hmw8 - f = cos πf X 4 f = 1 2 j 8 πf-8 π 2 f 2 Note...

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Due March 24, 2010 BME 171, Spring 2010 - HOMEWORK #8 Fourier Transform 1. The textbook (p. 702) lists the following Fourier transform pairs: 1 2 πδ ( ω ) Π ± t τ ² τ sinc ± ωτ 2 ² while the table posted on the syllabus lists: 1 δ ( f ) Π ± t τ ² τ sinc ( τf ) Show that these Fourier transforms are equivalent. 2. Use direct integration to obtain Fourier transform of the following signal: x ( t ) = ³ - 4 sin(100 πt ) , 0 t 0 . 01 s 0 , otherwise 3. Signal x ( t ) has the Fourier transform: X ( f ) = j 2 πf - (2 πf ) 2 + j 10 πf + 6 Find Fourier transforms of x (2 t - 6). Explain which properties you are using. verte
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4. Using the tables of Fourier pairs and properties posted on the syllabus, compute inverse Fourier transforms of the following signals: X 1 ( f ) = cos( πf ) 1 + j 2 πf e - jπf X 2 ( f ) = u ( f + 2) - u ( f + 1) - u ( f - 1) + u ( f - 2) X 3
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Unformatted text preview: ( f ) = cos( πf ) X 4 ( f ) = 1 2 + j 8 πf-8 π 2 f 2 Note which pairs and properties you are using. 5. Consider the following three signals: x ( t ) = 3 sinc ± t 2 ² y ( t ) = cos(4 πt ) z ( t ) = x ( t ) y ( t ) where the definition of sinc function is sinc( x ) = sin( πx ) πx . (a) Use Matlab to plot these three signals. (b) Determine Fourier transforms of these signals. (c) Sketch (by hand) amplitude spectra of X ( f ), Y ( f ), and Z ( f ). (d) Which of these three signals are energy versus power signals? Determine and sketch their energy spectral density G ( f ) or power spectral density S ( f ) (whichever applicable). 2...
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hmw8 - f = cos πf X 4 f = 1 2 j 8 πf-8 π 2 f 2 Note...

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