SIGNAL AND SYSTEM

CT Fourier Transform

Fourier’s Derivation of the CT Fourier Transform•

• x(t) -an aperiodic signal -view it as the limit of a periodic signal as T→ ∞

•For a periodic signal, the harmonic components are spaced ω0= 2π/T apart ...

•As T→ ∞, ω0→0, and harmonic components are spaced closer and closer in frequency

⇓ Fourier series Fourier integral

Motivating Example: Square wave

Derivation

Derivation(continued)

Derivation(continued)

For what kind of signals we do this

Example

CT Fourier Transforms of Periodic Signals

Example

Properties

Properties(continued)

Properties(continued)

Properties(continued)

5. Differentiation and Integration

6. Duality

( )( )

1( ) ( ) (0) ( )

t

dx tj X j

dt

x d X j Xj

( ) 2 ( )X t x

Properties(continued)7. Parseval’s Relation

8. Multiplication Property

9. Convolution Property

1 2 1 2

1( ) ( ) ( ) ( )

2x t x t X j X j

( ) ( ) ( ) ( ) ( ) ( )y t h t t Y j H j X j

LTI System characterized by Differential Equations

0

0

( )( )

( )( ) ( )

Mk

kkN

kk

k

b jY j

H jX j a j

(Some slides are referred from MIT )