DCT is not orthonormal, and the class name is misspelled

[gabrielfrasantos]

Severity: high
Domain: analysis
Status: VERIFIED — read + hand-traced against a210d34 on 2026-08-10
Suggested labels: bug, numerical-correctness, analysis, api

Summary

The forward DCT-II scales non-DC coefficients by 2/√N instead of the orthonormal √(2/N) — a
factor of √2 too large. Parseval's identity does not hold, and the transform does not match the
scaling documented in doc/analysis/DiscreteCosineTransform.md.

Separately, the public class name is misspelled DiscreteConsineTransform ("Consine").

Location

numerical/analysis/DiscreteCosineTransform.hpp

Evidence

output[0] = QNumberType(math::ToFloat(fftResult[0].Real()) / math::Sqrt(static_cast<float>(Length)));

for (std::size_t k = 1; k < Length; ++k)
{
    float angle = -static_cast<float>(k) * std::numbers::pi_v<float> / (2.0f * Length);
    float scale = 2.0f / math::Sqrt(static_cast<float>(Length));   // should be sqrt(2/N)
    ...
}

The DC term is correct (√(1/N)); the non-DC scale is not.

implemented:  2/√N
orthonormal:  √(2/N) = √2/√N
ratio:        2/√2 = √2  ≈ 1.414

Trace

N = 8, unit impulse at n = 0. Coefficient energy should be 1:

Σ c_k²  =  (1/N)·1  +  Σ_{k=1}^{N-1} (4/N)·cos²(πk/2N)
        =  1/N + (4/N)·(N-1)/2
        =  2 − 1/N  =  1.875        (should be 1)

Why tests do not catch it

Inverse() applies the reciprocal scale, so any test of the form
Inverse(Forward(x)) == x passes. The error is only visible to a test that checks a single
coefficient against an analytic reference, or that asserts Parseval.

Suggested fix

float scale = math::Sqrt(2.0f / static_cast<float>(Length));

Adjust Inverse() symmetrically. Add a Parseval test and an analytic single-coefficient test.

Rename DiscreteConsineTransform → DiscreteCosineTransform (breaking change; the file is already
named correctly).

Notes

The documentation describes an unnormalised DCT, which matches neither the current code nor the
orthonormal convention — decide on one contract and align all three.