What is the Mean Square of a Signal and Why is its Time Average Zero?

[likephysics]

I was reading about noise and noise voltage/root hertz here -
http://www.stanford.edu/class/me220/data/lectures/lect02/Noise_writeup.pdf
What exactly is mean square. why do we need Mean square.

How do we know time average of noise voltage is zero?

 

[analogdesign]

Mean square is when you square the signal and *then* take the mean. Otherwise you get a mean value of zero for signals that spend equal amounts of time above and below zero. This is not useful because signals that have different power would have the same mean value.

A non-zero time average is a DC level shift. Noise is defined as signal fluctuations around the DC level so therefore it must have a mean value of zero.

Another more physical point of view is that if noise had a non-zero time average you could use this time-average as a battery --> perpetual motion machine

Another way of seeing the same thing is that since thermal noise is white (equal power per frequency bin) if the average value were non-zero and you integrated over all time you would have infinite power.

A more technical point of view is that because noise an ergodic process, it must have zero mean. This refers to the fact that looking, for example, at the voltage across a resistor over time is equivalent to looking at an ensemble of resistors at a single time point. The noise must be uncorrelated between the resistors, so it must be zero mean in time.

 

[likephysics]

We are interested in squaring because power is V²/R or I² R.
Another way of thinking would be the negative voltage would become positive after squaring.

 

[sophiecentaur]

I was reading about noise and noise voltage/root hertz here -
http://www.stanford.edu/class/me220/data/lectures/lect02/Noise_writeup.pdf
What exactly is mean square. why do we need Mean square.

How do we know time average of noise voltage is zero?

First, the time average of the noise voltage will be zero - if it were not, it would just contain a DC offset. Now, that could actually represent an error in measurement (say a bit of electrolytic action on a joint) but it isn't usually considered, or t least, it's addressed differently.

If you have a resistor, value R and the Voltage across it is V, then the Power dissipated is V²/R.
If the voltage is constantly changing (in this case, randomly), the power at each instant will be the instantaneous V^2//R. So the average power will be the average (mean) of the instantaneous values of Power, which is Mean(V²/R). So the Equivalent DC voltage is the RMS voltage of the varying signal.
The RMS voltage is different for all different waveforms. For a sinusoidal waveform, the RMS voltage value happens to be the Peak volts/√2. If the waveform is a square wave (+Vand -V about zero), the RMS value is the same as the peak voltage V. For a noise signal, the RMS is the only way to describe its 'size' because its peak value can be 'anything' (massive noise spikes on rare occasions). People try to assess noise Voltage by looking on a 'scope but is isn't very reliable because what you see will depend on the way the scope is set up - even the brightness control and the triggering.

 

[likephysics]

sophiecentaur, that's a really nice explanation. Thanks.
Does the same theory apply to other random phenomenon like vibration?

But one more question - why is noise spec'd in V/rootHz
Noise power depends on frequency?

 

[sophiecentaur]

sophiecentaur, that's a really nice explanation. Thanks.

Does the same theory apply to other random phenomenon like vibration?
But one more question - why is noise spec'd in V/rootHz

Noise power depends on frequency?

Any varying physical quantity involving Energy in some way will have the same equivalent calculations for noise (the random fluctuations).
The bandwidth thing follows for noise that has a flat spectrum. If you double the bandwidth of a (say) receiver, then twice as much noise power will be admitted. The noise voltage depends on the Root of the Power so it must be proportional to the Root of the Bandwidth. Hence the 'per root Hz'. Simples.

 

[sophiecentaur]

Old RF Engineer's saying -"The wider you open the window, the more s**t flies in".