Ae104a_2010_handout_4

# Ae104a_2010_handout_4 - ( ) y ( t- )d } = X( ) * Y( ) F { x...

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Figure 1: Time histories: sine wave, sine wave plus random noise, narrow bandwidth random noise, wide bandwidth random noise . 16

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Figure 2: Autocorrelations: sine wave, sine wave plus random noise, narrow bandwidth random noise, wide bandwidth random noise . 17
Figure 3: Autospectral density functions: sine wave, sine wave plus random noise, narrow bandwidth random noise, wide bandwidth random noise . 18

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5.4 Properties of the Fourier Transform 5.4.1 Linearity If x ( t ) = ax 1 ( t ) + bx 2 ( t ) + ... then X ( ω ) = aX 1 ( ω ) + bX 2 ( ω ) + ... and X ( ω ) = a F { x 1 ( t ) } + b F { x 2 ( t ) } . .. 5.4.2 Symmetry F { X ( ω ) } = x ( t ) and FF { x(t) } = x( - t). 5.4.3 Scaling F { x ( at ) } = 1 | a | X ( ω a ) 5.4.4 Shifting F { x ( t - t 0 ) } = X ( ω ) e - iωt 0 F { x ( t ) e iωt 0 } = X ( ω - ω 0 ) 5.4.5 Diﬀerentiation F { d n x ( t ) dt n } = ( ) n X ( ω ) d n X ( ω ) n = F - 1 { ( - it ) n x ( t ) } 5.4.6 Convolution theorem F { x ( t ) .y ( t ) } = F { R -∞ x
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Unformatted text preview: ( ) y ( t- )d } = X( ) * Y( ) F { x ( t ) * y ( t ) } = F { 1 2 R - x ( ) y ( t- )d } = 1 2 X( ) . Y( ) 5.4.7 Parsevals theorem of energy conservation R - x ( t ) y ( t )d = 1 2 R - X * ( )Y( )d The special case x ( t ) = y ( t ) indicates that the total energy of a waveform summed in time is equal to the total energy of the Fourier transform summed over frequency: R - | x ( t ) | 2 d = 1 2 R - | X( ) | 2 d 5.4.8 Higher dimensions f ( x ,t ) = ( 1 2 ) 4 R k , F ( k , ) e i ( k . x-t ) d 3 t F ( k , ) = R x ,t f ( x ,t ) e i ( k . x-t ) d 3 t 19...
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## Ae104a_2010_handout_4 - ( ) y ( t- )d } = X( ) * Y( ) F { x...

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