04 fourier transforms

8/6/2019 04 Fourier Transforms

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Sep 13, 2005 CS477: Analog and Digital Communications 1

Fourier Transforms

Analog and Digital

CommunicationsAutumn 2005-2006


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Fourier Series: Example

cn = T01 R

T0v(t)e j 2nf 0tdt

= T0Asinc( nf 0)

2 2

T0 T0

Av(t)

t

= 4Asinc( 4

n) for = 4T0

-3 -2 -1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

nf 0= 4n

Amplitude norm

alizedby

A/4

Find Fourier Series of ?v(t)

Fourier series expansion is:v(t) =

P 1

1

4Asinc( 4

n)e j 2nf 0t

is sum of rotating phasorsv(t)


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Sinusoidal Fourier SeriesFor real signals

and can be expressed in terms of

= c0 +P

n=1

1(cne j 2nf 0t + cne j 2nf 0t)

cn = c n

v(t) =P

1

1cne j 2nf 0t

Pn=1

1 jcn je j (2nf 0t+

6 cn)

= c0 + 2P

n=1

1cn j j cos(2nf 0t + 6 cn)

= a 0 + 2Pn=1

1[a n cos(2 nf 0t) + bn sin(2 nf 0t )]

a n bn cn

= c0 +Pn=1

1 jcn je j (2nf 0t+

6 cn) +


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Parsevals theoremAverage power of a periodic signal can beobtained from its Fourier coefficients

To prove, write the Fourier series expansion

P = hj x(t)j2i = T1 R

T=2

T=2 jx(t)j2dt

= T1 R

T=2

T=2

x(t)x(t)dt

=P 1

1cncn =

P 1

1 jcn j2


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Fourier Transform

2 2 T0 T0

Av(t)

t

-3 -2 -1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

nf 0= 4n

Amplitude normalizedby

A/4

What if the x-axis in the Fourierseries represents frequency?

The spectral lines appear at integer

multiples of fundamental frequency! Separation between two consecutivespectral lines is equal to f 0 = T

01

How does increasing fundamental period affect theseparation between two consecutive spectral lines? Fourier transform


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Fourier TransformFourier series in the limitRepresents spectral components of a

signalSignal uniquely represented in time orfrequency domain

X(f ) =R

1

1x(t)e j2f t dt

x(t) =R

1

1X(f )e j2ft dt

x(t) $ X(f )


7/16Sep 13, 2005 CS477: Analog and Digital Communications 7

Shifting and Scaling

Review

a t

x(t)A

ab t

x(t=b)A

a=b t

x(tb)A

a + c t

x(t c)A

a c t

x(t + c)A

a t

x( t)A

d a t

x(d t)A

d + ak t

x( kt ) A



Fourier Transform:

Example

21 2

1

1

t

x(t) = rect( t)

X(f ) = sinc( f )

2 2

A

t

x(t) = Arect( t)

X(f ) = Asinc( f )



Properties of FTLinearityScalingDuality

Time shifting

Frequency shifting (modulation)

x(at ) $ a j j1 X(a

f )

X(t) $ x( f )

x(t ) $ X(f )e j2f

x(t)e j 2f ct $ X(f f c)

x1(t) + x2(t ) $ X 1(f ) + X 2(f )



Properties of FTModulation

Product and convolution

More on this under LTI systems

x(t)y(t) $ X(f ) Y(f )

x(t) y(t) $ X(f )Y(f )

x(t) cos(2 f ct + ) $ 2e j X(f f c) + 2

e j X(f + f c)



Properties of FTDifferentiation

Integration

dtd x(t) $ j2f X (f )

R 1

x()d $ j 2f1

X(f )

dt ndn x(t) $ ( j2f )nX(f )

R 1

1x()d = X(0)

R 1

1X(f )df = x(0)



The Dirac Delta FunctionA generalized function

R 1

1

(t )dt =R

(t )dt = 1R

1

1x(t) (t t 1)dt = x(t 1)

R 1

1

x(t t 2) (t t 1)dt = x(t 1 t 2)

x(t) (t ) =R

1

1x() (t )d = x(t)



FT of 1 and exponentialsR

1

1e j2f t (f f 0)df = e j 2f 0t

) e j 2f 0t $ (f f 0)

1 $ (f ) and (t) $ 1

Fourier transform of RF Pulse:

x(t) = Arect( t) cos2 f ct

X(f ) = 2Asinc( f f c) + 2

Asinc( f + f c)



Other FunctionsSignum Function:

sgn( t) = 1 t0

n$ jf

1t

1

1

Unit step Function:

u(t) = 21

(1+ sgn( t))u(t) $ 2

1h jf

1 + (f )i

t

1


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FT of Periodic Signalsxp(t) =

P 1

1cne j 2nf 0t; f 0 = T0

1

Let x(t) = 0 elsex p(t) jt j


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FT of Periodic Signalsxp(t ) =

Pm= 1

1 (t mT 0)

xp(t ) $ f 0P

n= 1

1X(nf 0) (f nf 0)

Let x(t) = 0 elsex p(t) jt j

04 fourier transforms

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