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Spring 2012

Discrete Time Fourier Series and Transform

© Ammar Abu-Hudrouss -Islamic University Gaza

Slide ٢Digital Signal Processing

Discrete Time Fourier Series

1

0

/2)(N

k

Nknjkecnx

1

0

/2)(1 N

n

Nknjk enx

Nc

Nkk cc

٢

Any periodic sequence x(n) with period N, can be represented as summation of exponentials or

This representation is named Fourier series where the series coefficients ck are given by

ck which us the amplitude of the frequency spectrum are repeated every N terms, or

Thus the spectrum of a periodic signal with period Nis also periodic with period N.

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Slide ٣Digital Signal Processing

Discrete Time Fourier Series

)5.0sin()( nnx

4

00 0))3()2()1()0((

41)(

41n

xxxxnxc

2/))1(0)1(0(41

)3()2()1()0((41)(

41 2/32/

3

0

4/21

jjj

exexexxenxc jjj

n

nj

2/)(41 3

0

2/33 jenxc

n

nj

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ExampleExample: Find the DTFS for the following signal

f = ¼ = k/N which means that N = 4

x (0) = 0, x (1) = 1, x (2) = 0, x (3) = -1

0)(41 3

02

n

njenxc

SolutionSolution

Slide ٤Digital Signal Processing

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ck

j /2

-j /2

1 2

3-1

-2-3k

kc

0.5

1 2 3-1-2-3 k

/2

-/2

1 2

3-1

-2-3k

kc

The frequency spectrum is sketched in the following diagrams. The magnitude and phase are shown as well.

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Slide ٥Digital Signal Processing

Discrete Time Fourier Series

1

0

2)(1 N

n

nxN

P

1

0

* )()(1 N

nnxnx

NP

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Power density Spectrum of Periodic Signal

Which leads to

The power of periodic signal is given by

1

0

1

0

/2*)(1 N

n

N

k

Nknjkecnx

NP

1

0

1

0

/2* )(1N

k

N

n

Nknjk enxN

cP

1

0

2N

kkcP

Slide ٦Digital Signal Processing

Discrete Time Fourier Series

1,....,111

0/

/2

/21

0

/2

Nkee

NA

kNALe

NAc

Nkj

NkLjL

n

nNkjk

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Example

Which leads to the general expression

Find the power spectrum density for the signal shown

The coefficients over one period of ck

otherwise

/sin/sin

2,,0/

/1 kNkNkLe

NA

NNkNALc NLkjk

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Slide ٧Digital Signal Processing

Discrete Time Fourier Series

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ExampleFind the power spectrum density for the signal shown

The power spectral density is given by

otherwise

/sin/sin

2,,0/22

2

2

kNkNkL

NA

NNkNALck

Slide ٨Digital Signal Processing

Discrete Time Fourier Transform

n

njenxX )()(

)()(

)(

)()2(

2

)2(

k

nj

k

knjnj

k

nkj

Xenx

eenx

enxkX

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A Fourier transform for a finite energy discrete time signal x(n) is defined as

X() is a periodic function with period 2

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Slide ٩Digital Signal Processing

Discrete Time Fourier Transform

deXnx nj)(21)(

)()()()( *2 XXXSxx

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An inverse Fourier transform is defined as

The energy spectrum of aperiodic signal is defined as

Slide ١٠Digital Signal Processing

Discrete Time FourierTransform

11)()( anuanx n

j

n n

njnjn

ae

aeeaX

11

)()(0 0

2

*

cos211

)()()(

aa

XXS XX

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Find the DTFT for the following signal

Solution

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Slide ١١Digital Signal Processing

Discrete Time Fourier Transform

11)()( anuanx n

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Find the DTFT for the following signal

Sxx(), a = 0.5 Sxx(), a = -0.5

Slide ١٢Digital Signal Processing

Discrete Time Fourier Transform

otherwise

LnAnx

,010,

)(

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Find the DTFT for the following signal

Solution as x (n) has a finite absolute sum, then

2/sin

2/sin11)(

12/

1

0

LAe

eeAAeX

Lj

L

nj

Ljnj

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Slide ١٣Digital Signal Processing

Discrete Time Fourier Transform

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The graph for both magnitude and phase is given by

2/sin

2/sin)( LX

ALLX

2/sin2/sin1

2)(

Slide ١٤Digital Signal Processing

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Slide ١٥Digital Signal Processing

Self Study

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Students are encouraged to solve the following questions from the textbook

4.4, 4.5, 4.6, and 4.7