Continuous-time Fourier Transform

Continuous-time Fourier Transform

Prof. Siripong Potisuk

Derivation of CTFT

Ttx

tx

as )(~ signal periodic a of

limit a asit viewslet' ),( signal aperiodican Given

and 2

ere wh

asgiven is )(~ ofexpansion seriesFourier The

0 T

tx

define wewhere

obtain we),(~ ofexpansion FS theinto ngSubstituti txak

CT Fourier Transform Pair

Conditions for Existence

Applicable for aperiodic signal of finite and infinite duration which satisfies:

)( of ansformFourier tr inverse theNote t

Examples

Example: Real Exponential Function

Example: Square Pulse

Example: Gaussian-shaped Signal

Example: Gaussian-shaped Signal (cont’d)

Example of ICTFT: An Ideal Lowpass Filter

Impulse Response Frequency Response

CTFT of Periodic Signals

Recall the following CTFT pair:

Represent periodic signal x(t) in terms of FS

Example: Sinusoidal Signal

where

Example: A Pulse Train (Sampling Function)