Communication System¶

$\displaystyle\int _{-\infty}^{\infty}ae^{-b|x|}dx=\dfrac{2a}{b}$

$\displaystyle\int _{0}^{\infty}e^{-x^2}dx=\dfrac{\sqrt{\pi}}{2}$

standard PDF for CRV
- uniform PDF
  
  X~U[a,b]
  
  $\mu_{X} = \frac{a+b}{2}$
  
  $\sigma _{X}^2=(\frac{b-a}{12})^2$
  
  $E[X^2]=\frac{a^2+b^2+ab}{3}$
- triangular PDF
  
  $X \sim \triangle[a,c,b]$
  
  peak value = $\dfrac{2}{b-a}$
  
  $E[X] = \mu _X= \dfrac{a+b+c}{3}$
  
  $\sigma _X^2 = \dfrac{a^2+b^2+c^2-ab-bc-ca}{18}$
- gaussian, normal distribution
  
  $X\sim N(\mu_X, \sigma_X ^2)$
  
  pdf : $\boxed{f_X(x)=\dfrac{1}{\sqrt{2\pi\sigma_X^2}}e^{\frac{-(x-\mu_X)^2}{2\sigma_X ^2}}}$
  
  cdf : $F_X(x)=P\{X\le x\} = 1-P\{X \gt x\}=1-Q [ \dfrac{x-\mu_X}{\sigma_X} ]$
  
  $P\{X=c\}=0$
  
  $\boxed{P\{X\gt a\}=Q [ \dfrac{a-\mu_X}{\sigma_X} ]}$
- rayleigh pdf
  
  $f_X(x)=\frac{x}{\sigma_X^2}e^{\frac{-x^2}{2\sigma_X^2}}u(x)$
- exponential distribution
  
  $f_X(x) = \begin{cases} \lambda e^{-\lambda x} &\text{if } x \ge 0 \\ 0 &\text{if } x \lt 0 \end{cases}$
- laplace distribution
  
  $f_X(x)=ae^{-b |x|} \space where \space\dfrac{2a}{b}=1$ ( a >0 , b>0)
linear transformation

Y=aX+b
- $E[Y]$
  
  $aE[X]+b$
- $E[Y^2]$
  
  $a^2E[X^2]+b^2+2abE[X]$
- $\sigma_Y^2$
  
  $a^2\sigma_X^2$
- $F_Y(y)$
  
  $P\{aX+b\le y\}$
  
  $P\{X\le \frac{y-b}{a}\}$
- $f_Y(y)$
  
  $\frac{1}{a}f_X(\frac{y-b}{a})$
- $X\sim U[-m,m]$
  
  $Y\sim U[-am+b,am+b]$
- $X \sim \triangle[-m,0,m]$
  
  $Y \sim \triangle[-am+b,b,am+b]$
- $X \sim N[\mu_X,\sigma_X^2]$
  
  $X \sim N[a\mu_X+b,a^2\sigma_X^2]$
bivariate RV
- joint CDF
  
  $F_{XY}(x,y)=P\{X\le x,Y\le y \}$
  - properties
    
    $0\le F_{XY}(x,y) \le 1$
    
    $F_{XY}(\infty,\infty)=1$
    
    $F_{XY}(-\infty,y)=F_{XY}(x,-\infty)=F_{XY}(-\infty,-\infty)=0$
- marginal CDF
  
  $F_{XY}(\infty,y)=F_{Y}(y)$
  
  $F_{XY}(x,\infty)=F_{X}(x)$
- conditional CDF
  
  $F_{\frac{Y}{X}}(\frac{y}{x})=\dfrac{F_{XY}(x,y)}{F_{X}(x)}$
  
  $F_{\frac{X}{Y}}(\frac{x}{y})=\dfrac{F_{XY}(x,y)}{F_{Y}(y)}$
$F_{XY}(x,y)=F_{\frac{X}{Y}}(\frac{x}{y})F_{Y}(y)=F_{\frac{Y}{X}}(\frac{y}{x})F_{X}(x)$
- if X and Y are independent then $F_{\frac{Y}{X}}(\frac{y}{x})=F_Y(y)$ and $F_{\frac{X}{Y}}(\frac{x}{y})=F_x(x)$
  
  $F_{XY}(x,y)=F_{X}(x)F_{Y}(y)$
  
  $f_{XY}(x,y)=f_{X}(x)f_{Y}(y)$
- joint PDF
  
  $f_{XY}(x,y)=\dfrac{\partial^2F_{XY}(x,y)}{\partial x\partial y}$
  
  $\displaystyle F_{XY}(x,y)=\int_{-\infty}^x\int_{-\infty}^yf_{XY}(x,y)dxdy$
  
  $\displaystyle \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f_{XY}(x,y)dxdy=1$
  - $\displaystyle P\{x_1\le X\le x_2,y_1\le Y\le y_2\}=\iint_R f_{XY}(x,y)dxdy$
    
    $R=\{x_1\le X\le x_2,y_1\le Y\le y_2\} \cap \{ region \space of \space pdf\}$
- marginal PDF
  
  $\displaystyle f_{X}(x)=\int_{-\infty}^{\infty}f_{XY}(x,y)dy$
  
  $\displaystyle f_{Y}(y)=\int_{-\infty}^{\infty}f_{XY}(x,y)dx$
- conditional PDF
  
  $f_{\frac{Y}{X}}(\frac{y}{x})=\dfrac{f_{XY}(x,y)}{f_{X}(x)}$
  
  $f_{\frac{X}{Y}}(\frac{x}{y})=\dfrac{f_{XY}(x,y)}{f_{Y}(y)}$
function of 2 RV
- $R_{XY}$ correlation
  
  $E[XY] = R_{XY}$
- $\sigma _{XY}$ cross-covariance
  
  $E[(X- \overline {X})(Y- \overline {Y})]=cov(X,Y)=R_{XY} - \mu_X \mu _Y$
- $\sigma ^2_{X}$ auto-covariance
  
  $E[(X- \overline {X})(X- \overline {X})]=\sigma _X ^2$
- orthogonal
  
  $E[XY] = R_{XY} =0$
- uncorrelated
  
  $cov(X,Y) = \sigma_{XY} = 0$
  
  $\rightarrow E[XY] = E[X]E[Y]$
- independent
  
  $E[X^k Y^r] = E[X^k]E[Y^r]$
  - if 2 RV are independent then they will be uncorrelated always but vice-versa may not be true
    
    $cov(X,Y) = \sigma_{XY} = 0$
    
    $\rightarrow E[XY] = E[X]E[Y]= \mu_X \mu_Y$
- correlation coefficient
  
  $\rho = \dfrac{\sigma _{XY}}{\sigma_X \sigma _Y}$
maximum and minimum of 2 independent RV

$P[max(X,Y)<z] = P[(X<z)(Y<z)]$

$P[max(X,Y)>z] = 1-P[max(X,Y)<z]$

$P[min(X,Y)>z] = P[(X>z)(Y>z)]$

$P[min(X,Y)<z] = 1-P[min(X,Y)>z]$
RP

$E[X(t_1)X(t_2)] = R_{X(t_1)X(t_2)}$

$\sigma _{XX} (t_1, t_2)=R_{XX} (t_1,t_2)-\mu _{X(t_1)}\mu_{X(t_2)}$
- types of RP
  - strict sense stationary (SSSRP)
    
    $k^{th}$ order PDF, CDF and PMF are independent of time shifting
    
    $k^{th}$ order joint PDF : $f_{X(t_1)X(t_2)\dots X(t_k)} (x_1,x_2\dots x_k)$
  - wide sense stationary (WSSRP)
    
    SSSRP up-to $2^{nd}$ order
    
    $E[X(t)]=\mu_X$
    
    $E[X^2(t)] = R_{XX}(0)= constant$
    
    $\sigma^2_{X(t)} = constant$
    
    $E[X(t_1) X(t_2)] = R_{XX} (t_1 \sim t_2)$
    
    $E[X(t) X(t+\tau)] = R_{XX} (\tau)= R_{XX} (-\tau)$
    
    $cov(X(t_1) X(t_2)) = R_{XX} (t_1 \sim t_2) - \mu_X ^2$
    - $X(t)=Acos(\omega_o t +\phi)$
      
      $E[X(t)] =0$
      
      $E[X^2(t)]=\dfrac{A^2}{2}$
      
      $E[X(t)X(t+\tau)]=\dfrac{A^2}{2}cos(\omega_o \tau)$
    - transmission through LTI
      
      $x(t) \longrightarrow \boxed{h(t)} \longrightarrow y(t)$
      
      $R_Y(\tau) =R_X(\tau)*h(\tau)*h(-\tau)$
      
      $S_{YY}(f) = S_{XX}(f)| H(f) |^2$
      
      $E[Y(t)] = \mu_X H(s) \Big|_{s=0}$
      
      $\displaystyle E[Y^2(t)]=\int _{-\infty}^{\infty}S_{YY}(f) df= \int _{-\infty}^{\infty}S_{XX}(f)| H(f) |^2 df$
  - ergodic
    
    statistical average of RP = time average of RP
    
    $\displaystyle\int _{-\infty}^{\infty} X(t) f(x)_{X(t)}dx=\lim \limits_{T\rightarrow \infty} \dfrac{1}{T} \int _{-\frac{1}{T}}^{\frac{1}{T}} X(t) dt$
    
    $R_{XX} (\infty) = \mu^2_{X}$ only when X(t) is non-periodic ergodic
  - cyclostationary
density functions
- ESD
  
  $G_{XX}(\omega)=|X(\omega)|^2 \space \text{or } G_{XX}(f)=|X(f)|^2 \space$$\dfrac{\text{Joule}}{\text{Hz}}$
  
  $R_{XX} (\tau) \longleftrightarrow G_{XX}(\omega)$
  
  $R_{XX}(0) = \text{energy} =\dfrac{1}{2\pi}\displaystyle\int_{-\infty}^{\infty} G_{XX}(\omega) d\omega=\int_{-\infty}^{\infty} G_{XX}(f) df$
- PSD
  
  $S_{XX}(f)=\lim \limits_{T\rightarrow\infty}\dfrac{1}{T}|X_T(f)|^2 \space$$\dfrac{\text{Watt}}{\text{Hz}}$
  
  $R_{XX} (\tau) \longleftrightarrow S_{XX}(\omega)$
  
  $R_{XX}(0) = \text{power}=\dfrac{1}{2\pi}\displaystyle\int_{-\infty}^{\infty} S_{XX}(\omega) d\omega=\int_{-\infty}^{\infty} S_{XX}(f) df$
- bandpass density functions
  
  \[A \cdot X(t) \cos(\omega_c t) \overset{\text{ESD}} \longleftrightarrow \dfrac{A^2}{4}[G_{X}(f-f_c)+G_X(f+f_c)]\]
  
  \[A \cdot X(t) \cos(\omega_c t) \overset{\text{PSD}} \longleftrightarrow \dfrac{A^2}{4}[S_{X}(f-f_c)+S_X(f+f_c)]\]
  - if $$\theta \sim [0,2\pi]$ and $X(t)$$ is WSSRP
    
    \[A \cdot X(t) \cos(\omega_c t+\theta) \overset{\text{PSD}} \longleftrightarrow \dfrac{A^2}{4}[S_{X}(f-f_c)+S_X(f+f_c)]\]
    
    \[A \cdot X(t) \cos(\omega_c t+\theta) \overset{\text{ACF}} \longleftrightarrow \dfrac{A^2}{2}R_X (\tau) \cos (\omega_c\tau)\]

analog communication¶

$voice \rightarrow [300 Hz - 3.5 kHz] \\ audio \rightarrow [20 Hz - 20 kHz] \\video \rightarrow [ 0 Hz - 4.5 MHz]$

hilbert transform

HT is a non-causal LTI system

$\boxed{h(t) =\dfrac{1}{\pi t}}$

$\hat {x} (t) =x(t)*h(t)$

$\boxed{\dfrac{1}{\pi t} \longleftrightarrow-j \cdot\text{sgn}(\omega)}$
- $A\cos(\omega_o t+\phi) \longrightarrow \boxed{H(\omega)} \longrightarrow A\sin(\omega_ot+\phi)$
  
  $y(t) = A|H(\omega_o)|\cos (\omega_ot+\phi+\angle H(\omega_o))$
  
  $|H(\omega_o)|=1$
  
  $\angle H(\omega_o) = -\dfrac{\pi}{2}$ , $\omega _o >0$
- for low pass signals
  
  $\text{HT} [m(t) \cos (\omega _ct)]=m(t)\sin (\omega _ct)$
  
  $\text{HT} [m(t) \sin (\omega _ct)]=-m(t)\cos (\omega _ct)$
envelope

pre-envelope : $\boxed{x_+(t) = x(t) +j \hat{x} (t)}$ → calculated for baseband(wideband) and bandpass (narrow band)

complex envelope: $\boxed{x_c(t) =x_+(t) e^{-j\omega_c t}}$ → calculated for bandpass (narrow band)
amplitude modulation
- DSB-FC
  
  $\mu =\dfrac{|m(t)|_{max}}{A_c}=k_a |m(t)|_{max}$
  
  $\boxed{S_{AM}(t)= [A_C + m(t)]\cos (2\pi f_c t)=A_C[1 + k_am(t)]\cos (2\pi f_c t)}$
  
  $P_{AM }=P_c +P_{SB}= P_c +\dfrac{P_m}{2}=P_c(1+k_a^2P_m)$
  
  $\eta =\dfrac{P_{SB}}{P_{AM}}$
  
  $P_{AM} = \overline I_{AM}^2(t)= \overline V_{AM}^2(t)$
  - sinusoidal
    
    $P_{AM }=P_c\Big(1+\dfrac{\mu_1^2+\mu_2^2+\dots}{2}\Big)$
  - triangular
    
    half-wave symmetric : $f_o,3f_o,5f_o,7f_o \dots$
    
    $P_{AM }=P_c\Big(1+\dfrac{\mu^2}{3}\Big)$
  - square wave
    
    $P_{AM }=P_c\Big(1+\mu^2\Big)$
  - AM modulator
    - SLD
      - BPF
        
        $2f_m < f_L<f_c-f_m$
        
        $f_c+f_m<f_H<2f_c$
      $A_c' =a_o A_c$
      
      $k_a =\dfrac{2a_1}{a_o}$
    - switching modulator
      
      $A_c' = \dfrac{A_c }{2}$
      
      $k_a = \dfrac{4}{\pi A_c}$
  - AM demodulator
    - envelop detector
      
      for detection of peaks : $\tau_c =R_s C << \dfrac{1}{f_c}$
      
      to avoid diagonal clipping : $\tau_d =R_L C << \dfrac{1}{f_m}$
      
      to avoid fluctuations or steep discharging : $\tau_d =R_L C >> \dfrac{1}{f_c}$
    - synchronous detector (costly)
      
      received signal is multiplied by locally generated carrier
      
      $c'(t)= A_c ' cos[(\omega_c + \Delta \omega )t + \Delta \phi]$ and then passed through LPF
      
      LPF output: $\frac{A_c A_c '}{2}\cos(\Delta \omega t +\Delta\phi)+\frac{ A_c ' m(t)}{2}\cos(\Delta \omega t +\Delta\phi)$
    - SLD (impractical method)
      
      $\Big(\dfrac{S}{I}\Big)_{min} = \dfrac{2}{\mu}$
- DSB-SC
  
  suppressed carrier impulse
  
  $\boxed{S_{\text{DSB-SC}}(t)= A_c m(t)\cos (2\pi f_c t)}$
  
  $P = P_{m}P_c = \overline {m^2(t)} \times \dfrac{A_c^2}{2}$
  
  DSB modulator --→ multiplier or product modulator
  - DSB-SC modulator
    - balanced modulator
      
      $s(t) =2A_c k_a m(t)\cos (2\pi f_c t)$
    - ring modulator (product modulating)
      
      BPF output : $y(t) =\dfrac{4}{\pi}m(t) \cos (\omega _c t)$
      
      $x(t) \propto m(t) c(t) + m(t)\cos (3\omega_c t)+ m(t)\cos (5\omega_c t)+\dots$
  - synchronous detector
    
    output : $y(t) = \dfrac{kA_c A_c '}{2}m(t)$ where k is filter gain
- SSB-SC
  
  $\boxed{S(t)= \dfrac{A_C}{2} m(t)\cos (2\pi f_c t)\pm \dfrac{A_C}{2}\hat{m}(t)\sin (2\pi f_c t)}$
  - SSB-SC modulator
    - phase discrimination
      - -90$^{\circ}$ hilbert wideband transform (phase shifter) is not practical
        
        $\phi = - \tan^{-1} (\omega RC)$ RC → High , $\omega$ → fixed
    - frequency discrimination (practical method)
  - SSB-SC demodulator (synchronous detector)
- VSB-SC
  
  SSB-SC signal + small portion of adjacent sideband
angle modulation
- $|x(t)|_{max}$
  
  \[\begin{split} \def\arraystretch{1.5}\begin{array}{c:c} A\cos (\omega_o t) + B\sin(\omega_o t)& \sqrt{A^2 + B^2} \\ \hdashline A\cos (\omega_o t) + B\cos(\omega_o t) & |A+B| \\ \hdashline A\cos (\omega_1 t) + B\cos(\omega_2 t) & |A+B| \\ \hdashline A\cos (\omega_1 t) + B\sin(\omega_2 t) & <|A+B| \text{ if $A\ne B$} \\ & =|A+B| \text{ if $A = B$} \end{array} \end{split}\]
$\omega _i (t) =\dfrac{d \theta_i(t)}{dt}$

$f _i (t) =\dfrac{1}{2\pi}\dfrac{d \theta_i(t)}{dt} = f_c +\Delta f(t)$

$\theta_i(t) = \displaystyle\int_{-\infty}^{t}\omega _i (t) dt=2\pi\int_{-\infty}^{t}f _i (t) dt$

$\boxed{s(t) =A_c \cos (\omega _c t +\Delta\phi(t))=A_c \cos (\theta_i(t))}$

$|\Delta\omega(t)|_{max} =\Big|\dfrac{d\Delta \phi}{dt} \Big| _{max}$ $|\Delta f(t)|_{max} = \dfrac{1}{2\pi}\Big|\dfrac{d\Delta \phi}{dt} \Big| _{max}$
- frequency modulation
  
  $\dfrac{d\Delta \phi}{dt} \propto m(t)$
  
  $`K_f: \frac{rad}{V-sec}`$ $\Delta \omega (t) = K_f m(t)$ $\boxed{\omega_i(t) = \omega_c + K_f m(t)}$
  
  $`K_f: \frac{Hz}{V}`$ $\Delta f(t) = K_f m(t)$ $\boxed{f_i(t) = f_c + K_f m(t) }$
  
  modulation index / deviation ratio $\beta$
  
  $\boxed{\beta _{FM} = \dfrac{|\Delta\omega(t)|_{max}}{\omega_{max}}=\dfrac{K_f|m(t)|_{max}}{\omega_{max}}}$
  
  $\boxed{\beta _{FM} = \dfrac{|\Delta f(t)|_{max}}{f_{max}}=\dfrac{K_f|m(t)|_{max}}{f_{max}}}$
- phase modulation
  
  $\Delta \phi\propto m(t)$
  
  $`K_p: \frac{rad}{V}`$ $\Delta \phi(t) = K_p m(t)$ $\boxed{\theta_i(t) = \omega_ct + K_p m(t)}$
  
  modulation index / deviation ratio $\beta$
  
  $\boxed{\beta _{PM} = \dfrac{|\Delta f(t)|_{max}}{f_{max}}=\dfrac{\dfrac{K_p}{2\pi}\Big|\dfrac{dm(t)}{dt}\Big|_{max}}{f_{max}}}$
- bessel’s function
  
  $J_{-n}(\beta) = (-1)^n J_n(\beta)$
  
  $\sum \limits_{n=-\infty}^{\infty} J^2_n (\beta) =1$
  
  $J_n (\beta) =0$ for $n>>\beta$
  
  $J_0 (\beta) =0$ for $\beta =2.4,5.5,8.6,11.8$
$\boxed{s(t) = A_c \cos (\omega_c t+\beta \sin \omega_m t) \equiv A_c \sum \limits_{n=-\infty}^{\infty}J_n (\beta)\cos [2\pi(f_c +nf_m) t]}$
- narrow band $(\beta <<1)$
  
  $\theta \rightarrow small$
  
  $\cos\theta \approx 1$
  
  $\sin\theta \approx 0$
  
  NBFM: $s(t)= A_c \cos (2\pi f_c t) - A_c \beta\sin (2\pi f_m t)\sin(2\pi f_c t)$
  
  NBPM: $s(t)= A_c \cos (2\pi f_c t) - A_c \beta\cos (2\pi f_m t)\sin(2\pi f_c t)$
- wide band
- BW
  
  $n^{th}$ order sideband ——→ BW = $n(2f_m)$
  
  carson's rule : $\boxed{BW=(\beta +1)2f_m}$ for 98% transmission ($\beta+1$ order sideband ) $J_{\beta+1} (\beta) \ne 0$
- power
  
  $n^{th}$ harmonic : $\overline {s^2 (t)} = P_c [J_o^2(\beta) +2(J_1^2(\beta)+J_2^2(\beta)+\dots +J_n^2(\beta))]$
  
  $J_o^2(\beta) +2(J_1^2(\beta)+J_2^2(\beta)+\dots +\infty = 1$
- frequency mixer
  
  frequency mixer : multiplier followed by BPF
  
  BPF → $f_c +f_l$ : upconverter
  
  BPF → $f_c \sim f_l$ : downconverter
- frequency multiplier or angle multiplier
  
  frequency multiplier or angle multiplier : square law device followed by BPF
  
  $A\cos(\theta) \longrightarrow \boxed {\times n} \longrightarrow A' \cos(n\theta)$
- wide band FM generation
  - armstrong method (indirect method)
  - direct method of FM generation
    
    VCO : modified hartley oscillator
    
    $L_{eq} =L_1 + L_2$
    
    $C_{junc} = C_o -km(t)$
    
    $\omega _i(t) = \dfrac{1}{\sqrt{LC_{junc}(t)}}$
    
    for VCO as FM generator : $\boxed{\dfrac{\Delta \omega}{\omega_o}<<<\dfrac{1}{2}}$ $\boxed{\dfrac{\Delta C}{C_o}<<<1}$
- FM demodulator
  
  $s(t) \rightarrow \boxed{\dfrac{d}{dt}} \xrightarrow{ \text{amplitude angle modulated}} \boxed{\text {ED}} \longrightarrow || \longrightarrow m(t)$ $|\Delta f_{max}| \le f_c$
  - practical FM demodulators
    - PLL
      - phase comparator
        
        multiplier followed by LPF : rejects the frequencies centered at $2f_c$
      - VCO
        
        sinusoidal waveform generator whose output frequency varies linearly with input message signal
        
        frequency of VCO → $f_c$
      phase lock : $|H(f)| \rightarrow \infty$ $v(t) = \dfrac{k_f}{k_v} m(t)$
      
      lock mode : $f_{s(t)} = f_{v(t)}$
      
      capture mode : $\phi_e (t) \rightarrow 0$
      
      $\text{lock range} \ge \text{capture range}$

J-Book

Communication System¶

analog communication¶