Signal processing

• The spectrum $[f_{\text{low}},f_{\text{high}}]$ of a signal is the range of frequencies it contains

• The center frequency $f_c$ of a channel is defined in two ways:

  • $f_c=\frac{f_{\text{high}}+f_{\text{low}}}{2}$ (arithmetic mean, most common)
  • $f_c=\sqrt{f_{\text{high}}\cdot f_{\text{low}}}$ (geometric mean)

• The (analog) bandwidth (or frequency bandwidth) (רוחב סרט) is the range of frequencies that a channel can transmit, defined as $B=f_{\text{high}}-f_{\text{low}}$ (unit: Hz)

  • The effective bandwidth refers to the range of frequencies within which a significant portion of the signal’s power or energy is concentrated.

• Fractional bandwidth: $B_{\text{frac}}=\frac{B}{f_c}$

• The symbol rate (or baud rate) $R_s$ is the number of symbols transmitted per unit time

  • the number of times the signal changes state per second
  • (unit: baud (Bd) = symbols per second)

• The symbol duration time $T_s$ is the time taken to transmit one symbol (unit: seconds)

  • $T_s=\frac{1}{R_s}$

• (Nyquist’s formula) $R=R_s\cdot {\log }_2(M)\le 2B\cdot {\log }_2(M)=C$

  • (for a noiseless channel)
  • $R_s$: symbol rate (in $\mathrm{baud}$)
  • $M$: modulation order (number of distinct symbols, or distinct amplitude (or phase, or frequency) levels)
  • $R$: bit rate (in $\text{bps}$)
  • $N={\log }_2(M)$ = number of bits encoded per symbol
  • $B$ is the bandwidth of the channel (in $\mathrm{Hz}$)
  • $R_{\text{max}}=2B$ is the Nyquist rate (in symbols per second (baud)), which is the maximum symbol rate
  • $C$ is the channel capacity (in bps) (maximum bit rate)

• The Nyquist rate of a signal is defined as $2f_{\text{max}}$ (in samples per second (Hz)), where $f_{\text{max}}$ is the highest frequency present in the signal (in Hz)

• The Nyquist frequency (in Hz) is defined as $f_s/2$, where $f_s$ is the sampling rate (in samples per second (Hz)), and is the highest frequency that can be accurately represented when sampling at $f_s$.

• (Shannon–Hartley theorem) $C=B{\log }_2(1+\mathrm{SNR})$ is the channel capacity (in bps) (maximum possible data rate) of a channel with bandwidth $B$ (in Hz) and signal-to-noise ratio $\mathrm{SNR}$

• $C/B$ is the spectral efficiency (in bps/Hz)

• $\mathrm{SNR}=\frac{S}{N}$: signal-to-noise ratio (SNR) (unitless)

  • $\mathrm{SN}{\mathrm{R}}_{\mathrm{dB}}=10{\log }_{10}\left(\frac{S}{N}\right)$: signal-to-noise ratio (in dB)
  • $S$: signal power (in watts)
  • $N$: noise power (in watts)

• Nyquist–Shannon sampling theoremtodo

• $\frac{\text{data}}{\text{data}+\text{overhead}}$

Modulation

• modulator, demodulator
• carrier signal

Analog Modulation

analog-analog

AM, FM, PM

Digital Modulation

analog carrier signal is modulated by a discrete signal

Digital data , Discrete analog signal ASK, FSK, PSK, QAM DSL modem

• DAC (digital-to-analog converter) is a device that converts a digital signal to an analog signal
  • Spectral band
  • frequency band
  • Digital data
  • A digital signal is a signal that represents data as a sequence of discrete values
  • analog signal
  • analog data

Pulse Modulation

analog-to-digital

$\underset{\text{Analog}}{\overset{s(t)}{\to }}\boxed{\genfrac{}{}{0ex}{}{\mathrm{Sampler}}{T}}\underset{\text{Sampled values}}{\overset{s[n]=s(nT)}{\to }}\boxed{\genfrac{}{}{0ex}{}{\mathrm{Quantizer}}{}}\underset{\genfrac{}{}{0ex}{}{\mathrm{Quantized}}{\mathrm{values}}}{\overset{\hat{s}[n]}{\to }}\boxed{\genfrac{}{}{0ex}{}{\mathrm{Encoder}}{}}\underset{\genfrac{}{}{0ex}{}{\mathrm{Digital}}{\mathrm{sequence}}}{\overset{b_1,b_2,\dots }{\to }}$

• ADC (analog-to-digital converter) is a device that converts a continuous analog signal to a discrete digital signal
  • Sampling converts a continuous-time signal $s(T)$ to a discrete-time signal, a sequence of numbers $s(nT)$, where:
    • $T$ is the sampling period (or sampling interval).
    • $f_s=1/T$ is the sampling frequency (or sampling rate) which is the number times per second the original analog voltage is measured (“sampled”)
  • Quantization replaces input values by an approximation from a finite set of values
    • The resolution (or bit depth) is the number of bits or values for the voltage of each sample (=measurement)
    • The difference between the original continuous analog signal and its digital approximation is called the quantization error (or quantization noise)
  • $\text{Sampling rate}\times \text{Bit depth}=\text{Bit rate}$

pulse amplitude modulation (PAM)

Multiplexing

• FDM
• TDM
• CDM

encoding

digital to digital

• baseline wander

An arbitrary bit pattern in various binary line code formats (source: commons.wikimedia.org)

Signal		1	0
NRZ–L	non-return to zero level	high	low
NRZ-I	non-return to zero inverted	transition	no transition
Manchester (IEEE 802.3)	midpoint transition	high-to-low	low-to-high
Manchester (G. E. Thomas)	midpoint transition	low-to-high	high-to-low
Differential Manchester	always midpoint transition	no change at the start	change at the start

Encoding data using exclusive or logic (802.3 convention)

Original data		Clock		Manchester value
0	XOR	0	=	0
		1		1
1		0		1
		1		0

4B/5B

• No code has more than one leading 0.
• No code has more than two trailing 0s.

Data	4B5B code
0000	11110
0001	01001
0010	10100
0011	10101
0100	01010
0101	01011
0110	01110
0111	01111
1000	10010
1001	10011
1010	10110
1011	10111
1100	11010
1101	11011
1110	11100
1111	11101