The question is:

Consider a continuous random number with a Gaussian distribution of mean $\mu$ and variance $\sigma$ . The RV is measured from time $t=-\infty$ to $t=\infty$. This time domain signal $x(t)$ is passed through a first order low pass filter with cutoff at $f_o$. What would the statistics of the output be? Would it still be Gaussian? If yes, then what is the new mean and variance?

I am unable to understand where to start with this. Any help is greatly appreciated.

  • 1
    $\begingroup$ This is a badly posed question. If it really is homework (that is, the OP is merely copying what has been asked, and not his/her understanding or translation of what has been asked), then the instructor deserves censure. Unfortunately, the accepted answer is even worse. $\endgroup$ Aug 30, 2017 at 14:37
  • $\begingroup$ Actually it is homework. I was hoping for some leads. Do u have any? $\endgroup$
    – Bootstrap
    Aug 30, 2017 at 14:38
  • $\begingroup$ @ dilip Actually professor I was hoping you would take a look at this. I have seen some of your answers on similar questions where people have asked to generate band limited noise and your lecture notes appendix on white noise. This was actually a discussion in class. What information do you think is needed and how should we proceed with this? $\endgroup$
    – Bootstrap
    Aug 30, 2017 at 15:05
  • $\begingroup$ @ dilip-sarwate i'm open to any feedback to improve my answers. I guess using random variable instead of random process is the problem. I tried to write the answer using random process concept, but then i taught it's not a common knowledge (even for some so called experts in the field) so i changed it to random variable, but at the end i give a hint that we must use random process. $\endgroup$
    – Mohammad M
    Aug 30, 2017 at 16:36
  • $\begingroup$ I've made some correction to my answer. $\endgroup$
    – Mohammad M
    Aug 30, 2017 at 17:46

1 Answer 1


at first read a book on stochastic process or stochastic signal processing. it's better that we say we have a random process which it's first order distribution (PDF of the process at each instant of time) is independent of time and is a Gaussian with known mean and variance. Now we want to obtain the first order properties of filtered process.

To begin we write the output of filter as a convolution. The output process at each instant of time is a linear combination (weighted average) of input process at different instant of time which all has the same Gaussian distribution, also we know linear combination of variables with Gaussian PDF leads to another variable with Gaussian distribution. So the first order distribution of output must be a Gaussian.

Now taking expectation from both side of convolution, we see expectation of filtered process at each instant of time becomes convolution of filter response and the mean of input process which is a constant over time, so it becomes the multiplication of input mean and the DC response of filter. To obtain the variance of process around its mean, we have to find the expectation of squared difference of the process and its mean or expectation of squared value of output minus the square of it's mean. To relate the squared value of output to the input we could convolve the filterd signal with its reversed time version at zero lag. At the end we see new variance is the variance of input times square of DC response of filter.


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