Download - 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.

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Page 1: 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 AND SYSTEM

CT Fourier Transform

Page 2: 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.

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

Page 3: 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.

Motivating Example: Square wave

Page 4: 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.

Derivation

Page 5: 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.

Derivation(continued)

Page 6: 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.

Derivation(continued)

Page 7: 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.

For what kind of signals we do this

Page 8: 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.

Example

Page 9: 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.

CT Fourier Transforms of Periodic Signals

Page 10: 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.

Example

Page 11: 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.

Properties

Page 12: 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.

Properties(continued)

Page 13: 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.

Properties(continued)

Page 14: 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.

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

Page 15: 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.

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

Page 16: 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.

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 )