Communication Systems Formulas

Signal Power

$$dBm=10{\log }_{10}\frac{P}{1mW}$$

Where P is the Signal Power in milliwatts typically

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

ADC Resolution:

$$N=2^M$$

Where $N$ is the number of voltage intervals and $M$ is the resolution of the ADC

ADC Voltage Resolution / Least Significant Bit:

$$V_{res}=\frac{\Delta V}{2^M}$$

Where $\Delta V$ is the ($V_{max}-V_{min}$) and M is as before. $V_{min}$ in most systems is referred to as the “noise floor” As noted, this tends to define your least significant bit voltage

Dynamic Range:

$$\text{Dynamic Range}=20{\log }_{10}\frac{V_{max}}{LSB}$$

Where $LSB$ is the least significant bit as specified earlier

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

Nyquist Frequency:

$$f_s\ge 2\ast f_N$$

Where $f_s$ is the sampling frequency and where $f_N$ = the highest frequency in the system you are trying to record.

Convolution:

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

Where $\mathcal{F}$ is the fourier transform of a function

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:

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

This can be filtered by mulitplying this all by a $sin(2\pi v_{LO}\frac{n}{N})$ term

Binary Phase Shift Key (PSK)

$$y_n=(f_n)\sin (2\pi v_1\frac{n}{N})+|f_n-1|\sin (2\pi v_1\frac{n}{N}+\varphi )+\text{AWGN}$$

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: $\varphi (t)=\text{atan2}(\frac{Q}{I}),A=\sqrt{I^2+Q^2}$

Demodulation :

In Phase: Multiply by $\cos (2\pi v_{c}t)$ Quadrature: Multiply by $sin(2\pi v_{c}t)$

Both of these will leave a constant x the arm, plus the other side which is frequency dependant, and therefore can be filtered out

Noise Power Distribution:

$$f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{\frac{1}{2}(\frac{x-DC}{\sigma })}$$

Where $\sigma$ is the standard deviation of the system

Fading Formula:

For Just Fading:

$$f(v)=\mathcal{F}(f(t))\ast h(v)$$

Where $h(v)$ is the transfer function for the enviroment

In general:

$$f_r(t)={\mathcal{F}}^{-1}(\mathcal{F}(f(t))\ast h(v))+AWGN$$

Where $f_r(t)$ is the received signal

Rayleigh Fading:

$$S(v_s)=\frac{1}{\pi v_d\sqrt{1-(\frac{v_s}{v_d}{)}^2}}$$

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:

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

Filter Response:

$$|H(\omega )|=\frac{1}{\sqrt{(\frac{\omega }{{\omega }_0}{)}^{2\Delta }+1}}$$

Where $\Delta$ is the order of the system, and ${\omega }_0$ is the cut off frequency

Bit Error Rate:

$$\text{BER}=0.5\text{erfc}(\text{SNR}\ast \frac{B}{v_s{\log }_{2}m})$$

where $\text{erfc(z)}=1-\frac{2}{\sqrt{\pi }}{\int }_0^{z}e-x^{2}dx$ and $B$ is bandwidth, $v_s$ is baud rate, and $m$ is the bit depth

Impedance Mismatch:

Standing Wave Ratio:

$$\text{SWR}=\frac{Z_L}{Z_O}$$

Reflection:

$$\Gamma =\frac{Z_L-Z_O}{Z_L+Z_O}$$

Quality Factor:

$$Q(\omega )=\omega \ast \frac{\text{maximum energy}}{\text{power loss}}$$

General Knowledge:

Transmitter Structure:

Encoder:

Takes data and encoders into a format suitable for transmission, typically with discrete high and discrete low voltages.

Carrier Wave:

These are created using oscillators, basic oscillators can be LC circuits, though for higher accuracy and higher frequency waves, we use crystals such as quartz. These crystals are used in Surface Acoustic Wave Devices (SAW) that generate physical waves upon an electrical stimulation

Modulator:

On off keying can be done using a transistor where:

• Collector : Carrier Wave
• Base : Digital Signal
• Emitter : Modulated Signal

Other types of Keying/Modulation are Frequency Shift Keying (FSK) and Multi Level (i.e. 4 level) ASKs, which are used in lower noise systems

Filter:

Since we pay for all the bandwidth we use, it’s typically a good idea to filter less important frequencies so we don’t pay for useless bandwidth.

We can use LCR circuits and convolution for this:

Amplification:

This is where we boost the signal so as to for it able to be received by the other side, this does however introduce noise which is why we have:

Output Filter:

This filter is used to mitigate any noise added by the amplification

Data Throughput:

• Optical Fibre : >Pbps
• Ethernet Cat 5 : > 1 Gpbs
• SuperSpeed USB : 10 Gbps
• Serial : 100 kpbs
• 5g : 1 Gbps
• Wifi : 250Mbps
• Bluetooth : 1 Gbps
• Freesat : 50Mbps

Types of Distortion

Diagrams:

FSK:

Modulation:

Demodulation:

Async: Synchronous:

Signal:

PSK:

Diagram:

Modulation:

Demodulation:

Quadrature Phase Shift Key (QPSK):

Diagram:

QPSK vs 8-QPSK or

Modulation:

Demodulation:

Quadrature Amplitude Modulation (QAM):

Diagram

QAM-4:

SuperHeterodyne Receivers:

$$\begin{array}{rl}{\text{Let SNR = 16dB: SNR}}_{linear}& =10^{\frac{16}{10}}=39.81\\ \text{Power at Recevier: }{P}_{receiver}& =\frac{(V_{rms}{)}^2}{R}=\frac{(V_{noise}\ast \sqrt{{\text{SNR}}_{linear}}{)}^2}{R}\\ & =\frac{(6.310\ast 10^{-3}{)}^2}{50}=7.962\ast 10^{-7}\\ \text{FPSL}& =(\frac{4\pi dv}{c}{)}^2\\ & =(\frac{4\pi \ast 400\ast 460\ast 10^6}{3\ast 10^8}{)}^2\\ & =59403613\\ \text{Power at Transmitter: }{P}_{transmitter}& =\text{FPSL}\ast P_{receiver}\\ & =59403613\ast (7.962\ast 10^{-7})\\ & =47.297W\end{array}$$