Convolution theorem

The convolution theorem is the most useful property of the Fourier transform (and the Laplace transform): convolution in one domain is multiplication in the other.

$x(t)\ast h(t)\stackrel{\mathcal{F}}{\leftrightarrow }X(f)\cdot H(f),$

$x(t)\cdot h(t)\stackrel{\mathcal{F}}{\leftrightarrow }X(f)\ast H(f).$

For the Laplace transform, only the first direction is standard:

$x(t)\ast h(t)\stackrel{\mathcal{L}}{\leftrightarrow }X(s)\cdot H(s).$

Why this matters

For an LTI system with input $x$ and impulse response $h$, the output is $y=x\ast h$. Taking the Fourier transform of both sides:

$Y(f)=X(f)H(f).$

The output spectrum is the input spectrum times the system’s frequency response. The convolution integral has become a pointwise multiplication.

To find the output:

1. Transform $x$ to $X$.
2. Multiply: $Y=XH$.
3. Inverse-transform $Y$ to $y$.

For most realistic signals and systems this beats evaluating the convolution integral directly, and it’s the reason we developed the Fourier and Laplace transforms in the first place.

Modulation: the dual

Multiplication in time is convolution in frequency. If we multiply two signals (say, modulate a message $x(t)$ with a carrier $\cos (2\pi f_{c}t)$), the spectrum of the product is the convolution of the two individual spectra.

Convolving the message spectrum $X(f)$ with $\frac{1}{2}[\delta (f-f_c)+\delta (f+f_c)]$ (the cosine spectrum) produces two shifted copies of $X(f)$, centered at $\pm f_c$. That’s amplitude modulation: the message spectrum gets shifted up to a high carrier frequency for transmission, then shifted back down at the receiver.

ω-form bookkeeping

In the ω-form, the multiplication direction is asymmetric:

$x(t)\cdot h(t)\stackrel{\mathcal{F}}{\leftrightarrow }\frac{1}{2\pi }X(j\omega )\ast H(j\omega ).$

A factor of $1/(2\pi )$ appears on the right. One reason the f-form is cleaner: no extra factors in either direction of the theorem.

Connection to Fourier series

For periodic signals, multiplication in time corresponds to discrete convolution of the Fourier-series coefficients:

$x(t)y(t)\stackrel{\text{FS}}{\leftrightarrow }{c}_x[k]\ast c_y[k]={\sum }_n{c}_x[n]c_y[k-n].$

And periodic convolution in time corresponds to multiplication of harmonic coefficients (with a factor of $T$).

So in every Fourier setting (series, transform, discrete) the theme holds: convolution in one domain corresponds to multiplication in the other.

Why convolution in time = multiplication in frequency

The handwave: convolution is “sliding integrate the product.” The Fourier transform decomposes a signal into complex exponentials, which are eigenfunctions of LTI systems. Each eigenfunction passes through the system multiplied by the constant $H(f)$. Summing eigenfunctions and applying the system means scaling each one separately, which is pointwise multiplication of the eigenfunction weights, i.e. pointwise multiplication of the spectra.

That’s the geometric reason. The algebraic proof is a change of variable in the double integral defining $\mathcal{F}\{x\ast h\}$.