| multiplication property |
| $ \mathcal{Z}(\omega)=\mathcal{X}(\omega)*\mathcal{Y}(\omega) \ $ |
| $ Z(f)=X(f)*Y(f) \ $ |
| $ Z(f)=\mathcal{Z}(2\pi f)=\frac{1}{2\pi} \int_{-\infty}^{\infty} \mathcal{X}(\theta)\mathcal{Y}(2\pi f-\theta)d\theta \ $ |
| $ Let\ \varphi =\frac{\theta}{2\pi},\ then\ \theta=2\pi \varphi \ $ |
$ \begin{align} Z(f) &= \frac{1}{2\pi} \int_{-\infty}^{\infty} \mathcal{X}(2\pi \varphi)\mathcal{Y}(2\pi f-2\pi \varphi)d2\pi \varphi \\ &= \int_{-\infty}^{\infty} \mathcal{X}(2\pi \varphi)\mathcal{Y}(2\pi (f-\varphi))d\varphi \\ &= \int_{-\infty}^{\infty} X(\varphi)Y(f-\varphi)d\varphi \end{align} $
$ Since\ X(\alpha)=\mathcal{X}(2\pi \alpha),Y(\alpha)=\mathcal{Y}(2\pi \alpha) $
