The chord sung by the three Roche sisters in the aliasing demonstration in class was B flat major, consisting of the notes Bflat4, F4, and D4 ("4" refers to the fourth octave on the piano). The frequencies corresponding to these notes are 466.16 Hz, 349.23Hz, and 293.66Hz. Assume each singer's voice contains very strong second and third harmonics in addition to the fundamental pitch. If sampled at 1600 samples per second (way too slow!), what frequencies would appear in the range 0Hz - 800Hz? This is the range that would be retained by a lowpass filter when reconstructing the signal from the samples. (Note: the 2nd harmonic is twice the fundamental frequency; the 3rd harmonic is three times the fundamental. This is a little different from the way "partials" often are numbered.)

Answer: a) 466.16 Hz b) 349.23 Hz c) 698.46 Hz d) 293.66 Hz e) 587.32 Hz

Explanation:

If the compound signal is sampled at 1600 samples/second, the Nyquist Theorem says that in order to be able to be reconstructed in full, the signal with the highest frequency must be sampled more than twice during a cycle, so the highest frequency in this case is 800 Hz.

So, for the first singer, only the fundamental (466.16 Hz) falls within the filter range.

For the second one, even we know that her voice contains very strong 2nd and 3rd harmonics in addition to the fundamental pitch, only the fundamental and the 2nd harmonic fall within the low-pass range, i. e., 349.23 Hz and 698.46 Hz.

Finally, for the 3rd singer, also only the fundamental (293.66 Hz) and the 2nd harmonic (587.32 Hz) can be reconstructed, as the 3rd harmonic is beyond 800 Hz mark.