Extracting precise frequencies from FFT Bins using phase change between frames

Question

I have been looking through this fantastic article: http://blogs.zynaptiq.com/bernsee/pitch-shifting-using-the-ft/

While being fantastic, it is extremely hard and heavy going. This material is really stretching me.

I have extracted the maths from Stefan's code module that calculates the exact frequency for a given bin. But I don't understand the last calculation. Can someone explain to me the mathematical construction at the end?

Before digging into the code, let me set the scene:

• Let's say we set fftFrameSize = 1024, so we are dealing with 512+1 bins

• As an example, Bin[1]'s ideal frequency fits a single wave in the frame. At a sample rate of 40KHz, tOneFrame = 1024/40K seconds = 1/40s, so Bin[1] would ideally be collecting a 40Hz signal.

• Setting osamp (overSample) = 4, we progress along our input signal in steps of 256. So the first analysis examines bytes zero through 1023, then 256 through 1279, etc. Note each float gets processed 4 times.

...

void calcBins( 
              long fftFrameSize, 
              long osamp, 
              float sampleRate, 
              float * floats, 
              BIN * bins
              )
{
    /* initialize our static arrays */
    static float gFFTworksp[2*MAX_FRAME_LENGTH];
    static float gLastPhase[MAX_FRAME_LENGTH/2+1];

    static long gInit = 0;
    if (! gInit) 
    {
        memset(gFFTworksp, 0, 2*MAX_FRAME_LENGTH*sizeof(float));
        memset(gLastPhase, 0, (MAX_FRAME_LENGTH/2+1)*sizeof(float));
        gInit = 1;
    }

    /* do windowing and re,im interleave */
    for (long k = 0; k < fftFrameSize; k++) 
    {
        double window = -.5*cos(2.*M_PI*(double)k/(double)fftFrameSize)+.5;
        gFFTworksp[2*k] = floats[k] * window;
        printf("sinValue: %f", gFFTworksp[2*k]);
        gFFTworksp[2*k+1] = 0.;
    }

    /* do transform */
    smbFft(gFFTworksp, fftFrameSize, -1);

    printf("\n");

    /* this is the analysis step */
    for (long k = 0; k <= fftFrameSize/2; k++) 
    {
        /* de-interlace FFT buffer */
        double real = gFFTworksp[2*k];
        double imag = gFFTworksp[2*k+1];

        /* compute magnitude and phase */
        double magn = 2.*sqrt(real*real + imag*imag);
        double phase = atan2(imag,real);

        /* compute phase difference */
        double phaseDiff = phase - gLastPhase[k];
        gLastPhase[k] = phase;

        /* subtract expected phase difference */
        double binPhaseOffset = M_TWOPI * (double)k / (double)osamp;
        double deltaPhase = phaseDiff - binPhaseOffset;

        /* map delta phase into [-Pi, Pi) interval */
        // better, but obfuscatory...
        //    deltaPhase -= M_TWOPI * floor(deltaPhase / M_TWOPI + .5);

        while (deltaPhase >= M_PI)
            deltaPhase -= M_TWOPI;
        while (deltaPhase < -M_PI)
            deltaPhase += M_TWOPI;

(EDIT:) Now the bit I don't get:

        // Get deviation from bin frequency from the +/- Pi interval 
        // Compute the k-th partials' true frequency    

        // Start with bin's ideal frequency
        double bin0Freq = (double)sampleRate / (double)fftFrameSize;
        bins[k].idealFreq = (double)k * bin0Freq;

        // Add deltaFreq
        double sampleTime = 1. / (double)sampleRate;
        double samplesInStep = (double)fftFrameSize / (double)osamp;
        double stepTime = sampleTime * samplesInStep;
        double deltaTime = stepTime;        

        // Definition of frequency is rate of change of phase, i.e. f = dϕ/dt
        // double deltaPhaseUnit = deltaPhase / M_TWOPI; // range [-.5, .5)
        double freqAdjust = (1. / M_TWOPI) * deltaPhase / deltaTime; 

        // Actual freq <-- WHY ???
        bins[k].freq = bins[k].idealFreq + freqAdjust;
    }
}

I just can't see it clearly, even though it seems to be staring in the face. Could someone please explain this process from scratch, step by step?

Answer 1

The basic principle is very simple. If a given component exactly matches a bin frequency then its phase will not change from one FT to the next. However if the frequency does not correspond exactly with the bin frequency then there will be a phase change between successive FTs. The frequency delta is just:

delta_freq = delta_phase / delta_time

and the refined estimate of the frequency of the component will then be:

freq_est = bin_freq + delta_freq