7.3 Convolution and Parseval

The Fourier transform of 7.2 is invertible, linear, and converts calculus into algebra. Those properties already make it useful. But the deep power of Fourier methods comes from two further identities — the convolution theorem and Parseval’s identity — that turn the transform from a computational tool into the language of linear-systems theory.

This lesson develops both.

Convolution

The convolution of two functions $f$ and $g$ is

\boxed{\;(f * g)(t) \;\equiv\; \int_{-\infty}^{\infty} f(\tau)\, g(t - \tau)\, d\tau.\;}

Convolution is the operation of sliding one function past another and integrating the product at each shift. The output is a function of the shift $t$ .

f(t):

g(t):

shift t = 0.00

The interactive shows $(f * g)(t)$ built up as $g$ slides past $f$ . The shaded area is the overlap between $f(\tau)$ and the flipped, shifted $g(t - \tau)$ at the current shift; the bottom curve plots that overlap as a function of $t$ . Hit sweep and watch the convolution waveform draw itself.

A few properties worth noting:

Commutative: $f * g = g * f$ . Sliding $f$ past $g$ produces the same result as sliding $g$ past $f$ .
Associative: $(f * g) * h = f * (g * h)$ . You can convolve in any order.
Distributive over addition: $f * (g + h) = f * g + f * h$ . Convolution is a linear operation.
Identity: $f * \delta = f$ . Convolving with a delta function gives back the original.

Why convolution matters: linear time-invariant systems

A linear time-invariant (LTI) system is one that satisfies (a) linearity (response to a sum of inputs is the sum of the responses) and (b) time invariance (delaying the input by $\tau$ delays the output by $\tau$ , but doesn’t change it otherwise). Almost every linear physical system is LTI: linear filters, room acoustics, electrical circuits, mechanical resonators, optical systems.

Define an LTI system’s impulse response $h(t)$ as its output when the input is a delta function $\delta(t)$ . By linearity and time-invariance, the response to any input $x(t)$ is the convolution of $x$ with $h$ :

y(t) \;=\; (h * x)(t) \;=\; \int_{-\infty}^{\infty} h(\tau)\, x(t - \tau)\, d\tau.

This is one of the central facts of signal processing. An LTI system is completely characterised by its impulse response — knowing $h(t)$ is enough to compute the output for any input. The impulse response of a room (recorded with a microphone after popping a balloon) lets you simulate that room’s effect on any sound by convolving the impulse response with the dry signal. The impulse response of an audio compressor lets you re-create its sonic character entirely in software. Convolution reverb — the technology behind most high-end audio plug-ins — is exactly this operation.

The convolution theorem

Convolution in the time domain corresponds to a stunningly simple operation in the frequency domain:

\boxed{\;(f * g)(t) \;\longleftrightarrow\; \tilde f(\omega) \cdot \tilde g(\omega).\;}

Convolution in time is multiplication in frequency. This is the convolution theorem, and it is the single most-used property of the Fourier transform.

▶ Why convolution becomes multiplication

Fourier-transform $(f * g)(t) = \int f(\tau)\, g(t - \tau)\, d\tau$ in $t$ :

\widetilde{f * g}(\omega) \;=\; \int dt\, e^{-i\omega t} \int d\tau\, f(\tau)\, g(t - \tau).

Swap the order of integration (justified by Fubini for $f, g \in L^1$ ):

\;=\; \int d\tau\, f(\tau) \int dt\, e^{-i\omega t}\, g(t - \tau).

In the inner integral, substitute $u = t - \tau$ , so $t = u + \tau$ and $dt = du$ :

\int dt\, e^{-i\omega t}\, g(t - \tau) \;=\; \int du\, e^{-i\omega(u + \tau)}\, g(u) \;=\; e^{-i\omega \tau} \int du\, e^{-i\omega u}\, g(u) \;=\; e^{-i\omega \tau}\, \tilde g(\omega).

Substitute back:

\widetilde{f * g}(\omega) \;=\; \int d\tau\, f(\tau)\, e^{-i\omega \tau}\, \tilde g(\omega) \;=\; \tilde g(\omega) \int d\tau\, f(\tau)\, e^{-i\omega \tau} \;=\; \tilde g(\omega)\, \tilde f(\omega).

The product separates because of the time-shift property — convolution is precisely the operation that the Fourier transform turns into pointwise multiplication.

Why this matters

The practical implications cascade:

LTI systems in frequency. If $y = h * x$ , then $\tilde y = \tilde h \cdot \tilde x$ . The system’s effect on an input is to multiply the input spectrum by the system’s transfer function $\tilde h(\omega)$ . The whole “Bode plot” / “frequency response” picture of electrical engineering and audio engineering is this multiplication.
Filtering by multiplication. A low-pass filter is one whose $\tilde h(\omega)$ is large for small $\omega$ and small for large $\omega$ . Multiply the input spectrum by such a $\tilde h$ , inverse-transform, and you have a low-passed signal. Equivalent to convolving with $h$ in time, but in frequency it’s literally just an element-wise multiply.
Fast convolution. Direct convolution of two signals of length $N$ takes $O(N^2)$ operations. Going through frequency — FFT, multiply, inverse FFT — takes $O(N \log N)$ (see Numerical Methods 10.5). For non-trivial $N$ this is the only practical way to do convolution.
Solving PDEs. Linear PDEs with translation-invariant coefficients become multiplication in Fourier space. The Green’s function approach is convolution with the impulse response of the PDE operator.

Parseval’s identity

A second fundamental relation: the energy of a signal is the same in both domains. With the physics convention $\tilde f = \int f\, e^{-i\omega t}\, dt$ and $f = \tfrac{1}{2\pi} \int \tilde f\, e^{i\omega t}\, d\omega$ :

\boxed{\;\int_{-\infty}^{\infty} |f(t)|^2\, dt \;=\; \frac{1}{2\pi} \int_{-\infty}^{\infty} |\tilde f(\omega)|^2\, d\omega.\;}

This is Parseval’s identity (or, in the more general form below, the Plancherel theorem).

▶ Parseval's identity from the inner product

Define the inner product $\langle f, g \rangle = \int f^*(t)\, g(t)\, dt$ . The Fourier transform preserves inner products up to a factor:

\langle f, g \rangle \;=\; \frac{1}{2\pi}\, \langle \tilde f, \tilde g \rangle.

To derive this, expand $\langle f, g \rangle = \int f^*(t)\, g(t)\, dt$ using the inverse Fourier representation of $f$ :

f(t) \;=\; \frac{1}{2\pi} \int d\omega\, \tilde f(\omega)\, e^{i \omega t}, \quad\text{so}\quad f^*(t) \;=\; \frac{1}{2\pi} \int d\omega'\, \tilde f^*(\omega')\, e^{-i \omega' t}.

Then

\langle f, g \rangle \;=\; \frac{1}{2\pi} \int dt\, \int d\omega'\, \tilde f^*(\omega')\, e^{-i \omega' t}\, g(t).

Swap the order of integration:

\;=\; \frac{1}{2\pi} \int d\omega'\, \tilde f^*(\omega') \int dt\, g(t)\, e^{-i \omega' t} \;=\; \frac{1}{2\pi} \int d\omega'\, \tilde f^*(\omega')\, \tilde g(\omega') \;=\; \frac{1}{2\pi} \langle \tilde f, \tilde g \rangle.

Setting $g = f$ gives Parseval’s identity.

The deep statement: the Fourier transform is a unitary map between the function space $L^2$ and itself (up to the convention-dependent factor of $2\pi$ ). It preserves inner products, hence preserves norms, hence preserves energy. The information content of $f$ and $\tilde f$ is identical — only the representation differs.

Why Parseval matters

A few of the cash applications:

Energy is energy. The total acoustic energy of a transient pulse is the same whether you compute it from the time signal $\int |p(t)|^2 dt$ or from the spectrum $\tfrac{1}{2\pi} \int |\tilde p(\omega)|^2 d\omega$ . Frequency-domain measurements (a spectrum analyser) give the same total energy as time-domain measurements (an oscilloscope), differing only in how that energy is partitioned.
Mean-square error in either domain. The mean-square approximation error of a band-limited reconstruction is computable in the frequency domain by integrating the squared spectrum outside the band — sometimes much easier than the time-domain integral.
Spectral density of noise. For random signals, the power spectral density $|\tilde f(\omega)|^2 / T$ is the natural object to measure, and its integral gives the total power. This is the basis of every noise-floor calculation in acoustics and electronics.

What we use this for

Acoustic filters and the room as transfer function — Sound 8.3 develops every linear acoustic element (an open window, a duct, a room mode, an HRTF) as $\tilde h(\omega)$ .
Diffraction patterns — Fourier transforms of aperture functions, computed efficiently via FFT-convolutions for complicated apertures.
Cochlear-amplifier filterbanks — each hair cell is approximately a gammatone filter; the cochlear output is the input convolved with a bank of these filters (Hearing Ch 4).
Convolutional neural networks — the “convolution” in CNN is exactly this operation, and the convolution theorem is what makes large-kernel CNNs trainable efficiently via FFT-based convolution.

The next lesson, 7.4, brings everything down to discrete signals: how the continuous Fourier transform becomes the DFT when sampling, why aliasing happens, and how the FFT actually computes the thing.