ECE-440_hw1

# ECE-440_hw1 - Spring 2011 ECE 440: HW # 1 Due: Beginning of...

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Spring 2011 ECE 440: HW # 1 Due: Beginning of Class Jan. 28, 2011 1.) Prove the following. Do not just state that it is true. Go through the formal steps (e.g., u -substitution) to show equality. i.) sifting: R -∞ x ( t ) δ ( t - t 0 ) dt = x ( t 0 ) . ii.) R -∞ x ( t ) δ ( at ) dt = R -∞ x ( t ) 1 | a | δ ( t ) dt iii.) x ( t ) δ ( t - a ) = x ( a ) δ ( t - a ) if x ( t ) continuous at t = a. (Hint: Show that this is true using the integral of each side) iv.) δ ( t - a ) * f ( t ) = f ( t - a ) v.) ( t - a ) * ( t - b ) = ABδ ( t - a - b ) 2.) Prob. 2.13, pg. 101 (Do parts (a), (b) and (c) only). Note that Π( t ) = rect ( t ) . 3.) For a real signal x ( t ), compute F [ F [ x ( t )]] . How does your answer change for F - 1 [ F - 1 [ x ( t )]]? 4.) Prob. 2.24, pg. 103, For each of the signals ﬁnd the Fourier Transform and then plot the magnitude and the phase of the Fourier Transform. 5.) Derive

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## This note was uploaded on 02/19/2012 for the course ECE 440 taught by Professor Staff during the Spring '08 term at Purdue.

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ECE-440_hw1 - Spring 2011 ECE 440: HW # 1 Due: Beginning of...

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