Multi User CDMA System Model

Let's have CDMA System with $K$ Users indexed by $j=1 \cdots K$. Let Each User Transmits BPSK Modulated Signal Simultaneously. Then the Transmitted Signal for one such User $j$ can be Represented as:

$$ S_j(t)= \sqrt{2 P_j}\,c_j(t)\,b_j(t)\,Cos(\omega_c t+\theta_j) $$ Where,

$$c_j(t)=\sum_{n=-\infty}^{\infty}\,c_j^{(n)}\,p_{T_c}(t-n\,T_c) $$ is the $j$th User Signature Waveform, with the Signature bits

$$c_j^{(n)} \in {-1,+1} $$

$$ b_j(t)=\sum_{n=-\infty}^{\infty}\,b_j^{(n)}\,p_T(t-n \, T) $$ is the BPSK Modulated Signal for User $j$


$$b_j^{(n)} \in {-1,+1} $$ and

$$ T=N\,T_c $$ Where $N$ is the Spreading Factor or Processing Gain. The Received Signal at the Receiver can be Represented as

$$ r(t)= \sum_{j=1}^{K} \sqrt{2 P_j}\,c_j(t-\tau_j)\,b_j(t-\tau_j)\,Cos(\omega_c t+\phi_j) + \eta(t) $$ Where $\tau_j$ is Relative Time offset and $$\tau_j \in \left [ 0 \: T \right ] $$ $\phi_j$ is Phase Offset such that $$\phi_j  \in \left [ 0 \: 2 \pi \right ]$$ and $$\phi_j=\theta_j-\omega_c \tau_j $$ $\eta(t)$ is Zero Mean AWGN Process with PSD $N_0$

A Good Introduction to MIMO Wireless Communications

http://www.comm.toronto.edu/~rsadve/Notes/MIMOSystems.pdf

Professor Raviraj Adve is a Notable Researcher in Wireless Communications and Signal Processing.

Gil-Palaez Theorem

It Helps us to Find the CDF of a Random Variable Directly from MGF or Characteristic Function .

if $X$ is any Random Variable with Characteristic Function (CF) given by


$$ \Phi_X(\omega)= \int_{-\infty}^{\infty} f_X(x) e^{j \omega x} dx $$, Then


$$ F_X(x)=0.5-\frac{1}{j\,\pi}\int_0^{\infty} \frac{\Phi_X(\omega)}{\omega} d \omega $$

Definition of Hypergeometric Function of Matrix Argument

$$_pF^q(a_1 \cdots a_p;b_1 \cdots b_q;X)=\sum_{k=0}^{\infty}\sum_{\kappa} \frac{(a_1)_{\kappa} \cdots (a_p)_{\kappa}}{(b_1)_{\kappa} \cdots (b_q)_{\kappa}}\,\frac{C_{\kappa}(X)}{k!}$$
where the Generalized Hypergeometric Coefficient is Given by

$$(a)_{\kappa}=\prod _{i=1}^m \left(a-\frac{1}{2}(i-1){} \right )_{k_i} $$, Where the Pochammer Symbol

$$(\alpha)_j=\alpha (\alpha+1) \cdots (\alpha+j-1),\;(\alpha)_0=1 $$

Definition of Zonal Polynomial

Let $Y$ be $m \times m $ Symmetric Matrix with Latent Roots(Eigen values) $y_1 \cdots y_m$.

Let $$\kappa=\left(k_1 \cdots k_m \right)$$

be the Partition of an Integer $k$ with Partition Size not more than $m$.

The Zonal Polynomial Corresponding to $\kappa$ Denoted by $C_{\kappa}(Y)$ is a Symmetric , Homogeneous Polynomial of Degree $k$ in Latent Roots $y_1 \cdots y_m$ Such that:

$$ C_{\kappa}(Y)= d_k\,y_1^{k_1} \cdots y_m^{k_m}+ \text{Terms of Lower Weight}$$, $d_k$ being a constant

$C_{\kappa}(Y)$ is an Eigen Function of Differential Operator $\Delta_{Y}$ given by:

$$ \Delta_Y=\sum_{i=1}^{m}y_i^2 \frac{\partial^2 }{\partial y_i^2}+\sum_{i=1}^m \sum_{j=1,j \neq i}^m \frac{y_i^2}{y_i-y_j}\frac{\partial }{\partial y_i}$$

As $\kappa$ varies over all Partitions of $k$,The Zonal Polynomials have Unit Coefficients in the Expansion of $Tr(Y)^k$ i.e.,

$$ Tr(Y)^k= \left(y_1+ \cdots +y_m \right)^k= \sum_{\kappa} C_{\kappa}(Y) $$

Definite Matrices

Can we Express an Indefinite Matrix as Sum of Positive Definite/Semi Definite and Negative Definite/Semi Definite matrices? I am working on it...

Conditional Gaussian Distribution

Let $X_1$ and $X_2$ are Jointly Gaussian Random Variables(Which Implies They are Individually Gaussian) i.e.,

$$ X_1 \sim \mathcal{N}(0,\sigma_1^2)$$ and
$$X_2 \sim \mathcal{N}(0,\sigma_2^2)$$, with Correlation Coefficient as $\rho$.

Then we Know that :

$$f_{X_1X_2}(x_1,x_2)=\\ \\ \left(\frac{1}{2 \pi \sigma_1 \sigma_2 \sqrt{1-\rho^2}} \right)\times
\mathbf{e^{\frac{-1}{2(1-\rho^2)}\left(\frac{x_1^2}{\sigma_1^2}-\frac{2 \rho x_1 x_2}{\sigma_1 \sigma_2}+\frac{x_2^2}{\sigma_2^2} \right )}}\\ \\ \;\;\;\; \forall\; x_1,x_2 \in (-\infty\; \infty)\\
\\ $$

Prove that :
$$ E(X_2|X_1)= \frac{\rho x_1 \sigma_2}{\sigma_1} $$
$$ Var(X_2|X_1)=\sigma_2^2\,(1-\rho^2) $$

Find the Digit

This is a Pretty Logical Question..

1216451*0408832000 is 19!, ! Denotes Factorial.  Find the Digit in the place of *

Unitary and Orthonormal Matrix

Let the Matrix $U \in \mathbb{C}^{N \times N}$ is Unitary. Define $\widetilde{U} \in \mathbb{R}^{2N \times 2N}$ as:

$$ \widetilde{U}=\begin{bmatrix}
Re(U) & -Im(U)\\
\\
 Im(U)& Re(U)
\end{bmatrix}$$

Prove that $\widetilde{U}$ is Orthonormal

Complex Gaussian Random Vector

Consider an $N$ Dimensional Complex Gaussian Random Vector $Z \in \mathbb{C}^N$, Such That

$$ Z=X+jY $$, where $X \in \mathbb{R}^N$ and $ Y \in \mathbb{R}^N$ are $N$ Dimensional Real Gaussian Random Vectors .

Covariance matrix $Q$ of $Z$ in terms of Auto Covariance and Cross Covariance Matrices of $X$ and $Y$ is Defined as:

$$ Q=E((Z-E(Z))(Z-E(Z))^H)$$

So:
$$ Q=(\Sigma_{XX}+\Sigma_{YY})-j(\Sigma_{XY}-\Sigma_{XY}^T)$$

Define $2N$ Dimensional Gaussian Random Vector $\widetilde{Z}$, as:

$$ \widetilde{Z}= \begin{bmatrix}
X\\
Y
\end{bmatrix}$$

The Covariance Matrix of $\widetilde{Z}$ is Denoted as $\widetilde{K}$, Defined as:

$$ \widetilde{K}=E((\widetilde{Z}-E(\widetilde{Z}))(\widetilde{Z}-E(\widetilde{Z}))^T)$$

So $\widetilde{K}$, In terms of Auto Covariance and Cross Covariance Matrices of $X$ and $Y$ is:

$$ \widetilde{K}=\begin{bmatrix}
\Sigma_{XX} & \Sigma_{XY} \\
\\
\Sigma_{XY}^T & \Sigma_{YY}
\end{bmatrix}$$

Now if $Z$ is Circularly Symmetric, Prove That:

$$ \widetilde{K}=\frac{1}{2}\begin{bmatrix}
Re(Q) & -Im(Q)\\
\\
Im(Q) & Re(Q)
\end{bmatrix} $$

Affine Transformation Does not Alter Correlation Coefficient

if $X$ and $Y$ are Two Random Variables with Correlation Coefficient $\rho$, Then the Correlation Coefficient of Random Variables
$$ a_1X+b_1\;\text{and}\; a_2Y+b_2$$, Where $a_1$ and $a_2$ are Non Zero Real Numbers with Same Sign is also $\rho$. if They have Opposite Signs, The Correlation Coefficient is $-\rho$.

Laplacian Distribution

Consider the Laplacian Distribution whose PDF is Given by:

$$ f_X(x)=\frac{1}{2}\,e^{-|x|}\,\, {-\infty}<x<{\infty} $$

If Random Variable $Y$ is Defined as

$$ Y=|X|+|X-3| $$,  Find

$$ Pr(Y\geq 3) $$

Expectation and Even PDF

By Definition of Expectation of Random Variable:

$$ E(X)=\int_{-\infty}^{\infty}\,x\,f_X(x)\,dx $$

If $f_X(x)$ is Even, Then $E(X)=0$, Provided The Integral Exists, An Exception Being $\textbf{Cauchy Distribution}$ , Whose Mean Doesn't Exist.

Now the Question is, Will the PDF of Random Variable is Even ,When $E(X)=0$, I am Unable to Solve/Prove This. I Encourage all of you to Solve This...

Conditional Distribution

Consider two random Variables $X$ and $Y$ whose Joint PDF is:
\begin{align*}
 f_{XY}(x,y) &=2(x+y),\; 0 \leq y \leq  1,\; 0 \leq x \leq y \\
&=0,\; Else
\end{align*}

Find $$f_{Y|X}(y|x) $$

Independence and UnCorrelated

We Know that if Two Random variables $X$ and $Y$ are  Independent Statistically, Then

$$ E(XY)=E(X)E(Y)$$, i.e., They are UnCorrelated. It Doesnt Mean that Dependent Random Variables are Always Correlated. Lets Think of Some Examples with Dependent Random Variables being UnCorrelated.

Coin Tossing and Mutual Information

There are Two Coins viz., A fair Coin and a Two Headed Coin. A Coin is Selected at Random for which the Random Variable is Denoted as X, and Selected Coin is Tossed Twice, and Number of Heads is Recorded which is Denoted as Random variable Y.

Find $$ I(X;Y)$$

Log Normal Distribution

if $X$ is Gaussian Distributed Random variable i.e., if
\begin{align*}
X \sim  \mathcal{N}(\mu,\sigma^2) \\
Y=e^X\\
\end{align*}
Y is said to be Log Normally Distributed i.e.,
\begin{align*}
Y \sim Log-\mathcal{N}(\mu,\sigma^2)
\end{align*}
 If MGF of X is Given i.e., $\Phi_X(s)$ is Given.

Find the nth Moment of Y, where $n \in \mathbb{Z}$, Without Finding the PDF of Y

A Very Interesting Probability Problem

Consider a Line Segment of length $L$. Randomly Select a Point on Each Side of the Midpoint of the Line Segment. What is The Probability that The Distance Between the Two Points Randomly Selected is Greater Than $\frac{L}{3}$

Testing Math

$$\sum_{k=1}^{N} k= \frac{N(N+1)}{2}$$

This Post is Basically to Test Usage of LaTeX in My Blog. Please Ignore it...