From d48ff87455714a742eda1036cbd6c8e15c668d5e Mon Sep 17 00:00:00 2001 From: Peter Tanner Date: Wed, 30 Oct 2024 19:16:15 +0800 Subject: [PATCH] Add new information on onesearch papers --- README.html | 130 ++++++++++++++++++++++++++++++++------ README.md | 178 ++++++++++++++++++++++++++++++++++++++++++++++++---- README.pdf | Bin 624140 -> 627778 bytes 3 files changed, 278 insertions(+), 30 deletions(-) diff --git a/README.html b/README.html index 3a1373d..33961cd 100644 --- a/README.html +++ b/README.html @@ -145,7 +145,7 @@

If you have issues or suggestions, raise them on GitHub. I accept pull requests for fixes or suggestions but the content must not be copyrighted under a non-GPL compatible license.

Download PDF 📄

It is recommended to refer to use the PDF copy instead of whatever GitHub renders.

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License and information

+

License and information

Notes are open-source and licensed under the GNU GPL-3.0. You must include the full-text of the license and follow its terms when using these notes or any diagrams in derivative works (but not when printing as notes)

Copyright (C) 2024 Peter Tanner

@@ -162,6 +162,40 @@ GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see http://www.gnu.org/licenses/.

+

Other advice for this unit

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Get more exam papers on OneSearch

+ +
+Listing of examination papers on OneSearch + +
+

Tests

+ +

Printable notes begins on next page (in PDF)

+

Fourier transform identities

@@ -273,20 +307,22 @@ along with this program. If not, see http g(t)*h(t)=(g*h)(t)&=\int_\infty^\infty g(\tau)h(t-\tau)d\tau&\text{Convolution}\\ \end{align*}

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Fourier transform of continuous time periodic signal

Required for some questions on sampling:

Transform a continuous time-periodic signal xp(t)=n=x(tnTs)x_p(t)=\sum_{n=-\infty}^\infty x(t-nT_s) with period TsT_s:

Xp(f)=n=Cnδ(fnfs)fs=1TsX_p(f)=\sum_{n=-\infty}^\infty C_n\delta(f-nf_s)\quad f_s=\frac{1}{T_s}

+

Calculate CnC_n coefficient as follows from xp(t)x_p(t):

Cn=1TsTsxp(t)exp(j2πfst)dt=1TsX(nfs)(TODO: Check)x(tnTs) is contained in the interval Ts\begin{align*} - % C_n&=X_p(nf_s)\\ C_n&=\frac{1}{T_s} \int_{T_s} x_p(t)\exp(-j2\pi f_s t)dt\\ &=\frac{1}{T_s} X(nf_s)\quad\color{red}\text{(TODO: Check)}\quad\color{white}\text{$x(t-nT_s)$ is contained in the interval $T_s$} \end{align*}

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rect\text{rect} function

rect

Bessel function

@@ -295,6 +331,7 @@ along with this program. If not, see
http 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)\\ @@ -304,6 +341,7 @@ G_y(f)&=|H(f)|^2G_w(f)\\ G_y(f)&=G(f)G_w(f)\\ \end{align*}

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WSS

μX(t)=μX ConstantRXX(t1,t2)=RX(t1t2)=RX(τ)E[X(t1)X(t2)]=E[X(t)X(t+τ)]\begin{align*} \mu_X(t) &= \mu_X\text{ Constant}\\ @@ -311,6 +349,7 @@ G_y(f)&=G(f)G_w(f)\\ E[X(t_1)X(t_2)]&=E[X(t)X(t+\tau)] \end{align*}

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Ergodicity

X(t)T=12TTTx(t)dtX(t+τ)X(t)T=12TTTx(t+τ)x(t)dtE[X(t)T]=12TTTx(t)dt=12TTTmXdt=mX\begin{align*} \braket{X(t)}_T&=\frac{1}{2T}\int_{-T}^{T}x(t)dt\\ @@ -318,6 +357,7 @@ G_y(f)&=G(f)G_w(f)\\ E[\braket{X(t)}_T]&=\frac{1}{2T}\int_{-T}^{T}x(t)dt=\frac{1}{2T}\int_{-T}^{T}m_Xdt=m_X\\ \end{align*}

+
@@ -351,6 +391,7 @@ G_y(f)&=G(f)G_w(f)\\ \sum_{x=-\infty}^\infty(f(x a)\delta(\omega-x b))&=f\left(\frac{\omega a}{b}\right) \end{align*}

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Other trig

cos2θ=2cos2θ1cos2θ+12=cos2θejαejα=2jsin(α)ejα+ejα=2cos(α)cos(A)=cos(A)sin(A)=sin(A)sin(A+π/2)=cos(A)sin(Aπ/2)=cos(A)cos(Aπ/2)=sin(A)cos(A+π/2)=sin(A)xRsinc(Ax)=1A\begin{align*} \cos2\theta=2 \cos^2 \theta-1&\Leftrightarrow\frac{\cos2\theta+1}{2}=\cos^2\theta\\ @@ -365,6 +406,7 @@ G_y(f)&=G(f)G_w(f)\\ \int_{x\in\mathbb{R}}\text{sinc}(A x) &= \frac{1}{|A|}\\ \end{align*}

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cos(A+B)=cos(A)cos(B)sin(A)sin(B)sin(A+B)=sin(A)cos(B)+cos(A)sin(B)cos(A)cos(B)=12(cos(AB)+cos(A+B))cos(A)sin(B)=12(sin(A+B)sin(AB))sin(A)sin(B)=12(cos(AB)cos(A+B))\begin{align*} \cos(A+B) &= \cos (A) \cos (B)-\sin (A) \sin (B) \\ \sin(A+B) &= \sin (A) \cos (B)+\cos (A) \sin (B) \\ @@ -373,6 +415,7 @@ G_y(f)&=G(f)G_w(f)\\ \sin(A)\sin(B) &= \frac{1}{2} (\cos (A-B)-\cos (A+B)) \\ \end{align*}

+

cos(A)+cos(B)=2cos(A2B2)cos(A2+B2)cos(A)cos(B)=2sin(A2B2)sin(A2+B2)sin(A)+sin(B)=2sin(A2+B2)cos(A2B2)sin(A)sin(B)=2sin(A2B2)cos(A2+B2)cos(A)+sin(B)=2sin(A2B2π4)sin(A2+B2+π4)cos(A)sin(B)=2sin(A2+B2π4)sin(A2B2+π4)\begin{align*} \cos(A)+\cos(B) &= 2 \cos \left(\frac{A}{2}-\frac{B}{2}\right) \cos \left(\frac{A}{2}+\frac{B}{2}\right) \\ \cos(A)-\cos(B) &= -2 \sin \left(\frac{A}{2}-\frac{B}{2}\right) \sin \left(\frac{A}{2}+\frac{B}{2}\right) \\ @@ -382,6 +425,7 @@ G_y(f)&=G(f)G_w(f)\\ \cos(A)-\sin(B)&= -2 \sin \left(\frac{A}{2}+\frac{B}{2}-\frac{\pi }{4}\right) \sin \left(\frac{A}{2}-\frac{B}{2}+\frac{\pi }{4}\right) \\ \end{align*}

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IQ/Complex envelope

Def. g~(t)=gI(t)+jgQ(t)\tilde{g}(t)=g_I(t)+jg_Q(t) as the complex envelope. Best to convert to ejθe^{j\theta} form.

Convert complex envelope representation to time-domain representation of signal

@@ -404,6 +448,7 @@ c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722 c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5 c53.7,-170.3,84.5,-266.8,92.5,-289.5z M1001 80h400000v40h-400000z"/>AmplitudePhase=A(t)cos(ϕ(t))In-phase component=A(t)sin(ϕ(t))Quadrature-phase component

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For transfer function

h(t)=hI(t)cos(2πfct)hQ(t)sin(2πfct)=2Re[h~(t)exp(j2πfct)]h~(t)=hI(t)/2+jhQ(t)/2=A(t)/2exp(jϕ(t))\begin{align*} h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\ @@ -411,6 +456,7 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\ \Rightarrow\tilde{h}(t)&=h_I(t)/2+jh_Q(t)/2=A(t)/2\exp{(j\phi(t))} \end{align*}

+

AM

CAM

ma=mintkam(t)Acka is the amplitude sensitivity (volt1), ma is the modulation index.ma=AmaxAminAmax+Amin (Symmetrical m(t))ma=kaAm (Symmetrical m(t))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 signalPc=Ac22Carrier powerPx=14ma2Ac2η=Signal PowerTotal Power=PxPx+PcBT=2fm=2B\begin{align*} @@ -425,6 +471,7 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\ B_T&=2f_m=2B \end{align*}

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BTB_T: Signal bandwidth BB: Bandwidth of modulating wave

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

@@ -434,6 +481,7 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\ B_T&=2f_m=2B \end{align*}

+

FM/PM

s(t)=Accos[2πfct+kpm(t)]Phase modulated (PM)s(t)=Accos[2πfct+2πkf0tm(τ)dτ]Frequency modulated (FM)s(t)=Accos[2πfct+βsin(2πfmt)]FM single toneβ=Δffm=kfAmModulation indexΔf=βfm=kfAmfm=maxt(kfm(t))mint(kfm(t))Maximum frequency deviationD=ΔfWmDeviation ratio, where Wm is bandwidth of m(t) (Use FT)\begin{align*} s(t) &= A_c\cos\left[2\pi f_c t + k_p m(t)\right]\quad\text{Phase modulated (PM)}\\ @@ -444,11 +492,13 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\ D&=\frac{\Delta f}{W_m}\quad\text{Deviation ratio, where $W_m$ is bandwidth of $m(t)$ (Use FT)} \end{align*}

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Bessel form and magnitude spectrum (single tone)

s(t)=Accos[2πfct+βsin(2πfmt)]s(t)=Acn=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] \end{align*}

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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}\\ @@ -457,6 +507,7 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\ \text{band\_index}&=1\implies f_c+1f_m,\dots\\ \end{align*}

+

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

B=2Mfm=2(β+1)fm=2(Δf+fm)=2(kfAm+fm)=2(D+1)WmB={2(Δf+fm)FM, sinusoidal message2(Δϕ+1)fmPM, sinusoidal message\begin{align*} B &= 2Mf_m = 2(\beta + 1)f_m\\ @@ -469,6 +520,7 @@ B &= \begin{cases} \end{cases}\\ \end{align*}

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Δf\Delta f of arbitrary modulating signal

Find instantaneous frequency fFMf_\text{FM}.

MM: Number of pairs of significant sidebands

@@ -483,6 +535,7 @@ D &= \frac{\Delta f}{W_m}\\ B_T &= 2(D+1)W_m \end{align*}

+

Complex envelope

s(t)=Accos(2πfct+βsin(2πfmt))s~(t)=Acexp(jβsin(2πfmt))s(t)=Re[s~(t)exp(j2πfct)]s~(t)=Acn=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))\\ @@ -490,6 +543,7 @@ B_T &= 2(D+1)W_m \tilde{s}(t) &= A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\exp(j2\pi f_m t) \end{align*}

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Band

@@ -519,6 +573,7 @@ B_T &= 2(D+1)W_m R_x(\tau) &= \mathfrak{F}(G_x(f))\quad\text{PSD to Autocorrelation} \end{align*}

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Noise performance

CNRin=PinPnoiseCNRin,FM=A22WN0SNRFM=3A2kf2P2N0W3SNR(dB)=10log10(SNR)Decibels from ratio\begin{align*} @@ -528,6 +583,7 @@ B_T &= 2(D+1)W_m \text{SNR(dB)} &= 10\log_{10}(\text{SNR}) \quad\text{Decibels from ratio} \end{align*}

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Sampling

t=nTsTs=1fsxs(t)=x(t)δs(t)=x(t)nZδ(tnTs)=nZx(nTs)δ(tnTs)Xs(f)=X(f)nZδ(fnTs)=X(f)nZδ(fnfs)B>12fs,2B>fsAliasing\begin{align*} t&=nT_s\\ @@ -537,6 +593,7 @@ B_T &= 2(D+1)W_m B&>\frac{1}{2}f_s, 2B>f_s\rightarrow\text{Aliasing}\\ \end{align*}

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Procedure to reconstruct sampled signal

Analog signal x(t)x'(t) which can be reconstructed from a sampled signal xs(t)x_s(t): Put xs(t)x_s(t) through LPF with maximum frequency of fs/2f_s/2 and minimum frequency of fs/2-f_s/2. Anything outside of the BPF will be attenuated, therefore nn which results in frequencies outside the BPF will evaluate to 00 and can be ignored.

Example: fs=5000    LPF[2500,2500]f_s=5000\implies \text{LPF}\in[-2500,2500]

@@ -549,14 +606,15 @@ B_T &= 2(D+1)W_m

Transform a continuous time-periodic signal xp(t)=n=x(tnTs)x_p(t)=\sum_{n=-\infty}^\infty x(t-nT_s) with period TsT_s:

Xp(f)=n=Cnδ(fnfs)fs=1TsX_p(f)=\sum_{n=-\infty}^\infty C_n\delta(f-nf_s)\quad f_s=\frac{1}{T_s}

+

Calculate CnC_n coefficient as follows from xp(t)x_p(t):

Cn=1TsTsxp(t)exp(j2πfst)dt=1TsX(nfs)(TODO: Check)x(tnTs) is contained in the interval Ts\begin{align*} - % C_n&=X_p(nf_s)\\ C_n&=\frac{1}{T_s} \int_{T_s} x_p(t)\exp(-j2\pi f_s t)dt\\ &=\frac{1}{T_s} X(nf_s)\quad\color{red}\text{(TODO: Check)}\quad\color{white}\text{$x(t-nT_s)$ is contained in the interval $T_s$} \end{align*}

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Nyquist criterion for zero-ISI

Do not transmit more than 2B2B samples per second over a channel of BB bandwidth.

@@ -570,6 +628,7 @@ B_T &= 2(D+1)W_m \Delta &= \frac{x_\text{Max}-x_\text{Min}}{2^k} \quad\text{for $k$-bit quantizer (V/lsb)}\\ \end{align*}

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Quantization noise

e:=yxQuantization errorμE=E[E]=0Zero meanσE2=E[E2]02=Δ/2Δ/2e2×(1Δ)deWhere E1/Δ uniform over (Δ/2,Δ/2)SQNR=Signal powerQuantization noiseSQNR(dB)=10log10(SQNR)\begin{align*} e &:= y-x\quad\text{Quantization error}\\ @@ -579,6 +638,7 @@ B_T &= 2(D+1)W_m \text{SQNR(dB)}&=10\log_{10}(\text{SQNR}) \end{align*}

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

Cannot add directly due to copyright!

quantizer @@ -599,6 +659,7 @@ B_T &= 2(D+1)W_m 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*}

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Modulation and basis functions

Constellation diagrams

BASK

@@ -617,9 +678,11 @@ s-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185 c-2,6,-10,9,-24,9 c-8,0,-12,-0.7,-12,-2z M1001 80 h400000v40h-400000z"/>cos(2πfct)0tTb

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Symbol mapping

bn:{1,0}an:{1,0}b_n:\{1,0\}\to a_n:\{1,0\}

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2 possible waveforms

s1(t)=AcTb2φ1(t)=2Ebφ1(t)s1(t)=0Since Eb=Eaverage=12(Ac22×Tb+0)=Ac24Tb\begin{align*} s_1(t)&=A_c\sqrt{\frac{T_b}{2}}\varphi_1(t)=\sqrt{2E_b}\varphi_1(t)\\ @@ -647,6 +710,7 @@ c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z M1001 80h400000v40h-400000z"/>φ1(t)=0Since Eb=Eaverage=21(2Ac2×Tb+0)=4Ac2Tb

+

Distance is d=2Ebd=\sqrt{2E_b}

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Symbol mapping

bn:{1,0}an:{1,1}b_n:\{1,0\}\to a_n:\{1,\color{green}-1\color{white}\}

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2 possible waveforms

s1(t)=AcTb2φ1(t)=Ebφ1(t)s1(t)=AcTb2φ1(t)=Ebφ2(t)Since Eb=Eaverage=12(Ac22×Tb+Ac22×Tb)=Ac22Tb\begin{align*} s_1(t)&=A_c\sqrt{\frac{T_b}{2}}\varphi_1(t)=\sqrt{E_b}\varphi_1(t)\\ @@ -725,6 +791,7 @@ c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z M1001 80h400000v40h-400000z"/>φ2(t)Since Eb=Eaverage=21(2Ac2×Tb+2Ac2×Tb)=2Ac2Tb

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Distance is d=2Ebd=2\sqrt{E_b}

+

4 possible waveforms

s1(t)=Es/2[φ1(t)+φ2(t)]s2(t)=Es/2[φ1(t)φ2(t)]s3(t)=Es/2[φ1(t)+φ2(t)]s4(t)=Es/2[φ1(t)φ2(t)]\begin{align*} s_1(t)&=\sqrt{E_s/2}\left[\varphi_1(t)+\varphi_2(t)\right]\\ @@ -813,6 +881,7 @@ c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z M1001 80h400000v40h-400000z"/>[φ1(t)φ2(t)]

+

Note on energy per symbol: Since si(t)=Ac|s_i(t)|=A_c, have to normalize distance as follows:

si(t)=AcT/2/2×[α1iφ1(t)+α2iφ2(t)]=TAc2/4[α1iφ1(t)+α2iφ2(t)]=Es/2[α1iφ1(t)+α2iφ2(t)]\begin{align*} s_i(t)&=A_c\sqrt{T/2}/\sqrt{2}\times\left[\alpha_{1i}\varphi_1(t)+\alpha_{2i}\varphi_2(t)\right]\\ @@ -860,6 +929,7 @@ c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1 s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26 c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z M1001 80h400000v40h-400000z"/>[α1iφ1(t)+α2iφ2(t)]

+

Signal

Symbol mapping: {1,0}{1,1}I(t)=b2nφ1(t)Even bitsQ(t)=b2n+1φ2(t)Odd bitsx(t)=Ac[I(t)cos(2πfct)Q(t)sin(2πfct)]\begin{align*} \text{Symbol mapping: }& \left\{1,0\right\}\to\left\{1,-1\right\}\\ @@ -868,6 +938,7 @@ M1001 80h400000v40h-400000z"/> x(t) &= A_c[I(t)\cos(2\pi f_c t)-Q(t)\sin(2\pi f_c t)] \end{align*}

+

Example of waveform

Code @@ -927,23 +998,22 @@ tBitstream[{1, 1, -1, -1, -1, -1, 1, 1, -1, -1}, 1, "Q(t)"] n_o(t)&=h(t)*n(t)\quad\text{Noise at filter output} \end{align*}

+

2. Bit error rate

Bit error rate (BER) from matched filter outputs and filter output noise

-

Q(x)=1212erf(x2)erf(x2)=12Q(x)Eb=d2=s1(t)s2(t)2dtEnergy per bit/DistanceT=1/RbRb: BitrateEb=PT=Pav/RbEnergy per bitP(W)=10P(dB)10PRX(W)=PTX(W)10Ploss(dB)10Ploss is expressed with negative sign e.g. "-130 dB"BERMatchedFilter=Q(d22N0)=Q(Eb2N0)BERunipolarNRZ|BASK=Q(d2N0)=Q(EbN0)BERpolarNRZ|BPSK=Q(2d2N0)=Q(2EbN0)\begin{align*} - % H_\text{opt}(f)&=\max_{H(f)}\left(\frac{s_{o1}-s_{o2}}{2\sigma_o}\right) - - % \text{BER}_\text{bin}&=p Q\left(\frac{s_{o1}-V_T}{\sigma_o}\right)+(1-p)Q\left(\frac{V_T-s_{o2}}{\sigma_o}\right)\text{, $p\rightarrow$Probability $s_1(t)$ sent, $V_T\rightarrow$Threshold voltage} +

Q(x)=1212erf(x2)erf(x2)=12Q(x)Eb=d2=s1(t)s2(t)2dtEnergy per bit/DistanceT=1/RbRb: BitrateEb=PavT=Pav/RbEnergy per bitPav=Eb/T=EbRbAverage powerP(W)=10P(dB)10PRX(W)=PTX(W)10Ploss(dB)10Ploss is expressed with negative sign e.g. "-130 dB"BERMatchedFilter=Q(d22N0)=Q(Eb2N0)BERunipolarNRZ|BASK=Q(d2N0)=Q(EbN0)BERpolarNRZ|BPSK=Q(2d2N0)=Q(2EbN0)\begin{align*} Q(x)&=\frac{1}{2}-\frac{1}{2}\text{erf}\left(\frac{x}{\sqrt{2}}\right)\Leftrightarrow\text{erf}\left(\frac{x}{\sqrt{2}}\right)=1-2Q(x)\\ E_b&=d^2=\int_{-\infty}^\infty|s_1(t)-s_2(t)|^2dt\quad\text{Energy per bit/Distance}\\ T&=1/R_b\quad\text{$R_b$: Bitrate}\\ - E_b&=PT=P_\text{av}/R_b\quad\text{Energy per bit}\\ + E_b&=P_\text{av}T=P_\text{av}/R_b\quad\text{Energy per bit}\\ + P_\text{av}&=E_b/T=E_bR_b\quad\text{Average power}\\ P(\text{W})&=10^{\frac{P(\text{dB})}{10}}\\ P_\text{RX}(W)&=P_\text{TX}(W)\cdot10^{\frac{P_\text{loss}(\text{dB})}{10}}\quad \text{$P_\text{loss}$ is expressed with negative sign e.g. "-130 dB"}\\ \text{BER}_\text{MatchedFilter}&=Q\left(\sqrt{\frac{d^2}{2N_0}}\right)=Q\left(\sqrt{\frac{E_b}{2N_0}}\right)\\ \text{BER}_\text{unipolarNRZ|BASK}&=Q\left(\sqrt{\frac{d^2}{N_0}}\right)=Q\left(\sqrt{\frac{E_b}{N_0}}\right)\\ \text{BER}_\text{polarNRZ|BPSK}&=Q\left(\sqrt{\frac{2d^2}{N_0}}\right)=Q\left(\sqrt{\frac{2E_b}{N_0}}\right)\\ \end{align*} -

+h400000v40h-400000z"/>)

+

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

Q(x)Q(x) function

@@ -1688,6 +1759,7 @@ h400000v40h-400000z"/>

+

Raised cosine (RC) pulse

Raised cosine pulse

0α10\leq\alpha\leq1

+

⚠ NOTE might not be safe to assume T=TT'=T, if you can solve the question without TT then use that method.

To solve this type of question:

    @@ -1718,8 +1792,8 @@ h400000v40h-400000z"/>
- - + + @@ -1737,14 +1811,13 @@ h400000v40h-400000z"/>
Linear modulation (MM-PSK, MM-QAM)NRZ unipolar encodingLinear modulation (MM-PSK, BPSK, MM-QAM)NRZ unipolar encoding, BASK
-

Symbol set size MM

-

D symbol/s=2W Hz1+αRb bit/s=(D symbol/s)×(k bit/symbol)M symbol/set=2kEb=PT=Pav/RbEnergy per bit\begin{align*} - D\text{ symbol/s}&=\frac{2W\text{ Hz}}{1+\alpha}\\ +

Rb bit/s=(D symbol/s)×(k bit/symbol)M symbol/set=2kEb=PT=Pav/RbEnergy per bit\begin{align*} R_b\text{ bit/s}&=(D\text{ symbol/s})\times(k\text{ bit/symbol})\\ M\text{ symbol/set}&=2^k\\ E_b&=PT=P_\text{av}/R_b\quad\text{Energy per bit}\\ \end{align*} -

+

+

Nyquist stuff

TODO: Condition for 0 ISI

Pr(kT)={1k=00k0P_r(kT)=\begin{cases} @@ -1752,6 +1825,7 @@ h400000v40h-400000z"/>

+

Other

Excess BW=BabsBNyquist=1+α2T12T=α2TFOR NRZ (Use correct Babs)α=Excess BWBNyquist=BabsBNyquistBNyquistT=1/D\begin{align*} \text{Excess BW}&=B_\text{abs}-B_\text{Nyquist}=\frac{1+\alpha}{2T}-\frac{1}{2T}=\frac{\alpha}{2T}\quad\text{FOR NRZ (Use correct $B_\text{abs}$)}\\ @@ -1759,6 +1833,7 @@ h400000v40h-400000z"/>

+

Table of bandpass signalling and BER

@@ -1987,6 +2062,7 @@ M1001 80h400000v40h-400000z"/> H(x,y) &= H(x) + H(y|x) = H(y) + H(x|y)\\ \end{align*}

+

Entropy is maximized when all have an equal probability.

Differential entropy for continuous random variables

TODO: Cut out if not required

@@ -1994,12 +2070,14 @@ M1001 80h400000v40h-400000z"/> h(x) &= -\int_\mathbb{R}f_X(x)\log_2(f_X(x))dx \end{align*}

+

Mutual information

Amount of entropy decrease of xx after observation by yy.

I(x;y)=H(x)H(xy)=H(y)H(yx)\begin{align*} I(x;y) &= H(x)-H(x|y)=H(y)-H(y|x)\\ \end{align*}

+

Channel model

Vertical, xx: input
Horizontal, yy: output

@@ -2012,6 +2090,7 @@ Horizontal, P=p11p21pM1p12p22pM2p1Np2NpMN

+

P(yjxi)y1y2yNx1p11p12p1Nx2p21p22p2NxMpM1pM2pMN\begin{array}{c|cccc} P(y_j|x_i)& y_1 & y_2 & \dots & y_N \\\hline x_1 & p_{11} & p_{12} & \dots & p_{1N} \\ @@ -2020,6 +2099,7 @@ M347 1759 V0 H263 V1759 v1800 v1759 h84z"/>

+

Input has probability distribution pX(ai)=P(X=ai)p_X(a_i)=P(X=a_i)

Channel maps alphabet {a1,,aM}{b1,,bN}`\{a_1,\dots,a_M\} \to \{b_1,\dots,b_N\}`

Output has probabiltiy distribution pY(bj)=P(y=bj)p_Y(b_j)=P(y=b_j)

@@ -2029,6 +2109,7 @@ 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(yixi)P(xi)4. Find H(x,y) using each element from (3.)5. Find H(xy)=H(x,y)H(y)6. Find I(y;x)=H(x)H(xy)\begin{align*} &\text{1. Find }H(x)\\ @@ -2039,6 +2120,7 @@ M347 1759 V0 H263 V1759 v1800 v1759 h84z"/>

+

Channel types

@@ -2067,11 +2149,14 @@ M347 1759 V0 H263 V1759 v1800 v1759 h84z"/>

+

Channel capacity of an AWGN channel

yi=xi+niniN(0,N0/2)y_i=x_i+n_i\quad n_i\thicksim N(0,N_0/2)

+

C=12log2(1+PavN0/2)C=\frac{1}{2}\log_2\left(1+\frac{P_\text{av}}{N_0/2}\right)

+

Channel capacity of a bandwidth AWGN channel

Note: Define XOR (\oplus) as exclusive OR, or modulo-2 addition.

PsBandwidth limited average poweryi=bandpassW(xi)+niniN(0,N0/2)C=Wlog2(1+PsN0W)C=Wlog2(1+SNR)SNR=Ps/(N0W)\begin{align*} @@ -2081,6 +2166,7 @@ M347 1759 V0 H263 V1759 v1800 v1759 h84z"/>

+

Channel code

@@ -2142,6 +2228,7 @@ H403z M403 1759 V0 H319 V1759 v1800 v1759 h84z"/>=g0,0g1,0gk1,0g0,n2g1,n2gk1,n2g0,n1g1,n1gk1,n1

+

A message block m\mathbf{m} is coded as x\mathbf{x} using the generation codewords in G\mathbf{G}:

m=[m0mn2mk1]m=[101001]Example for k=6x=mG=m0g0+m1g1++mk1gk1\begin{align*} \mathbf{m}&=[\begin{matrix} @@ -2151,6 +2238,7 @@ M347 1759 V0 H263 V1759 v1800 v1759 h84z"/>

+

Systemic linear block code

Contains kk message bits (Copy m\mathbf{m} as-is) and (nk)(n-k) parity bits after the message bits.

G=[IkP]=[100010001p0,0p0,n2p0,n1p1,0p1,n2p1,n1pk1,0pk1,n2pk1,n1]m=[m0mn2mk1]x=mG=m[IkP]=[mIkmP]=[mb]b=mPParity bits of x\begin{align*} @@ -2178,6 +2266,7 @@ M347 1759 V0 H263 V1759 v1800 v1759 h84z"/>

+

Parity check matrix H\mathbf{H}

Transpose P\mathbf{P} for the parity check matrix

H=[PTInk]=[p0Tp1Tpk1TInk]=[p0,0p0,k2p0,k1p1,0p1,k2p1,k1pn1,0pn1,k2pn1,k1100010001]xHT=0    Codeword is valid\begin{align*} @@ -2209,6 +2298,7 @@ M347 1759 V0 H263 V1759 v1800 v1759 h84z"/>

+

Procedure to find parity check matrix from list of codewords

  1. From the number of codewords, find k=log2(N)k=\log_2(N)
  2. @@ -2225,6 +2315,7 @@ M347 1759 V0 H263 V1759 v1800 v1759 h84z"/>

    +

    Set x1,x2x_1,x_2 as information bits. Express x3,x4,x5x_3,x_4,x_5 in terms of x1,x2x_1,x_2.

    x3=x1x2x4=x1x2x5=x2    P=x1x2x311x411x501H=[111101100010001]\begin{align*} \begin{aligned} @@ -2256,6 +2347,7 @@ M347 1759 V0 H263 V1759 v1800 v1759 h84z"/>

    +

    Error detection and correction

    Detection of ss errors: dmins+1d_\text{min}\geq s+1

    Correction of uu errors: dmin2u+1d_\text{min}\geq 2u+1

    diff --git a/README.md b/README.md index 843ace9..47b14eb 100644 --- a/README.md +++ b/README.md @@ -10,7 +10,7 @@ If you have issues or suggestions, [raise them on GitHub](https://github.com/pet It is recommended to refer to use [the PDF copy](https://raw.githubusercontent.com/peter-tanner/IDIOTS-GUIDE-TO-ELEC4402-communication-systems/refs/heads/master/README.pdf) instead of whatever GitHub renders. -## License and information +### License and information Notes are open-source and licensed under the GNU GPL-3.0. **You must include the [full-text of the license](/COPYING.txt) and follow its terms when using these notes or any diagrams in derivative works** (but not when printing as notes) @@ -35,6 +35,41 @@ along with this program. If not, see . +## Other advice for this unit + +### Get more exam papers on OneSearch + +- You can access up to **6** more papers with this method (You normally only get the previous year's paper on LMS in week 12). +- Either [search "Communications" and filter by type "Examination Papers"](https://onesearch.library.uwa.edu.au/discovery/search?query=any%2Ccontains%2Ccommunications&tab=Everything&search_scope=MyInst_and_CI&vid=61UWA_INST%3AUWA&facet=rtype%2Cinclude%2Cexampaper&lang=en&offset=0) +- Or search old unit codes + - ELEC4301 Digital Communications and Networking + - ENGT4301 Digital Communications and Networking + - ELEC3302 Communications Systems + - Note that ELEC5501 Advanced Communications is a different unit. + +
    +Listing of examination papers on OneSearch +
      +
    • Communications Systems ELEC3302 Examination paper [2008 Supplementary]
    • +
    • Communications Systems ELEC4402 Examination paper [2014 Semester 2]
    • +
    • Communications Systems ELEC3302 Examination paper [2014 Semester 2]
    • +
    • Communications Systems ELEC3302 Examination paper [2008 Semester 1]
    • +
    • Digital Communications and Networking ENGT4301 Examination paper [2005 Supplementary]
    • +
    • Digital Communications and Networking ELEC4301 Examination paper [2009 Supplementary]
    • +
    +
    + +### Tests + +- A lot of the unit requires you to learn processes and apply them. This is quite time consuming to do during the semester and the marking of the tests will destroy your wam if you do not know the process (especially compared to signal processing and signals and systems), I do not recommend doing this unit during thesis year. +- This formula sheet will attempt to condense all processes/formulas you may need in this unit. +- **You do not get given a formula sheet**, so you are entirely dependent on your own notes (except for some exceptions, such as the $\text{erf}(x)$ table). So bring good notes. +- Doing this unit after signal processing is a good idea. + +## Printable notes begins on next page (in PDF) + +
    + ## Fourier transform identities | **Time Function** | **Fourier Transform** | @@ -73,6 +108,8 @@ along with this program. If not, see . \end{align*} ``` + + ### Fourier transform of continuous time periodic signal Required for some questions on **sampling**: @@ -85,18 +122,21 @@ Transform a continuous time-periodic signal $x_p(t)=\sum_{n=-\infty}^\infty x(t- X_p(f)=\sum_{n=-\infty}^\infty C_n\delta(f-nf_s)\quad f_s=\frac{1}{T_s} ``` + + Calculate $C_n$ coefficient as follows from $x_p(t)$: ```math \begin{align*} - % C_n&=X_p(nf_s)\\ C_n&=\frac{1}{T_s} \int_{T_s} x_p(t)\exp(-j2\pi f_s t)dt\\ &=\frac{1}{T_s} X(nf_s)\quad\color{red}\text{(TODO: Check)}\quad\color{white}\text{$x(t-nT_s)$ is contained in the interval $T_s$} \end{align*} ``` + + ### $\text{rect}$ function ![rect](images/rect.drawio.svg) @@ -110,6 +150,8 @@ Calculate $C_n$ coefficient as follows from $x_p(t)$: \end{align*} ``` + + ### White noise ```math @@ -122,6 +164,8 @@ G_y(f)&=G(f)G_w(f)\\ \end{align*} ``` + + ### WSS ```math @@ -132,6 +176,8 @@ G_y(f)&=G(f)G_w(f)\\ \end{align*} ``` + + ### Ergodicity ```math @@ -142,6 +188,8 @@ G_y(f)&=G(f)G_w(f)\\ \end{align*} ``` + + | Type | Normal | Mean square sense | | ----------------------------------- | ------------------------------------------------------- | ----------------------------------------------------------- | | ergodic in mean | $$\lim_{T\to\infty}\braket{X(t)}_T=m_X(t)=m_X$$ | $$\lim_{T\to\infty}\text{VAR}[\braket{X(t)}_T]=0$$ | @@ -159,6 +207,8 @@ Note: **A WSS random process needs to be both ergodic in mean and autocorrelatio \end{align*} ``` + + ### Other trig ```math @@ -176,6 +226,8 @@ Note: **A WSS random process needs to be both ergodic in mean and autocorrelatio \end{align*} ``` + + ```math \begin{align*} \cos(A+B) &= \cos (A) \cos (B)-\sin (A) \sin (B) \\ @@ -186,6 +238,8 @@ Note: **A WSS random process needs to be both ergodic in mean and autocorrelatio \end{align*} ``` + + ```math \begin{align*} \cos(A)+\cos(B) &= 2 \cos \left(\frac{A}{2}-\frac{B}{2}\right) \cos \left(\frac{A}{2}+\frac{B}{2}\right) \\ @@ -197,6 +251,8 @@ Note: **A WSS random process needs to be both ergodic in mean and autocorrelatio \end{align*} ``` + + ## IQ/Complex envelope Def. $\tilde{g}(t)=g_I(t)+jg_Q(t)$ as the complex envelope. Best to convert to $e^{j\theta}$ form. @@ -215,6 +271,8 @@ g_Q(t)&=A(t)\sin(\phi(t))\quad\text{Quadrature-phase component}\\ \end{align*} ``` + + ### For transfer function ```math @@ -225,6 +283,8 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\ \end{align*} ``` + + ## AM ### CAM @@ -243,6 +303,8 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\ \end{align*} ``` + + $B_T$: Signal bandwidth $B$: Bandwidth of modulating wave @@ -257,6 +319,8 @@ Overmodulation (resulting in phase reversals at crossing points): $m_a>1$ \end{align*} ``` + + ## FM/PM ```math @@ -270,6 +334,8 @@ Overmodulation (resulting in phase reversals at crossing points): $m_a>1$ \end{align*} ``` + + ### Bessel form and magnitude spectrum (single tone) ```math @@ -278,6 +344,8 @@ Overmodulation (resulting in phase reversals at crossing points): $m_a>1$ \end{align*} ``` + + ### FM signal power ```math @@ -289,6 +357,8 @@ Overmodulation (resulting in phase reversals at crossing points): $m_a>1$ \end{align*} ``` + + ### Carson's rule to find $B$ (98% power bandwidth rule) ```math @@ -304,6 +374,8 @@ B &= \begin{cases} \end{align*} ``` + + #### $\Delta f$ of arbitrary modulating signal Find instantaneous frequency $f_\text{FM}$. @@ -323,6 +395,8 @@ B_T &= 2(D+1)W_m \end{align*} ``` + + ### Complex envelope ```math @@ -333,6 +407,8 @@ B_T &= 2(D+1)W_m \end{align*} ``` + + ### Band | Narrowband | Wideband | @@ -356,6 +432,8 @@ B_T &= 2(D+1)W_m \end{align*} ``` + + ## ## Noise performance @@ -369,6 +447,8 @@ B_T &= 2(D+1)W_m \end{align*} ``` + + ## Sampling ```math @@ -381,6 +461,8 @@ B_T &= 2(D+1)W_m \end{align*} ``` + + ### Procedure to reconstruct sampled signal Analog signal $x'(t)$ which can be reconstructed from a sampled signal $x_s(t)$: Put $x_s(t)$ through LPF with maximum frequency of $f_s/2$ and minimum frequency of $-f_s/2$. Anything outside of the BPF will be attenuated, therefore $n$ which results in frequencies outside the BPF will evaluate to $0$ and can be ignored. @@ -405,18 +487,21 @@ Transform a continuous time-periodic signal $x_p(t)=\sum_{n=-\infty}^\infty x(t- X_p(f)=\sum_{n=-\infty}^\infty C_n\delta(f-nf_s)\quad f_s=\frac{1}{T_s} ``` + + Calculate $C_n$ coefficient as follows from $x_p(t)$: ```math \begin{align*} - % C_n&=X_p(nf_s)\\ C_n&=\frac{1}{T_s} \int_{T_s} x_p(t)\exp(-j2\pi f_s t)dt\\ &=\frac{1}{T_s} X(nf_s)\quad\color{red}\text{(TODO: Check)}\quad\color{white}\text{$x(t-nT_s)$ is contained in the interval $T_s$} \end{align*} ``` + + ### Nyquist criterion for zero-ISI @@ -440,6 +525,8 @@ Cannot add directly due to copyright! \end{align*} ``` + + ### Quantization noise ```math @@ -452,6 +539,8 @@ Cannot add directly due to copyright! \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! @@ -479,6 +568,8 @@ Cannot add directly due to copyright! \end{align*} ``` + + ## Modulation and basis functions ![Constellation diagrams](./images/Constellation.drawio.svg) @@ -493,12 +584,16 @@ Cannot add directly due to copyright! \end{align*} ``` + + #### Symbol mapping ```math b_n:\{1,0\}\to a_n:\{1,0\} ``` + + #### 2 possible waveforms ```math @@ -509,6 +604,8 @@ b_n:\{1,0\}\to a_n:\{1,0\} \end{align*} ``` + + Distance is $d=\sqrt{2E_b}$ ### BPSK @@ -521,12 +618,16 @@ Distance is $d=\sqrt{2E_b}$ \end{align*} ``` + + #### Symbol mapping ```math b_n:\{1,0\}\to a_n:\{1,\color{green}-1\color{white}\} ``` + + #### 2 possible waveforms ```math @@ -537,6 +638,8 @@ b_n:\{1,0\}\to a_n:\{1,\color{green}-1\color{white}\} \end{align*} ``` + + Distance is $d=2\sqrt{E_b}$ ### QPSK ($M=4$ PSK) @@ -551,6 +654,8 @@ Distance is $d=2\sqrt{E_b}$ \end{align*} ``` + + ### 4 possible waveforms ```math @@ -562,6 +667,8 @@ Distance is $d=2\sqrt{E_b}$ \end{align*} ``` + + Note on energy per symbol: Since $|s_i(t)|=A_c$, have to normalize distance as follows: ```math @@ -572,6 +679,8 @@ Note on energy per symbol: Since $|s_i(t)|=A_c$, have to normalize distance as f \end{align*} ``` + + #### Signal ```math @@ -583,6 +692,8 @@ Note on energy per symbol: Since $|s_i(t)|=A_c$, have to normalize distance as f \end{align*} ``` + + ### Example of waveform
    @@ -636,19 +747,19 @@ Find transfer function $h(t)$ of matched filter and apply to an input: \end{align*} ``` + + ### 2. Bit error rate Bit error rate (BER) from matched filter outputs and filter output noise ```math \begin{align*} - % H_\text{opt}(f)&=\max_{H(f)}\left(\frac{s_{o1}-s_{o2}}{2\sigma_o}\right) - - % \text{BER}_\text{bin}&=p Q\left(\frac{s_{o1}-V_T}{\sigma_o}\right)+(1-p)Q\left(\frac{V_T-s_{o2}}{\sigma_o}\right)\text{, $p\rightarrow$Probability $s_1(t)$ sent, $V_T\rightarrow$Threshold voltage} Q(x)&=\frac{1}{2}-\frac{1}{2}\text{erf}\left(\frac{x}{\sqrt{2}}\right)\Leftrightarrow\text{erf}\left(\frac{x}{\sqrt{2}}\right)=1-2Q(x)\\ E_b&=d^2=\int_{-\infty}^\infty|s_1(t)-s_2(t)|^2dt\quad\text{Energy per bit/Distance}\\ T&=1/R_b\quad\text{$R_b$: Bitrate}\\ - E_b&=PT=P_\text{av}/R_b\quad\text{Energy per bit}\\ + E_b&=P_\text{av}T=P_\text{av}/R_b\quad\text{Energy per bit}\\ + P_\text{av}&=E_b/T=E_bR_b\quad\text{Average power}\\ P(\text{W})&=10^{\frac{P(\text{dB})}{10}}\\ P_\text{RX}(W)&=P_\text{TX}(W)\cdot10^{\frac{P_\text{loss}(\text{dB})}{10}}\quad \text{$P_\text{loss}$ is expressed with negative sign e.g. "-130 dB"}\\ \text{BER}_\text{MatchedFilter}&=Q\left(\sqrt{\frac{d^2}{2N_0}}\right)=Q\left(\sqrt{\frac{E_b}{2N_0}}\right)\\ @@ -657,6 +768,8 @@ Bit error rate (BER) from matched filter outputs and filter output noise \end{align*} ``` + +
    ## Value tables for $\text{erf}(x)$ and $Q(x)$ @@ -750,6 +863,8 @@ Adapted from table 6.1 M F Mesiya - Contemporary Communication Systems \end{align*} ``` + + + ### Raised cosine (RC) pulse ![Raised cosine pulse](images/RC.drawio.svg) @@ -782,6 +899,8 @@ TODO: 0\leq\alpha\leq1 ``` + + ⚠ NOTE might not be safe to assume $T'=T$, if you can solve the question without $T$ then use that method. To solve this type of question: @@ -789,23 +908,22 @@ To solve this type of question: 1. Use the formula for $D$ below 2. Consult the BER table below to get the BER which relates the noise of the channel $N_0$ to $E_b$ and to $R_b$. -| Linear modulation ($M$-PSK, $M$-QAM) | NRZ unipolar encoding | +| Linear modulation ($M$-PSK, BPSK, $M$-QAM) | NRZ unipolar encoding, BASK | | --------------------------------------------------- | -------------------------------------------------- | | $W=B_\text{\color{green}abs-abs}$ | $W=B_\text{\color{green}abs}$ | | $W=B_\text{abs-abs}=\frac{1+\alpha}{T}=(1+\alpha)D$ | $W=B_\text{abs}=\frac{1+\alpha}{2T}=(1+\alpha)D/2$ | | $D=\frac{W\text{ symbol/s}}{1+\alpha}$ | $D=\frac{2W\text{ symbol/s}}{1+\alpha}$ | -#### Symbol set size $M$ - ```math \begin{align*} - D\text{ symbol/s}&=\frac{2W\text{ Hz}}{1+\alpha}\\ R_b\text{ bit/s}&=(D\text{ symbol/s})\times(k\text{ bit/symbol})\\ M\text{ symbol/set}&=2^k\\ E_b&=PT=P_\text{av}/R_b\quad\text{Energy per bit}\\ \end{align*} ``` + + ### Nyquist stuff #### TODO: Condition for 0 ISI @@ -817,6 +935,8 @@ P_r(kT)=\begin{cases} \end{cases} ``` + + #### Other ```math @@ -827,6 +947,8 @@ P_r(kT)=\begin{cases} \end{align*} ``` + + ### Table of bandpass signalling and BER | **Binary Bandpass Signaling** | **$B_\text{null-null}$ (Hz)** | **$B_\text{abs-abs}\color{red}=2B_\text{abs}$ (Hz)** | **BER with Coherent Detection** | **BER with Noncoherent Detection** | @@ -875,6 +997,8 @@ Adapted from table 11.4 M F Mesiya - Contemporary Communication Systems \end{align*} ``` + + Entropy is **maximized** when all have an equal probability. ### Differential entropy for continuous random variables @@ -887,6 +1011,8 @@ TODO: Cut out if not required \end{align*} ``` + + ### Mutual information Amount of entropy decrease of $x$ after observation by $y$. @@ -897,6 +1023,8 @@ Amount of entropy decrease of $x$ after observation by $y$. \end{align*} ``` + + ### Channel model Vertical, $x$: input\ @@ -911,6 +1039,8 @@ Horizontal, $y$: output \end{matrix}\right] ``` + + ```math \begin{array}{c|cccc} P(y_j|x_i)& y_1 & y_2 & \dots & y_N \\\hline @@ -921,6 +1051,8 @@ Horizontal, $y$: output \end{array} ``` + + Input has probability distribution $p_X(a_i)=P(X=a_i)$ Channel maps alphabet $`\{a_1,\dots,a_M\} \to \{b_1,\dots,b_N\}`$ @@ -935,6 +1067,8 @@ Output has probabiltiy distribution $p_Y(b_j)=P(y=b_j)$ \end{align*} ``` + + #### Fast procedure to calculate $I(y;x)$ ```math @@ -948,6 +1082,8 @@ Output has probabiltiy distribution $p_Y(b_j)=P(y=b_j)$ \end{align*} ``` + + ### Channel types | Type | Definition | @@ -967,16 +1103,22 @@ Output has probabiltiy distribution $p_Y(b_j)=P(y=b_j)$ \end{align*} ``` + + #### Channel capacity of an AWGN channel ```math y_i=x_i+n_i\quad n_i\thicksim N(0,N_0/2) ``` + + ```math C=\frac{1}{2}\log_2\left(1+\frac{P_\text{av}}{N_0/2}\right) ``` + + #### Channel capacity of a bandwidth AWGN channel Note: Define XOR ($\oplus$) as exclusive OR, or modulo-2 addition. @@ -990,6 +1132,8 @@ Note: Define XOR ($\oplus$) as exclusive OR, or modulo-2 addition. \end{align*} ``` + + ## Channel code | | | | @@ -1037,6 +1181,8 @@ Each generator vector is a binary string of size $n$. There are $k$ generator ve \end{align*} ``` + + A message block $\mathbf{m}$ is coded as $\mathbf{x}$ using the generation codewords in $\mathbf{G}$: ```math @@ -1049,6 +1195,8 @@ A message block $\mathbf{m}$ is coded as $\mathbf{x}$ using the generation codew \end{align*} ``` + + ### Systemic linear block code Contains $k$ message bits (Copy $\mathbf{m}$ as-is) and $(n-k)$ parity bits after the message bits. @@ -1078,6 +1226,8 @@ Contains $k$ message bits (Copy $\mathbf{m}$ as-is) and $(n-k)$ parity bits afte \end{align*} ``` + + #### Parity check matrix $\mathbf{H}$ Transpose $\mathbf{P}$ for the parity check matrix @@ -1111,6 +1261,8 @@ Transpose $\mathbf{P}$ for the parity check matrix \end{align*} ``` + + #### Procedure to find parity check matrix from list of codewords 1. From the number of codewords, find $k=\log_2(N)$ @@ -1130,6 +1282,8 @@ Example: \end{array} ``` + + Set $x_1,x_2$ as information bits. Express $x_3,x_4,x_5$ in terms of $x_1,x_2$. ```math @@ -1162,6 +1316,8 @@ Set $x_1,x_2$ as information bits. Express $x_3,x_4,x_5$ in terms of $x_1,x_2$. \end{align*} ``` + + #### Error detection and correction **Detection** of $s$ errors: $d_\text{min}\geq s+1$ diff --git a/README.pdf b/README.pdf index abe030784f90f7e4f565f9beadb0276a15fab84f..40eb3ad86fb60b97b4f7bccf1b7b3877b498cf3c 100644 GIT binary patch delta 360925 zcmZshV|1oX(4b@6&XY`R+qP|+6Z45}V`AI3Z95Y?nPg(^`|j?yAI{$Yy3|#DPM_-g zTon^TF<4H4O0Fa(LC?&<2}j;?UHcBl!j*`M4$7VGj08yytkJr4+~Pv}yy*)%OP3Lv ziNcVU?*ARoQ@V9sxaPhlFueiB1f`3naSqF zpd09G=#STv=FpDojsG&*lcYIR!H}cWGlS}bSkq|=DWfpYF$XqSq`YW$ar@{~*xe}f zuxuG>?t3HxPZf1u20U)6e;q7|30VIQH|S`pWl+h5{nM<(_g6v7>y}>F@$=G`j7>3A zx|nFOv8x!`{@wX-ETP1*xusX zTTY5Qk&PkY(f<8BvR9$o-BN9~I)xN}Zj~&!TkLfoO|=%MF=@?=BEZxfb1MIn~&vAKKP{)G?r;?<4-F z8n5%XW^mh9=L0i%>h5dO{Y$xzwD3pULS&k{D*Y6X!1o@qa#2m!&nErG){;x^_Zoc? z=V-V#0^N1*;}`ezBukkS9(Y|pK~wzpW~w?mU{O0yn*>1jV}rRZcdkLSc%uhS*V{(` z?BfI5i3&{_2ZMi2hEiMs0P#i?5Wo~<{Ufm52$HD1ntl?Nld{!0O%RVp#3|MC#R^clb4+h& 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