# Statistics of random processes passed through an LTI system

1. May 27, 2012

### physics baws

Hello,

I apologize in advance if I have missed the right place to ask. I'd be grateful if you could forward me to the right place, if that is the case.

Google didn't help, so maybe someone here can point me in the right direction:

1) "If the input to a LTI system is a Gaussian random process, the output is a Gaussian random process" <- How do we know this?
All I see on the internet is people showing how to compute stochastic parameters (e.g. correlations and SPDs), but I have no idea how would I manage to show or to derive the statistics of the output process.

2) In most cases, people often interchange the expectation operator and time integral (usually when convolution occurs) without saying anything. I was wondering when exactly is that allowed (and when is not), and what conditions need to be met for that to happen.

2. Jul 12, 2012

### carlgrace

I know this is an old question but someone might like to see the answer.

1. Putting a process through an LTI system is equivalent to convolving it with a differential equation. This process makes the output more Gaussian. If it is already Gaussian it can't be "more" Gaussian, so it just stays Gaussian. The reason for this is the Central Limit Theorem.

For fun, if you have Matlab or something, try Convolving a simple boxcar transfer function with itself multiple times. At first you will get a triangle for the output spectrum, then it will get more and more Gaussian. After 3 or 4 times it will look like a bell curve.

2. Finding an expectation (e.g. 1st moment is the mean) is a linear operation. So, it can be done inside or outside the integral. It is similar to factoring out a multiplicative factor before doing an integration.

3. Jul 14, 2012

### rbj

it's as gooduva place to ask the question as any. you might also want to check out the USENET newsgroup comp.dsp.

carlgrace's answer to question 1 is sufficient, in fact more than sufficient. while it is true that under most conditions, the Central Limit Theorem says that adding more and more random variables together tends to make the sum more and more Gaussian, you don't actually need that fact. all you need is that the sum of two independent Gaussian random variables (of known mean and variance) results in another Gaussian random variable (with the mean and variance summed from the two).

in an LTI system, the output is equal to a weighted summation of the present and past inputs, all of which are Gaussian.

there is another issue that i thought you would ask, but another statistical property of a process is its power spectrum (which is related to its auto-correlation). the properties of the LTI system (its frequency response or its impulse response) affects the power spectrum and the auto-correlation of the process the LTI system is working on.

about question 2, when you sample a random process at any specific time, you have a random variable with a mean, variance, and higher moments. the expectation operator of that random value (at that sampled time of a random process) is the probabilistic mean or average of the random number. the expectation is a weighted integral over all possible values of the process and integrals over time are adding up over different and independent domains. you can one up first and then the other, or you can add the other up first and the one up second. they add up the same either way.

now, you might end up replacing averages over time with averages from an expectation operator. you can do that when you assume or decide that the process is "ergodic". that might be a different issue.