Final_Tables

# Final_Tables - jω Multiplication x t p t 1 2 π X jω P...

This preview shows pages 1–2. Sign up to view the full content.

Table of Fourier Transform Pairs Signal Name Time-Domain: x ( t ) Frequency-Domain: X ( ) Right-sided exponential e - at u ( t ) ( a > 0) 1 a + Left-sided exponential e bt u ( - t ) ( b > 0) 1 b - Square pulse [ u ( t + T/ 2) - u ( t - T/ 2)] sin( ωT/ 2) ω/ 2 “sinc” function sin( ω 0 t ) πt [ u ( ω + ω 0 ) - u ( ω - ω 0 )] Impulse δ ( t ) 1 Shifted impulse δ ( t - t 0 ) e - jωt 0 Complex exponential e 0 t 2 πδ ( ω - ω 0 ) General cosine A cos( ω 0 t + φ ) πAe δ ( ω - ω 0 ) + πAe - δ ( ω + ω 0 ) Cosine cos( ω 0 t ) πδ ( ω - ω 0 ) + πδ ( ω + ω 0 ) Sine sin( ω 0 t ) - jπδ ( ω - ω 0 ) + jπδ ( ω + ω 0 ) General periodic signal X k = -∞ a k e jkω 0 t X k = -∞ 2 πa k δ ( ω - 0 ) Impulse Train X n = -∞ δ ( t - nT ) 2 π T X k = -∞ δ ( ω - 2 πk/T ) Table of Fourier Transform Properties Property Name Time-Domain x ( t ) Frequency-Domain X ( ) Linearity ax 1 ( t ) + bx 2 ( t ) aX 1 ( ) + bX 2 ( ) Conjugation x * ( t ) X * ( - ) Time-Reversal x ( - t ) X ( - ) Scaling x ( at ) 1 | a | X ( j ( ω/a )) Delay x ( t - t d ) e - jωt d X ( ) Modulation x ( t ) e 0 t X ( j ( ω - ω 0 )) Modulation x ( t ) cos( ω 0 t ) 1 2 X ( j ( ω - ω 0 )) + 1 2 X ( j ( ω + ω 0 )) Diﬀerentiation dx k ( t ) dt k ( ) k X ( ) Convolution x ( t ) * h ( t ) X ( ) H (

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: jω ) Multiplication x ( t ) p ( t ) 1 2 π X ( jω ) * P ( jω ) Table of z-Transform Pairs Signal Name Time-Domain: x [ n ] z-Domain: X ( z ) Impulse δ [ n ] 1 Shifted impulse δ [ n-n ] z-n Right-sided exponential a n u [ n ] 1 1-az-1 General cosine r n cos(ˆ ω n ) u [ n ] 1-r cos(ˆ ω ) z-1 1-2 r cos(ˆ ω ) z-1 + r 2 z-2 Table of z-Transform Properties Property Name Time-Domain x [ n ] z-Domain X ( z ) Linearity ax 1 [ n ] + bx 2 [ n ] aX 1 ( z ) + bX 2 ( z ) Delay x [ n-n d ] z-n d X ( z ) Convolution x [ n ] * h [ n ] X ( z ) H ( z )...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

Final_Tables - jω Multiplication x t p t 1 2 π X jω P...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online