Scaling Theorem

The scaling theorem (or similarity theorem) provides that if you horizontally ``stretch'' a signal by the factor $\alpha$ in the time domain, you ``squeeze'' and amplify its Fourier transform by the same factor in the frequency domain. This is an important general Fourier duality relationship.

Theorem: For all continuous-time functions $x(t)$ possessing a Fourier transform,

$$\zbox {\hbox{\sc Stretch}_\alpha(x) \;\longleftrightarrow\;\left\vert\alpha\right\vert\hbox{\sc Stretch}_{(1/\alpha)}(X)}$$ (B.9)

where

$$\hbox{\sc Stretch}_{\alpha,t}(x) \isdefs x\left(\frac{t}{\alpha}\right)$$ (B.10)

and $\alpha$ is any nonzero real number (the abscissa stretch factor). A more commonly used notation is the following:

$$\zbox {x\left(\frac{t}{\alpha}\right) \;\longleftrightarrow\; \left\vert\alpha\right\vert\cdot X(\alpha\omega)}$$ (B.11)

Proof: Taking the Fourier transform of the stretched signal gives

$$\begin{eqnarray*} \hbox{\sc FT}_{\omega}(\hbox{\sc Stretch}_\alpha(x)) &\isdef & \int_{-\infty}^\infty x\left(\frac{t}{\alpha}\right) e^{-j\omega t} dt\qquad\hbox{(let $\tau=t/\alpha$)}\\ &=& \int_{-\infty}^\infty x(\tau) e^{-j\omega (\alpha\tau)} d (\alpha\tau) \\ &=& \left\vert\alpha\right\vert\int_{-\infty}^\infty x(\tau) e^{-j(\alpha\omega)\tau} d \tau \\ &\isdef & \left\vert\alpha\right\vert X(\alpha\omega). \end{eqnarray*}$$

The absolute value appears above because, when $\alpha<0$ , $d (\alpha\tau) < 0$ , which brings out a minus sign in front of the integral from $-\infty$ to $\infty$ .