fourier transform lecture presentation

8/9/2019 Fourier Transform Lecture Presentation

1/35

Fourier Transform

Yi ChengCal Poly Pomona

1


2/35

Contents

1. Fourier Transform of AperiodicContinuous-time signals (Slide3

!. Fourier Transform of Aperiodic"iscrete-time signals (Slide!3

!


3/35

Fourier Transform

The fre#uency-domain function$%(&$ or spectrum$ of anaperiodic continuous-time signal$

'(t$ can e e'pressed as

%(& )

3


4/35

+n,erse Fourier Transform

The aperiodic continuous-timesignal$ '(t$ of a fre#uency-domain function$ %(&$ can e

e'pressed as

)


5/35

Fourier Series to Fourier Transform

C) (1/T

0et T ) $ and ω ) ω2

C T ) %(&)


6/35

4'ample 15 Aperiodic pulses &ith apulse &idth of

'(t ) 1 for 26t6

) 2 other&ise

7

'(t

2 t

1


7/35

Fourier Transform of AperiodicS#uare 8a,e

%(& )

)

) (e-9& : 1/(-9&

) e-9&/!(e-9&/! : e 9&/!/(-9&) e-9&/! (e-9&/! : e 9&/!/(-9&

) (!/&e-9&/! (e 9&/! : e-9&/!/(!9

) (!/&e-9&/! sin(& /!) (/!/(& /!e-9&/! sin(& /!

) ( e-9&/! sinc(& /!

;

sinc(' )sin('/'


8/35

Sinc function sin('/'

<


9/35

4'ample !5 Aperiodic +mpulse

'(t ) (t

=

'(t

2 t

1


10/35

Fourier Transform of +mpulse

%(& )

)

) 1

12

) f(t1

%(&

&

1


11/35

4'ample 35 Fourier Transform of+mpulse at t1$ '(t ) >

%1(& )

)

) e-9&

%!(& )

)) e 9&

11

) f(t1


12/35

Fourier Transform is linear

0et '(t ) >%(& ) %1(& > %!(& ) e-9&t > e

9&t

) ! cos( &t1

"uality of Fourier Transform

0et '(t ) ! cos( &1

t

%(& ) >

1!


13/35

'(t ) >

13

%(& )

'(t

tt 1-t 1

%(&

-!?/t1 2 !?/t1 &


14/35

'(&) >

1

%(t )'(&

&&1-&

1

%(t

-!?/&1 2 !?/&1t

'amp e 5 our er rans orm oSinusoidals


15/35

"uality of Fourier Transform0et %(& is the Fourier Transform

of '(t$ Then '(& is the Fourier

Transform of %(t$

1


16/35

4'ample 5 Aperiodic Triangular8a,e

'(t ) (t>T/!/(T/! for :T/!6t6 2

)(T/! : t/(T/! for 2 6t6 : T/!

17

'(t

- -T/!2 t

1


17/35

4'ample a5 "eri,ati,e of Aperiodic Triangular 8a,e

'1(t) '@(t ) (T/! for :T/!6t6 2

)-(T/! for 2 6t6 :T/!

1;

'@(t

- -T/!2 t

T/!

-T/!


18/35

4'ample 5 "oule "eri,ati,e ofAperiodic Triangular 8a,e

'!(t)'@@(t) (t- T/!-! (t> (t>T/!B(T/!

%!(&)e :9&T/!-! > e :9&T/!B(T/!

)cos(&T/!-1BT

1<

%!(&

-!

2 ?/! ? 3?/! &

-1


19/35

"eri,ati,e of Aperiodic Triangular8a,e

%!(& ) e :9&T/! -! > e :9&T/!B(T/!'1

(t ) d'!(t/dt %

1(& ) 9&%

!(&)(9&TAcos(&T/!-1B

1=

'@(t

- -T/!2 t

T/!

-T


20/35

Aperiodic Triangular 8a,e

'(t ) d'1(t/dt ) d! '!(t/dt!

%(& ) 9&%1(&)9& 9& %!(&B

) (9&! Tcos(&T/!-1B

) -&! T cos(&T/! -1B

!2


21/35

4'ample 75 Aperiodic alf 8a,e

'(t ) cos(t for :T/6t6T/

)2 other&ise

) !?/T

!1

'(t

- -T/ 2 t

1


22/35

Fourier Transform of Aperiodic alf8a,e

%(& )

)

)

)

) (- /!9(B

-(- /-!9(B

!!


23/35

Fourier Transform of Aperiodic alf8a,e

%(&) (- /!9(B-(- /-!9(B

) sin (-&T/B/( -

sin (>&T/B/(

!3


24/35


25/35

Fourier Transform of aperiodicdiscrete-time signals y E-Transform

%(&)

%()

Gote ) e sT &here T is the

sampling period0et s ) 9&

E ) e 9&T

!


26/35

4'ample 15 T&-sample Ho,inga,erage

h(n ) I2.$ 2.Jy(n ) 2. K '(n > 2. K '(n-1

(&)

) 2. > 2. e -9&

) (1 > e -9& /!) e :9&/! (e 9&/! > e :9&/! /!

) e:9&/! cos(&/!

!7

A lit d S t f (


27/35

Amplitude Spectrum of (&for a !-sample a,erage F+L$ a

lo&pass Dlter.

!;

M(&M

&2 ?2

1

2.

"C fre#uency$&hich is thelo&est fre#uency

alf of thesamplingfre#uency$&hich is thehighestfre#uency

M(&M ) M e :9&/! cos (&/!M

) (cos(&/!M

Ph S t f ( f


28/35

Phase Spectrum of (& for a!-sample a,erage F+L$ a lo&pass

Dlter.

!<

(& ) -&/!

&2 ?

-?/!

2



(& ) e :9&/! cos (&/!

) -&/!


29/35

4'ample !5 T&-sample diNerenceF+L

h(n ) I2.$ -2.Jy(n ) 2. K '(n - 2. K '(n-1

(&)

) 2. - 2. e -9&

) (1 - e -9& /!) e :9&/! (e 9&/! - e :9&/! /!

) 9e:9&/! sin(&/!

!=


30/35

Phase Spectrum of (& for a


31/35

Phase Spectrum of (& for a!-sample diNerence F+L$ a

highpass Dlter.

31

(& )?/! -&/!

&2 ?2



?/!


32/35

Gth-order F+L system

y(n ) n-0 Tae E-transform of h(n$

( )Leplace y e 9&

(& )

)h(2 >h(1 >h(! >h(3 > ---

>h(G-! >h(G-1 >h(G

)h(2 >h(G B> h(1 >h(G-1 B> h(! >h(G-! B > -------

) h(2 >h(G B

> h(1 >h(G-1 B>---

3!


33/35

4'ample 35 3rd-order 0inear-PhaseF+L systems

0et G ) 3(& )

)h(2 >h(1 >h(! >h(3

)h(2 >h(3 B> h(1 >h(! B) h(2 >h(3 B

> h(1 >h(! B

)h(2 > B> h(1 >B

) !h(2 cos(&/! > !h(1cos(3&/!B

33


34/35

4'ample 5 First-order ++L system

h(n ) an u(n $ ( ) 1/(1-a-1(&)

) 1/( 1- a e-9& ) ( ) e 9&

M(&M ) 1/

O (&) -tan-1(aKsin(&/(1- aK cos(&B

3

Spectrum of (& of a Frist


35/35

Spectrum of (& of a Frist-order ++L system$ &hich is a

lo&pass Dlter.

M(&M

?2 2 ! 2 2 7 2 < 1 1 !

2

!

7

<

12

1!

1/sqrt(1-a * cos(w))^2 + a*sin(w))^2)

fourier transform lecture presentation

Documents