Fourier transform property: Scaling

Statement: if $g(t)\leftrightarrow G(f)$, then for a real constant $a$

$$g(at)\leftrightarrow\frac{1}{\lvert a\rvert}G\left(\frac{f}{a}\right)$$

Explanation: if $\lvert a\rvert>1$ the signal is compressed in time, and its spectrum is expanded in frequency (with reduced height); if $\lvert a\rvert<1$ the signal is stretched and its spectrum shrinks. The factor $1/\lvert a\rvert$ keeps the energy relation consistent. For $a=-1$, $g(-t)\leftrightarrow G(-f)$ (time reversal).

Example: a rectangular pulse of width $T$ has spectrum $T\,\text{sinc}(fT)$ with first null at $1/T$. Halving the width to $T/2$ moves the first null to $2/T$, doubling the bandwidth.

Significance: this is the inverse relation between pulse duration and bandwidth: faster data (shorter pulses) needs more bandwidth. Playing a tape at double speed raises all pitches by an octave.