Archived from groups: rec.audio.tech (
More info?)
On Fri, 03 Jun 2005 10:47:13 GMT, "Tim Martin"
<tim2718281@ntlworld.com> wrote:
>
>"Stewart Pinkerton" <patent3@dircon.co.uk> wrote in message
>news:mhvv9118jips49mc63o6c90c1lr0qt9ng0@4ax.com...
>
>Tim
>
>> >Nyquist's Theoerem is about representation of periodic signals; most
>sounds
>> >are not periodic signals.
>
>Stewart
>
>> Actually, it's not, except in so far as it does have a bandwidth limit
>> of less than half the sampling frequency.
>
>Nyquist's Theorem tells us we can exactly represent the information in a
>waveform by sampling it at a rate at least twice as high as the highest
>frequency in the waveform
>
>I think there's an implication here that by "highest frequency", we are
>talking about the highest frequency in a Fourier transform of the original
>signal. And the Fourier transform applies only to periodic signals.
It does however apply to every frequency required for the synthesis of
an impulse, however short that impulse may be. To assume otherwise is
to misunderstand the nature of the Fourier transform.
>Take a waveform consisting of, say a 1000Hz sine wave that is repeatedly
>switched on and off at random times. This is not a periodic waveform, and
>cannot be represented exactly by a Fourier transform. It has infinite
>bandwidth. (Conceptually at least. As you've previously remarked, there
>are physical constraints imposed by the transmission medium.)
And as such, it *can* be represented by a Fourier transform containing
frequencies up to any arbitrarily high limit which you feel the medium
justifies. Infinity is never involved.
>So what happens when we try to represent that wavefom by a digital
>representation (and I'm thinking here of a general-purpose digital
>representation.) To keep things simple, let's suppose the amplitude of the
>sine wave is small, so gives rise to only two different digital values, 0
>and 1.
>If we are sampling 48000 times a second, and if the sine wave is long enough
>when on, our digital signal will, after some start-up sequence, consist of a
>repeating pattern of 24 ones followed by 24 zeroes. Once we're in this part
>of the signal, we can reproduce the original sine wave exactly (within the
>resolution limits, which are not the issue here.)
No, it won't. You really don't understand the basics of A/D
conversion, do you? The waveform will exactly replicate the original
sine wave, so there will be less ones than zeroes, in the appropriate
ratio (can't be arsed to work it out offhand, but it's around 10:14
ratio, rather than 24:24)
>But when does our reproduction start?
>
>Our digital representation will have an initial series of values
>representing the silence. Again leaving aside start-up considerations, a
>series of 48000 identical values will represent one second's initial
>silence, and 48001 will represent 1.000021 seconds of silence.
>
>All silent intervals between 1 and 1.000021 seconds will be represented by
>one of these values. And as there are more than 2 different analog signals
>starting with between 1 and 1.000021 seconds of silence, we are losing
>information in the digital representation ... that is, we are compressing
>it.
You don't understand the basics, do you? What about dither? And are
you *assuming* that the signal starts from zero? You simply have no
basis for your above statement.
>All this is straightforward maths.
Indeed it is, and you're hopelessly wrong again!
>I think where you went wrong is failing to distinguish between the audio
>frequencies contained in the signal, and the timing information contained in
>the signal. In hi-fi, this doesn't usually matter - correctly reproducing
>the audio frequencies gives us more than enough timing precision - but in
>information theory it does.
If you actually understood information theory, we wouldn't be having
this debate..............
--
Stewart Pinkerton | Music is Art - Audio is Engineering