From: mike@CEB.McMaster.CA (Mike Anderson)
Subject: Re: HDTV Question
Organization: McMaster University, Hamilton, Ontario, Canada
Date: Tue, 12 Jan 1993 17:57:57 GMT

>Bzzt, try again.  Being digital does indeed limit the number of available
>pitches.  There is quite a variety of them, and certainly more than the ear
>can pick out, but it's not continuous, since time is being quantized when
>the sampling is taking place.

According to Nyquist's sampling theory: If a signal that is band limited to
frequencies of less than f is sampled discretely at a frequency of 2f then 
the original (analog) bandwidth limited signal can be EXACTLY reproduced from
the discrete samples.

This is the basic theory that CDs, digital telephone systems, etc. are
designed around, and will apply to any new digital HDTV system.

For example if a voltage signal represents a HI-FI audio passage that
contains frequencies less than 20 kHz, then if we "sample" the signal by 
taking discrete voltage measurements at intervals of 40kHz.  The theory
says that the discrete bundle of samples we have measured contain
ALL THERE IS TO KNOW about the original continuously varying "infinite
resolution" analog signal.

Since the original analog signal can be EXACTLY reproduced from its samples,
then there is no theoretical obstacle to infinitely fine (continuous) tonal
reproduction.

In practice, as always, there are practical engineering considerations that 
limit the performance of actual devices.  These are due to the crudeness of
our devices rather than due to theoretical limitations.  The main limitations
are:

1. Filters.  Perfect filters are required for perfect reconstruction of
the signal from the samples.  Filters are used both before the original signal 
is digitized and during playback to reconstruct the original signal.  The pre-
filtering is required to ensure that the signal to be recorded is indeed band
limited as required by sampling theory.  Any "leak through" of higher
frequencies by these pre-filters will result in what is called "aliasing".
These high frequencies will show up as frequency shifted images in the
baseband area of the signal, thus causing interference.  The post filtering
is required to eliminate spurious high frequency images of the baseband signal
that are present in the sampled version of the signal.  

Assuming linear filters, which is reasonable, the filtering issue in no way 
detracts from out ability to reproduce infinite range of tone with the
digital system.  We have to live with some level of additional tones or
"noise" introduced into the signal due to aliasing, and some leak through of
high frequency modulation images due to non ideal filtering.

2. Quantization errors.  To store or transmit the samples, they are digitized.
A limited precision is used to represent the sample value.  Such as 16 bits
for CD audio.  In this case there is a +- 1/2 Least Significant Bit uncertainty.
So, for example, to digitize a signal that is 1V peak to peak with 16 bits, 
there will always be +-7.63 micro Volt uncertainty in the sample values.  
Most analysis treat this uncertainty as a small noise signal due to
quantization.  So the CD system is said to reproduce the analog signal that
was recorded EXACTLY but in addition it introduces a small noise signal.

In fact, I would argue, that the quantization error is not "white" noise, as
it is correlated to the input signal.  In any case it can be treated an 
unwanted signal that is unfortunately introduced by the digitization process.

Again, this in no way interferes with the ability to produce any desired
range of continuous pure tone, or any arbitrary signal that is within the
bandwidth of the system.  One just has to live with the small noise signal 
that is introduced.

AND this noise signal is EXTREMELY small!  In CD audio the oft quoted figure
can be derived from the previous example:

1V signal, quantization "noise" is +- 7.63E-6V or 1.53e-5 V peak to peak.
SNR = noise/signal = 1.53e-5
the power dB ratio is 20 log(1.53e-5) = -96.33 dB

Newsgroups: sci.electronics