DIGIAL COMMUNICAIONS SYSEMS MSc in Electronic echnologies and Communications
Scheme o a communication system
Spectrum o electromagnetic requencies Wavelength c Speed o light 3. km/s Frequency Audio khz Radio FM MHz Microwaves GHz Inrared 3 Hz Audio 3 km Radio FM 3 m Microwaves 3 cm Inrared nm 3
Radiorequency bands Frequency band Denomination ropagation characteristics ypical use 3-3 khz Very low requency VLF errestrial waves Long distance navigation, underwater communication 3-3 khz Low requency LF Similar to LF less reliable Long distance navigation and radiolamps or marine communication 3-3 khz Medium requency MF errestrial wave and nocturnal aerial wave 3-3 MHz High requency HF he ionospheric relection changes with the time o the day, season and requency 3-3 MHz Very high requency VHF Almost line-o-sight LOS wave propagation Sea radio, location o routes and AM broadcasting Radio hams, international broadcasting, military communication, long distance aerial and marine communication, telephony VHF television, FM radio, AM aerial communication, navigation aids to aircrats,3-3 GHz Ultra high requency UHF LOS propagation UHF television, cellular telephony, navigation aids, radar, microwaves links, personal communication systems 3-3 GHz Super high requency SHF LOS propagation, attenuation due to rainall beyond GHz Communication via satellite, radar links via microwaves 3-3 GHz Etremely high requency EHF Similar to the previous one Radar, satellite and eperimental communications 4
Channel capacity heorem o Shannon-Hartley: Signal power W Channel capacity bits/s C B log S N Bandwidth Hz Noise power W Energy per bitw-s S N Eb / N b B Bit period s E N b Rb B E b N ower spectral density o white noise W/Hz Keeps on being dimensionless 5
Shannon limit C What is the minimum S/N ratio to which we still have error-ree communication? - his is the same as asking what is the value o S/N or which C or, equivalently, B. lim B log Finally: / E b N e S N CR b C B log C, B Eb C N B e log loge log C B log E N b C B,693,59 db E N C B / b log e 6
Shannon s limit Eample: Using BSK modulation or a digital signal, i we wanted a error probability o -5, we would need a E b /N ratio o 9.6 db. aking into account the Shannon s limit, it would be possible to design some kind o codiication scheme which reduced the E b /N as ar as. db. Evidently, Shannon did not tell us how to design those so good codiication schemes. Nowadays, turbo codes oer E b /N gains in the order o db. Claude E. Shannon, A mathematical theory o communication, he Bell System echnical Journal, vol. 7, pp. 379-43, 63-657, Julio, Octubre 948. Dimensionality theorem Nyquist Nyquist demonstrated that it is possible to send non-interering pulses at a maimum data rate o B pulses/s, where B is the transmission bandwidth Maimum spectral eiciency = symbols/s/hz Harry Nyquist, Certain topics on telegraph transmission theory, ransactions o the American Institute o Electrical Engineers, vol. 47, pp. 67-644, Abril 98. 7
Signals and noise Signal: Desirable part o the received waveorm Noise: Non-desirable part o the received waveorm he physically realizable waveorms ollow the net rules:. he waveorm has signiicant values along a inite time period.. he spectrum o the waveorm has signiicant values along a inite requency interval. 3. he waveorm is a continuous unction o time. 4. he waveorm has a maimum inite value. 5. he waveorm only has real values, that is, it cannot be a comple number a+jb, being b dierent to zero, in any time instant. he real signals are, thereore, energy signals their total energy is inite and non-zero, although, in general, power signals with mean power inite and non-zero are used to model real signals in order to simpliy the analysis. 8
Some interesting relations Mean value DC o a signal: t Root mean square rms value o a signal: lim / / rms t dt t otal normalized energy: E lim / / t dt Mean or average normalized power: t lim output Gain in decibels o a system: G db log input Signal to noise ratio: s t signal S / R log log noise n t ower level in db with respect to mw: power level W dbm log 3 / / t dt 9
COMMUNICAION SYSEMS Fourier transorms and spectra Spectral symmetry o real signals: arseval s theorem: Energy spectral density: ower spectral density: Wienner-Khintchine s theorem: W W d W W dt t w t w W E d E E W w lim t w t t w t t w elsewhere / / d t w w lim / / w w w w w R R dt w t w t w t w t R
Signals bandwidth Bandwidth at 3 db or hal-power bandwidth: It is, where < < is the requency band where the magnitude spectrum is not reduced more than / times the maimum value o H, which is located inside this band. Null-to-nulll bandwidth is, where is the irst null in the magnitude spectrum envelope beyond and, in bandpass systems, is the irst null below, being this last requency the point where the magnitude spectrum reaches its maimum value. In baseband systems, is generally zero.
Signals bandwidth Noise equivalent bandwidth is the bandwidth o a ictitious rectangular spectrum in such a way that the power in this rectangular band is the same as the power associated with the signal spectrum at positive requencies: B eq H H d ower bandwidth deines the requency band which contains the 99% o the total power o the signal.
3 COMMUNICAION SYSEMS Random signals Random variables: Represent a unctional relation between a random event A and a real number : =A Distribution unction: roperties o the distribution unction: robability density unction pd: roperties: F F F F F F 4. 3... si d df p d p F F d p F F d p p..
Random signals robability density unction Mean value m or epected value o : m E p d n-th moment: Mean square value: E n n p d E p d Variance o : var E m E m m m m p p d d E m E m 4
Random signals Random processes: hey can be seen as a unction o two variables: an event A and the time t, A,t. Mean value in t = t k : Autocorrelation: R E tk p d m tk k E t t, t t It is said that a random process is wide-sense stationary i: E t m constant R t t, t R t t R E t, roperties o the autocorrelation:. R R symmetrical in about zero. 3. 4. R R R R E para todo maimum value occursat the origin the autocorrelation and the power espectraldensity orm a Fourier transorm pair at the origin is equal to the the value t average power o the signal 5
Random signals Random processes It is said that a process is ergodic in the mean i: / m lim t dt / It is said that a process is ergodic in the autocorrelation unction i: / R lim / t t dt ower spectral density o a random process. and isalways real valued. 3. 4. R d or t real- valued the SD and the autocorrelation unction orma Fourier transorm pair relationship between the average normalized power and the SD 6
Random signals Autocorrelation and SD o a random binary signal R or sin sinc or 7
8 COMMUNICAION SYSEMS Random signals he noise in the communication systems hermal noise: It is characterized by being a Gaussian random process o zero mean. For z=a+n a=constant, n Gaussian: ep n n p ep a z z p
Random signals he noise in the communication systems White noise: he thermal noise behaves like white noise, presenting a plain power spectral density at all the requencies. N N n W/Hz Rn n 9