# 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?

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