From c62d1f02f524bb247c70f51c9e4849c4f05c63c6 Mon Sep 17 00:00:00 2001 From: Peter Tanner Date: Tue, 5 Nov 2024 02:18:47 +0800 Subject: [PATCH] Final update 2024. Anyone may add PRs to suggest changes --- README.html | 343 +++++++++++++++++++----------- README.md | 228 ++++++++++++-------- README.pdf | Bin 666095 -> 706173 bytes images/Binary_erasure_channel.svg | 34 +++ images/rect.drawio.svg | 2 +- 5 files changed, 403 insertions(+), 204 deletions(-) create mode 100644 images/Binary_erasure_channel.svg diff --git a/README.html b/README.html index c71d0ab..1b5f148 100644 --- a/README.html +++ b/README.html @@ -383,15 +383,15 @@ along with this program. If not, see http -

u(t)={1,t>012,t=00,t<0Unit Step Functionsgn(t)={+1,t>00,t=0−1,t<0Signum Functionsinc(2Wt)=sin⁡(2πWt)2πWtsinc Functionrect(t)=Π(t)={1,−0.5<t<0.50,∣t∣>0.5Rectangular/Gate Functiontri(t/T)={1−∣t∣T,∣t∣<T0,∣t∣>T=Π(t/T)∗Π(t/T)Triangle Functiong(t)∗h(t)=(g∗h)(t)=∫∞∞g(τ)h(t−τ)dτConvolution\begin{align*} +

u(t)={1,t>012,t=00,t<0Unit Step Functionsgn(t)={+1,t>00,t=0−1,t<0Signum Functionsinc(2Wt)=sin⁡(2πWt)2πWtsinc Functionrect(t)=Π(t)={1,−0.5<t<0.50,∣t∣>0.5Rectangular/Gate Functiontri(t/T)={1−∣t∣T,∣t∣<T0,∣t∣≥T=1TΠ(t/T)∗Π(t/T)Triangle Functiong(t)∗h(t)=(g∗h)(t)=∫∞∞g(τ)h(t−τ)dτConvolution\begin{align*} u(t) &= \begin{cases} 1, & t > 0 \\ \frac{1}{2}, & t = 0 \\ 0, & t < 0 \end{cases}&\text{Unit Step Function}\\ \text{sgn}(t) &= \begin{cases} +1, & t > 0 \\ 0, & t = 0 \\ -1, & t < 0 \end{cases}&\text{Signum Function}\\ \text{sinc}(2Wt) &= \frac{\sin(2\pi W t)}{2\pi W t}&\text{sinc Function}\\ \text{rect}(t) = \Pi(t) &= \begin{cases} 1, & -0.5 < t < 0.5 \\ 0, & \lvert t \rvert > 0.5 \end{cases}&\text{Rectangular/Gate Function}\\ - \text{tri}(t/T) &= \begin{cases} 1 - \frac{|t|}{T}, & \lvert t\rvert < T \\ 0, & \lvert t \rvert > T \end{cases}=\Pi(t/T)*\Pi(t/T)&\text{Triangle Function}\\ + \text{tri}(t/T) &= \begin{cases} 1 - \frac{|t|}{T}, & \lvert t\rvert < T \\ 0, & \lvert t \rvert \geq T \end{cases}=\frac{1}{T}\Pi(t/T)*\Pi(t/T)&\text{Triangle Function}\\ g(t)*h(t)=(g*h)(t)&=\int_\infty^\infty g(\tau)h(t-\tau)d\tau&\text{Convolution}\\ \end{align*} -

+

Fourier transform of continuous time periodic signal

Required for some questions on sampling:

@@ -408,46 +408,46 @@ along with this program. If not, see http

Shape functions

+

rect and tri functions

+

Random processes examples

+

Example: separate RV from expressionX(t)=Acos⁡(2πfct)A∼N(μ=5,σ2=1)  ⟹  E[X(t)]=E[Acos⁡(2πfct)]=E[A]cos⁡(2πfct)=5cos⁡(2πfct)Example: random phaseX(t)=Bcos⁡(2πfct+θ)θ∼U(0,2π)  ⟹  E[X(t)]=E[Bcos⁡(2πfct+θ)]=B∫02π12π⏟uniformcos⁡(2πfct+θ)dθ=0\begin{align*} + &\text{Example: separate RV from expression}\\ + X(t) &= A\cos(2\pi f_c t)\quad A\thicksim \mathcal{N}(\mu=5,\sigma^2=1)\\ + \implies E[X(t)] &= E[A\cos(2\pi f_c t)] = E[A]\cos(2\pi f_c t) = 5\cos(2\pi f_c t)\\ + &\text{Example: random phase}\\ + X(t) &= B\cos(2\pi f_c t+\theta)\quad \theta\thicksim \mathcal{U}(0,2\pi)\\ + \implies E[X(t)] &= E[B\cos(2\pi f_c t+\theta)] = B\int_0^{2\pi}\underbrace{\frac{1}{2\pi}}_{\text{uniform}}\cos(2\pi f_c t+\theta)d\theta=0 +\end{align*} +

+

Wide sense stationary (WSS)

+

Two conditions for WSS:

- - + + - - + + + + + +
rect\text{rect} functiontri\text{tri} functionConstant meanAutocorrelation only dependent on time difference
rectTODO: Add graphic.μX(t)=μX Constant\mu_X(t) = \mu_X\text{ Constant}RXX(t1,t2)=RX(t1−t2)=RX(τ)R_{XX}(t_1,t_2)=R_X(t_1-t_2)=R_X(\tau)
μX(t)=E[X(t)]\mu_X(t)=E[X(t)]E[X(t1)X(t2)]=E[X(t)X(t+τ)]E[X(t_1)X(t_2)]=E[X(t)X(t+\tau)]
-

Tri placeholder: For tri(t/T)=1−∥t∥/T\text{tri}(t/T)=1-\|t\|/T, Intersects xx axis at −T-T and TT and yy axis at 11.

-

Bessel function

-

∑n∈ZJn2(β)=1Jn(β)=(−1)nJ−n(β)\begin{align*} - \sum_{n\in\mathbb{Z}}{J_n}^2(\beta)&=1\\ - J_n(\beta)&=(-1)^nJ_{-n}(\beta) -\end{align*} -

- -

White noise

-

RW(τ)=N02δ(τ)=kT2δ(τ)=σ2δ(τ)Gw(f)=N02N0=kTGy(f)=∣H(f)∣2Gw(f)Gy(f)=G(f)Gw(f)\begin{align*} -R_W(\tau)&=\frac{N_0}{2}\delta(\tau)=\frac{kT}{2}\delta(\tau)=\sigma^2\delta(\tau)\\ -G_w(f)&=\frac{N_0}{2}\\ -N_0&=kT\\ -G_y(f)&=|H(f)|^2G_w(f)\\ -G_y(f)&=G(f)G_w(f)\\ -\end{align*} -

- -

WSS

-

μX(t)=μX ConstantRXX(t1,t2)=RX(t1−t2)=RX(τ)E[X(t1)X(t2)]=E[X(t)X(t+τ)]\begin{align*} - \mu_X(t) &= \mu_X\text{ Constant}\\ - R_{XX}(t_1,t_2)&=R_X(t_1-t_2)=R_X(\tau)\\ - E[X(t_1)X(t_2)]&=E[X(t)X(t+\tau)] -\end{align*} -

-

Ergodicity

⟨X(t)⟩T=12T∫−TTx(t)dt⟨X(t+τ)X(t)⟩T=12T∫−TTx(t+τ)x(t)dtE[⟨X(t)⟩T]=12T∫−TTx(t)dt=12T∫−TTmXdt=mX\begin{align*} \braket{X(t)}_T&=\frac{1}{2T}\int_{-T}^{T}x(t)dt\\ @@ -467,17 +467,13 @@ G_y(f)&=G(f)G_w(f)\\
ergodic in mean -

lim⁡T→∞⟨X(t)⟩T=mX(t)=mX\lim_{T\to\infty}\braket{X(t)}_T=m_X(t)=m_X

- -

lim⁡T→∞VAR[⟨X(t)⟩T]=0\lim_{T\to\infty}\text{VAR}[\braket{X(t)}_T]=0

- +lim⁡T→∞⟨X(t)⟩T=mX(t)=mX\lim_{T\to\infty}\braket{X(t)}_T=m_X(t)=m_X +lim⁡T→∞VAR[⟨X(t)⟩T]=0\lim_{T\to\infty}\text{VAR}[\braket{X(t)}_T]=0 ergodic in autocorrelation function -

lim⁡T→∞⟨X(t+τ)X(t)⟩T=RX(τ)\lim_{T\to\infty}\braket{X(t+\tau)X(t)}_T=R_X(\tau)

- -

lim⁡T→∞VAR[⟨X(t+τ)X(t)⟩T]=0\lim_{T\to\infty}\text{VAR}[\braket{X(t+\tau)X(t)}_T]=0

- +lim⁡T→∞⟨X(t+τ)X(t)⟩T=RX(τ)\lim_{T\to\infty}\braket{X(t+\tau)X(t)}_T=R_X(\tau) +lim⁡T→∞VAR[⟨X(t+τ)X(t)⟩T]=0\lim_{T\to\infty}\text{VAR}[\braket{X(t+\tau)X(t)}_T]=0 @@ -556,24 +552,24 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\

AM

-

CAM

-

x(t)=Accos⁡(2πfct)[1+kam(t)]=Accos⁡(2πfct)[1+mam(t)/Ac],where m(t)=Amm^(t) and m^(t) is the normalized modulating signalma=∣min⁡t(kam(t))∣Acka is the amplitude sensitivity (volt−1), ma is the modulation index.ma=Amax−AminAmax+Amin (Symmetrical m(t))ma=kaAm (Symmetrical m(t))Pc=Ac22Carrier powerPx=14ma2Ac2η=Signal PowerTotal Power=PxPx+PcBT=2fm=2B\begin{align*} - x(t)&=A_c\cos(2\pi f_c t)\left[1+k_a m(t)\right]=A_c\cos(2\pi f_c t)\left[1+m_a m(t)/A_c\right], \\ +

Conventional AM modulation (CAM)

+

x(t)=Accos⁡(2πfct)[1+kam(t)]=Accos⁡(2πfct)[1+mam(t)/Ac]CAM signalwhere m(t)=Amm^(t) and m^(t) is the normalized modulating signalma=∣min⁡t(kam(t))∣Acka is the amplitude sensitivity (volt−1), ma is the modulation index.ma=Amax−AminAmax+Amin (Symmetrical m(t))ma=kaAm (Symmetrical m(t))Pc=Ac22Carrier powerPs=14ma2Ac2Signal power, total of all 4 sideband power, single-tone caseη=Signal PowerTotal Power=PsPs+Pc=PsPxPower efficiencyBT=2fm=2B\begin{align*} + x(t)&=A_c\cos(2\pi f_c t)\left[1+k_a m(t)\right]=A_c\cos(2\pi f_c t)\left[1+m_a m(t)/A_c\right]\quad\text{CAM signal}\\ &\text{where $m(t)=A_m\hat m(t)$ and $\hat m(t)$ is the normalized modulating signal}\\ m_a &= \frac{|\min_t(k_a m(t))|}{A_c} \quad\text{$k_a$ is the amplitude sensitivity ($\text{volt}^{-1}$), $m_a$ is the modulation index.}\\ m_a &= \frac{A_\text{max}-A_\text{min}}{A_\text{max}+A_\text{min}}\quad\text{ (Symmetrical $m(t)$)}\\ m_a&=k_a A_m \quad\text{ (Symmetrical $m(t)$)}\\ P_c &=\frac{ {A_c}^2}{2}\quad\text{Carrier power}\\ - P_x &=\frac{1}{4}{m_a}^2{A_c}^2\\ - \eta&=\frac{\text{Signal Power}}{\text{Total Power}}=\frac{P_x}{P_x+P_c}\\ - B_T&=2f_m=2B + P_s &=\frac{1}{4}{m_a}^2{A_c}^2\quad\text{Signal power, \textbf{total} of all 4 sideband power, \textbf{single-tone} case}\\ + \eta&=\frac{\text{Signal Power}}{\text{Total Power}}=\frac{P_s}{P_s+P_c}=\frac{P_s}{P_x}\quad\text{Power efficiency}\\ + B_T&=2f_m=2B\\ \end{align*} -

+

BTB_T: Signal bandwidth BB: Bandwidth of modulating wave

Overmodulation (resulting in phase reversals at crossing points): ma>1m_a>1

-

DSB-SC

+

Double sideband suppressed carrier (DSB-SC)

xDSB(t)=Accos⁡(2πfct)m(t)BT=2fm=2B\begin{align*} x_\text{DSB}(t) &= A_c \cos{(2\pi f_c t)} m(t)\\ B_T&=2f_m=2B @@ -593,58 +589,58 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\ \end{align*}

-

Bessel form and magnitude spectrum (single tone)

-

s(t)=Accos⁡[2πfct+βsin⁡(2πfmt)]⇔s(t)=Ac∑n=−∞∞Jn(β)cos⁡[2π(fc+nfm)t]\begin{align*} - s(t) &= A_c\cos\left[2\pi f_c t + \beta \sin(2\pi f_m t)\right] \Leftrightarrow s(t)= A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[2\pi(f_c+nf_m)t] +

Bessel function

+

Jn(β)={J−n(β)n is even−J−n(β)n is odd1=∑n∈ZJn2(β)Conservation of power\begin{align*} + J_n(\beta)&=\begin{cases} + J_{-n}(\beta) & \text{$n$ is even}\\ + -J_{-n}(\beta) & \text{$n$ is odd} + \end{cases}\\ + 1&=\sum_{n\in\mathbb{Z}}{J_n}^2(\beta)&\text{Conservation of power}\\ \end{align*} -

+

+ +

Bessel form of FM signal

+

s(t)=Accos⁡[2πfct+βsin⁡(2πfmt)]⟺s(t)=Ac∑n=−∞∞Jn(β)cos⁡[2π(fc+nfm)t]\begin{align*} + s(t) &= A_c\cos\left[2\pi f_c t + \beta \sin(2\pi f_m t)\right]\\ + \Longleftrightarrow s(t) &= A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[2\pi(f_c+nf_m)t] +\end{align*} +

FM signal power

-

Pav=Ac22Pband_index=Ac2Jband_index2(β)2band_index=0  ⟹  fc+0fmband_index=1  ⟹  fc+1fm,…\begin{align*} - P_\text{av}&=\frac{ {A_c}^2}{2}\\ - P_\text{band\_index}&=\frac{ {A_c}^2{J_\text{band\_index}}^2(\beta)}{2}\\ - \text{band\_index}&=0\implies f_c+0f_m\\ - \text{band\_index}&=1\implies f_c+1f_m,\dots\\ +

Pav=Ac22Av. power of full signalPi=Ac2∣Ji(β)∣22Av. power of band ii=0  ⟹  fc+0fmMiddle bandi=1  ⟹  fc+1fm1st sidebandi=−1  ⟹  fc−1fm-1st sideband…\begin{align*} + P_\text{av}&=\frac{ {A_c}^2}{2}&\text{Av. power of full signal}\\ + P_\text{i}&=\frac{ {A_c}^2|{J_\text{i}}(\beta)|^2}{2}&\text{Av. power of band $i$}\\ + i=0&\implies f_c+0f_m&\text{Middle band}\\ + i=1&\implies f_c+1f_m&\text{1st sideband}\\ + i=-1&\implies f_c-1f_m&\text{-1st sideband}\\ + &\dots\\ \end{align*} -

+

Carson's rule to find BB (98% power bandwidth rule)

-

B=2Mfm=2(β+1)fm=2(Δfmax+fm)=2(D+1)WmB={2(Δfmax+fm)=2(Δfmax+Wm)FM, sinusoidal message2(Δϕmax+1)fm=2(Δϕmax+1)WmPM, sinusoidal message\begin{align*} -B &= 2Mf_m = 2(\beta + 1)f_m\\ - &= 2(\Delta f_\text{max}+f_m)\\ - &= 2(D+1)W_m\\ +

B=2(β+1)fmB=2(Δfmax+fm)B=2(D+1)WmB={2(Δfmax+fm)=2(Δfmax+Wm)FM, sinusoidal message2(Δϕmax+1)fm=2(Δϕmax+1)WmPM, sinusoidal messageD<1,β<1  ⟹  NarrowbandD>1,β>1  ⟹  Wideband\begin{align*} +B &= 2(\beta + 1)f_m\\ +B &= 2(\Delta f_\text{max}+f_m)\\ +B &= 2(D+1)W_m\\ B &= \begin{cases} 2(\Delta f_\text{max}+f_m)=2(\Delta f_\text{max}+W_m) & \text{FM, sinusoidal message}\\ 2(\Delta\phi_\text{max} + 1)f_m=2(\Delta \phi_\text{max}+1)W_m & \text{PM, sinusoidal message} -\end{cases}\\ +\end{cases}\\\\ + & D<1,\beta<1 \implies \text{Narrowband}\quad D>1,\beta>1\implies \text{Wideband} \end{align*} -

+

-

Complex envelope

-

s(t)=Accos⁡(2πfct+βsin⁡(2πfmt))⇔s~(t)=Acexp⁡(jβsin⁡(2πfmt))s(t)=Re[s~(t)exp⁡(j2πfct)]s~(t)=Ac∑n=−∞∞Jn(β)exp⁡(j2πfmt)\begin{align*} - s(t)&=A_c\cos(2\pi f_c t+\beta\sin(2\pi f_m t)) \Leftrightarrow \tilde{s}(t) = A_c\exp(j\beta\sin(2\pi f_m t))\\ +

Complex envelope of a FM signal

+

s(t)=Accos⁡(2πfct+βsin⁡(2πfmt))⟺s~(t)=Acexp⁡(jβsin⁡(2πfmt))s(t)=Re[s~(t)exp⁡(j2πfct)]s~(t)=Ac∑n=−∞∞Jn(β)exp⁡(j2πfmt)\begin{align*} + s(t)&=A_c\cos(2\pi f_c t+\beta\sin(2\pi f_m t))\\ + \Longleftrightarrow \tilde{s}(t) &= A_c\exp(j\beta\sin(2\pi f_m t))\\ s(t)&=\text{Re}[\tilde{s}(t)\exp{(j2\pi f_c t)}]\\ \tilde{s}(t) &= A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\exp(j2\pi f_m t) \end{align*} -

+

-

Band

- - - - - - - - - - - - - -
NarrowbandWideband
D<1,β<1D<1,\beta<1D>1,β>1D>1,\beta>1

Power, energy and autocorrelation

-

GWGN(f)=N02Gx(f)=∣H(f)∣2Gw(f) (PSD)Gx(f)=G(f)Gw(f) (PSD)Gx(f)=lim⁡T→∞∣XT(f)∣2T (PSD)Gx(f)=F[Rx(τ)] (WSS)Px=σx2=∫RGx(f)dfFor zero meanPx=σx2=lim⁡t→∞1T∫−T/2T/2∣x(t)∣2dtFor zero meanP[Acos⁡(2πft+ϕ)]=A22Power of sinusoid Ex=∫−∞∞∣x(t)∣2dt=∫−∞∞∣X(f)∣2dfParseval’s theoremRx(τ)=F(Gx(f))PSD to Autocorrelation\begin{align*} +

GWGN(f)=N02Gx(f)=∣H(f)∣2Gw(f) (PSD)Gx(f)=G(f)Gw(f) (PSD)Gx(f)=lim⁡T→∞∣XT(f)∣2T (PSD)Gx(f)=F[Rx(τ)] (WSS)Px=σx2=∫RGx(f)dfFor zero meanPx=σx2=lim⁡t→∞1T∫−T/2T/2∣x(t)∣2dtFor zero meanP[Acos⁡(2πft+ϕ)]=A22Power of sinusoid Ex=∫−∞∞∣x(t)∣2dt=∫−∞∞∣X(f)∣2dfParseval’s theoremRx(τ)=F(Gx(f))PSD to AutocorrelationPx=Rx(0)Average power of WSS process x(t)\begin{align*} G_\text{WGN}(f)&=\frac{N_0}{2}\\ G_x(f)&=|H(f)|^2G_w(f)\text{ (PSD)}\\ G_x(f)&=G(f)G_w(f)\text{ (PSD)}\\ @@ -654,21 +650,29 @@ B &= \begin{cases} P_x&={\sigma_x}^2=\lim_{t\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}|x(t)|^2dt\quad\text{For zero mean}\\ P[A\cos(2\pi f t+\phi)]&=\frac{A^2}{2}\quad\text{Power of sinusoid }\\ E_x&=\int_{-\infty}^{\infty}|x(t)|^2dt=\int_{-\infty}^{\infty}|X(f)|^2df\quad\text{Parseval's theorem}\\ - R_x(\tau) &= \mathfrak{F}(G_x(f))\quad\text{PSD to Autocorrelation} + R_x(\tau) &= \mathfrak{F}(G_x(f))\quad\text{PSD to Autocorrelation}\\ + P_x &= R_x(0)\quad\text{Average power of WSS process $x(t)$}\\ \end{align*} -

+

- -
-

Noise performance

-

Coherent detection system.

-

y(t)=m(t)cos⁡(2πfct+θ)=m(t)cos⁡2(2πfct+θ) After IF mixing.=m(t)12(1+cos⁡(4fct+2θ))=12m(t)+12cos⁡(4fct+2θ)  ⟹  Su(f)=(12)2Su(f)After LPF.\begin{align*} - y(t) &= m(t)\cos(2\pi f_c t+\theta) = m(t)\cos^2(2\pi f_c t+\theta)\text{ After IF mixing.}\\ - &= m(t)\frac{1}{2}(1+\cos(4 f_c t+2\theta))\\ - &= \frac{1}{2}m(t)+\frac{1}{2}\cos(4 f_c t+2\theta)\\ - \implies S_u(f) &= \left(\frac{1}{2}\right)^2S_u(f)\quad\text{After LPF.} +

White noise

+

RW(τ)=N02δ(τ)=kT2δ(τ)=σ2δ(τ)Gw(f)=N02\begin{align*} +R_W(\tau)&=\frac{N_0}{2}\delta(\tau)=\frac{kT}{2}\delta(\tau)=\sigma^2\delta(\tau)\\ +G_w(f)&=\frac{N_0}{2}\\ \end{align*} -

+

+ +

Noise performance

+

Use formualas from previous section, Power, energy and autocorrelation.
Use these formulas in particular:

@@ -727,27 +731,30 @@ Use these formulas in particular:

By Bob K - Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=94674142

Insert here figure 8.3 from M F Mesiya - Contemporary Communication Systems (Add image to images/sampling.png)

-

Cannot add directly due to copyright!

+

Cannot add directly due to copyright! +TODO: Make an open source replacement for this diagram Send a PR to GitHub.

sampling

sampling

Quantizer

-

Δ=xMax−xMin2kfor k-bit quantizer (V/lsb)\begin{align*} - \Delta &= \frac{x_\text{Max}-x_\text{Min}}{2^k} \quad\text{for $k$-bit quantizer (V/lsb)}\\ +

Δ=xMax−xMin2kfor k-bit quantizer (V/lsb)Quantizer step size Δ\begin{align*} + \Delta &= \frac{x_\text{Max}-x_\text{Min}}{2^k} \quad\text{for $k$-bit quantizer (V/lsb)}\quad\text{Quantizer step size $\Delta$}\\ \end{align*} -

+

Quantization noise

-

e:=y−xQuantization errorμE=E[E]=0Zero meanσE2=E[E2]−02=∫−Δ/2Δ/2e2×(1Δ)deWhere E∼1/Δ uniform over (−Δ/2,Δ/2)SQNR=Signal powerQuantization noiseSQNR(dB)=10log⁡10(SQNR)\begin{align*} +

e:=y−xQuantization errorμE=E[E]=0Zero meanPE=σE2=Δ212=2−2mV2/3Uniformly distributed errorSQNR=Signal powerQuantization noise power=PxPESQNR(dB)=10log⁡10(SQNR)m→m+A bits  ⟹  newSQNR(dB)=SQNR(dB)+6A dB\begin{align*} e &:= y-x\quad\text{Quantization error}\\ \mu_E &= E[E] = 0\quad\text{Zero mean}\\ - {\sigma_E}^2&=E[E^2]-0^2=\int_{-\Delta/2}^{\Delta/2}e^2\times\left(\frac{1}{\Delta}\right) de\quad\text{Where $E\thicksim 1/\Delta$ uniform over $(-\Delta/2,\Delta/2)$}\\ - \text{SQNR}&=\frac{\text{Signal power}}{\text{Quantization noise}}\\ - \text{SQNR(dB)}&=10\log_{10}(\text{SQNR}) + P_E&={\sigma_E}^2=\frac{\Delta^2}{12}=2^{-2m}V^2/3\quad\text{Uniformly distributed error}\\ + \text{SQNR}&=\frac{\text{Signal power}}{\text{Quantization noise power}}=\frac{P_x}{P_E}\\ + \text{SQNR(dB)}&=10\log_{10}(\text{SQNR})\\ + m\to m+A\text{ bits}&\implies \text{newSQNR(dB)}=\text{SQNR(dB)}+6A\text{ dB} \end{align*} -

+

Insert here figure 8.17 from M F Mesiya - Contemporary Communication Systems (Add image to images/quantizer.png)

-

Cannot add directly due to copyright!

+

Cannot add directly due to copyright! +TODO: Make an open source replacement for this diagram Send a PR to GitHub.

quantizer

quantizer

Line codes

@@ -766,6 +773,7 @@ Use these formulas in particular:

G_\text{unipolarRZ}(f)&=\frac{A^2}{16} \left(\sum _{l=-\infty }^{\infty } \delta \left(f-\frac{l}{T_b}\right) \left| \text{sinc}(\text{duty} \times l) \right| {}^2+T_b \left| \text{sinc}\left(\text{duty} \times f T_b\right) \right| {}^2\right), \text{NB}_0=2R_b \end{align*}

+

TODO: Someone please make plots of the PSD for all line code types in Mathematica or Python! Send a PR to GitHub.

Modulation and basis functions

Constellation diagrams

@@ -1246,6 +1254,18 @@ h400000v40h-400000z"/>​

Value tables for erf(x)\text{erf}(x) and Q(x)Q(x)

Q(x)Q(x) function

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You should use erf\text{erf} function table instead in exams using the identity Q(x)=12−12erf(x2)Q(x)=\frac{1}{2}-\frac{1}{2}\text{erf}\left(\frac{x}{\sqrt{2}}\right). Use this for validation.

@@ -1724,6 +1744,19 @@ h400000v40h-400000z"/>​

Adapted from table 6.1 M F Mesiya - Contemporary Communication Systems

erf(x)\text{erf}(x) function

+

Q(x)=12−12erf(x2)Q(x)=\frac{1}{2}-\frac{1}{2}\text{erf}(\frac{x}{\sqrt{2}}) +

@@ -1859,6 +1892,48 @@ h400000v40h-400000z"/>​

**The value of erf(3.30)\text{erf}(3.30) should be ≈0.999997\approx0.999997 instead, but this value is quoted in the formula table.

+

Q(x)Q(x) fast reference

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Using identity.

+ + + + + + + + + + + + + + + + + +
xxQ(x)Q(x)
2\sqrt{2}0.078650.07865
222\sqrt{2}0.002340.00234

Receiver output shit

ro(t)={so1(t)+no(t)code 1so2(t)+no(t)code 0n:AWGN with σo2\begin{align*} @@ -2165,6 +2240,12 @@ M1001 80h400000v40h-400000z"/>

Information theory

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Stats

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P(A∣B)=P(B∣A)P(A)P(B)=P(A,B)P(B)\begin{align*} + P(A|B) &= \frac{P(B|A)P(A)}{P(B)} = \frac{P(A,B)}{P(B)}\\ +\end{align*} +

+

Entropy for discrete random variables

H(x)≥0H(x)=−∑xi∈AxpX(xi)log⁡2(pX(xi))H(x,y)=−∑xi∈Ax∑yi∈AypXY(xi,yi)log⁡2(pXY(xi,yi))Joint entropyH(x,y)=H(x)+H(y)Joint entropy if x and y independentH(x∣y=yj)=−∑xi∈AxpX(xi∣y=yj)log⁡2(pX(xi∣y=yj))Conditional entropyH(x∣y)=−∑yj∈AypY(yj)H(x∣y=yj)Average conditional entropy, equivocationH(x∣y)=−∑xi∈Ax∑yi∈AypX(xi,yj)log⁡2(pX(xi∣y=yj))H(x∣y)=H(x,y)−H(y)H(x,y)=H(x)+H(y∣x)=H(y)+H(x∣y)\begin{align*} H(x) &\geq 0\\ @@ -2180,13 +2261,19 @@ M1001 80h400000v40h-400000z"/>

Entropy is maximized when all have an equal probability.

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Differential entropy for continuous random variables

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TODO: Cut out if not required

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h(x)=−∫RfX(x)log⁡2(fX(x))dx\begin{align*} - h(x) &= -\int_\mathbb{R}f_X(x)\log_2(f_X(x))dx +

Transition probability diagram

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Example for binary erasure channel where XX is input and YY is output:

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Binary erasure channel David Eppstein, Public domain, via Wikimedia Commons

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Equivalent to:

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P[Y=0∣X=0]=1−pP[Y=e∣X=0]=pP[Y=1∣X=1]=1−pP[Y=e∣X=1]=pP[X=0∣Y=0]=0Note the directionP[Y=0]=P[Y=0∣X=0]P[X=0]\begin{align*} + P[Y=0|X=0] &= 1-p\\ + P[Y=e|X=0] &= p\\ + P[Y=1|X=1] &= 1-p\\ + P[Y=e|X=1] &= p\\ + P[X=0|Y=0] &= 0\quad\text{Note the direction}\\ + P[Y=0] &= P[Y=0|X=0] P[X=0] \end{align*} -

- +

Mutual information

Mutual information

Amount of entropy decrease of xx after observation by yy.

@@ -2197,7 +2284,8 @@ M1001 80h400000v40h-400000z"/>

Channel model

Vertical, xx: input
-Horizontal, yy: output

+Horizontal, yy: output
+Remember P\mathbf{P} is a matrix where each element is P(yj∣xi)P(y_j|x_i)

P=[p11p12…p1Np21p22…p2N⋮⋮⋱⋮pM1pM2…pMN]\mathbf{P}=\left[\begin{matrix} p_{11} & p_{12} &\dots & p_{1N}\\ p_{21} & p_{22} &\dots & p_{2N}\\ @@ -2228,16 +2316,25 @@ M347 1759 V0 H263 V1759 v1800 v1759 h84z"/>

Fast procedure to calculate I(y;x)I(y;x)

-

1. Find H(x)2. Find [pY(b0)pY(b1)…pY(bj)]=[pX(a0)pX(a1)…pX(ai)]×P3. Multiply each row in P by pX(ai) since pXY(xi,yi)=P(yi∣xi)P(xi)4. Find H(x,y) using each element from (3.)5. Find H(x∣y)=H(x,y)−H(y)6. Find I(y;x)=H(x)−H(x∣y)\begin{align*} +

1. Find H(x)2. Find [pY(b0)pY(b1)…pY(bj)]=[pX(a0)pX(a1)…pX(ai)]×P3. Multiply each row in P by pX(ai) since pXY(ai,bi)=P(bi∣ai)P(ai)4. Find H(x,y) using each element from (3.)5. Find H(x∣y)=H(x,y)−H(y)6. Find I(x;y)=H(x)−H(x∣y)\begin{align*} &\text{1. Find }H(x)\\ &\text{2. Find }[\begin{matrix}p_Y(b_0)&p_Y(b_1)&\dots&p_Y(b_j)\end{matrix}] = [\begin{matrix}p_X(a_0)&p_X(a_1)&\dots&p_X(a_i)\end{matrix}]\times\mathbf{P}\\ - &\text{3. Multiply each row in $\textbf{P}$ by $p_X(a_i)$ since $p_{XY}(x_i,y_i)=P(y_i|x_i)P(x_i)$}\\ + &\text{3. Multiply each row in $\textbf{P}$ by $p_X(a_i)$ since $p_{XY}(a_i,b_i)=P(b_i|a_i)P(a_i)$}\\ &\text{4. Find $H(x,y)$ using each element from (3.)}\\ &\text{5. Find }H(x|y)=H(x,y)-H(y)\\ - &\text{6. Find }I(y;x)=H(x)-H(x|y)\\ + &\text{6. Find }I(x;y)=H(x)-H(x|y)\\ \end{align*} -

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Example of step 3:

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PXY=[P(y1∣x1)P(x1)P(y2∣x1)P(x1)…P(y1∣x2)P(x2)P(y2∣x2)P(x2)…⋮⋮⋱]\mathbf{P_{XY}}=\left[\begin{matrix} + P(y_1|x_1) P(x_1) & P(y_2|x_1) P(x_1) & \dots\\ + P(y_1|x_2) P(x_2) & P(y_2|x_2) P(x_2) & \dots\\ + \vdots & \vdots &\ddots +\end{matrix}\right] +

Channel types

@@ -2266,6 +2363,7 @@ M347 1759 V0 H263 V1759 v1800 v1759 h84z"/>

+

Note that the channel capacity is realized when the channel inputs are uniformly distributed (i.e. P(x1)=P(x2)=⋯=P(xN)=1NP(x_1)=P(x_2)=\dots=P(x_N)=\frac{1}{N})

Channel capacity of an AWGN channel

yi=xi+nini∼N(0,N0/2)y_i=x_i+n_i\quad n_i\thicksim N(0,N_0/2) @@ -2486,7 +2584,14 @@ M347 1759 V0 H263 V1759 v0 v1759 h84z"/>h~(t)\tilde{h}(t) should be divided by two.

  • Convolutions: do not forget width when using graphical method
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  • todo: add more items to check
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  • 2W2W for rectangle functions
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  • Scale sampled spectrum by fsf_s
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  • 2fc2f_c for spectrum after IF mixing.
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  • Square transfer function for PSD Gy(f)=∣H(f)∣2Gx(f)G_y(f)=|H(f)|^\mathbf{2}G_x(f)
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  • Square besselJ function for FM power ∣Jn(β)∣2|J_n(\beta)|^\mathbf{2}
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  • Bandwidth: only consider positive frequencies (so the bandwidth of an AM signal will be the range from the lowest to greatest sideband frequency. For a rectangular function, it will be from 0 to W).
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  • TODO: add more items to check
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  • TODO: add some graphics for these checklist items
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