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I have a basic question about stochastic processes:

When some informations such as wss, uncorraleted sampled, white about random signal (say x[n]) are given, what do we exactly have?

  • For example white information: I know if x[n] is white, its power density function $ P_{x}(e^{j\omega}) $ is constant value independent of $ \omega $. I suppose it means autocorrelation function of x[n] consists of dirac? (I am not sure here, If I am wrong, please correct me).
  • Uncorraleted samples: I know if the random signal is uncorrelated, its covariance equals to zero which means its autocorrelation equals to square of its mean. But uncorraleted samples means this, or something else?
  • What about wss?: I know if the signal is wss, mean (average) of x[n] is constant which is useful information when $ x_{c}(t) $ (constinuous time form of x[n]) and its means are given and we can say directly mean of x[n] is the same as mean of $ x_{c}(t) $. Moreover, we have $ \phi_{x}[m] = \phi_{x_{c}}[mT] $ where T is the sampling period and $ \mu_{x_{c}}^{2} = \mu_{x}^{2} $ where $ \mu $ is the power of the signals (second moment).

Any correction, addition, comment would be appreciated.

Pass through an example may be more convenient:

Suppose we have a system like this:

enter image description here

where, x[n]: wss, $ \eta_{x} = 0 $ (mean value), uncorrelated samples, $ \sigma_{x}^{2} $ and h[n] are given.

  1. How to calculate $ \eta_{y}[m] $ the mean, $ \phi_{y}[m] $ the autocorrelation and $ \mu_{y}[m] $ the average power of y[n]?
  2. How to calculate $ \eta_{v}[m] $ the mean, $ \phi_{v}[m] $ the autocorrelation and $ \mu_{v}[m] $ the average power of v[n]?
mehmet
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  • There are far too many separate queries lumped into a single question, and any answer will be far too long and detailed: you are essentially asking that people write a whole chapter of a book for your edification. So, I am voting to close your question as too broad. Some of your questions are answered here and perhaps you can edit your question to something more reasonable after reading and understanding that answer. – Dilip Sarwate May 03 '15 at 13:19
  • @DilipSarwate, You are right it is a bit broad question, I guess. In fact, I am looking for the information of if x[n] wss, then ...; if x[n] white, then ...; if x[n] uncorrelated samples, then ... I think they are basic properties to solve question. I feel like I can't use these informations and I fail to solve question because of unused infromations. – mehmet May 03 '15 at 13:28

1 Answers1

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As pointed out in the comments, this question is rather broad, so here I will just answer your last question about the autocorrelation of the output signal $y[n]$. You might know the following relation concerning the power spectra of input and output sequences:

$$S_y(\omega)=S_x(\omega)|H(\omega)|^2\tag{1}$$

where $H(\omega)$ is the frequency response of the LTI system. From the inverse Fourier transform of (1) you get a corresponding relation between the autocorrelation functions:

$$\phi_y[m]=\phi_x[m]\star h[m]\star h^*[-m]\tag{2}$$

where $h[m]$ is the system's impulse response, and $\star$ denotes convolution.

Matt L.
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  • And even $(2)$ is shown as the last displayed equation $R_y = h\tilde{h}R_X$ in the answer that I had referred the OP to. – Dilip Sarwate May 03 '15 at 13:55
  • Thank you Matt L. for the equations. At least, can you tell me if my inferences about wss, white and uncorrelated samples are correct? – mehmet May 03 '15 at 14:08