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# hw3 - ECE 301 HW3 Due Problem 1 For parts a-c nd the...

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ECE 301 HW3 Due: 6/29/11 Problem 1 For parts a-c, find the impulse response of the continuous-time system. Part a: y ( t ) = integraltext t +2 t - 2 x ( τ ) Part b: y ( t ) = x ( t ) + x ( t - 1) Part c: d dt y ( t ) + 4 y ( t ) = x ( t ) Problem 2 For parts a-c, find the impulse response of the discrete-time system. Part a: y [ n ] = n k = -∞ x [ k ] Part b: y [ n ] = x [ n ] - 1 3 y [ n - 2] Part c: y [ n ] = x [ n +1] - 2 x [ n ]+ x [ n - 1] 4 Problem 3 Find the output of the system given the impulse response, h ( t ), and input, x ( t ), and sketch the output, y ( t ). 1

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Part a: h ( t ) = e -| t | x ( t ) = cos ( πt ) Part b: h ( t ) = U ( t ) - U ( t - 2) x ( t ) = U ( t + 1) - U ( t - 1) Problem 4 Find the output of the system given the impulse response, h ( t ), and input, x ( t ), and sketch the output, y ( t ). Part a: h ( t ) = δ ( t ) - δ ( t - 1) x ( t ) = U ( t ) - U ( t - 3) Part b: h ( t ) = cos ( t ) U ( t ) x ( t ) = δ ( t ) - δ ( t - π ) Problem 5 Find the output of the system given the impulse response, h [ n ], and input, x [ n ], and sketch the output, y [ n ]. Part a: h [ n ] = parenleftbigg 1 2 parenrightbigg | n | x [ n ] = U [ n ] - U [ n - 5] 2
Part b: h [ n ] = δ [ n + 1] - δ [ n ] + δ [ n - 1] x [ n ] = U [ n ] - U [ n - 4] Problem 6 Find the output of the system given the impulse response, h [ n ], and input, x [ n ], and sketch the output, y [ n ]. Part a: h [ n ] = ( - 1) n x [ n ] = δ [ n ] + 2 δ [ n - 1] + δ [ n - 2] Part b: h [ n ] = cos parenleftbigg π 4 n parenrightbigg x [ n ] = U [ n ] - U [ n - 9] Problem 7 For the impulse responses below, determine if the system is causal, memoryless, and/or stable.

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