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Unformatted text preview: 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 < q ( t ) = sin( t ) if t 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 ) q ( ) d = sin( ) d = cos( ) | t = 1 cos t. 2. h ( t ) = sin( t ), where is a fixed positive number. Solution: t y ( t ) = h ( ) q ( t ) d t = sin( ) sin( t ) d t = [sin( ) sin t cos sin( ) cos t sin ] d (from a trig. identity) t t = sin t sin( ) cos d cos t sin( ) sin d 1 t = sin t [sin(( + 1) ) + sin(( 1) )] d 2 1 t cos t [cos(( 1) ) cos(( + 1) )] d (two more trig. identities) 2 1 1 1 1 1 = sin t cos(( + 1) t )...
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This note was uploaded on 11/29/2011 for the course CIVIL 1.00 taught by Professor Georgekocur during the Spring '05 term at MIT.
- Spring '05