Standard deviation for a signal with a noise component

[ForceBoard]

Hello,

I've been trying to find the answer to this question on the internet but no real luck so here goes:

Imagine a signal My_signal which has two components, the actual signal A which is not constant over time and a noise component, we can call it N, where said noise component has a standard deviation which is more or less constant over time regardless of the amplitude of A. How do I calculate the standard deviation for A?

If sigma is the standard deviation, is it:

My_signal = A + N

sigma(My_signal) = sqrt( (sigma(A)^2) + (sigma(N)^2))
-> sigma(A) = sqrt (sigma(My_signal)^2-sigma(N)^2))Thanks,

Marcus

 

[chiro]

Hey ForceBoard and welcome to the forums.

The first thing to ask is whether the signal is discrete time or continuous time.

If it is discrete time then you will need to consider how to sum the variances for each step to get the total variance over a time interval.

The easiest case happens when each individual part of the signal is independent, but this is not a feasible assumption in many cases because there are dependencies between the values of each part of the signal over the interval.

If it is continuous time, you do the same thing but depending on the properties it can be complex.

The simplest form of a continuous-time stochastic process is known as a thing called Brownian motion.

Brownian motion assumes that every disjoint interval is independent which simplifies things (like in the discrete case above) quite a bit: in fact all you have to do to find the variance of a typical Weiner Process is to look at the length of the interval. So if it's say an interval [0,1], then the variance of that process if the Weiner process starts at 0 with a value of 0 is 1 - 0 = 1.

Again though, this is not often feasible for signal analysis because signals often carry dependencies between different portions of the signals.

You will need to understand the context of the actual signal and the intrinsic structural properties of the signal with regard to what is actually being transported before you can answer this question.

To give you an idea think about the following situation:

You are trying to transmit digital data through some communication medium and suppose that a packet of data only gets transmitted once in one frame (basically a bunch of undivided data guaranteed to get to the destination by the requirements of the hardware, the channel, and so on).

Now in transmission, typically what happens is that you don't just send the minimum 1' and 0's: you add what is called an error correcting code. It can make the process send literally twice as much data but the point is to send these structures so that errors can be detected and in some cases corrected.

Depending on the noise model for the channel, you construct specific codes that minimize the errors by exploiting the noise models for the channel.

In the above case, things are not dependent: it's not just the noise model but the actual structure of the data since some of the data is dependent on the others.

This is a case where you can't just use the easiest assumption and this is more along the lines of a real situation than an ideal one.

 

[ForceBoard]

Chiro, thank you for your elaborate response...

I'm actually looking at a signal from a force transducer where I can control the sampling rate between 10-50Hz, so I would say discrete time with an independent noise component.

As for the pure signal itself I think I would have to look at its segmens as being independent even if that is not completely true depending on the speed of the varying force I'm measuring and the physical nature of the force transducer.

More or less, no matter how much load I put on (using control weights) the standard deviation in the noise is maybe around 0.04. (Important to say that I don't know what type of statistical distribution the noise has, looks to be randomly jumping around a baseline)

As you already understand I'm interested in what the deviation of the pure signal is when the noise portion is canceled out mathematically since that potentially can tell me a lot about the test object interacting with the force tranducer.

 

[chiro]

I am going to sleep soon, but you might want to check out the Weiner process first to get an idea of continuous time, independent normally distributed noise.

 

[Stephen Tashi]

-> sigma(A) = sqrt (sigma(My_signal)^2-sigma(N)^2))

That is a correct formula. It's your general conception of the problem that needs improvement.

For a "random variable" X , the terms "standard deviation" can have several different meanings. Among these are

1. "Standard deviation" can refer to the standard deviation as a parameter of the probability distribution for X. This type of standard deviation can be called "the population standard deviation".

2. "Standard deviation" can refer to a number (like 0.43) that you calculate from a set of data for X, in which case it should be called a "sample standard deviation".

3. Both "standard deviation" and "sample standard deviation" can refer to a formula you use to calculate the "sample standard deviation" (rather than referring to just one specific number). From this point of view, the "sample standard deviation" becomes a random variable also since it depends on random outcomes.

4. "Standard deviation" can refer to an estimate (like 0.43) of the "population standard deviation" (usually an estimate that is based on a particular sample of data). There are are a variety of ways to make estimates. There is no law that says you can only use the "sample standard deviation" as your estimate of the the "population standard deviation".

5. "Standard deviation" can refer to a formula or algorithm for making an estimate of the "population standard deviation" as a function of the data. This is called an "estimator of the standard deviation" and such an estimator is a random variable since the data is random.

There is no law that says than an estimate or an estimator must use the same formula as the formula for the population standard deviation. In fact, that would be impossible since the formula for the "population standard deviation" uses inputs that are probabilities and a formula for estimating the standard deviation must use inputs that are data values. The data might involve observed frequencies of values, but observed frequencies are not the same as probabilities. (Think of tossing a fair coin an odd number of times. The fraction of heads can't turn out to be 1/2.)

The formula you gave is correct for 1) if the signal and noise are independent random variables.

if you talk about specific numbers from an experiment, your talking about 2) or 4). If you are discussing general statistical concepts, you usually refer to 1), 3) or 5). If you have lots of data, It is likely that the formula you gave is useful for 2),3),4),5) also.

A further complication of your problem is that you have a variable sample rate. If you have a random variable X that varies continuously in time, you probably won't get the same standard deviation ( in any sense of that phrase) if you change sample rates. If you want to talk about "the" standard deviation of the random variable X, you must make an even more specialized definitions of "standard deviation" that doesn't depend on the particular sample rate. The basic idea is that the definition must incorporate the concept of "per unit time". This is why chiro suggests you look at the theory of Brownian motion.

If you use only 1 sample rate in your experiments, then the formula you gave is OK.