Re: Summing Noise Sources

In article <5mb66dFcle75U1@xxxxxxxxxxxxxxxxxx>, "Phil Allison"
<philallison@xxxxxxxxxx> wrote:

** Hi all,

Following on from the " Familiar formula ? " thread:

If we have two or more random ( band limited) noise sources ( be they pink,
white or whatever ) and we sum them, then the TOTAL noise is found by
either summing the individual power levels OR by taking the RMS voltage of
each noise source, squaring the values, summing the results and then taking
the square root of that sum.

The latter gives a total RMS noise voltage while the former gives the total
noise power.

OK ??

But what about the peak value ?????

Any steady noise source will have a "peak to average ratio" or Crest
Factor ( CF) - which is the number ratio of the magnitude of the peak
value to the steady RMS voltage level.

The CF for band limited pink noise is often quoted as being about 4 or 12
dB.

But if you sum two pink noise sources of the same average amplitude, the
peak voltage value should double. I say this because there will be regular
points in time when BOTH noise sources attain maximum ( or near maximum)
values and have the same sign.

So, for the sum, the average power is double but the peak power is four
times that of a single source.

Sounds like the CF of the sum has increased by a factor of sq rt 2 - ie
from 4 to 5.65.

With more independent sources it gets even worse.

Is anomalous - no ??




....... Phil

The effect Phil describes is evident when summing a few pure tones, but
not for noise signals.

The peak voltage doubles for two sine waves with common amplitude but
different frequencies and thus the peak increases by 6 dB; of course the
rms voltage goes up by 3 dB, so the crest factor increases, too. Similar
results hold for three of four, or any finite number. But there are
multiple "peak" voltages---the global peak, and also local peaks we know
to ignore if we are looking at an oscilloscope trace. The sum of a finite
number of sinusoids is periodic, and if we sample long enough for a full
cycle, we will observe the true peak. Such a sample is essentially a
perfect representation of the signal.

Now consider a single noise source, which may be thought of as an infinite
sum of individual sinusoidal components. There is an overall peak, but
that's *not* what we observe when we sample that noise source.

The probability that a finite-length sample will capture the global peak
voltage of a random noise signal is zero.

What we do see are the lesser peaks; the typically-quoted crest factor for
noise is that of a sample, not the entire noise signal. Recall that in
the case of a finite number of sine components the sum was was periodic,
so a sufficiently large sample would have the sample peak matching the
true global peak. In contrast, we do not observe the true peak voltage
for a noise signal.

Now suppose we sum two independent noise signals. We end up with an
infinite sum of sinusoidal components---much like *each* of the things we
added.

Phil is correct that the true peak voltage would double when we sum two
independent noise signals, but we'll never see that. What we see is still
the collection of lesser peaks in a sample, and that does not increase as
the overall peak would.


That's what the theory says, and it is easy to demonstrate that it is correct.

I used a Dorrough test set whose meters simultaneously show both average
and peak levels on the same scale.

I first used four signal sources, each generating a single frequency. I
set each to the same level, and verified that combining two or three or
all four did result in the predicted change of crest factor. The average
level and the peak level both increased, but at different rates, and the
change in crest factor is easily seen with the Dorrough meter.

Then I used six independent noise generators; each produced pink noise
with about 10 dB difference between peak and average values. Combining
some or all of those always produced a noise signal with the same 10 dB
difference between peak and average values. The overall level increased
but peak and average increase at the same rate, so there is no change in
crest factor.
.

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