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copied!<p>Finally I have figured this out; really I had to derive it from scratch. I knew there would be some simple way to derive it, my (usual) mistake was to attempt to follow other people's logic rather than use my own common sense.</p> <p>This puzzle takes <strong>two keys</strong> to unlock it.</p> <ul> <li><p><strong>The first key</strong> is to understand how over-sampling introduces a rotation on bin phase.</p></li> <li><p><strong>The second key</strong> comes from Graph 3.3 & 3.4 here: <a href="http://www.dspdimension.com/admin/pitch-shifting-using-the-ft/">http://www.dspdimension.com/admin/pitch-shifting-using-the-ft/</a></p></li> </ul> <p>...</p> <pre><code>for (int k = 0; k <= fftFrameSize/2; k++) { // compute magnitude and phase bins[k].mag = 2.*sqrt(fftBins[k].real*fftBins[k].real + fftBins[k].imag*fftBins[k].imag); bins[k].phase = atan2(fftBins[k].imag, fftBins[k].real); // Compute phase difference Δϕ fo bin[k] double deltaPhase; { double measuredPhaseDiff = bins[k].phase - gLastPhase[k]; gLastPhase[k] = bins[k].phase; // Subtract expected phase difference <-- FIRST KEY // Think of a single wave in a 1024 float frame, with osamp = 4 // if the first sample catches it at phase = 0, the next will // catch it at pi/2 ie 1/4 * 2pi double binPhaseExpectedDiscrepancy = M_TWOPI * (double)k / (double)osamp; deltaPhase = measuredPhaseDiff - binPhaseExpectedDiscrepancy; // Wrap delta phase into [-Pi, Pi) interval deltaPhase -= M_TWOPI * floor(deltaPhase / M_TWOPI + .5); } // say sampleRate = 40K samps/sec, fftFrameSize = 1024 samps in FFT giving bin[0] thru bin[512] // then bin[1] holds one whole wave in the frame, ie 44 waves in 1s ie 44Hz ie sampleRate / fftFrameSize double bin0Freq = (double)sampleRate / (double)fftFrameSize; bins[k].idealFreq = (double)k * bin0Freq; // Consider Δϕ for bin[k] between hops. // write as 2π / m. // so after m hops, Δϕ = 2π, ie 1 extra cycle has occurred <-- SECOND KEY double m = M_TWOPI / deltaPhase; // so, m hops should have bin[k].idealFreq * t_mHops cycles. plus this extra 1. // // bin[k].idealFreq * t_mHops + 1 cycles in t_mHops seconds // => bins[k].actualFreq = bin[k].idealFreq + 1 / t_mHops double tFrame = fftFrameSize / sampleRate; double tHop = tFrame / osamp; double t_mHops = m * tHop; bins[k].freq = bins[k].idealFreq + 1. / t_mHops; } </code></pre>

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