ex)
$\displaystyle f_1(t)*f_2(t)=\int\nolimits_{-\infty}^{\infty} f_1(\tau) f_2(t-\tau) d\tau$
성질
1. 교환성 commutative
$\displaystyle f_1(t)*f_2(t)=f_2(t)*f_1(t)$
sol)
$\displaystyle =\int\nolimits_{-\infty}^{\infty}f_1(\tau)*f_2(t-\tau)d\tau$
$\displaystyle x=t-\tau,\;\; \tau=t-x,\;\; d\tau=-dx$
$\displaystyle =-\int\nolimits_{\infty}^{-\infty}f_2(x)f_1(t-x)dx$
$\displaystyle =\int\nolimits_{-\infty}^{\infty}f_2(x)f_1(t-x)dx$
$\displaystyle =f_2(t)*f_1(t)$
2. 분배성 distributive
$\displaystyle f_1(t)*[f_2(t)+f_3(t)]=f_1(t)*f_2(t)+f_1(t)*f_3(t)$
3. 결합성 associative
$\displaystyle f_1(t)*[f_2(t)*f_3(t)]=[f_1(t)*f_2(t)]*f_3(t)$
4. 추이성 (shift property)
$\displaystyle f_1(t)*f_2(t)=C(t)$ 일 때
$\displaystyle f_1(t)*f_2(t-T)=C(t-T)$
$\displaystyle f_1(t-T)*f_2(t)=C(t-T)$
$\displaystyle f_1(t-T_1)*f_2(t-T_2)=C(t-T_1-T_2)$
proof)
$\displaystyle f_1(t)*f_2(t)=\int\nolimits_{-\infty}^{\infty}f_1(\tau)f_2(t-\tau)d\tau=C(t)$
$\displaystyle f_1(t)*f_2(t-T)=\int\nolimits_{-\infty}^{\infty}f_1(\tau)f_2(t-T-\tau)d\tau =C(t-T)$
convolution operator는 시불변이다. time-invariant이다.
5. impulse와의 convolution
$\displaystyle f(t)*\delta(t)=\int\nolimits_{-\infty}^{\infty}f(\tau)\delta(t-\tau)d\tau=f(t)$
6. 폭(width)
$\displaystyle \left.\begin{matrix} f_1(t)\to T_1,\\ f_2(t)\to T_2\end{matrix}\right\rbrace f_1(t)\ast f_2(t)\to T_1+T_2$
합성곱,convolution