Optimize forward DFT with Cooley-Tukey FFT

[thomas-vilte]

Description

forwardComplexDFT in the encoder path was doing an O(N²) naive DFT, at N=480 thats ~230k multiply-accumulates per call.

Added a Cooley-Tukey mixed-radix FFT in fft.go. CELT frame sizes factor as 2^k × 15, so it handles the radix-2 part with bit-reversal +
butterfly and uses a direct O(K²) DFT for the odd factor 15 — small enough that it doesn't matter.

Benchmarks on my machine (N=480): ~6080µs naive vs ~233µs with FFT, roughly 26x.

The decoder side (inverseComplexDFT) already had the butterfly FFT from a previous pass, this just closes the gap on the encoder side.

Reference issue

Fixes #...

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 82.88%. Comparing base (4a3f1a6) to head (fd5be78).

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #125      +/-   ##
==========================================
+ Coverage   82.66%   82.88%   +0.21%     
==========================================
  Files          28       29       +1     
  Lines        6046     6110      +64     
==========================================
+ Hits         4998     5064      +66     
+ Misses        812      811       -1     
+ Partials      236      235       -1     

Flag	Coverage Δ
go	82.88% <100.00%> (+0.21%)	⬆️

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[JoTurk]

Thank you it would be cool to add a few benchmarks.

[Copilot]

Pull request overview

This PR speeds up the CELT encoder’s forward complex DFT (used in forwardMDCT) by replacing the previous O(N²) implementation with a Cooley–Tukey mixed-radix FFT optimized for CELT frame sizes (2^k × 15). This aligns the encoder path with the existing FFT-based approach already used on the decoder side.

Changes:

• Replaced forwardComplexDFT’s naive DFT with an FFT call plus 1/N scaling.
• Added a mixed-radix FFT implementation (radix-2 stages + direct DFT for the odd factor).
• Added unit tests and benchmarks validating FFT correctness vs a naive DFT and round-trip properties.

Reviewed changes

Copilot reviewed 3 out of 3 changed files in this pull request and generated 3 comments.

File	Description
internal/celt/mdct.go	Switches encoder-side forward complex DFT to FFT + scaling.
internal/celt/fft.go	Introduces mixed-radix FFT implementation used by forwardComplexDFT.
internal/celt/fft_test.go	Adds correctness tests (vs naive DFT / round-trip) and benchmarks for the new FFT.

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