Prof. Nizamettin AYDIN

naydin@yildiz.edu.tr

http://www.yildiz.edu.tr/~naydin

Advanced Digital Signal Processing

1

Introduction to the Fourier Introduction to the Fourier TransformTransform

2

• Review– Frequency Response– Fourier Series

• Definition of Fourier transform

Relation to Fourier Series

• Examples of Fourier transform pairs

dtetxjX tj )()(

Everything = Sum of Sinusoids

• One Square Pulse = Sum of Sinusoids– ???????????

• Finite Length

• Not Periodic

• Limit of Square Wave as Period infinity– Intuitive Argument

Fourier Series: Periodic x(t)

x(t) x(t T0 )

T0 2T0 T0 2T00 t

x(t) akej 0k t

k

ak 1

T0

x(t)e j0ktdt T0/ 2

T0 / 2

Fundamental Freq.

0 2 / T0 2f0

Fourier Synthesis

Fourier Analysis

Square Wave Signal

ak e j0kt

j0kT0 T0 /4

T0 / 4

e jk / 2 e jk / 2

j2k

sin(k / 2)

k

x(t) x(t T0 )

T0 2T0 T0 2T00 t

4/

4/0

0

0

0)1(1

T

T

tkjk dte

Ta

Spectrum from Fourier Series

,4,20

,3,1,00)2/sin(

k

k

k

kak

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 integralintegral) that involves

ALLALL frequencies– can represent signals that are identically zero for

negative t. !!!!!!!!!

Limiting Behavior of FS

T0=2T

T0=4T

T0=8T

Limiting Behavior of Spectrum

T0=2T

T0=4T

T0=8T

)(Plot

0 kaT

FS in the LIMIT (long period)

Fourier Synthesis

Fourier Analysis

dejXtxeaTtx tj

Ttkj

kkT )()()( 2

1202

10

0

0

dtetxjXdtetxaT tjT

T

tkjTk

)()()(2/

2/

0

0

0

0

0

d

TT

0

2lim0

k

TT 0

2lim0

)(lim 00

jXaT kT

Fourier Transform Defined

• For non-periodic signals

Fourier Synthesis

Fourier Analysis

dtetxjX tj )()(

dejXtx tj)()( 21

Example 1: x(t) e atu(t)

X( j) 1

a j

X( j ) e at

0

e j tdt 0

e (a j )tdt

X( j )e ate j t

a j0

1

a j

a 0

Frequency Response

• Fourier Transform of h(t) is the Frequency Response

jjHtueth t

1

1)()()(

)()( tueth t

Magnitude and Phase Plots

jajH

1)(

ajH

1tan)(

22

11

aja

)()( jHjH

X( j) sin(T / 2)

/ 2

Example 2:x(t)

1 t T / 2

0 t T / 2

X( j ) e j t

j T / 2

T /2

e jT / 2 e jT /2

j

X( j ) (1)e jtdt T / 2

T /2

e jtdt T / 2

T /2

x(t)

1 t T / 2

0 t T / 2

X( j ) sin(T / 2)

/ 2

Example 3:

b

bjX

0

1)(

t

ttx b

)sin(

)(

b

b

dedejXtx tjtj

12

1)(

2

1)(

jt

ee

jt

etx

tjtjtj bbb

b

2

1

2

1)(

b

bb jX

t

ttx

0

1)(

)sin()(

Example 4:

X( j ) (t)e jtdt

1

Shifting Property of the Impulse

)()( 0tttx

0)()( 0tjtj edtettjX

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

Example 5: X( j ) 2 ( 0 )

x(t) 1

22 ( 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 )

x(t) cos(0t)

X( j) ( 0 ) ( 0 )

Table of Fourier Transforms

x(t) e atu(t) X( j ) 1

a j

x(t) 1 t T / 2

0 t T / 2

X( j ) sin(T / 2)

/ 2

x(t) sin(0t)

t X( j )

1 0

0 0

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

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

Fourier TransformFourier Transform

PropertiesProperties

25

• The Fourier transform

• More examples of Fourier transform pairs• Basic properties of Fourier transforms

– Convolution property

– Multiplication property

dtetxjX tj )()(

Fourier Transform

Fourier Analysis(Forward Transform)

dtetxjX tj )()(

Fourier Synthesis(Inverse Transform)

dejXtx tj)(2

1)(

)()(

Domain-FrequencyDomain-Time

jXtx

WHY use the Fourier transform?

• Manipulate the “Frequency Spectrum”

• Analog Communication Systems– AM: Amplitude Modulation; FM

– What are the “Building Blocks” ?• Abstract Layer, not implementation

• Ideal Filters: mostly BPFs

• Frequency Shifters– aka Modulators, Mixers or Multipliers: x(t)p(t)

Frequency Response

• Fourier Transform of h(t) is the Frequency Response

jjHtueth t

1

1)()()(

)()( tueth t

2/

)2/sin()(

2/0

2/1)(

T

jXTt

Tttx

b

bb jX

t

ttx

0

1)(

)sin()(

0)()()( 0tjejXtttx

00 t

Table of Fourier Transforms

)(2)()( ctj jXetx c

0)()()( 0tjejXtttx

b

bb jX

t

ttx

0

1)(

)sin()(

2/

)2/sin()(

2/0

2/1)(

T

jXTt

Tttx

jjXtuetx t

1

1)()()(

)()()()cos()( ccc jXttx

Fourier Transform of a General Periodic Signal

• If x(t) is periodic with period T0 ,

0

00

00

)(1

)(T

tjkk

k

tjkk dtetx

Taeatx

)(2 since Therefore, 00 ke tjk

k

k kajX )(2)( 0

Square Wave Signal

x(t) x(t T0 )

T0 2T0 T0 2T00 t

ak e j0kt

j0kT0 0

T0 / 2

e j 0kt

j0kT0 T0 /2

T0

1 e jk

jk

ak 1

T0

(1)e j0 ktdt 1

T0

( 1)e j 0ktdtT0 / 2

T0

0

T0 / 2

Square Wave Fourier Transform

X( j ) 2 ak( k0 )k

x(t) x(t T0 )

T0 2T0 T0 2T00 t

Table of Easy FT Properties

ax1(t) bx2 (t) aX1( j ) bX2 ( j )

x(t td ) e jtd X( j )

x(t)e j0t X( j( 0 ))

Delay Property

Frequency Shifting

Linearity Property

x(at) 1|a | X( j(a ))

Scaling

Scaling Property

expands)(shrinks;)2( 221 jXtx

)(

)()(

1

)/(

aa

adajtj

jX

exdteatx

)()( 1aa

jXatx

Scaling Property

)()( 1aa

jXatx

)2()( 12 txtx

Uncertainty Principle

• Try to make x(t) shorter– Then X(j) will get wider– Narrow pulses have wide bandwidth

• Try to make X(j) narrower– Then x(t) will have longer duration

• Cannot simultaneously reduce time Cannot simultaneously reduce time duration and bandwidthduration and bandwidth

Significant FT Properties

x(t)h(t) H( j )X( j )

x(t)e j0t X( j( 0 ))

x(t)p(t) 1

2X( j )P( j )

dx(t)

dt ( j)X( j)

Differentiation Property

Convolution Property

• Convolution in the time-domain

corresponds to MULTIPLICATIONMULTIPLICATION in the frequency-

domain

y(t) h(t) x(t) h( )

x(t )d

Y( j ) H( j )X( j )

y(t) h(t) x(t)x(t)

Y( j ) H( j )X( j )X( j )

Convolution Example

• Bandlimited Input Signal– “sinc” function

• Ideal LPF (Lowpass Filter)– h(t) is a “sinc”

• Output is Bandlimited– Convolve “sincs”

Ideally Bandlimited Signal

1000

1001)(

)100sin()( jX

t

ttx

100b

Convolution Example

sin(100 t)

t

sin(200t)

t

x(t)h(t) H( j )X( j )

sin(100 t)

t

Cosine Input to LTI System

Y ( j) H( j )X( j)

H( j )[( 0 )( 0)]

H( j0 ) ( 0 ) H( j0 ) ( 0 )

y(t) H (j0 ) 12 e j0t H( j0 ) 1

2 e j 0t

H( j0 ) 12 e j0t H *( j 0)

12 e j0t

H( j0 ) cos( 0t H( j0 ))

Ideal Lowpass Filter

Hlp( j )

co co

y(t) x(t) if 0 co

y(t) 0 if 0 co

Ideal Lowpass Filter

y(t) 4

sin 50t 4

3sin 150t

fco "cutoff freq."

H( j ) 1 co

0 co

Signal Multiplier (Modulator)

• Multiplication in the time-domain corresponds to convolution in the frequency-domain.

Y( j ) 1

2X( j )P( j )

y(t) p(t)x(t)

X( j)

x(t)

p(t)

Y( j ) 1

2X( j )

P( j( ))d

Frequency Shifting Property

x(t)e j0t X( j( 0 ))

y(t) sin 7t

te j 0 t Y ( j )

1 0 7 07

0 elsewhere

))((

)()(

0

)( 00

jX

dtetxdtetxe tjtjtj

y(t) x(t)cos(0t)

Y( j ) 12

X( j( 0 )) 12

X( j( 0 ))

x(t)

Differentiation Property

dx(t)

dt

d

dt1

2X( j )e j td

1

2( j )X( j )e j td

d

dte atu(t) ae atu(t) e at (t)

(t) ae atu(t)

ja j

Multiply by j