## Thursday, 18 April 2013

### Functions

Can we Find any two Distinct Functions
$$f:\Re \rightarrow \Re$$ such that

$$f\left ( x \right )=f\left (\frac{x}{2}\right )$$
Necessary and Sufficient Conditions for the General Second Degree Equation
$$Ax^2+2Hxy+By^2+2Gx+2Fy+C=0$$
to Represent Pair of Straight Lines:

$$\Delta =\begin{vmatrix} A & H &G\\ H& B & F\\ G& F& C \end{vmatrix} =0$$

Case 1 :if $$H^2 > AB$$ Then they are Intersecting pair of Straight Lines

Case 2: if $$H^2 = AB$$ Then they are pair of Parallel Straight Lines

Case 3: if $$H^2 < AB$$ Then they Represent a Point in a Plane

## Friday, 8 February 2013

### Remainder Problem

Find the Remainder when the number $N$ is divided by 7, where

$$N=2222^{5555}+5555^{2222}$$

## Thursday, 12 July 2012

### 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 $N_0$

## Tuesday, 10 July 2012

### A Good Introduction to MIMO Wireless Communications

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

## Monday, 2 July 2012

### 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

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

## Friday, 22 June 2012

### 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

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