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s08_ps09_sol

# s08_ps09_sol - Unied Engineering II Spring 2004 Problem S8...

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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 ) (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 ) −∞ t = sin( 2 π ( t τ )) sin(2 πτ ) 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 ) 2 2 So the integral is given by t t t cos(2 πt ) cos(2 πt ) sin(2 πt ) y ( t ) = cos(4 πτ ) sin(4

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