3.2 Fourier Transform Pairs and Properties

Fourier Transform Pairs and Applications

The Fourier Transform converts a time-domain signal into its frequency-domain representation, revealing which frequencies are present and how strong each one is. Knowing the standard transform pairs and properties lets you move fluidly between domains, which is the foundation for filter design, modulation, and system analysis.

Common Fourier Transform Pairs

A Fourier Transform pair links a time-domain function to its frequency-domain counterpart. If you memorize the most common pairs, you can often read off a transform (or inverse transform) without computing an integral.

Dirac Delta and Constant Function

The delta-constant pair deserves special attention because it anchors many derivations.

• The Dirac delta $\delta(t)$ transforms to a flat spectrum of value 1. Physically, an ideal impulse contains all frequencies in equal measure.
• A constant signal (DC) transforms to $2\pi\,\delta(\omega)$. That makes sense: a signal that never oscillates has energy only at zero frequency.

These two pairs are duals of each other (more on duality below), and they show up constantly when you derive other transform pairs or verify properties.

Fourier Transform Properties

Properties let you handle operations like shifting, scaling, and combining signals without re-deriving the transform integral every time. Each property connects a time-domain operation to a specific frequency-domain effect.

Linearity

If $f(t) \Leftrightarrow F(\omega)$ and $g(t) \Leftrightarrow G(\omega)$, then for any constants $a$ and $b$:

$a\,f(t) + b\,g(t) \Leftrightarrow a\,F(\omega) + b\,G(\omega)$

This means you can break a complicated signal into simpler pieces, transform each one separately, and add the results. It's the reason superposition works in the frequency domain.

Time-Shifting Property

Delaying a signal by $t_0$ seconds does not change its magnitude spectrum. It only adds a linear phase shift:

$f(t - t_0) \Leftrightarrow e^{-j\omega t_0} \cdot F(\omega)$

The factor $e^{-j\omega t_0}$ rotates the phase of each frequency component by an amount proportional to $\omega$. The magnitude $|F(\omega)|$ stays the same, which matches intuition: sliding a signal in time doesn't change what frequencies are present, only when they arrive.

Frequency-Shifting Property

Multiplying by a complex exponential in time shifts the entire spectrum:

$f(t)\,e^{j\omega_0 t} \Leftrightarrow F(\omega - \omega_0)$

This is the mathematical basis of modulation. When you multiply a baseband signal by a carrier at frequency $\omega_0$, you move its spectrum up to center on $\omega_0$.

Scaling Property

Compressing a signal in time spreads it out in frequency, and vice versa:

$f(at) \Leftrightarrow \frac{1}{|a|}\,F\!\left(\frac{\omega}{a}\right)$

For $|a| > 1$ the signal gets shorter in time but wider in frequency. For $|a| < 1$ it stretches in time and narrows in frequency. This inverse relationship between time duration and bandwidth is a recurring theme in signal processing and is closely related to the uncertainty principle.

Convolution and Multiplication Properties

These two properties are duals of each other and together form the most practically powerful tools in Fourier analysis.

Convolution Property

Convolution in time becomes multiplication in frequency:

$f(t) * g(t) \Leftrightarrow F(\omega) \cdot G(\omega)$

Why this matters: the output of any linear time-invariant (LTI) system equals the convolution of the input with the system's impulse response $h(t)$. In the frequency domain that convolution reduces to simple multiplication:

1. Transform the input: $f(t) \rightarrow F(\omega)$.
2. Transform the impulse response: $h(t) \rightarrow H(\omega)$ (the transfer function).
3. Multiply: $Y(\omega) = F(\omega) \cdot H(\omega)$.
4. Inverse transform to get the output: $Y(\omega) \rightarrow y(t)$.

This is far easier than evaluating the convolution integral directly, and it's exactly how frequency-domain filter design works. You specify the desired $H(\omega)$ and let multiplication do the filtering.

Multiplication Property

Multiplication in time becomes convolution in frequency (scaled by $\frac{1}{2\pi}$):

$f(t) \cdot g(t) \Leftrightarrow \frac{1}{2\pi}\,F(\omega) * G(\omega)$

Two key applications:

• Amplitude modulation (AM): Multiplying a message signal by a cosine carrier in time convolves their spectra, creating the familiar sidebands around the carrier frequency.
• Windowing: Multiplying a signal by a finite-length window function in time convolves the signal's spectrum with the window's spectrum. That's why different window shapes (Hamming, Hanning, etc.) produce different spectral leakage characteristics.

Duality Property and Implications

Duality Property Explained

Duality says that the Fourier Transform "works both ways" in a very specific sense. If you already know a transform pair, you can swap the roles of time and frequency to get a second pair for free.

Formally, given $f(t) \Leftrightarrow F(\omega)$:

$F(t) \Leftrightarrow 2\pi\,f(-\omega)$

In words: take the frequency-domain function $F$, treat it as a function of time, and its Fourier Transform is $2\pi$ times the original time-domain function evaluated at $-\omega$.

Quick example: You know $\delta(t) \Leftrightarrow 1$. By duality, $1 \Leftrightarrow 2\pi\,\delta(-\omega) = 2\pi\,\delta(\omega)$, which confirms the constant-to-delta pair from the table above.

Implications for Signal Analysis

The duality property reinforces the inverse relationship between time duration and frequency bandwidth. A signal that is narrow in time must be broad in frequency, and a signal that is narrow in frequency must be broad in time. This isn't just a mathematical curiosity; it has direct engineering consequences:

• Short pulses (like radar chirps) occupy wide bandwidths.
• Narrowband filters require long impulse responses.
• Time-frequency analysis methods (spectrograms, wavelet transforms) are all constrained by this tradeoff.

Duality also serves as a useful sanity check. Whenever you derive a new transform pair or verify a property, you can apply duality to see if the "swapped" version also makes physical sense. If it doesn't, something went wrong in the derivation.