# Noise & Electric Circuits

• Apteronotus
So a Wiener process is nowhere differentiable. So the derivative of a Wiener process is not a function.http://www.maths.qmul.ac.uk/~thomas/Notes/Markov.pdfIn summary, the conversation is about understanding the behavior of simple electric circuits when the input is a random signal or noise. The participants discuss the characteristics of random noise and how it can be filtered to improve signal to noise ratio. They also touch on the issue of differentiating a random voltage function and how it relates to capacitors. The conversation ends with a discussion on the theoretical concept of white noise and its infinite bandwidth, making it non-differentiable.

#### Apteronotus

I'm trying to find how simple electric circuits behave when the input is a random signal or noise.

I'm wondering if anyone is familiar with this area or know of any resources they can point me to.

There isn't anything particular special about random noise. You can often just regard it as a voltage waveform. It will tend to have a low peak to average power and a flat spectrum. Are you interested in improving the signal to noise ratio of a received signal? In which case you will be wanting to filter appropriately - as much as possible. Or are you interested in getting good noise performance from your input stages?
A few more specific details would make it easier to make a more useful comment.

The problem in specific is considering the current which would flow across a capacitor due a random voltage difference across the capacitor. For a deterministic voltage V, current is described by
$$I_C=C\frac{dV}{dt}$$
But when V is random, we cannot evaluate this equation.

Any ideas?

Apteronotus said:

The problem in specific is considering the current which would flow across a capacitor due a random voltage difference across the capacitor. For a deterministic voltage V, current is described by
$$I_C=C\frac{dV}{dt}$$
But when V is random, we cannot evaluate this equation.

Any ideas?

It still holds. It's best to write it out like this:

$$I_C (t) = C\frac{dV(t)}{dt}$$

That makes it more clear that the instantaneous current depends on the (change in the) instantaneous voltage. As sophie mentioned, even though the input voltage is random, you can still see the waveform on an oscilloscope. If you use a current probe on one channel of the 'scope, and a voltage probe on the other, you will see that at all times (all points in the waveforms), the above relation holds.

Hi Berkeman,

Thanks for your reply. It seems like you always come to my rescue.

I guess the issue that I'm having is that the term on the right of the equation
$$I_C(t)=C\frac{dV(t)}{dt}$$
is not defined when V is a random function; the function V(t) simply does not have a well defined slope.

:s

Apteronotus said:
Hi Berkeman,

Thanks for your reply. It seems like you always come to my rescue.

I guess the issue that I'm having is that the term on the right of the equation
$$I_C(t)=C\frac{dV(t)}{dt}$$
is not defined when V is a random function; the function V(t) simply does not have a well defined slope.

:s

Actually it does, you just have to think more "time domain", instead of frequency domain or other view. Show me a single-shot of a random V(t) waveform on an oscilloscope, and I can show you the slope at any point on that waveform -- it's just the tangent to that waveform at that point, right?

You may have to zoom in pretty far to see the slopes, but it's a continuous V(t) trace, so it has to have a slope (hence a derivative) everywhere.

http://www.moviesonline.ca/movie-gallery/albums/userpics//whitenoiseposter.jpeg

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lol... I love the poster. Very befitting!

I definitely agree with you about continuity, but strictly speaking a noise signal is not differentiable. Just as in Brownian motion, the more you zoom in the more you'll see the same jagged peaks. In fact I'm pretty sure you can prove its non-differentiability. I'll try to find a reference and post.

Now this is in the realm of theory. I would certainly not make the same claims for a "real" generated noise signal.

Apteronotus said:
lol... I love the poster. Very befitting!

I definitely agree with you about continuity, but strictly speaking a noise signal is not differentiable. Just as in Brownian motion, the more you zoom in the more you'll see the same jagged peaks. In fact I'm pretty sure you can prove its non-differentiability. I'll try to find a reference and post.

Now this is in the realm of theory. I would certainly not make the same claims for a "real" generated noise signal.

Interesting thought. But in the real world, the noise signal will have some maximum bandwidth, so we should be able to get a slope or differentiate. But I see now what you were saying about not being differentiable. Yeah, for theoretical white noise, it would have infinite bandwidth, and would not be differentiable. So theoretical white noise and capacitors don't mix?

But in system of infinite BW you would also e.g. have an infinite amount of Johnson noise; it is an unphysical system so it is hardly surprising that the math does not work out.
Even a theoretical noise signal will have a finite BW, at least if you want sensible answers.

This problem is usually avoided by only using integration. I am really interested if u can find the link about not differentiating noise theoretically, id imagine you would end up with a lot of singularities.

Here's a quick overview.

For a Wiener W and small t

$$W(t+s)-W(s)=\sqrt t Z_t\\$$
where Z is N(0,1)
$$lim_{t \rightarrow 0}\frac{W(t+s)-W(s)}{t}=lim_{t \rightarrow 0}\frac{Z_t}{\sqrt t} \rightarrow \infty$$