Communication Technology Laboratory Fundamentals of Wireless Communications Homework 2 Solutions Problem 1 Linear Filters and Gaussian Noise

1. By using the sampling theorem to expand N (t − τ, ω) and h(τ) with respect to τ we obtain: Z(t, ω) = τ k∈Z h(kT) sinc π τ − kT T k∈Z N (t − kT, ω) sinc π τ − kT T dτ = T k∈Z h(kT)N (t − kT, ω) where we used orthogonality of the set of functions {sinc (π(τ − kT)/T)} k∈Z. 2. Since N (τ) is a complex Gaussian random process, for each finite set of epochs t 1 , ..., t K , the random variables {N (t 1), ..., N (t K)} are jointly Gaussian. Hence, for a given t, any finite sum K k=1 h k N (t − kT), h k ∈ C (1.1) is a Gaussian random variable. From part 1 we have that for a given t, Z(t) = T k∈Z h(kT)N (t − kT) (1.2) which is not a finite sum. To justify why, for a given t, the sum in (1.2) converges to a zero-mean Gaussian random variable, observe that since h(t) is square-integrable, the coefficients h(kT) decrease with k. 3. To justify why Z(t) is a zero-mean Gaussian random process, we have to show that for each set of epochs t 1 , ..., t n , the random variables {Z(t 1), ..., Z(t n)} are jointly Gaussian. This follows using the same argument as in part (2). Furthermore, note that if Z is a zero-mean, proper jointly Gaussian random vector, then Z = HN is also zero-mean, proper and jointly Gaussian, where H is a complex matrix (see e.g. Gallager's notes on Circularly-Symetric Gaussian random vectors, which you can find on the course website).