepresentation (continuous function of freq.)!

Discrete Fourier TransformDTFT of Finite Length Signal


2

1

nj j n

n n

X e x n e

11 211 1 21 ...j nj n j njX e x n e x n e x n e

Discrete Fourier TransformFrequency-Domain Sampling


2

2

21

0

, 0,1,..., 1

j kj N

kN

N j knN

n

X k X e X e

x n e k N

for example:

2 4 6 8 10 12 140, , , , , , , for 8

8 8 8 8 8 8 8N

Discrete Fourier Transform (DFT)

Discrete Fourier TransformFrequency-Domain Sampling


2 4 6 8 10 12 140, , , , , , , for 8

8 8 8 8 8 8 8

3 30, , , , , , ,

4 2 4 4 2 4

N

jX e

......

4

2

3

4

3

4

2

4

0

2

8

j

kX e

Discrete Fourier Transform


• DTFT: 2

0

1

2j j nx n X e e d

2 2k d dk

N N

21

0

1 , 0,1,..., 1

N j knN

k

x n X k e n NN

Inverse Discrete Fourier Transform (IDFT)



• Alternative representation of DFT and IDFT:

1

0

, 0,1,..., 1N

knN

n

X k x n W k N

1

0

1 , 0,1,..., 1

Nkn

Nk

x n X k W n NN




• If we write n→n+N or k→k+N in DFT and IDFT,are the results change?

2 21 12

0 0

(periodic with !)

N Nj k n N j kn j kN N

n n

x n e x n e e

X k N

2 21 1

0 0

1 1

(periodic with !)

N Nj k n N j knN N

k k

X k e X k eN N

x n N



• Example: Find the DFT of [2 3 1]x n

223

0

, 0,1, 2j kn

n

X k x n e k

2

0

0

0 0 1 2 6j

n

X x n e x x x

2 2 423 3 3

0

1 0 1 2 1.732j n j j

n

X x n e x x e x e j

4 4 823 3 3

0

2 0 1 2 1.732j n j j

n

X x n e x x e x e j

Discrete Fourier TransformPeriodic Sequences


• Given a periodic sequence with period N

• The Fourier series representation:

• The Fourier series representation of continuous-time periodic signals require infinite manycomplex exponentials.

[ ] [ ]x n x n rN

2 /1[ ] j N kn

k

x n X k eN

[ ]x n

Discrete Fourier TransformPeriodic Sequences


• Not that for discrete-time periodic signals

• Due to the periodicity of the complexexponential we only need N exponentials fordiscrete time Fourier series

1

2 /

0

1[ ]

Nj N kn

k

x n X k eN

2 / 2 / 2 2 /j N k mN n j N kn j mn j N kne e e e

Discrete Fourier TransformDiscrete Fourier Series


• A periodic sequence in terms of Fourier seriescoefficients

• The Fourier series coefficients can be obtainedvia

1

2 /

0

[ ]N

j N kn

n

X k x n e

1

2 /

0

1[ ]

Nj N kn

k

x n X k eN



• Analysis eq.:

• Synthesis eq.:

2 /j NNW e

1

0

1[ ]

Nkn

Nk

x n X k WN

1

0

[ ]N

knN

n

X k x n W



• Example: DFS of periodic impulse train

1[ ]

0r

n rNx n n rN

else

1 1

2 / 2 / 2 / 0

0 0

[ ] [ ] 1N N

j N kn j N kn j N k

n n

X k x n e n e e

1

2 /

0

1[ ]

Nj N kn

r k

x n n rN eN



• For N=4:

32 /4

0

30

0

32 /4 2 /4 4 /4 6 /4

0

32 /4 2 4 /4 8 /4 12 /4

0

32 /4 3 6 /4 12 /4 18 /4

0

2 /4 4

1[ ]

4

1[0] 1

4

1 1[1] 1 0

4 4

1 1[2] 1 0

4 4

1 1[3] 1 0

4 4

1[4]

4

j kn

k

j

k

j k j j j

k

j k j j j

k

j k j j j

k

j

x n e

x e

x e e e e

x e e e e

x e e e e

x e

3

8 /4 16 /4 24 /4

0

11 1

4k j j j

k

e e e







• Example: DFS of periodic rectangular pulse train

• The DFS coefficients

2 /10 54

2 /10 4 /10

2 /100

sin / 21

sin /101

j kj kn j k

j kn

keX k e e

ke



Discrete Fourier TransformProperties of Discrete Fourier Series


• Linearity:

• Shift:

• Duality:

1 1

2 2

1 2 1 2

DFS

DFS

DFS

x n X k

x n X k

ax n bx n aX k bX k

2 /

2 /

DFS

DFS j km N

DFSj nm N

x n X k

x n m e X k

e x n X k m

DFS

DFS

x n X k

X n Nx k



• Proof (Duality):

n↔k:

1

2 /

0

[ ]N

j N kn

n

X k x n e

1

2 /

0

1[ ]

Nj N kn

k

x n X k eN

12 /

0

12 /

0

[ ]

[ ]

Nj N kn

k

Nj N kn

n

Nx n X k e

Nx k X n e







• Periodic Convolution:

1 1

2 2

DFS

DFS

x n X k

x n X k

3 1 2X k X k X k

1

3 1 20

1

2 10

N

m

N

m

x n x m x n m

x m x n m



Discrete Fourier TransformDFS – DFT Relation



1Nn0 of outside 0nx

r

x n x n rN

• The DFS coefficients of the periodic sequence aresamples of the DTFT of x[n].

• Since x[n] is of length N there is no overlap betweenterms of x[n-rN] and we can write the periodicsequence as

mod NN

x n x n x n



• We choose one period of as the Fouriertransform of x[n]

kX~

mod NN

X k X k X k

0 1

0

X k k NX k

else



1

2 /

0

[ ]N

j N kn

n

X k x n e

1

2 /

0

1[ ]

Nj N kn

k

x n X k eN

,0 1[ ]

0 ,otherwise

x n n Nx n

,0 1[ ]

0 ,otherwise

X k k NX k

1

2 /

0

[ ]N

j N kn

n

X k x n e

1

2 /

0

1[ ]

Nj N kn

k

x n X k eN

DFT Pair

Discrete Fourier TransformDiscrete Fourier Transform


• To evaluate the relation between size of asignal (L) and the number of frequency sample(N).

• N must be equal or greater than L toreconstruct the signal without a loss.

• If L is bigger than N, aliasing occurs in the timedomain!



Discrete Fourier TransformProperties of DFT


• Linearity:

1 1

2 2

1 2 1 2

DFT

DFT

DFT

x n X k

x n X k

ax n bx n aX k bX k




• Circular Shift:

2 / 0 n N-1

DFT

j k N mDFT

N

x n X k

x n m X k e

The n Modulo N Operation


Circular Shift of a Sequence








Circular folding (or reversal)


Circular folding (or reversal)


Circular Convolution:




• Circular Convolution:

1

3 1 20

N

Nm

x n x m x n m

1

3 2 10

N

Nm

x n x m x n m



• Example: Circular convolution

• DFT of each sequence

• Multiplication of DFTs

• the inverse DFT

1 2

1 0 1

0

n Lx n x n

else

21

1 20

0

0

N j knN

n

N kX k X k e

else

2

3 1 2

0

0

N kX k X k X k

else

3

0 1

0

N n Nx n

else

Circular Convolution-Graphical Interpretation


Linear Convolution (Review)


Linear Convolution Using DFT








• Example: if N=2L=12• DFT of each sequence

• Multiplication of DFTs

2

1 2 2

1

1

Lkj

N

kjN

eX k X k

e

22

3 2

1

1

Lkj

N

kjN

eX k

e

Result of the linearconvolution!

Duality Property


Discrete Fourier TransformWindowing

54

• Periodicity causes unwanted spectral effects infrequency domain

• This issue is called as spectral leakage.

0 2 4 6 8 10 12 14 16-1

0

1x[n]

0 2 4 6 8 10 12 14 160

5

10DFT Coefficients

0 5 10 15 20 25 30 35 40 45-1

0

1x~[n]

0 5 10 15 20-1

0

1x[n]

0 5 10 15 200

5

10DFT Coefficients

0 10 20 30 40 50 60 70-1

0

1x~[n]

Spectral leakage

Windowing Property



56

• Windowing is utilized to overcome this effect.• Well known window functions:

• Triangular• Trapezoid• Hamming• Hanning• Blackman• Parzen• Welch• Nuttall• Kaiser

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

Window Function (Hamming)


57

• The effect of windowing:

0 5 10 15 20-1

0

1x[n]

0 5 10 15 200

5

10DFT Coefficients

0 10 20 30 40 50 60 70-1

0

1x~[n]

0 5 10 15 20-1

0

1Windowed Signal

0 5 10 15 200

5

10DFT Coefficients

0 10 20 30 40 50 60 70-1

0

1Periodic form of windowed function

Discrete Fourier TransformLinear Transform Perspective


• Computation of each point of the DFT can beaccomplished by N complex multiplications and(N-1) complex additions.

• The N point DFT coefficients can be computed in atotal of N^2 complex multiplications and (N-1)Ncomplex additions.

1

0

[ ] , 0,1,..., 1N

knN

n

X k x n W k N

1

0

1[ ] , n 0,1,..., 1

Nkn

Nk

x n X k W NN



N=4 point DFT calculation:

10 0 0 0 0

0

10 1 2 3

0

10 2 4 6

0

10 3 6 9

0

0 [ ] 0 1 2 3

1 [ ] 0 1 2 3

2 [ ] 0 1 2 3

3 [ ] 0 1 2 3

Nn

N N N N Nn

NnN N N N N

n

NnN N N N N

n

NnN N N N N

n

X x n W x W x W x W x W






Alternative representation:

0

1

1

N

x

x

x N

x

0

1

1

N

X

X

X N

X

1 2 1

2 12 4

1 2 1 1 1

1 1 1 1

1

1

1

NN N N

NN N N N

N N N NN N N

W W W

W W W

W W W

W

1 1N N N N NN

x W X W X

N N NX W x



Example:0

1

2

3

N

x1 2 3 1 2 3 1 2 1

4 4 4 4 4 4 4 4 42 4 6 2 0 2 2 0 2

4 4 4 4 4 4 4 4 43 6 9 3 2 1 1 2 1

4 4 4 4 4 4 4 4 4

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1

1 1 1

1 1 1

N

W W W W W W W W W

W W W W W W W W W

W W W W W W W W W

W

1 1 1 1 0 6

1 1 1 2 2

1 1 1 1 2 2

1 1 3 2 2

N N N

j j j

j j j

X W x

Documents

presentation-8 discrete fourier transformehm.kocaeli.edu.tr/upload/duyurular/121118081844827aa.pdf · d z ( ( } ( Á ] v } Á ] v p w [>q@ ')7 &rhiilflhqwv [a>q@ :lqgrzhg 6ljqdo