intro

• A link is

• Connection-oriented service

• Connection-less service

• Link Classification

  • Last-Mile
  • Backbone
  • LAN

Signal processing

• 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)

• 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

• 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 $\text{baud}$)
  • $M$: 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 (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

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

Performance

Bandwidth

• $\text{data rate}=\frac{\text{data transmitted}}{\text{time taken}}$
• $\text{Bit rate}=\frac{\text{ \# bits transmitted}}{\text{Time taken}}$ (unit: bits per second, bps)
• The bit rate $R$ (קצב נתונים) is the number of bits transmitted per a unit of time (unit: bps, bits per second)
• The data bandwidth (or digital bandwidth or simply bandwidth) is the maximum data rate that can be transmitted over a communication channel

Throughput

• Network throughput (or just throughput) (in bps) is a measurement of the average amount of data that actually passes through a network in a specific time frame, taking into account the impact of latency

Latency

$\text{Latency}=t_{\text{prop}}+t_{\text{tx}}+t_{\text{queue}}$

• latency is the amount of time it takes for a packet of data to travel between two points across a network connection
• latency (also called delay (?)todo) is the time it takes for data to pass from one point on a network to another
• (Propagation delay) $t_{\text{prop}}=\frac{d}{s}=\frac{\text{distance}}{\text{speed of signal}}$
• (Transmission delay) $t_{\text{tx}}=\frac{L}{R}=\frac{\text{length}\text{[bits]}}{\text{transmission rate}\text{[bps]}}$
• (Queueing delay)
• (Processing delay)
• round-trip time (RTT) (or (round-trip delay (RTD)): the time it takes for a signal to travel from the source to the destination and back again
  • $\text{RTT}\approx 2\times t_{\text{prop}}$

Bandwidth-delay product

• The bandwidth-delay product, $\text{RTT}\times \text{bandwidth}$, in bits, is the amount of data that can be in transit in the network at any given time

Stop-and-wait

$\text{Throughput}\text{[bps]}=\frac{\text{frame size}\text{[bits]}}{\text{RTT}}$

File transfer time

• Minimum transfer time: $\text{RTT}+t_{\text{tx}}$

• $T_{\text{total}}=t_{\text{handshake}}+t_{\text{data}}$

  • $t_{\text{handshake}}=k\times \text{RTT}$, where $k$ is the number of RTTs needed for handshaking
    • $2\times \text{RTT}$todo
  • (Continuous pipeline) $t_{\text{data}}=N\cdot t_{\text{tx}}+t_{\text{prop}}$
  • (Stop-and-wait) $t_{\text{data}}=(N-1)\cdot (t_{\text{tx}}+\text{RTT})+t_{\text{tx}}+t_{\text{prop}}$
  • (window-limited) $t_{\text{data}}=(K-1)\cdot \text{RTT}+t_{\text{prop}}$
    • $K=\lceil \frac{N}{W}\rceil$
    • $W$ is the window size (packets per RTT)
  • $N=\lceil \frac{\text{File Size}}{\text{Packet Size}}\rceil$ is the number of packets needed to transmit the file

encoding

• baseline wander

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

• non-return to zero (NRZ or NRZ‑L): low: 0, high: 1
• non-return to zero inverted (NRZI): change at the start: 1, no change at the start: 0
• Manchester: transition at the midpoint.
  • (G. E. Thomas) low-to-high: 1, high-to-low: 0
  • (IEEE 802.3) low-to-high: 0, high-to-low: 1
• Differential Manchester: transition at the midpoint. change at the start: 0, no change at the start: 1

modulation

Data	Signal	Encoding/Conversion Technique
Analog	Analog	AM, FM
Digital	(Square-wave) digital	NRZ, NRZI, Manchester, Differential Manchester
Digital	(Discrete) analog
Analog	Digital

Framing

• A frame

errors

• A bit error is when a bit is received incorrectly (0 instead of 1 or vice versa)

• A burst error is when a sequence of bits is received incorrectly

• The bit error ratio is defined as $\text{BER}=\frac{\text{\# bit errors}}{\text{Total transmitted bits}}$

• The bit error probability is defined as $p_e=\mathrm{E}[\text{BER}]$

• error detection:

  • goal: transmit a message $M$ of length $n$ with $k\ll n$ redundant bits $R=f(M)$
  • the sender transmits $(M,R)$. The receiver receives $(M^{\prime },R^{\prime })$ and checks if $R^{\prime }=f(M^{\prime })$, if yes, assume no error with high probability, else, error detected

• internet checksum

  • message is divided into words of 16 bits: $w_1,w_2,\dots ,w_m$
  • the checksum is $R=\sim (w_1+w_2+\dots +w_m)$ (where $\sim$ is the bitwise NOT operation. The sum is done using ones’ complement addition)
  • the sender sends $(M,R)$
  • the receiver computes $S=w_1^{\prime }+w_2^{\prime }+\dots +w_m^{\prime }+R^{\prime }$ (using ones’ complement addition) and checks if $\sim S=0$, if yes, assume no error, else, error detected