# Question: Suppose Z X jY is a circular Gaussian

Suppose Z = X + jY is a circular Gaussian random variable whose PDF is described by Equation (5.70),

(a) Find the PDF of the magnitude, R = |Z|, and phase angle, θ =∠ Z, for the special case when μZ = 0.

(b) Find the PDF of the magnitude, R = |Z|, and phase angle, θ =∠ Z, for the general case when μZ ≠ 0. Hint: In this case, you will have to leave the PDF of the phase angle in terms of a Q- function.

(c) For the case when μZ » σ, show that the PDF of the phase angle is well approximated by a Gaussian PDF. What is the variance of the Gaussian PDF that approximates the PDF of the phase angle?

(a) Find the PDF of the magnitude, R = |Z|, and phase angle, θ =∠ Z, for the special case when μZ = 0.

(b) Find the PDF of the magnitude, R = |Z|, and phase angle, θ =∠ Z, for the general case when μZ ≠ 0. Hint: In this case, you will have to leave the PDF of the phase angle in terms of a Q- function.

(c) For the case when μZ » σ, show that the PDF of the phase angle is well approximated by a Gaussian PDF. What is the variance of the Gaussian PDF that approximates the PDF of the phase angle?

**View Solution:**## Answer to relevant Questions

Suppose In figure 5.7 and P i = 1/3, i = 1, 2, 3. Determine the mutual information for this channel. For the transition matrix Q, prove that the equally likely source distribution, Pi = 1/3, i = 1, 2, 3, is the one that maximizes mutual information and hence the mutual information of the capacity associated with the channel ...Suppose we flip a coin three times, thereby forming a sequence of heads and tails. Form a random vector by mapping each outcome in the sequence to 0 if a head occurs or to 1 if a tail occurs. (a) How many realizations of ...Define the N- dimensional characteristic function for a random vector, X = [X1, X2,…..XN] T , according to ΦX (Ω) = E[ejΩTX] where Ω = [ω1, ω2, ωN] T. Show that the - dimensional characteristic function for a zero- ...Find the PDF of Z = X1X2 + X3X4 + X5X6 + X7X8 assuming that all of the Xi are independent zero- mean, unit- variance, Gaussian random variables. Hint: Use the result of Special Case # 2 in Section 6.4.2.1 to help.Post your question