## Re: Nyquist double sample rate rule

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Reply #3 –

A very interesting question which I also wondered about some time ago.

I wonder what happens when you approach the limit frequency, I cannot see how 2 samples per period could faithfully reconstruct the wave. The sampling could fall away from the peaks.

From a mathematical (or information theoretical) perspective, it's absolutely irrelevant whether you do or don't hit the waveform peaks during the sampling process. The problem "only" lies in the upsampling and/or D/A conversion during playback.

Using Audition, I created the attached example (48-kHz WAV file). Load that into the audio software of your choice. The **first part** (0-1 sec.) contains an accurately generated 11990-Hz sine wave. You'll notice that, when zooming in to the individual samples, that the waveform samples follow a joint high-frequency (11990 Hz) and low-frequency (10 Hz) pattern. Downsampling that file by two to 24 kHz, by taking only every second sample, will preserve the sine wave. Now, if you upsample that result back to 48 kHz *without* interpolation filtering (by just adding a zero sample after every sample value), you get the waveform in the **second part** (1-2 sec.) of the file. You'll notice the obvious 10-Hz amplitude modulation, and if you do a fine spectral analysis of that part, you'll see that the aliasing introduced by the simple upsampling is a sine tone at 12010 Hz. The problem now is to design, and implement, an upsampling anti-aliasing filter steep enough around the Nyquist frequency (12000 Hz) that it will *a)* fully preserve the 11990-Hz tone and *b)* fully remove the 12010-Hz tone. For that you'd need a very long 12-kHz lowpass filter, and I haven't seen such a filter to date (it would also cause quite some temporal ringing on transients, I would guess).

If you try to downsample a 12000-Hz sine wave, sampled at 48 kHz, to 24 kHz, then yes, the sine phase matters since you might "accidentally" sample the waveform at the zero crossings. That's, I guess why the theorem says that "you must sample at a frequency **greater than** 2f."

Chris