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BACKGROUND:

Equation (3.6) of Wireless Communications by Goldsmith gives the baseband impulse response of a time-varying channel as: $$ c(\tau,t) = \sum_{n=0}^{N(t)}\alpha_n(t)e^{-j\phi_n(t)}\delta(\tau-\tau_n(t)), $$ where $N(t)$ is the number of multipath components ($n=0$ is the line of sight path), $\alpha_n$, $\phi_n$, and $\tau_n$ are the amplitude, phase, and lag of the $n^\text{th}$ path, respectively. If $u(t)$ is the baseband transmitted signal, then the received signal $r(t)$ (at baseband) would be: $$ r(t) = \int_{-\infty}^\infty c(\tau, t) u(t-\tau) d\tau $$

Frequently the impulse response $c(\tau,t)$ is modeled as a zero-mean wide-sense stationary uncorrelated scattering (WSSUS) stochastic process. Coupled with this is frequently the assumption that $c(\tau,t)$ is a complex Gaussian random process, which gives a Rayleigh fading channel as the result.

Normally when a stochastic process $x(t)$ is called a Gaussian process it is because the marginal distribution of $x$ at any time $t$ follows a Gaussian distribution.

In the case of $c(\tau,t)$, the "value" of the random process is "infinite", due to the multiplication with the Dirac delta above (which is not actually a function, but a distribution).

QUESTION:

If a Gaussian random variable is multiplied by a Dirac delta distribution, is it still fair to call it a Gaussian random process? Does it still have all the "nice" properties that a Gaussian random process has, such as:

  • Being completely specified by its first and second-order statistics
  • Any linear transformation being done on it still resulting in a Gaussian distribution at the output
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