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Search options questions only not deleted user 103291

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, …