# `dct` — type-II discrete cosine transform

## Purpose

The DCT-II module is the cepstral projection step of the MFCC pipeline: it takes the log mel-spectrum of one frame and produces a compact vector of cepstral coefficients. The same routine is reusable anywhere a closed-form orthonormal-with-scale DCT is needed.

This implementation is unscaled (no `sqrt(2/N)` orthonormality factor) because downstream MFCC analysis is unit-agnostic: every coefficient is later compared to peers, not to an external reference. Callers that need an orthonormal form can post-multiply by the standard scaling factor.

## Features

- Closed-form `X_k = Σ x_n · cos((π/N)(n + ½)k)` for `0 ≤ k < coefficients`.
- Cached per `(inputSize, coefficients)` so repeated frames reuse the kernel matrix.
- Pure portable TypeScript over `Float32Array` inputs and outputs.

## Usage: Basic

```ts
import { dctII } from "@suaveplan/audio-analysis-spectrum/algorithms/dct";

const logMel = new Float32Array(40); // log mel-spectrum
const mfcc = dctII(logMel, 13);
```

## Usage: Advanced

```ts
import {
  clearDctCache,
  dctII,
} from "@suaveplan/audio-analysis-spectrum/algorithms/dct";

const coeffs = dctII(logMel, 20);
// later, before a benchmark sweep
clearDctCache();
```

## API Reference

- `dctII(input, coefficients): Float32Array` — first `coefficients` cepstral values.
- `clearDctCache(): void` — evict every cached kernel.

## Implementation Notes

The kernel is stored as a flat `Float64Array` of `coefficients * inputSize` doubles so the per-frame inner loop is a single linear pass with no nested allocation. Kernel construction is `O(coefficients · inputSize)` and is cached on the `(inputSize, coefficients)` pair; runtime cost is then a single matrix-vector multiply per frame. Boundary checks on `coefficients` (positive, `≤ melBands`) are enforced at the analyzer layer before this routine is called, so this module assumes a validated input.