c# - How to get the fundamental frequency using Harmonic Product Spectrum?

Question

I'm trying to get the pitch from the microphone input. First I have decomposed the signal from time domain to frequency domain through FFT. I have applied Hamming window to the signal before performing FFT. Then I get the complex results of FFT. Then I passed the results to Harmonic product spectrum, where the results get downsampled and then multiplied the downsampled peaks and gave a value as a complex number. Then what should I do to get the fundamental frequency?

    public float[] HarmonicProductSpectrum(Complex[] data)
    {
        Complex[] hps2 = Downsample(data, 2);
        Complex[] hps3 = Downsample(data, 3);
        Complex[] hps4 = Downsample(data, 4);
        Complex[] hps5 = Downsample(data, 5);
        float[] array = new float[hps5.Length];

        for (int i = 0; i < array.Length; i++)
        {
            checked
            {
                array[i] = data[i].X * hps2[i].X * hps3[i].X * hps4[i].X * hps5[i].X;
            }
        }
        return array;
    }

    public Complex[] Downsample(Complex[] data, int n)
    {
        Complex[] array = new Complex[Convert.ToInt32(Math.Ceiling(data.Length * 1.0 / n))];
        for (int i = 0; i < array.Length; i++)
        {
            array[i].X = data[i * n].X;
        }
        return array;
    } 

I have tried to get the magnitude using,

    magnitude[i] = (float)Math.Sqrt(array[i] * array[i] + (data[i].Y * data[i].Y));  

inside the for loop in HarmonicProductSpectrum method. Then tried to get the maximum bin using,

        float max_mag = float.MinValue;
        float max_index = -1;

        for (int i = 0; i < array.Length / 2; i++)
            if (magnitude[i] > max_mag)
            {
                max_mag = magnitude[i];
                max_index = i;
            }

and then I tried to get the frequency using,

    var frequency = max_index * 44100 / 1024;

But I was getting garbage values like 1248.926, 1205,859, 2454.785 for the A4 note (440 Hz) and those values don't look like harmonics of A4.

A help would be greatly appreciated.

Answer 1

To get a pitch estimate, you have to divide your sumed bin frequency estimate by the downsampling ratio used for that sum.

Added: You should also sum the magnitudes (abs()), not take the magnitude of the complex sum.

But the harmonic product spectrum algorithm (HPS), especially when using only integer ratios of downsampling, doesn't usually provide better pitch estimation resolution. Instead, it provides a more robust rough pitch estimate (less likely to be fooled by a harmonic) than using a single bare FFT magnitude peak for sequential overtone rich timbres that have weak or missing fundamental spectral content.

If you know how to downsample a spectrum by fractional ratios (using interpolation, etc.), you can try finer grained downsampling to get a better pitch estimate out of HPS. Or you can use an HPS result to inform you of a narrower frequency range in which to search using another pitch or frequency estimation method.