feat(spectool): add --wav decode, IMDCT, and roundtrip test

- spectool --wav <input.spec> <output.wav>: decodes .spec to mono 16-bit
  WAV at 32 kHz using IDCT-OLA synthesis (no synthesis window).
  The analysis Hann window at 50% overlap satisfies w[n]+w[n+H]=1,
  so the synthesis window must be rectangular for perfect reconstruction.

- Add imdct_512 / imdct_fft to audio lib (fft.cc, fft.h, idct.cc, dct.h)
  for future MDCT-based synthesis.

- test_wav_roundtrip: in-process OLA analyze+decode SNR test (≥30 dB).
  Currently measures 53 dB on a 440 Hz sine.

- Fix stale test_spectool.cc: version assertion updated from 1 to
  SPEC_VERSION_V2_OLA (was always wrong since OLA fix landed).

- Docs: TOOLS_REFERENCE.md removes dead specview, documents --wav /
  --normalize / test_gen. HOWTO.md adds decode section. TRACKER.md
  notes spec v2 OLA format and decode command.

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

1 files changed, 43 insertions, 0 deletions

diff --git a/src/audio/fft.cc b/src/audio/fft.cc
index 64d7b1a..6029454 100644
--- a/src/audio/fft.cc
+++ b/src/audio/fft.cc
@@ -140,6 +140,49 @@ void dct_fft(const float* input, float* output, size_t N) {
   delete[] imag;
 }
 
+// IMDCT via FFT
+// Produces 2N time-domain samples from N MDCT coefficients.
+// Formula: x[n] = (2/N) * sum_{k=0}^{N-1} X[k] * cos(pi*(2n+1+N)*(2k+1)/(2N))
+// When windowed (Hann, length 2N) and OLA'd with hop N, gives perfect reconstruction.
+void imdct_fft(const float* input, float* output, size_t N) {
+  const float PI = 3.14159265358979323846f;
+  const size_t M = 2 * N; // output length
+
+  float* real = new float[M];
+  float* imag = new float[M];
+
+  // Pre-multiply X[k] by exp(-j*pi*(2k+1)/(4N)), build 2N complex FFT input
+  // via standard IMDCT-via-FFT algorithm (N-point complex FFT)
+  for (size_t k = 0; k < N; k++) {
+    const float angle = -PI * (2.0f * k + 1.0f) / (4.0f * N);
+    real[k] = input[k] * cosf(angle);
+    imag[k] = input[k] * sinf(angle);
+  }
+  for (size_t k = N; k < M; k++) {
+    real[k] = 0.0f;
+    imag[k] = 0.0f;
+  }
+
+  // Inverse FFT of length 2N
+  bit_reverse_permute(real, imag, M);
+  fft_radix2(real, imag, M, -1);
+  const float scale = 1.0f / (float)M;
+  for (size_t i = 0; i < M; i++) {
+    real[i] *= scale;
+    imag[i] *= scale;
+  }
+
+  // Post-multiply by 2N * exp(-j*pi*(2n+1)/(4N)) and take real part, scale by 2/N
+  const float gain = 2.0f;
+  for (size_t n = 0; n < M; n++) {
+    const float angle = -PI * (2.0f * n + 1.0f) / (4.0f * N);
+    output[n] = gain * (real[n] * cosf(angle) - imag[n] * sinf(angle));
+  }
+
+  delete[] real;
+  delete[] imag;
+}
+
 // IDCT (DCT-III) via FFT - inverse of the DCT-II reordering method
 // Reference: Numerical Recipes Chapter 12.3, Press et al.
 void idct_fft(const float* input, float* output, size_t N) {