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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 )