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For questions that specifically ask for determining a closed form of equations, integrals etc.
1
vote
2
answers
71
views
Expectation of $\left| \frac{(\textbf{x}+\textbf{y})^{H} \textbf{x} }{\| \textbf{x} + \textb...
Given that following two random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ ar …
3
votes
2
answers
419
views
PDF of $ | \sum_{k=1}^{n}{|h_k||g_k|\exp\left( j \theta_k \right)} |^2$ for small values of ...
Given the following function of random variables
$$f = \left|\sum_{k=1}^{n}{|h_k||g_k|\exp\left( j \theta_k \right)} \right|^2,$$
where $h_1, \cdots, h_n$ and $g_1, \cdots, h_n$ are i.i.d. random var …
1
vote
2
answers
188
views
PDF of $g = \frac{1}{n} \sum_{k=1}^{n}{|h_k|\exp\left( j \theta_k \right)}$?
Given the following function of random variables
$$g = \frac{1}{n} \sum_{k=1}^{n}{|h_k|\exp\left( j \theta_k \right)},$$
where $h_1, \cdots, h_n$ are i.i.d. random variables following the complex Gau …
0
votes
1
answer
112
views
PDF of $z = \exp(j\varphi)$, where $\varphi \sim \mathcal{U}[-a, +a]$ [closed]
How can I find the PDF of $z = \exp(j\varphi)$, where $\varphi \sim \mathcal{U}[-a, +a]$, $i.e.$, a uniformly distributed r.v.?
My difficulty here is that it involves complex numbers and I don't know …
0
votes
2
answers
255
views
PDF of $R$ given that $R^2 = C^2 + S^2$, with $C = \sum_{j=1}^{n}{\cos \theta_j}$ and $S = \...
Suppose that $\theta_1, \cdots, \theta_n$ are distributed independently and that $\theta_j$ has probability density function (PDF) $f_j = \frac{1}{2\pi}$ ($i.e.$, the uniform distribution) for $j = 1, …
2
votes
0
answers
504
views
Closed form expression for $Tr\left[ (\mathbf{DW})^k \right]$
Given the $N \times N$ diagonal matrices $\mathbf{D}$ and $\mathbf{W}$ as defined below
$
\begin{split}
\mathbf{DW} &=
\left[
\begin{array}{cccc}
\beta_{1} & 0 & \cdots & 0 \\
0 & \beta_{ …
0
votes
1
answer
119
views
CDF of a RV that is the ratio between a complex Gaussian and a Chi-squared RVs
Given the following p.d.f., which is the p.d.f. of the real and imaginary parts of a random variable that is the ratio between a complex Gaussian and a Chi-squared RVs:
\begin{equation*}
f_U(u)=\exp …
2
votes
2
answers
146
views
Expectation of $\left| \frac{\textbf{x}^{H} \textbf{y} }{\| \textbf{x} \|^2} \right|^2$, whe...
Given that following two random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ ar …
2
votes
1
answer
98
views
p.d.f. of $\left| \frac{\textbf{x}^{H} \textbf{y} }{\| \textbf{x} \|^2} \right|^2$, where $\...
Given that the random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ are independ …
1
vote
2
answers
273
views
Closed expression for $\mathbb{E} \left\lbrace \Re \frac{(\textbf{x} + \textbf{y})^{H}\textb...
Given the random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ are independent, …