sinusoidal.space

Nonlinearities create frequencies—and aliases

Follow newly created harmonics across the sampling boundary.

Audio · Sampling

RoPE: relative position as phase difference

Rotate content vectors and inspect relative attention scores.

Representations · Machine learning

Why fewer operations can still run slower

Follow FFT stages into memory traffic and hardware utilisation.

GPU computing · Performance

Memory as a dynamical system

Connect decaying, rotating states to filters and sequence models.

Signals · Machine learning

The Fourier transform, made visible

Compose a signal, then inspect the magnitude and phase that describe it.

Transforms · Signals

Aliasing is everywhere

See why different continuous signals can become identical after sampling.

Sampling · Machine learning

Convolution: FIR filters → CNNs → fast computation

Follow one operation across filtering, learned kernels and frequency space.

Convolution · Models

Sinusoidal positional encodings, from first principles

Build a vector representation of position from paired periodic coordinates.

Representations · Machine learning

Building a GPU STFT pipeline

Map a time–frequency transform from equations to buffers and execution.

GPU computing · Performance

Nonlinear harmonic folding

Compare direct cubic sampling with an ideal band-limited reference.

Rotary position explorer

Move query and key positions and inspect their dot product.

FFT traffic calculator

Compare stage-by-stage transfers with an ideal I/O floor.

Rotating state memory

Inspect how decay and rotation shape an impulse response.