[HAM] Vedr. Re: Vedr. Re: The New B-3

[Brad Baker]

On Feb 1, 2008 5:56 PM, Carl Mal <carl_mal at hotmail.com> wrote:
> How can digital not be an approximation? Digital is discreet steps
> -vs- analog's continuously varying voltage.

First about sampling, then about quantization.  From the
Nyquist-Shannon sampling theorem:
http://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem

"Exact reconstruction of a continuous-time baseband signal from its
samples is possible if the signal is bandlimited and the sampling
frequency is greater than twice the signal bandwidth."

This is a theorem that describes sampling, not quantizing.  This could
be achieved for example if one used a D/A converter with an infinite
number of bits.  Each sample would equal the actual voltage of the
input signal at the sampling instant - not a quantized version of that
voltage.  This is not a realistic system.

> By approximation, I mean the anti-aliasing filter that interpolates
> between the steps is "approximating".

The anti-aliasing filter does not interpolate between steps.  Its
purpose is to band limit the input signal so that aliasing does not
occur.  Any signal that is lower in frequency than its cutoff will be
unaffected in amplitude, frequency and, if it is a proper filter,
phase.  In real systems which quantize the input signal, any signal
within the Nyquist bandwidth (less than half the sampling frequency)
will be sampled at the sampling instant and its value will be
quantized or rounded to the nearest quantization level, dependent upon
the resolution (number of bits) of the A/D converter.  The result of
this quantization is an error.  The error is the difference between
the actual level of the input signal and the level to which it is
quantized.  The result is that for each sample, the digitized signal
is equal to the actual input value at the sample point plus the error
value.  The error shows up as a pattern-dependent noise component.
This noise can be reduced to an arbitrarily low level by various
techniques, for example increasing the number of bits in the converter
(dynamic range is roughly 6 dB per bit).  Also there are other
techniques, dither, noise shaping, that can be used to make this noise
inaudible in practice.

Sorry for the pedantry.  I'll go back to sleep now.

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