Better Comms Formula Sheet

Channel Capacity

$$C=B{\log }_2(1+\frac{S}{N})$$

Where $B$ is the Bandwidth (Hz), $S$ is the signal power, $N$ is the noise power, and $C$ is the channel capacity in bits (bps).

Sometimes you will see this formula without $B$ specified, in which case $B=1$ and this is specified for each frequency

Effective Number of Bits (ENOB):

$$\text{ENOB}=\frac{SNR-10{\log }_{10}(\frac{3}{2})}{20{\log }_{10}(2)}=\frac{SNR-1.76}{6.02}$$

Where $6.02$ is a term that converts decibels to bits, and $1.76$ is the Quantisation Error In systems with lots of noise, not all of our voltages are “useful”, so we use ENOB to calculate how many bits are useful

Euler Identities

$$\sin (\theta )=\frac{e^{i\theta }-e^{-i\theta }}{2}$$ $$\cos (\theta )=\frac{e^{i\theta }+e^{-i\theta }}{2}$$

Fourier Transform

$$f(\omega )={\int }_{-\infty }^{\infty }f(t)e^{-i2\pi \omega t}$$

Convolution:

$$f\ast g={\mathcal{F}}^{-1}(\mathcal{F}(f)\ast \mathcal{F}(g))$$

Where F if the Fourier Transform

Speed of Light:

$$c=v\lambda =3\ast 10^{8}m{s}^{-1}$$

Miles to Kilometer:

$$1km=0.62miles$$

$$v=\frac{V\cos \left(\theta \right)}{\lambda }$$

Amplitude Modulation:

$$y_n=[\text{DC}+m\ast f_n]Asin(2\pi v_{c}t)=[\text{DC}+m\ast f_n]Asin(2\pi v_c\frac{n}{N})$$

Where $n$ is the sample number, and $N$ is the total number of samples $v_c$ is the carrier frequency $m$ is the modulation amplitude $A$ is the carrier amplitude

It is also to note, you need to create $2\ast f_c$ samples per bit, where $f_c$ is the carrier frequency (see: Nyquist)

Frequency Modulation:

$$y(t)=A_c\cos (2\pi v_{c}t+\frac{v_{\Delta }}{v_m}sin(2\pi v_{m}t))$$

IQ Modulation:

$$f(t)=A(t)\ast \cos (2\pi v_{c}t+\varphi (t))=\cos (2\pi v_{c}t)\ast A_I(t)cos(\varphi (t))-\sin (2\pi v_{c}t)\ast A_Q(t)sin(\varphi (t))$$

Where the green section is the $\text{In Phase}$ and the red section is the $\text{Quadrature}$ Where: $\varphi (t)=\text{atan2}(\frac{Q}{I}),A=\sqrt{I^2+Q^2}$

Free Space Path Loss:

$$\frac{P_r}{P_t}={\Phi }_t{\Phi }_r(\frac{\lambda }{4\pi d}{)}^2$$

Where $P$ is the powers of the receiver ${}_r$ and transmitter ${}_t$ and $\Phi$ is the directionality at both.

In cases where you don’t have directional receivers we use:

$$FSPL=(\frac{4\pi d}{\lambda }{)}^2=(\frac{4\pi dv}{c}{)}^2$$

Hata Loss Model for 200-1500MHz

Open Enviroments:

L_o = L_u - 4.78(\log_{10}v)^2 + 18.3(\log_{10}v) - 40.94$$ Where $v$ is the frequency in MHz

L_U = 69.55 + 26.16\log_{10}v - 13.82\log_{10}h_b - C_H + (44.9 - 6.55\log_{10}h_b)\log_{10}d

Where $h_b$ is the height of the transmitter Small/Medium Cities:

C_H = 0.8 + (1.1\log_{10}v -0.7)h_M - 1.56\log_{10}v

Where $h_M$ is the height of the receiver Large Cities:

C_H = 3.2(log_{10}(11.75h_M))^2-4.97

#### Bandwidth of Cables, Speed of Cable Transmission: Speed of Transmission:

v = \frac{c}{\sqrt{\mu_r\epsilon_r}}

Where: - $\mu_r$ is the Relative Permeability, a dimensionless measure of the circuits formation of magnetic fields within itself, simply, a measure of how much better or worse than a vacuum the material is at magnetizing itself - $\epsilon_r$ is the Relative Permittivity, like before, but with electric fields Bandwidth:

f_c \approx \frac{c}{\pi(\frac{D+d}{2})\sqrt{\mu_r\epsilon_r}}

Where : - $D$ represents the inner diameter of the outer conductor of the coaxial cable. - $d$ represents the outer diameter of the inner conductor of the coaxial cable. - $\pi(\frac{D+d}{2})$ represents the geometric mean of the two diameters #### Capacitance and Inductance of a Coaxial Cable:

(\frac{C}{h}) = \frac{2\pi\epsilon_0\epsilon_r}{ln(D/d)}

Where: - C is the capacitance of the coaxial cable. - $h$ is the length of the cable. - $\epsilon_0$​ is the permittivity of free space. - $\epsilon_r$​ is the relative permittivity of the dielectric material between the conductors. - $D$ is the diameter of the outer conductor. - $d$ is the diameter of the inner conductor.

(\frac{L}{h}) = \frac{\mu_0\mu_r}{2\pi}ln(D/d)

- L is the inductance of the cable - $\mu_0$​ is the permeability of free space. - $\mu_r$​ is the relative permeability of the dielectric material between the conductors. #### Effective Impedance: ![[Pasted image 20231202162140.png|300]]

Z = \sqrt{\frac{R+sL}{G+sC}} = \sqrt{\frac{R+j\omega L}{G+j\omega C}}

Z_{DC} = \sqrt{\frac{R}{G}}, Z_{LowFreq} = \sqrt{\frac{R}{j\omega C}}, Z_{HighFreq\text{ or 0}} = \sqrt{\frac{L}{C}}

#### Generator and Parity Check Matrices for Error Correction: ###### 7/3 Hamming Code: Generator: ![[Pasted image 20231202162848.png]] You can see that it is 7 digits long and therefore a 7/3 hamming code Parity Check: ![[Pasted image 20231202162935.png]] It is 3 bits high for the 3 parity bits ###### 8/4 Hamming Code: Generator: ![[Pasted image 20231202163003.png]] Parity Check: ![[Pasted image 20231202163011.png]] Again notice how it is 4 digits high for the 4 parity bits #### Reed Solomon Coding For: - $n_{block}$ being the number of bits in a block - $N_{message}$ being the number of bits storing actual data

p = n_{block} - N_{message}

The approach can detect up to $p$ bits of error and correct up to $\frac{p}{2}$ errors The generator matrix is simply: ![[Pasted image 20231202163435.png|300]]

\int^{t_n}{t_0} f(t) \approx \sum^{k}{n=0}F[z_i] + 4F[\frac{z_i+z_{i-1}}{x-}]