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,

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

where

$$\hbox{\sc Stretch}_{\alpha,t}(x) \isdef x\left(\frac{t}{\alpha}\right)$$

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

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

Proof: Taking the Fourier transform of the stretched signal gives

$$\begin{eqnarray*} \hbox{\sc FT}_{\omega}(\hbox{\sc Stretch}_\alpha(x)) &\isdef ... ...\tau \\ &\isdef & \left\vert\alpha\right\vert X(\alpha\omega). \end{eqnarray*}$$

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 $\infty$.

The scaling theorem is fundamentally restricted to the continuous-time, continuous-frequency (Fourier transform) case. The closest we will come to the scaling theorem among the DTFT theorems (§2.3) is the stretch (repeat) theorem (page ).