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Signal Processing First

Chapter 11Continuous-TimeFourier Transform

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LECTURE OBJECTIVES Review Frequency Response Fourier Series

Definition of Fourier transform

Relation to Fourier Series Examples of Fourier transform pairs

dtetxjX tj )()(

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11.1 Definition of Fourier Transform

Forward Fourier transform

Inverse Fourier transform

dtetxjX tj )()(

dejXtx tj)()( 21

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11.2 Fourier Transform and the Spectrum

Everything = Sum of sinusoids (?) One Square Pulse = Sum of Sinusoids (?)

Finite Length Not Periodic Limit of Square Wave as Period infinity

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Fourier Series: Periodic x(t)x(t) x(t T0 )

T02T0 T0 2T00 t

x(t) akej 0k t

k

ak 1T0

x(t)e j0ktdtT0/ 2

T0 / 2

Fundamental Freq.0 2 / T0 2f0

Fourier Synthesis

Fourier Analysis

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Square wave signal

ak e j0kt

j0kT0 T0 /4

T0 / 4

e jk / 2 ejk / 2

j2k

sin(k / 2)k

x(t) x(t T0 )

T02T0 T0 2T00 t

4/

4/0

0

0

0)1(1 T

T

tkjk dte

Ta

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Spectrum from Fourier Series

,4,20

,3,1,00)2/sin(k

k

kkak

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What if x(t) is not periodic?

Sum of Sinusoids? Non-harmonically related sinusoids Would not be periodic, but would probably be

non-zero for all t. Fourier transform gives a “sum” (actually an integral) that

involves ALL frequencies

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Limiting Behavior of FS

T0=2T

T0=4T

T0=8T

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Limiting Behavior of Spectrum

T0=2T

T0=4T

T0=8T

)(Plot

0 kaT

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Fourier Transform Defined

For non-periodic signalsFourier Synthesis

Fourier Analysis

dtetxjX tj )()(

dejXtx tj)()( 21

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11.4 Examples of Fourier Transform Pairs

X( j) 1a j

x(t) eatu(t)

Example 1:

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Example 1: x(t) eatu(t)

X( j) 1a j

X( j ) eat

0

e j tdt 0

e (a j )tdt

X( j ) eate j t

a j 0

1

a j

a 0

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Frequency Response

Fourier Transform of h(t) is the Frequency Response

jjHtueth t

11)()()(

)()( tueth t

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Magnitude and Phase Plots

jajH

1)(

ajH 1tan)(

22

11

aja

)()( jHjH

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X( j) sin(T / 2) / 2

Example 2: x(t) 1 t T / 20 t T / 2

X( j ) e j t

j T / 2

T /2

e jT / 2 e jT /2

j

X( j) (1)e jtdtT / 2

T /2 e jtdt

T / 2

T /2

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x(t) 1 t T / 20 t T / 2

X( j ) sin(T / 2) / 2

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Example 3:

b

bjX

0

1)(

tttx b

)sin()(

b

b

dedejXtx tjtj

121)(

21)(

jtee

jtetx

tjtjtj bbb

b

21

21)(

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b

bb jXt

ttx

0

1)()sin()(

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Example 4:

X( j ) (t)e jtdt

1

Shifting Property of the Impulse

)()( 0tttx

0)()( 0tjtj edtettjX

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x(t) (t) X( j ) 1

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Example 5: X( j ) 2 ( 0 )

x(t) 12

2 ( 0 )e jtd

e j0t

x(t) 1 X( j ) 2 ()

x(t) e j0 t X( j) 2 ( 0 )

x(t) cos(0t) X( j) ( 0) ( 0)

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x(t) cos(0t) X( j) ( 0) ( 0)

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Table of Fourier Transforms

x(t) eatu(t) X( j ) 1a j

x(t) 1 t T / 20 t T / 2

X( j ) sin(T / 2) / 2

x(t) sin(0t) t

X( j ) 1 00 0

x(t) (t t0) X( j ) e jt0

x(t) e j0 t X( j ) 2 ( 0 )