Optimal value of sampling frequency for guitar notes detection

Question

I am running FFT algorithm to detect the music note played on a guitar.

The frequencies that I am interested are in the range 65.41Hz (C2) to 1864.7Hz (A#6).

If I set the sampling frequency of the input to 16KHz, the output of FFT would yield N points from 0Hz to 16KHz linearly. All the input I am interested would be in the first N/8 points approximately. The other N*7/8 points are of no use to me. They actually are decreasing my resolution.

From Nyquist's theory (https://en.wikipedia.org/wiki/Nyquist_frequency), the sampling frequency that is needed is just twice the maximum frequency one desires. In my case, this would be about 4KHz.

Is 4KHz really the ideal sampling frequency for a guitar tuning app?

Intuitively, one would feel a better sampling frequency would give you more accurate results. However, in this case, it seems having a lesser sampling frequency is better for improving the resolution. Regards.

Answer 1

You are confusing the pitch of a guitar note with spectral frequency. A guitar generates lots of overtones and harmonics at a much higher frequency than the pitch of a played note. Those higher harmonics and overtones, more than the possibly weak fundamental frequency in some cases, is what the human ear hears and interprets as the lower perceived pitch.

Any of the overtones and harmonics around or above 2 kHz that are not completely low pass filtered out before sampling at 4 kHz will cause aliasing and thus corruption of your sampled data and its spectrum.

If you want to create an accurate tuner, use a pitch estimation algorithm, not an FFT peak frequency bin estimator. And depending on which pitch estimation method you choose, a higher density of samples per unit time might allow finer accuracy or greater reliability under background noise or more prompt responsiveness.

Answer 2

Is 4KHz really the ideal sampling frequency for a guitar tuning app?

You've been mis-reading Nyquist's theorem if you ask it like that.

f_sample > 2*f_max http://latex.codecogs.com/png.download?%24f_%5Ctext%7Bsample%7D%3E2%20f_%5Ctext%7Bsignal%2Cmax%7D%24

States that every sampling frequency above twice your maximum signal frequency will allow you to perfectly reconstruct your original signal. So there's no "ideal" frequency. Just a set of frequencies that are sufficient. What is ideal hence depends on a lot of other things: mainly, what your digitizer really supports (hint: most sound cards can do 44.1kHz, but not 4kHz), what kind of margin you want to have for filters etc to work on, and what kind of processing power you can spend (hint: modern smart phones, PCs and even pocket calculators don't really have a hard time processing a couple hundred kHz in real time).

Also note that @hotpaw2 is right, the harmonics are important, and are multiples of the base tone frequency.

However, in this case, it seems having a lesser sampling frequency is better for improving the resolution.

no. No matter where that comes from, it's wrong. Information theory's first and foremost result is that based upon more information, you can't make worse estimates. An oversampled signal is simply more information on the same signal.

Answer 3

Yes, if all you are interested in is frequencies up to 2 kHz then you only need a sampling frequency of 4 kHz. This should include an anti-aliasing filter in front of the ADC or any downconverter to prevent any higher frequency components from aliasing into a lower frequency.

If all you are interested in is specific frequencies (one or two) then you may want to look at the Goertzel algorithm which is more efficient than an FFT for a single frequency. Also, the chirp-Z transform can be used to effectively get a zoomed FFT (resulting in a higher resolution over a smaller bandwidth without the computational complexity of an FFT with the same resolution). You may want to check out this CZT tutorial