Noise reduction through multi-tracking

Hi. I'm vaguely aware that noise in a recording can be reduced by
recording a sound (say on analogue) tape multiple times, and then
mixing the multiple recordings together on playback. The noise should,
to a degree, cancel, while the signal reinforces.

I'm just wondering if there is a more technical description of this.
Assuming that the noise is random, then adding together multiple
sources of noise should cancel so that the noise level is not

I simulated this with a very short program that generated multiple
"tracks" of uniformly distributed (around zero) noise, then added them
back together again. By the central limit theorem, the summed noise
will approach normally distributed noise the more tracks are added
together. I wasn't sure what measure to use to measure the scale of the
resultant noise, so used the average of the absolute values of the
noise. In any case, as the number of tracks rose, the noise (as
measured by me) rose at a slower than linear rate. Which measurement
would be better to use?

Just like to warn people that I'm not intending to do any dsp-type
things with this info. I'm actually using the multiple recordings/noise
reduction as an analogy when talking about Information Fusion, and I
want to check that the basis for my analogy is more-or-less correct.

Thanks in anticipation,

Ross-c

if the noise is random, it adds as power so you get a 2x times noise
POWER = 3dB for every doubling the number of tracks.

The signal is not random and if the tracks in in correct phase, the
signal will voltage add and you will get 2x the VOLTAGE  = 6 dB for
every doubling of tracks.

So when you go from 1 track to 2 tracks, you get 3 dB more noise and 6
dB more signal for a net SNR improvement of 3 dB.

Same when you go from 2 tracks to 4 tracks etc.

Note the signals from all the tracks must be IN phase at all
frequencies for this to work.  If you are talking about audio signals
and tape tracks,  the azmuth setting of the tape heads must be
CRITICALLY aligned to maintain this correct phase realtionship at the
higher audio frequencies.    In fact  this is a good way to align the
azmuth.

Before we go into more detail on this, are you talking about audio
signals and tape tracks?

Mark

Well, I was thinking in terms of audio signals and tape tracks. But,
I'm not going to apply this in practice, I'm only trying to think about
the abstract problem as the basis for analogies for thinking about
other problems. I wanted to get the theoretical version of the audio
noise-reduction problem clear in my head before thinking about quite
unrelated problems where the "signal" and "noise" are very different
physical properties and the application has no relation to audio or
DSP. Since I only wished a theoretical understanding of the problem, I
don't need to worry about practical details such as making sure that
making sure that the tape heads are aligned at all frequencies.

Thanks for the reply. Apologies if my question was a bit abstract and
should have been stated more clearly.

Cheers,

Ross-c

clemenr@wmin.ac.uk wrote:
> Well, I was thinking in terms of audio signals and tape tracks. But,
> I'm not going to apply this in practice, I'm only trying to think about
> the abstract problem as the basis for analogies for thinking about
> other problems. I wanted to get the theoretical version of the audio
> noise-reduction problem clear in my head before thinking about quite
> unrelated problems where the "signal" and "noise" are very different
> physical properties and the application has no relation to audio or
> DSP. Since I only wished a theoretical understanding of the problem, I
> don't need to worry about practical details such as making sure that
> making sure that the tape heads are aligned at all frequencies.
> 
> Thanks for the reply. Apologies if my question was a bit abstract and
> should have been stated more clearly.

A simple way to make many recordings that are certain to be played back 
in phase is to put all the tracks on one piece of tape. If you do that, 
there is no reason to keep them separate: just use a wide head. You 
might be amazed at the SNR of a single mono track on half-inch tape.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
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clemenr@wmin.ac.uk wrote:
> Since I only wished a theoretical understanding of the problem, I
> don't need to worry about practical details such as making sure that
> making sure that the tape heads are aligned at all frequencies.

The theory behind improving SNR via averaging is quite simple. Assume
you have a digital signal s ( = s[n], but I'll drop the time indeces
for clarity). Now you take N copies of s and add zero-mean
independently and identically distributed random variables e_k
(k=1,2,...,N) with variance sigma^2 to each copy, ie you have

s_k = s + e_k,      k=1, 2, ..., N.

After averaging the N copies together you get

s' = s + 1/N sum_k e_k = s + e',

where e' is the new error. It is also zero-mean and its variance is 1/N
sigma^2 - this new error is more concentrated around its mean value.

Regards,
Andor