20Window - MAE 591 RANDOM DATA Time Windows Windowing is...

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MAE 591 RANDOM DATA Time Windows Windowing is multiplying the time series by a ‘data window’, which is equivalent to applying a convolution operation to the ‘raw’ Fourier transform. The purpose of windowing or tapering is to suppress large side lobes in the effective filter obtained with the raw transform in the frequency domain and to round off potential discontinuities at each end of the finite segment of the time series in the time domain. The simplest window is the rectangular (box car) function given by otherwise T t T for t w 2 2 0 1 ) ( 0 = w 0 ( t ) t 2 T 2 T 1 y ( t ) x ( t ) The Fourier transform of the windowed data becomes then ) ( f X ) ( t x ∫∫ = = = dt ft j t w d t j Y dt ft j t w t y dt ft j t x f X T T ) exp( ) ( ] ) exp( ) ( [ ) exp( ) ( ) ( ) exp( ) ( ) ( 0 0 2 2 π ξ πξ
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MAE 591 RANDOM DATA ) ( ) ( ) ( ) ( ] } ) ( exp{ ) ( )[ ( 0 0 0 f W f Y d f W Y d dt t f j t w Y = = = ∫∫ ξ π where is the untruncated original data and Y are the Fourier transforms of , respectively. Note that the above relation holds for any window functions and that Fourier transform of the raw data Y
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This note was uploaded on 11/21/2009 for the course ME . taught by Professor . during the Spring '09 term at Korea University.

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20Window - MAE 591 RANDOM DATA Time Windows Windowing is...

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