Final_Tables

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

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

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