Can we Find any two Distinct Functions

$$ f:\Re \rightarrow \Re $$ such that

$$ f\left ( x \right )=f\left (\frac{x}{2}\right ) $$

$$ f:\Re \rightarrow \Re $$ such that

$$ f\left ( x \right )=f\left (\frac{x}{2}\right ) $$

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Can we Find any two Distinct Functions

$$ f:\Re \rightarrow \Re $$ such that

$$ f\left ( x \right )=f\left (\frac{x}{2}\right ) $$

$$ f:\Re \rightarrow \Re $$ such that

$$ f\left ( x \right )=f\left (\frac{x}{2}\right ) $$

$$ Ax^2+2Hxy+By^2+2Gx+2Fy+C=0 $$

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

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

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

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

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 $N_0$

$$ S_j(t)= \sqrt{2 P_j}\,c_j(t)\,b_j(t)\,Cos(\omega_c t+\theta_j) $$ 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 $N_0$

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

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

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

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

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

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

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

$$_pF^q(a_1 \cdots a_p;b_1 \cdots b_q;X)=\sum_{k=0}^{\infty}\sum_{\kappa} \frac{(a_1)_{\kappa} \cdots (a_p)_{\kappa}}{(b_1)_{\kappa} \cdots (b_q)_{\kappa}}\,\frac{C_{\kappa}(X)}{k!}$$

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

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

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

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

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