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.
It is recommended to refer to use the PDF copy 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 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
+
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).
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 erf(x) table). So bring good notes.
+
Doing this unit after signal processing is a good idea.
mamamax(t)PcPxηBT=Acmint∣kam(t)∣ka is the amplitude sensitivity (volt−1), ma is the modulation index.=Amax+AminAmax−Amin (Symmetrical m(t))=kaAm (Symmetrical m(t))=Accos(2πfct)[1+kam(t)]=Accos(2πfct)[1+mam(t)/Ac],where m(t)=Amm^(t) and m^(t) is the normalized modulating signal=2Ac2Carrier power=41ma2Ac2=Total PowerSignal Power=Px+PcPx=2fm=2B
+
BT: Signal bandwidth
B: Bandwidth of modulating wave
Overmodulation (resulting in phase reversals at crossing points): ma>1
s(t)s(t)s(t)βΔfD=Accos[2πfct+kpm(t)]Phase modulated (PM)=Accos[2πfct+2πkf∫0tm(τ)dτ]Frequency modulated (FM)=Accos[2πfct+βsin(2πfmt)]FM single tone=fmΔf=kfAmModulation index=βfm=kfAmfm=tmax(kfm(t))−tmin(kfm(t))Maximum frequency deviation=WmΔfDeviation ratio, where Wm is bandwidth of m(t) (Use FT)
Analog signal x′(t) which can be reconstructed from a sampled signal xs(t): Put xs(t) through LPF with maximum frequency of fs/2 and minimum frequency of −fs/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.
Example: fs=5000⟹LPF∈[−2500,2500]
@@ -549,14 +606,15 @@ B_T &= 2(D+1)W_m
Transform a continuous time-periodic signal xp(t)=∑n=−∞∞x(t−nTs) with period Ts:
Xp(f)=n=−∞∑∞Cnδ(f−nfs)fs=Ts1
+
Calculate Cn coefficient as follows from xp(t):
Cn=Ts1∫Tsxp(t)exp(−j2πfst)dt=Ts1X(nfs)(TODO: Check)x(t−nTs) is contained in the interval Ts
+
Nyquist criterion for zero-ISI
Do not transmit more than 2B samples per second over a channel of B bandwidth.
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)
+
Channel types
@@ -2067,11 +2149,14 @@ M347 1759 V0 H263 V1759 v1800 v1759 h84z"/>CNpCR→Channel capacity (bits/channels used)→Output alphabet size→Probability vector, any row of the transition matrix=log2(N)−H(p)Capacity for weakly symmetric and symmetric channels<C for error-free transmission
+
Channel capacity of an AWGN channel
yi=xi+nini∼N(0,N0/2)
+
C=21log2(1+N0/2Pav)
+
Channel capacity of a bandwidth AWGN channel
Note: Define XOR (⊕) as exclusive OR, or modulo-2 addition.
PsyiCC→Bandwidth limited average power=bandpassW(xi)+nini∼N(0,N0/2)=Wlog2(1+N0WPs)=Wlog2(1+SNR)SNR=Ps/(N0W)
A message block m is coded as x using the generation codewords in G:
mmx=[m0…mn−2mk−1]=[101001]Example for k=6=mG=m0g0+m1g1+⋯+mk−1gk−1
+
Systemic linear block code
Contains k message bits (Copy m as-is) and (n−k) parity bits after the message bits.
Gmxb=[IkP]=10⋮001⋮0……⋱…00⋮1p0,0p1,0⋮pk−1,0……⋱…p0,n−2p1,n−2⋮pk−1,n−2p0,n−1p1,n−1⋮pk−1,n−1=[m0…mn−2mk−1]=mG=m[IkP]=[mIkmP]=[mb]=mPParity bits of x
+
Parity check matrix H
Transpose P for the parity check matrix
HxHT=[PTIn−k]=[p0Tp1T…pk−1TIn−k]=p0,0p1,0⋮pn−1,0……⋱…p0,k−2p1,k−2⋮pn−1,k−2p0,k−1p1,k−1⋮pn−1,k−110⋮001⋮0……⋱…00⋮1=0⟹Codeword is valid
+
Procedure to find parity check matrix from list of codewords
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
@@ -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
@@ -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
@@ -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 abe0307..40eb3ad 100644
Binary files a/README.pdf and b/README.pdf differ
@@ -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*}
