s08_ps09_sol

s08_ps09_sol - Unified Engineering II Spring 2004 Problem...

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Unformatted text preview: Unified Engineering II Spring 2004 Problem S8 Solution 1. The convolution is given by ∞ y ( t ) = g ( t ) ∗ u ( t ) = g ( t − τ ) u ( τ ) dτ (1) −∞ Note that u ( τ ) is nonzero only for − 3 ≤ τ ≤ 0, and g ( t − τ ) is nonzero only for 0 ≤ t − τ ≤ 3, that is, for − 3 + t ≤ τ ≤ t . So there are four distinct regimes: (a) t < − 3 (b) − 3 ≤ t ≤ 0 (c) 0 ≤ t ≤ 3 (d) t > 3 For cases (a) and (d), there is no overlap between g ( t − τ ) and u ( τ ), so y ( t ) = 0. For case (b), the overlap is for − 3 ≤ τ ≤ t . So ∞ y ( t ) = g ( t − τ ) u ( τ ) dτ −∞ t = sin( − 2 π ( t − τ )) sin(2 πτ ) dτ − 3 At this point, we have to do a little trig: sin( − 2 π ( t − τ )) sin(2 πτ ) = sin(2 π ( τ − t )) sin(2 πτ ) = [sin(2 πτ ) cos(2 πt ) − cos(2 πτ ) sin(2 πt )] sin(2 πτ ) = cos(2 πt ) sin 2 (2 πτ ) − sin(2 πt ) cos(2 πτ ) sin(2 πτ ) sin(4 πτ ) = cos(2 πt ) 1 − cos(4 πτ ) − sin(2 πt...
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This note was uploaded on 01/28/2012 for the course AERO 16.01 taught by Professor Markdrela during the Fall '05 term at MIT.

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s08_ps09_sol - Unified Engineering II Spring 2004 Problem...

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