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What is meant by a continuous-time white noise process?

In a discussion following a question a few months ago, I stated that as an engineer, I am used to thinking of a continuous-time wide-sense-stationary white noise process $\{X(t) \colon -\infty < t < \infty\}$ as a zero-mean process having autocorrelation function $R_X(\tau) = E[X(t)X(t+\tau)] = \sigma^2\delta(\tau)$ where $\delta(\tau)$ is the Dirac delta or impulse, and power spectral density $S_X(f) = \sigma^2, -\infty < f < \infty$. At that time, several people with very high reputation on Math.SE assured me that this was an unduly restrictive notion, and that no difficulties arise if one takes the autocorrelation function to be $$E[X(t)X(t+\tau)] = \begin{cases}\sigma^2, & \tau = 0,\\ 0, & \tau \neq 0. \end{cases}$$

What engineers like to call a white noise process is a hypothetical beast that is never observed directly in any physical system, but which can be used to account for the fact that the output of a linear time-invariant system whose input is thermal noise is well-modeled by a wide-sense-stationary Gaussian process whose power spectral density is proportional to $|H(f)|^2$ where $H(f)$ is the transfer function of the linear system. Standard second-order random process theory says that the input and output power spectral densities $S_X(f)$ nd $S_Y(f)$ are related as $$S_Y(f) = S_X(f)|H(f)|^2.$$ Thus, pretending that thermal noise is a white Gaussian noise process in the engineering sense and pretending that the second-order theory extends to white noise processes (even though their variance is not finite) allows us to get to the result that the output power spectral density is proportional to $|H(f)|^2$.

My query about the definition of a white noise process is occasioned by a more recent question regarding the variance of a random variable $Y$ defined as $$Y = \int_0^T h(t)X(t)\ \mathrm dt$$ where $\{X(t)\}$ is a white Gaussian noise process. The answer given by Nate Eldredge leads to $$\operatorname{var}(Y) = \sigma^2 \int_0^T |h(t)|^2\ \mathrm dt$$ (as I pointed out in a comment on the answer) if the autocorrelation function is taken to be $R_X(\tau) = \sigma^2\delta(\tau)$ (the engineering definition). However, the OP on that question specified $R_X(0) = \sigma^2$, not $\sigma^2\delta(\tau)$, that is, the definition accepted by mathematicians. For this autocorrelation function, the variance is $$\int_0^T \int_0^T E[X(t)X(s)]h(t)h(s)\mathrm dt\mathrm ds = 0$$ since the integrand is nonzero only on a set of measure $0$.

So, what is the variance of the random variable $Y$? and what do readers of Math.SE understand by the phrase white noise process?

Perhaps this question should be converted to a Community wiki?

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CW-hammered per OP request. – Willie Wong Apr 24 '12 at 12:38
Short answer: (1.) $Y$ is undefined; (2.) the RHS of $Y$ is a shorthand for a mathematically elaborate object called stochastic integral; (3.) applying to this object operations valid for classical (deterministic) integrals can lead to chaos. – Did Jan 15 '13 at 7:16
I’m not the best at this, but I feel that there may be a confusion about $\delta$. My understanding is that $\delta(x)$ is infinity if $x=0$ and $0$ otherwise, such that $\int_a^b\delta(x)dx=1$ if $a<0<b$. This function should be consistent with the mathematical definition, but is usually not used by mathematicians. – Teepeemm Jun 8 '14 at 20:27

1 Answer 1

Actually X is the digital signal and Y is the average analog signal power generated in the time-domain. The variance shows how deviation goes if signal Y goes for a long time.

Moreover, you should learn more about the delta function (your function becomes one if x = 0 ) . The engineering and mathematical aspects to explain the autocorrelation function are also correct. There is no contracdiction between them.

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