Channel Capacity Graham Knight ([email protected]) Signals and signal representation Analogue and digital: introduction Signal representation Signal spectra Signal bandwidth Channel capacity: noise and bandwidth 01/27/20 GC15/GA07/6007 01 2 Questions What is the difference between analogue and digital representations? why use digital? How do we describe and characterise signals? Characteristics of a communication channel: constraints: noise and bandwidth why can we get 1Gb/s on Ethernet, but not on ADSL? 01/27/20 GC15/GA07/6007 01 3 Analogue and Digital Representations [1] Analogue

Signal level is analogue of information value E.g. microphone electrical wave mirrors sound wave. Typically continuous, smoothly varying voice/speech, video Easy to capture: mature (old!) technology 01/27/20 time Digital Information represented by an encoding Discrete signal levels: E.g. written text, numbers, morse code Sampled speech Anything can be converted to a digital representation New(er) technology GC15/GA07/6007 01 time 4 Analogue and Digital Representations[2]

Analogue: continuous, infinite range Transmission: subject to noise in media interference, e.g. radio Digital discrete symbols E.g. Binary numbers: 1010102 42 Only have to transmit and receive ones and zeros: e.g. use two approximate voltage levels time 01/27/20 GC15/GA07/6007 01 5 Digital Representation - Summary High fidelity: better error control (detection and correction) Source independence: anything (audio, video, etc.) can be digitised Time independence: transmission rate recording/capture rate Encoding: security compression 01/27/20 GC15/GA07/6007 01 6

Communications Channels noise (+distortion) Tx communication channel Rx Communications Channel abstract, generic concept Transmitter (Tx), Receiver (Rx) Signal something that represents information Noise anything that interferes Electrical interference, cosmic rays etc. Attenuation signal gets weaker with distance Distortion frequency dependent attenuation Bandwidth difference between highest and lowest frequencies carried by a channel (Cycles/sec or Hertz (Hz)) Capacity - maximum rate at which information can pass (Bits/sec or symbols/sec (baud)) 01/27/20 GC15/GA07/6007 01 7 Channels and Signals Signals may be analysed -> signal bandwidth Channel must be suitable for signal it carries In particular channel bandwidth > signal bandwidth Suppose signal has frequency range from 400 Hz to 3400Hz signal b/w 3kHz

Suppose channel has effective frequency range from 1kHz to 9kHz channel b/w 8kHz channel bandwidth > signal bandwidth But we need to modify signal so it fits the channel Modulation e.g. radio 01/27/20 GC15/GA07/6007 01 8 Signals and spectra [1] Time domain: variation of signal in time Frequency domain spectrum: discrete zero bandwidth sinusoidal wave 1 A 0.5 0 sinusoidal wave 1.2 -0.5 1 -1 0.8 -0.4

-0.2 0 0.2 0.4 time 0.6 T = 1/f s (t ) A cos(t ) 2f 01/27/20 0.8 1 0.6 0.4 0.2 0 0 GC15/GA07/6007 01 0.2 0.4 0.6 0.8 1 frequency [Hz]

1.2 1.4 9 Signals and spectra [2] square wave square wave 1.2 1 0.8 0.6 0.4 0.2 0 0 2 4 6 time 8 10 12 0 1

2 3 2 5 2 7 2 9 2 frequency [Hz] s (t ) 1; 0 t ,2 t 3 , 0; t 2 ,3 t 4 , Fourier Theorem: sinusoidal components 01/27/20 1 1 sin( t ) sin( 3 t ) sin(

5 t ) 1 2 3 5 s(t ) 1 2 1 sin(7t ) sin(9t ) 7 9 GC15/GA07/6007 01 10 Signals and spectra [3] 4 components 2 components 1.2 1.2 1 1 0.8 0.8 0.6

0.6 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 0 2 4 1 2 s (t ) 2 6 time 8 10 0 12 2

1 2 s (t ) 2 1 sin(t ) 3 sin(3t ) 4 6 time 8 10 12 1 1 1 sin(t ) 3 sin(3t ) 5 sin(5t ) 7 sin(7t ) 3 components 5 components 1.2 1.2

1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 0 2 1 2 s(t ) 2 01/27/20 4

6 time 8 10 1 1 sin(t ) 3 sin(3t ) 5 sin(5t ) 12 0 1 2 s (t ) 2 2 4 6 time 8 10 12 1 1

1 1 sin(t ) 3 sin(3t ) 5 sin(5t ) 7 sin(7t ) 9 sin(9t ) GC15/GA07/6007 01 11 Channel Bandwidth and Digital Signals 1 2 1 1 1 1 s (t ) sin(t ) sin(3t ) sin(5t ) sin(7t ) sin(9t ) 2 3 5 7 9 3 components square wave 1.2 1.2 1 1 0.8

0.8 0.6 0.6 0.4 0.4 0.2 0 0.2 0 -0.2 0 2 4 6 time 8 10 12 0 2

4 6 time 8 10 12 High-frequency components are lost Symbols (hence information) still recoverable 01/27/20 GC15/GA07/6007 01 12 Analogue to digital conversion (ADC) Audio, video need to convert analogue signal to digital Measure height of wave at regular intervals signal Problem 1 are we sampling often enough? Nyquist sampling theorem: sampling signal (freqency s) s 2F Problem 2 - precision Must round measurements quantisation error (noise) b bits/sample => Q = 2b quantisation levels b = log2(Q)

01/27/20 quantisation levels 8 7 6 5 4 3 2 1 0 GC15/GA07/6007 01 13 Channel Capacity - Nyquist Nyquist says: a) Sample rate must be at least s = 2F to retrieve all the information in the signal b) Sampling faster than 2F does not yield extra information We can regard the samples as a set of symbols of a digital data stream How much information can this stream carry? There cannot be more than 2F symbols per sec (baud) (Otherwise we would be extracting more information than Nyquist says we can)

Suppose we have Q quantisation levels log2Q bits/sample bit rate is slog2Q = 2Flog2Q bps If we know a signal has a max. frequency F then Its maximum information rate is 2Flog2Q bps Also, if a channel carries a maximum capacity of F then Its capacity C = 2Flog2Q bps 01/27/20 GC15/GA07/6007 01 14 Channel Capacity with Noise - Shannon How many quantisation levels can there be? i.e. how many bits per symbol? Number of distinct symbols is limited by noise E.g. suppose 8 symbols at 1v intervals If noise adds 0.25v we will interpret voltage correctly if noise adds 0.75v we wont Shannon: no. of distinct symbols = 1 S N S, N are signal and noise power No. of bits/symbol = log 2 1 S N 1 2 log 2 (1 S N ) Thus Channel capacity C = 2 F 1 2 log 2 (1 S N ) F log 2 (1 S N ) bits/sec 01/27/20 GC15/GA07/6007 01 15 Channel Capacity Shannon - Finally Suppose a channel passes a max. frequency F and a min. frequency f The lower limit means a capacity of flog2(1+S/N) is

unavailable Remaining capacity C = (F-f)log2(1+S/N) C = Blog2(1+S/N), where B is the bandwidth This is the Shannon-Hartley Theorem 01/27/20 GC15/GA07/6007 01 16 Signal and noise SNR Noise always present in a communication system Meaningful assessment of noise needed Signal to noise ratio SNR: ratio of signal power (S) to noise power (N) usually quoted in dB (deciBels) SNRdB = 10log10(S/N) Typical SNR is 1000 30dB: 3dB factor of two increase/decrease 01/27/20 GC15/GA07/6007 01 17 Channel capacity example For example local loop (copper wire from home to telephone exchange. B = 1.5MHz, SNR = 20dB (typical) 20 = 10log10(S/N) S/N = 102 = 100 Shannon: S C B log 2 1 N

log10 101 1.5 106 log 2 100 1 1.5 106 9.99 106 bps (3.s.f) log10 2 9.99 Mbps Channel capacity increased by: greater (more) bandwidth greater (better) SNR 01/27/20 GC15/GA07/6007 01 18 A Noisy Example Wireless channel: plenty of bandwidth but lots of noise Shannon sy! i o n VERY ln(1 S / N ) C B log 2 (1 S / N ) B 1.44 B ln(1 S / N ) ln 2 1.44 B S (for S N ). So B 0.7C N N S Suppose we want C=1Mbps but SNR is 10 dB SNR 10 10 log10 ( S / N ), log10 ( S / N ) 1 S / N 10 1 , N / S 10 B 0.7 1000000 10 7 MHz 01/27/20 GC15/GA07/6007 01

19 Properties of different media Copper: twisted pair, co-ax, parallel Radio, microwaves: lower frequencies, higher frequencies, satellite Light: free space and optical fibre Why do we use many different media? What are their relative advantages and disadvantages? 01/27/20 GC15/GA07/6007 01 20 Copper cabling twisted pair Cheap widely available Versatile Easy to connect Subject to interference Signal leakage Relatively high attenuation telephone wire Widely used: smart Tx/Rx electronics at termination speeds upto 1Gbps 01/27/20 Ethernet

cable (CAT5 UTP) with plug (RJ45) GC15/GA07/6007 01 21 Radio, Microwaves etc. Easy to set-up network: no wiring High data rates possible Mature technology: Local and nationwide Global satellites Interference Security Spectrum regulation Safety 01/27/20 Description High frequency VHF UHF Microwaves Millimetre waves Infra-red Visible light Ultra-violet X-rays Gamma rays Frequency

3 - 30MHz Wavelength 100 - 10m 50 - 100MHz 400 -1000MHz 3109 - 1011Hz 1011 - 1012Hz 6 - 3m 75 - 30cm 10cm - 3mm 3mm - 0.3mm 1012 - 61014Hz 61014 - 81014Hz 81014 - 1017Hz 1017 - 1019Hz > 1019Hz 0.3mm - 0.5m 0.5m - 0.4m 0.4m - 10-9m 10-9m - 10-13m < 10-13m GC15/GA07/6007 01 22 Light and Infra-red free space High data rates possible No interference from radio/ electrical signals No regulations Safety: laser sources need care

Line-of-sight: clear path may need to use diffusers/reflectors 01/27/20 Mainly infra-red Uses: (remote control handsets) connections between laptops and peripherals small mobile systems, e.g. PDAs floor-floor communication (e.g. within lift-shafts) GC15/GA07/6007 01 23 Optical fibre [1] stepped index Cladding Multiple paths take different times Core Pulses spread restricting distance and bit-rate Multiple paths take similar times graded index

Less spread -> greater distance and bit-rate Single path mono mode 01/27/20 Greatest distance and bitrates GC15/GA07/6007 01 24 Optical fibre [2] Very high data rates (Gbps and Tbps) No RF interference Secure Low signal loss Small cable dimensions Relatively expensive Hard to connect cabling 01/27/20 Multi-mode + LED: most common LAN, a few Km need repeaters for long runs Mono-mode + laser diode: higher data rates longer cable lengths possible more expensive harder to connect Costs ~25% more than quality copper: connections still more

expensive GC15/GA07/6007 01 25 Summary Analogue vs. digital Waves and signals Bandwidth Noise Sampling Nyquist-Shannon: maximum rate of change of signal Channel capacity Hartley-Shannon: capacity and bandwidth are directly related other factors affect channel data rate Physical media Copper wire, radio, microwaves, light and fibre 01/27/20 GC15/GA07/6007 01 26