Klarinet Archive - Posting 000731.txt from 1997/11

From: Jonathan Cohler <cohler@-----.net>
Subj: Re: Nyquist
Date: Thu, 20 Nov 1997 11:18:25 -0500

Jerry Korten writes:

>Well, I'm no expert but I can immediately see a case where this is not
>> going to be true. Take as an example a 20 KHz sine wave sampled at 40
>> KHz. According to the above definition the samples will contain enough
>> information to perfectly reconstruct the 20 KHz sine wave.
>>
>> Surely you are going to get different results depending on the relative
>> phases of the sampling and the sine wave itself. In one extreme case
>> you could get a set of samples that are all 0. This occurs because the
>> wave crosses the 0 point every 1/2 it's period. From that you aren't
>> going to be able to reconstruct very much at all! At the other extreme
>> you can get a series of +n, -n, +n, -n ... where n is the amplitude of
>> the signal.
>>
>> Jerry, Jonathon - am I missing something obvious?
>>
>> Ian Dilley
>>
>
>This is the issue I was hung up on, the Nyquist theorem as stated on the
>list is missing the part about reconstruction. The digitized waveform does
>look nothing like the sampled waveform. However upon reconstruction (using
>a filter function to interpolate the values) you can get back the original
>waveform... If an A/D is not used in the process. If an A/D is used the
>quantization effects will not allow us to perfectly reconstruct the
>waveform.
>
>Jerry Korten
>NYC

The samples that the Nyquist theorem refers to are idealized digital
samples (real numbers, not quantized numbers). And an idealized A/D does
precisely this. And therefore the waveform is reconstructed perfectly.

In real-life, A/D converters do the job better or worse, depending on their
quality, which involves many factors, one of which is the bit-resolution of
the converter, which in turn determines the level of the quantization
error. The good high-end, 20-bit (or more) A/D converters on the market
today do a job that is better than the human ear can detect, and far better
than the level of the errors introduced in typical analog
recording/playback equipment.

---------------------
Jonathan Cohler
cohler@-----.net