Re: Statistical signal processing question
- From: Tim Wescott <tim@xxxxxxxxxxxxxxxx>
- Date: Sat, 05 Nov 2005 12:50:50 -0800
taras.di@xxxxxxxxx wrote:
Sometimes you don't know the assumptions of the problem until you read the answer...Hi everyone,
I'm a final year engineering student doing a bit of statistical signal processing, and I have a pretty easy question for anyone that cares to answer (stats isn't my strong point :) ). Ok, here goes:
"If x[k] = cos(kw_d + P) is a random process with P uniformly distributed on [0,pi], then find an expression for the covariance function"
I've gone about this by calculating:
EV{x{k1]x[k2]}, where EV stands for 'expected value'.
However, I'm a bit confused about how the expected value should be calculated. If the you're doing the expected value of the function of two random variables, then this is equal to the double integral over infinity of the product of the function and the joint pdf. If the two random vars are independant, then this is equal to the double integral over infinity of the product of (the function, and the two pdfs). EG:
EV{XY} = int{ int{ X*Y*p_X*p_Y}} (assuming independance between X & Y)
In the question above, is the random variable 'P' which corresponds to the random process at time k1 different to the random variable 'P' which corresponds to the random process at time k2? Although they have the same statistical characteristics, I'm thinking that because they correspond to different random variables(the distribution of the random variable corresponding to the value of the random process at time k1 versus the distribution of the random variable corresponding to the value of the process at time k2). However, the solutions to the tutorial treats the two 'P' s as equal.
TIA
Taras
Once could see valid real-world problems either for the case with P being a randomly selected constant, or with P being the output of a random process. Usually if P were the output of a random process, however, it would be indicated in the problem formulation, and a conscientious author would explicitly indicate it's dependence on time, usually by subscripting it.
--
Tim Wescott Wescott Design Services http://www.wescottdesign.com .
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