【正文】 ntegral. ? ? ? ? ? ? ??? dthfty ?? ? ??? Why? The superposition integral for CT systems Graphic view of staircase approximation to the superposition integral CT convolution )(tf )(tyDerivation of staircase approximation to superposition integral ?)(tf)( ?? ?t)()( ?? ?thf)()( ??? ?? tf)()( ?? ?ttf)(ty)()( ?? ?? thfMechanics (步驟 、 技巧 ) To pute the superposition integral ? ? ? ? ? ? ? ? ? ? ??? dthfthtfty ???? ? ???Step 1 Plot f and h vs since the convolution integral is on . Step 2 Flip h ( ? ) around the vertical axis to obtain h ( ? ) Step 3 Shift h ( ? ) by t to obtain h ( t ? ) Step 4 Multiply to obtain f ( ? ) h ( t ? ) Step 5 Integrate on ? to pute Step 6 Increase t and repeat step 3 to 6 ? ? ? ? ??? dthf ?????)(?h )(180 2 ?????? ?? ff l i p ? )())(( ?? ???????? ?? ththts hi f t　CT convolution solution Example1 f1(?) ? t 1 1 f2(t?) t1 f1(?) ? t 1 1 f2(t?) t1 f1(?) ? t 1 1 f2(t?) t1 f1(?) ? t 1 1 f2(t?) t1 f1(?) ? t 1 1 f2(t?) t1 f(t) t 1 1 1 2 )(*)()( 21 tftftg ?Example2: y(t) = p1(t) ? p1(t) )(1 ?p? .?11),(1 ???? ttp ??t??? 50 .1a) ?? t ? ?1 y(t)=0 01),(1 ??? ttp ??t??? 50 .1b) ?1 ? t 0 y(t)=1+t 10),(1 ?? ttp ??t??? 50 .1c) 0 ? t 1 y(t)=1?t ? ttp 1),(1 ??t??? 50 .1d)1 ? t ? y(t)=0 )()( 11 tptp ?t1?11CT convolution solution CT convolution properties The convolution integral has properties similar to those of the convolution sum. The last property implies that differentiating the input n times and the impulse response m times results in an output that is differentiated n + m times. )()()()()( tfththtfty ????)]()([)()()()]()([)(2121ththtftyththtfty??????)()()()()()]()([)()(2121thtfthtftyththtfty???????)()()( 2121 ???? ?????? thtxty)()()( ][][][ thtxty mnmn ???Communitive Associative Distributive Delay accumulation Derivative accumulation )()()()()]()([:.1 212121 tfdtdtftftfdtdtftfdtda t i o nd i f f e r e n t i ?????)()()()()]()([i 212121 tfdfdftfdffe gr at i on ttt ????? ??? ?????? ???????：)()()( )()()( )(2)(1)(21tftftyt h entftftyw h eninin ????? ，)1(21)1(212)1(121))()(39。 ttf ?? )()(39。求例：)()()(,)(1tytekthjsAetfnitist i?????? ??? ?? ?????????? ???? dtekAethtftynititsfi ])([)()()(:1)(解][10)(? ???? ??nisist dekAe i ??????? )(,]R e []R e [ tys fi 則① ??spststni iistfiphAesHAeskAetys???????? )()(][)(,]R e []R e [1 ??? 則②???? ??tectynitixi 　　　1)( ?忽略)(]R e []R e [ tys xi ???)()(]R e []R e [)(Ae)(m a x sHAetyspHtfstist???響應滿足時，系統(tǒng)為無時限指數(shù)信號當輸入?