Module 4. Signal Representation and Baseband Processing. Version 2 ECE IIT, Kharagpur

Transcription

1 Module 4 Sigal Represetatio ad Basebad Processig ersio ECE IIT, Kharagpur

2 Lesso 17 Noise ersio ECE IIT, Kharagpur

3 After readig this lesso, you will lear about Basic features of Short Noise; Thermal (Johso) Noise; arious other forms of Noise; Shao s chael capacity equatio ad its iterpretatio; As oted earlier, whe sed some iformatio-bearig sigal through a physical chael, the sigal udergoes chages i several ways. Some of the ways are the followig: The sigal is usually reduced or atteuated i stregth (measured i terms of received power or eergy per symbol) The sigal propagates at a speed comparable to the speed of light, which is high but after all, fiite. This meas, the chael delays the trasmissio The physical chael may, occasioally itroduce additive oise. A trasmissio cable, for example, may be a source of oise. The physical chael may also allow some iterferig sigals, which are udesired The chael itself may have a limitatio i badwidth which may lead to some kid of distortio of the trasmitted sigal. Usually the stregth of the sigal at the receiver iput is so low that it eeds amplificatio before ay sigificat sigal processig ca be carried out. However, the amplifier, while tryig to boost the stregth of the weak sigal, also geerates oise withi. The power of this oise (ad i some other modules dow the lie such as the frequecy dow coverter i a heterodye radio receiver) is ot egligible. This iterally geerated oise is always preset i a commuicatio receiver. arious mathematical models exist to depict differet oise processes that origiate i the receiver ad affect the trasmitted sigal. We will cosider a simple additive oise model wherei a equivalet oise source will be assumed ahead of a oise-less receiver [(t) i Fig ]. This oise, sometimes referred as chael oise, is additive i the sese that the istataeous oise amplitude is added with the istataeous amplitude of the received sigal. s(t) r(t) s(t) chael Delay Atteuatio + + τ α r(t) Iterefece i(t) Noise (t) Fig : A equivalet model for the physical propagatio chael, icludig the oise geerated by the receiver frot ed If s(t) is the trasmitted sigal ad α is the atteuatio itroduced by the chael, the received sigal r(t) ca be expressed as, r(t) = αs(t τ) +I(t) +(t) I(t) represets the iterferig sigal, if ay. ersio ECE IIT, Kharagpur

4 I this lesso, we briefly discuss about the physical features of oise ad a short discussio o a basebad chael model for additive white Gaussia oise (AWGN) uder certai assumptios. It is a commo kowledge that movable electros withi a passive or active electroic compoet are resposible for curret whe excited by exteral voltage. However, eve whe o voltage is applied exterally, electros are always i radom motio, iteractig with other electros ad the material s lattice sites ad impurities. The average velocity i ay directio remais zero. This statistically radom electro motio creates a oise voltage. Noise is a very importat factor that degrades the trasmitted sigal i a receiver. It is ecessary to kow the oise level. Two importat ad relevat forms of oise are, a) thermal oise produced by radom, thermally produced, motios of carriers i metals ad semicoductors ad b) shot oise produced by particle-like behavior of electros ad photos whe a exteral excitatio is available to produce curret. Shot oise is avoidable oly if we reduce all curret to zero. Shot Noise Let us cosider a steady or dc electric curret I betwee two poits A ad B with each electro carryig a charge q. O a average, the umber of charges movig from A to B durig time t is I. t av = q Now, at the microscopic level, the electros do ot move i a perfectly regular fashio. The rate of flow varies upredictably withi short spas of time. This meas that the istataeous curret is usually differet from I. This fluctuatio aroud a omial average value of I is modeled as a oise curret (i ). It has bee established that the observed mea squared value of this fluctuatig curret is, E i =.q.i.b where B is the badwidth of the system used for measuremet. Iterestigly, the mea squared value of the oise curret is proportioal to the gross curret I. So, if the average (bias) curret i a photo detector is high, there is a possibility of cosiderable shot oise. This is a importat issue i the desig of optical detectors i fiber optic commuicatio. Shot oise i optical devices is widely called as quatum oise. Low oise active electroic amplifiers for wireless receivers are itelligetly desiged to suppress the shot oise by electrostatic repulsio of charge carriers. Shot oise is closely described ad modeled as a Poisso process. The charge carriers resposible for the shot oise follow Poisso distributio [Lesso #7]. Aalytically, the oise power may be obtaied from the Fourier trasform of the auto-correlatio of this radom process. ersio ECE IIT, Kharagpur

5 Thermal Noise ( also kow as Johso-Nyquist oise ad Johso oise) : Thermal oise is geerated by the equilibrium fluctuatios of the carriers i electroic compoets, eve i absece of a applied voltage. It origiates due to radom thermal motio of the charge carriers. It was first measured by J. B. Johso i 198 ad theoretically established by H. Nyquist through a fluctuatio dissipatio relatioship of statistical thermodyamics. Thermal oise is differet from shot oise, which is due to curret fluctuatios that occur oly whe a macroscopic curret exists. The thermal oise power P, i watts, is give by P = 4kTΔf, where k is Boltzma s Costat [ k = (4) 10 3 J/K ], T is the compoet temperature i Kelvi ad Δf is the badwidth i Hz. Thermal oise power spectral desity, Watt per Hz, is costat throughout the frequecy spectrum of iterest ( typically upto 300 GHz). It depeds oly o k ad T. That is why thermal oise is ofte said to be a white oise i the cotext of radio commuicatio. A quick ad good estimate of thermal oise, i dbm [ 0 dbm = 1 mwatt], at room temperature (about 7 0 C) is: P = log(Δf) A quick calculatio reveals that the total oise power i a receiver, with a badwidth of 1 MHz ad equivalet oise temperature of 7 0 C, may be about -114 dbm. The thermal oise voltage, v, that is expected across a R Ohm resistor at a absolute temperature of T K is give by: v = 4kTΔf So, thermal oise i a receiver ca be made very low by coolig the receiver subsystems, which is a costly propositio. Colour of oise Several possible forms of oise with various frequecy characteristics are some times amed i terms of colors. It is assumed that such oise has compoets at all frequecies, with a spectral desity proportioal to 1 f α. White oise It is a oise process with a flat spectrum vs. frequecy, i.e. with same power spectral No desity, W/Hz. This meas, a 1 KHz frequecy rage betwee KHz ad 3KHz cotais the same amout of power as the rage betwee MHz ad.001 MHz. Let us ote here that the cocept of a ifiite-badwidth white oise is oly theoretical as the oise power is after all, fiite i a physically realizable receiver. The additive Gaussia oise process is white. ersio ECE IIT, Kharagpur

6 Pik oise [flicker oise, 1/f oise] The frequecy spectrum of flicker oise is flat i logarithmic space, i.e., it has same power i frequecy bads that are proportioally wide. For example, flicker oise i a system will maifest equal power i the rage from 30 to 50 Hz ad i the bad from 3KHz to 5KHz. Iterestigly, the huma auditory system perceives approximately equal magitude o all frequecies. Brow oise Similar to pik oise, but with a power desity decrease of 6 db per octave with 1 icreasig frequecy (desity proportioal to f ) over a frequecy rage which does ot iclude DC. It ca be geerated by simulatig Browia motio ad by itegratioblue oise: Power spectral desity of blue oise icreases 3 db per octave with icreasig frequecy (α = -1) over a fiite frequecy rage. This kid of oise is sometimes useful for ditherig. Shao s Chael Capacity Equatio The amout of oise preset i the receiver ca be represeted i terms of its power N =, R ch where R ch is the characteristic impedace of the chael, as see by the receiver ad v is the rms oise voltage. Similarly, the message bearig sigal ca be represeted by its power we ca represet a typical message i terms of its average sigal power S = v s v, where v s is the rms R ch voltage of the sigal. Now, it is reasoable to assume that the sigal ad oise are ucorrelated i.e., they are ot related i ay way ad we caot predict oe from the other. If P r is the total power received due to the combiatio of sigal ad oise, which are ucorrelated radom processes, we ca write v r = v s + v P = S + N r, i.e., Now, let the received sigal with rms voltage v s cotai b bits of iformatio per uit time ad oise with rms voltage v. If, for the sake of simplicity, we decide to sample the received sigal oce per uit time, we ca hope to recover the b bits of iformatio correctly from the received sigal sample by adoptig the followig strategy: We quatize the sample i a maer such that the oise is ot likely to make our decisio about b-bits of iformatio wrog. This is achievable if we adopt a b-bit quatizer(i.e. b quatizer levels) ad the oise sample voltage is less tha half the step size. The idea the, is simply to read the quatizer output as the received b-bit iformatio. So, the limitig coditio may be stated as: b r max =, where r max is the maximum allowable received sigal amplitude ad max is the max maximum allowable oise amplitude. With this quatizer, our decisio will be correct whe ersio ECE IIT, Kharagpur

7 b r ad our decisio will be erroeous if b r. So, the limitig coditio for extractig b-bits of iformatio from oise-corrupted received sigal is, b r = Now, we ca write, b r = = v v r = v + v v s S = 1 + N S Or, equivaletly, log 1 + N Now, from Nyquist s samplig theorem, we kow that, for a sigal of badwidth B, the maximum umber of such samples that ca be obtaied per uit time is B ad hece, the maximum amout of iformatio (i bits) that ca be obtaied per uit time, is, S S Imax = Bb = B log 1 + = B log 1. N N Eq is popularly expressed as, S C = Blog N C idicates the capacity of the waveform chael, i.e. the maximum amout of iformatio that ca be trasmitted through a chael with badwidth B ad ejoyig sigal-to-oise ratio of S/N. Eq is popularly kow as Shao-Hartley Chael Capacity Equatio for additive white Gaussia oise waveform chael. Iterpretatio of Shao-Hartley Chael Capacity Equatio a) We observe that the capacity of a chael ca be icreased by either i) icreasig the chael badwidth or ii) icreasig the sigal power or iii) reducig the i-bad oise power or iv) ay judicious combiatio of the three. Each approach i practice has its ow merits ad demerits. It is ideed, iterestig to ote that, all practical digital commuicatio systems, desiged so far, operate far below the capacity promised by Shao-Hartley equatio ad utilizes oly a fractio of the capacity. There are multiple yet iterestig reasos for this. Oe of the overridig requiremets i a practical system is sustaied ad reliable performace withi the regulatios i force. However, advaces i codig theory (especially turbo codig), sigal processig techiques ad LSI techiques are ow makig it feasible to push the operatig poit closer to the Shao limit. ersio ECE IIT, Kharagpur

8 b) If, B, we apparetly have ifiite capacity but it is ot true. As B, the ibad oise power, N also teds to ifiity [N = N o.b, N o : sigle-sided oise power spectral desity, a costat for AWGN] ad hece, S/N 0 for ay fiite sigal S power S ad log 1+ also teds to zero. So, it eeds some more careful N iterpretatio ad we ca expect a asymptotic limit. 1 1 At capacity, the bit rate of trasmissio R b = C ad the duratio of a bit = T b = R = b C. If the eergy received per iformatio bit is E b, the sigal power S ca be expressed as, S = eergy received per uit time = E b.r b = E b.c. So, the sigal-to-oise ratio S N ca be expressed as, S N E C b = N0B Now, from Eq , we ca write, C EbC = log 1+ B NB This implies, C E B B b = - 1 N C 0 B C 1 l - 1 C + B, for B >> C = log e, for B >> C = -1.6 db E b So, the limitig N 0 errorless trasmissio oly whe the badwidth., i db is -1.6 db. So, ideally, a system desiger ca expect to achieve almost E b N 0 is more tha -1.6 db ad there is o costrait i c) I the above observatio, we set R b = C to appreciate the limit i ad we also saw N0 that if R b > C, the oise v is capable of distortig the group of b iformatio bits. We say that the bit rate has exceeded the capacity of the chael ad hece errors are ot cotrollable by ay meas. To reiterate, all practical systems obey the iequality R b < C ad most of the civilia digital trasmissio systems utilize the available badwidth efficietly, which meas B (i Hz) ad C (i bits per secod) are comparable. For badwidth efficiet trasmissio, the strategy is to E b ersio ECE IIT, Kharagpur

9 Rb icrease the badwidth factor B while R b < C. This is achieved by adoptig suitable modulatio ad receptio strategies, some of which will be discussed i Module #5. Problems Q ) Name two passive electroic compoets, which may produce oise. Q4. 17.) If a resistor geerates 1 ao-watt/hz, determie the temperature of the resistor. Q ) Determie the capacity of a waveform chael whose badwidth is 10 MHz ad sigal to oise rotatio is 10dB. ersio ECE IIT, Kharagpur