ℱ

Transforms

Fourier, Laplace and Z: changing the point of view from time to frequency, where convolutions become products and differential equations become algebra. The mathematics of audio, images, compression and telecommunications.

6 topics

Topics

Fourier series

Any reasonable periodic function is a sum of sines and cosines of multiples of a base frequency: $f(t) = \sum_n c_n e^{2\pi i n t/T}$ . The coefficients $c_n$ — the spectrum — say how much of each harmonic it contains.

Advanced

Fourier transform

The continuous version for non-periodic signals: $\hat f(\xi) = \int f(t)\,e^{-2\pi i\xi t}\,\dd t$ gives the amount of each frequency $\xi$ . It turns convolution into multiplication and differentiation into multiplication by $2\pi i\xi$ — which is why filtering, compression and solving linear PDEs are easier in frequency space.

Advanced

Convolution

$(f * g)(t) = \int f(\tau)\,g(t - \tau)\,\dd\tau$ : a sliding weighted average of one function by another. Blur, echo, smoothing, the output of any linear time-invariant system — and, discretized, the operation that gives convolutional neural networks their name.

Advanced

Laplace transform

$F(s) = \int_0^\infty f(t)\,e^{-st}\,\dd t$ : like Fourier but with complex $s$ , it handles growth, decay and initial conditions. Linear ODEs become algebraic equations in $s$ , and a system becomes its transfer function $G(s)$ .

Advanced

Z-transform

The discrete-time Laplace transform: $X(z) = \sum_n x[n]z^{-n}$ , a power series in $z^{-1}$ . Difference equations (digital filters, discrete controllers) become rational functions of $z$ ; stability means poles inside the unit circle. It is the bridge from calculus to digital signal processing.

Specialization

Wavelets

Localized oscillations at several scales: unlike sines, they say when a frequency occurs. Multiresolution analysis underlies JPEG 2000, denoising and some feature extractors.

Specialization

Where this area leads in computing

Topics

Fourier series

Fourier transform

Convolution

Laplace transform

Z-transform

Wavelets

Where this area leads in computing

ℒ AI and machine learning ★★★★★

⚙ Robotics and control ★★★★★

⚛ Physics and simulation ★★★★★

∿ Signals, media and vision ★★★★★

ψ Quantum computing and physics ★★★★★