4.2 Scaling and Duality

Scaling Property of Fourier Transform

Scaling and duality are two properties of the Fourier transform that let you find transforms of modified signals without re-deriving everything from the integral definition. Scaling describes what happens to the spectrum when you stretch or compress a signal in time. Duality reveals a deep symmetry between the time and frequency domains, letting you swap their roles to get new transform pairs for free.

Time Domain Scaling and Frequency Domain Effects

The scaling property captures a fundamental tradeoff: compressing a signal in time spreads it out in frequency, and vice versa. If you know the transform of $x(t)$, you can immediately write down the transform of $x(at)$ for any nonzero constant $a$.

The formal statement is:

$x(t) \leftrightarrow X(\omega) \implies x(at) \leftrightarrow \frac{1}{|a|}X\!\left(\frac{\omega}{a}\right)$

Notice two things happen simultaneously:

• The frequency axis scales by $1/a$ (frequencies shift outward or inward)
• The amplitude scales by $1/|a|$ (the peak gets shorter or taller)

The absolute value $|a|$ handles the case where $a$ is negative (which also flips the signal in time).

How compression and expansion play out:

• Time compression ($|a| > 1$): The signal gets squeezed in time, so it changes faster. Its spectrum spreads to higher frequencies, but the spectral amplitude drops by $1/|a|$.
• Time expansion ($|a| < 1$): The signal stretches in time, so it changes more slowly. Its spectrum compresses toward lower frequencies, and the spectral amplitude increases by $1/|a|$.

Concrete example: Suppose $x(t) \leftrightarrow X(\omega)$. The signal $x(2t)$ is compressed by a factor of 2 in time. Applying the property:

$x(2t) \leftrightarrow \frac{1}{2}X\!\left(\frac{\omega}{2}\right)$

The spectrum is now twice as wide ($\omega/2$ stretches the frequency axis), but half as tall (the $1/2$ out front).

Why the $1/|a|$ Factor?

The amplitude factor $1/|a|$ isn't arbitrary. It comes directly from the substitution $u = at$ inside the Fourier integral, which produces a factor of $1/|a|$ from the differential $dt = du/|a|$. Physically, it ensures that Parseval's theorem still holds: the total energy of the signal stays the same whether you measure it in time or frequency.

Gaussian Example

The Gaussian $e^{-\pi t^2}$ is its own Fourier transform: $e^{-\pi t^2} \leftrightarrow e^{-\pi \omega^2}$. For a scaled Gaussian $e^{-\pi (at)^2}$, apply the scaling property directly:

$e^{-\pi(at)^2} \leftrightarrow \frac{1}{|a|}\,e^{-\pi(\omega/a)^2}$

A narrower Gaussian in time (large $|a|$) gives a wider, shorter Gaussian in frequency.

Time vs. Frequency Domain Duality

The Duality Property

Duality says that the Fourier transform and its inverse are nearly the same operation. If you already know one transform pair, you can flip the roles of $t$ and $\omega$ to get a second pair essentially for free.

The formal statement is:

$x(t) \leftrightarrow X(\omega) \implies X(t) \leftrightarrow 2\pi\, x(-\omega)$

In words: take the frequency-domain function $X(\omega)$, plug in $t$ instead of $\omega$ to make it a time-domain signal $X(t)$, and its Fourier transform is $2\pi$ times the original time-domain signal evaluated at $-\omega$.

A few things to keep straight:

• The $2\pi$ factor comes from the asymmetry in the standard Fourier transform definition (the forward transform has no $2\pi$, but the inverse has $1/2\pi$). If your course uses the unitary convention with $1/\sqrt{2\pi}$ on both sides, this factor becomes 1.
• The negation $x(-\omega)$ matters for signals that aren't even. For even signals ($x(t) = x(-t)$), the negation disappears and the result simplifies to $2\pi\, x(\omega)$.

Deriving New Transforms with Duality

The real payoff of duality is turning one known pair into another without touching an integral.

Step-by-step process:

1. Start with a known pair: $x(t) \leftrightarrow X(\omega)$
2. Form the new time-domain signal $X(t)$ by replacing $\omega$ with $t$ in the frequency-domain expression
3. Write the new transform as $2\pi\, x(-\omega)$

Classic example (rect and sinc): The standard pair is:

$\operatorname{rect}(t) \leftrightarrow \operatorname{sinc}\!\left(\frac{\omega}{2\pi}\right)$

(The exact form of the sinc depends on your textbook's convention. Many signal processing texts use $\operatorname{sinc}(\omega/2\pi)$ or write the transform as $\operatorname{Sa}(\omega/2)$. Use whatever convention your course defines.)

Applying duality: treat the sinc as a time-domain signal. Its Fourier transform is:

$\operatorname{sinc}(t) \leftrightarrow 2\pi\,\operatorname{rect}(-\omega)$

Since $\operatorname{rect}$ is an even function, $\operatorname{rect}(-\omega) = \operatorname{rect}(\omega)$, so:

$\operatorname{sinc}(t) \leftrightarrow 2\pi\,\operatorname{rect}(\omega)$

You just derived the transform of a sinc pulse without computing any integrals.

Simplifying Fourier Transform Calculations

Using Scaling to Shortcut Calculations

Whenever you see a signal that looks like a known signal with a scaled argument, reach for the scaling property before setting up an integral.

1. Identify the "base" signal $x(t)$ whose transform $X(\omega)$ you already know
2. Determine the scaling factor $a$ such that your signal is $x(at)$
3. Write the result: $\frac{1}{|a|}X(\omega/a)$

This works for any value of $a$, including negative values (which flip the signal in time).

Using Duality to Shortcut Calculations

Duality is most useful when you encounter a time-domain signal that looks like a known frequency-domain function. Instead of integrating, just apply the duality relation.

For instance, if you need the transform of a triangular pulse $\Lambda(t)$ and you already know that a rect pulse's transform squared gives a triangle in frequency (because convolution of two rects in time gives a triangle, and convolution becomes multiplication in frequency), duality lets you go the other direction and write down the transform of $\Lambda(t)$ as a $\operatorname{sinc}^2$ function scaled by $2\pi$.

Combining Both Properties

For more complex signals, you can chain these properties together. The general strategy:

1. Recognize the base shape. Strip away scaling factors and identify a signal whose transform you know.
2. Apply scaling to account for any compression or expansion in the argument.
3. Apply duality if the signal's shape matches a known frequency-domain function rather than a time-domain one.
4. Check your result by verifying that the dimensions and limiting behavior make sense (e.g., a narrower pulse in time should have a wider spectrum).

These two properties, combined with linearity and the time-shift property, let you build up a large table of Fourier transform pairs from just a handful of base pairs. That's much of what makes the Fourier transform practical: you rarely need to evaluate the integral directly.