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MIT2_017JF09_p06

MIT2_017JF09_p06 - 6 CONVOLUTION OF SINE AND UNIT STEP 6 9...

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6 CONVOLUTION OF SINE AND UNIT STEP 9 6 Convolution of Sine and Unit Step The sine function q ( t ) has a zero value before zero time, and then is a unit sine wave afterwards: 0 if t < 0 q ( t ) = sin( t ) if t 0 For the LTI systems whose impulse responses h ( t ) are given below, use convolution to de- termine the system responses to a sine function input, i.e., u ( t ) = q ( t ). 1. h ( t ) = 1 Solution: t t y ( t ) = h ( t τ ) q ( τ ) = sin( τ ) = cos( τ ) | t 0 = 1 cos t. 0 0 2. h ( t ) = sin( αt ), where α is a fixed positive number. Solution: t y ( t ) = h ( τ ) q ( t τ ) 0 t = sin( ατ ) sin( t τ ) 0 t = [sin( ατ ) sin t cos τ sin( ατ ) cos t sin τ ] (from a trig. identity) 0 t t = sin t sin( ατ ) cos τdτ cos t sin( ατ ) sin τdτ 0 0 1 t = sin t [sin(( α + 1) τ ) + sin(( α 1) τ )] 2 0 1 t cos t [cos(( α 1) τ ) cos(( α + 1) τ )] (two more trig. identities) 2 0 1 1 1 1 1 = sin t cos(( α + 1) t ) cos(( α 1) t ) + + 2 α + 1 α 1 α + 1 α 1 1 1 1 cos t sin(( α 1) t ) sin(( α + 1) t ) 2 α
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