ECE 333: Itroductio to Commuicatio Networks Fall 22 Lecture : Physical layer II Impairmets - distortio, oise Fudametal limits Examples Notes: his lecture cotiues the discussio of the physical layer. Recall, the basic problem addressed at the physical layer is sedig a sequece of bits over a aalog commuicatio chael. his is the topic of digital commuicatio theory. Last time we cosidered a simple techique for mappig bits ito a aalog sigal - amely mappig a ito a square pulse with a positive amplitude ad mappig a ito a square pulse with a egative amplitude. Whe each pulse has duratio of sec., ad oe pulse is set every sec., this is sometimes called No Retur to Zero (NRZ) sigalig. he receiver ca the sample the received sigal every secods to determie if or was set. his aalog sigal could represet a voltage or curret waveform. Recall, the power i such a sigal is proportioal to the square of its amplitude. he above descriptio is complicated by the fact that the aalog sigal arrivig at the receiver will differ from the sigal set at the trasmitter due to a variety of chael impairmets. Last time we discussed oe impairmet - atteuatio; atteuatio is the loss i power a sigal experieces as a fuctio of the distace. For the above system, atteuatio will cause the differece i amplitude betwee a ad a to decrease. he hardware i the receiver will oly be able to reliably distiguish betwee the sigals that are separated by a adequate differece - the required differece is referred to as the sesitivity of the receiver. (he effect of atteuatio is eve more importat, whe cosidered i the cotext of oise, which we will discuss i the followig.) I this lecture, we will look at two other chael impairmets - distortio ad oise. 2
Distortio (Filterig) wo properties:. Most aalog chaels are well modeled as a liear system. 2. Chael's atteuatio ad delay varies with frequecy. cos( 2 ft ) H ( f ) cos(2 ft H( f )) Chael H(f) H ( f ) = frequecy respose of the chael. 3 Frequecy respose H(f) Frequecy Depeds o physics of medium ad filters iserted by commuicatio egieers. How does this impact trasmitted sigal?
Fourier Aalysis Fourier aalysis provides frequecy cotet of sigals: Suppose s(t) is periodic with period. frequecy cotet give by Fourier series Where f a a si(2 ft) s ( t) c b cos(2 ft) is the fudametal frequecy, 2 s( t)si(2 ft) dt b c s( t) dt 2 s( t)cos(2 ft) dt 5 Fourier series ca also be writte as s( t) c c cos(2 ft ) where c a b ta a / b 2 2 Give Fourier series of trasmitted sigal ad frequecy respose, H(f), of the chael, what is the output? 6
Notes: Slide 3: A liear system is a system with the followig property. If iput s ( t ) geerates output r ( t ) ad iput s 2 ( t ) geerates output r 2 ( t ), the iput s ( t ) s2 ( t) geerates the output r ( t) r2 ( t), where is ay real umber. Our emphasis is o liear distortio- o-liear chael effects are importat i some applicatios, but will ot be addressed here. Slide : For example, i the telephoe etwork, low pass filters are employed as part of aalog to digital coversio of speech sigals. he badwidth of a chael is the differece betwee the highest ad lowest frequecies that the chael will pass. For real chaels the cutoff is somewhat arbitrary. Warig: he term badwidth is also used to idicate the trasmissio rate of a lik i bits/sec. his is ot the same as the badwidth of the chael defied above. Slide 5: A fuctio is periodic with period if s(t+) = s(t) for all t. here are some mathematical requiremets eeded for a periodic fuctio to be represeted by its Fourier series; a complete discussio of this requires some fairly advaced mathematics ad will ot be covered here. he frequecy cotet of a large class of o-periodic fuctios is give by the Fourier trasform of the fuctio. his is covered i ECE 222. 7 Fourier Aalysis Example Cosider sedig ASCII b symbol () as a sequece of square pulses. Assume it is set over ad over (so its periodic). (ASCII "b") g t t 3 else or 6 t 7
/ Calculatio of Fourier Series: C O g t dt 3 dt 7 6 dt 3 7 6 3 3 a 2 2 3 g t si 2 ft 2 t si cos dt dt 2 7 6 2 t si cos 3 cos 6 dt cos 7 Similarly b )(*! "#%$! "#&! "#'$ +,.- si 3 si si 7! "# si 6 9 Example Amplitude, (C ) 2 3 5 6 7 9 2 Harmoic With 2 bps modem, what is fudametal frequecy? At 96 bps?
Suppose chael is modeled as perfect low-pass filter with cutoff at 3 Hz. (st order model of voice grade telephoe circuit.) 3 KHz Chael badwidth f What is the frequecy cotet of the output sigals? / For 2 bps modem? / For 96 bps? Resultig Sigals IME DOMAIN FREQUENCY DOMAIN 2 3 5 6 7 9 2 2 2 3 2 3 5 6 7 2
/ / / Notes A similar example ca be foud i Sec. 3.3 of Leo-Garcia. At 2 bits/sec (bps), we ca sed the b symbol 3 times per secod. his results i a fudametal frequecy of 3 Hz, the other harmoics will the be at 6 Hz, 9 Hz, etc. At 96 bits/sec (bps), the b symbol ca be set 2 times per secod, resultig i a fudametal frequecy of 2 Hz. For the 3 bps modem, the first harmoics will get through the ideal low pass filter, i.e., 3 Hz, 6 Hz,,3 Hz. For the 2 bps modem, oly the first 2 harmoics get through - oe at 2 Hz, ad oe at 2 Hz. For this example, we clearly caot sed this waveform reliably through the chael at a rate higher tha 2kbps, for at such a rate ot eve the first harmoic would get through. As show i this example, the effect of filterig is a smoothig ad spreadig out of the trasmitted sigals. he spreadig i time causes the pulses associated with differet bits to iterfere with each other. his pheomeo is called itersymbol iterferece. Digital commuicatio systems geerally ecode bits as a sequece of pulses, but ot ecessarily square pulses as i this example. Better pulse shapes ca be chose to reduce the effect of itersymbol iterferece. Our discussio has focused o low-pass chaels. May realistic chaels are bad-pass - that is they have low atteuatio oly i a bad of frequecies away from zero. I these cases some type of modulatio is eeded to shift the frequecy cotet of the trasmitted sigal ito the desired bad (see Sect. 3.6 of Leo-Garcia) 3 Nyquist's heorem Basic digital commuicatio system: Map bits ito sequece of pulses. How fast ca pulses be set ad data be recovered? Number of pulses per secod = baud rate. Nyquist: Maximum baud rate = 2 x badwidth of chael. How may bits per secod?
/ Symbol rate If each pulse correspods to bit the bit rate = baud rate. I geeral, each pulse may is chose from a set of V discrete values. Ex: V =, set of represetatios = {-3 volts, - volt, + volt, +3 volts}, time iterval = I secods. +3 + - -3 I 2I 3I I 5I 6I 7I I t Bit rate =? 5 Notes More precisely Nyquist's heorem gives a limit o the rate at which pulses ca be set without ay itersymbol iterferece. he pulse shapes that achieve this limit are sometimes referred to as Nyquist pulses. Each allowable pulse is ofte called a symbol ad the set of all symbols is called the (iput) alphabet to the chael. Usually the alphabet will cotai V 2 symbols, i which case each symbol represets bits. I this case the bit rate is equal to B, where B deotes the baud rate. For example if V=, the we are sedig 3 bits per symbol ad the bit rate is three times the baud rate. Nyquist's theorem ca be iterpreted as sayig with a alphabet of V symbols, the maximum bit rate over a chael with Badwidth B is 2Blog2(V). Ex: A stadard 3 bps modem uses a alphabet with 2 symbols, V = 2, so it is also a 3 baud modem. A stadard 96 modem uses a alphabet with 6 symbols, each symbol represets bits, so it is a 2 baud modem. 6
Noise Noise A H(f) + Noise is due to thermal agitatio of electros & iterferece from other electromagetic waves. Amout of oise preset is measured by Sigal-to-Noise Ratio (SNR) SNR typically measured i db's 7 Noise limits maximum data rate: Shao's Result Shao: Maximum umber of bits/secod = W log 2 ( + S/N) W = chaels Badwidth. S/N = sigal-to-oise ratio (liear) Ex: W = 3 Hz, SNR = 3 db => S/N = Max = 3 log 2 ( + ) ~ 3, bps
Notes: he sigal-to-oise ratio i db's is log of the liear value. Shao's result as stated here applies to a ideal bad-pass (or low-pass) chael with a certai type of oise called additive white Gaussia oise. However this result ca be geeralized to may other types of chaels. More precisely it refers to the maximum rate that ca be attaied with arbitrary small bit error rates. Shao's result is very deep; we have ot developed the backgroud here to eve state it precisely or justify it. he poit you should take away from this is that there is a provable upper boud o the bit rate for a commuicatio chael. (If you wat to lear more about this take ECE 2 - Iformatio heory) Approachig this boud requires the use of very sophisticated error correctig codes (more about this i a couple of lectures.) Note o voice grade telephoe chaels V.3 modems have a rate of 2. kbps, which is ot far off from the 3kbps approximated i the above example. he higher rates o Note for a give alphabet size a upper boud o the maximum data rate (assumig Nyquist pulses are used) is the miimum of the Nyquist ad Shao limits. 9 Examples of commo rasmissio media rasmissio media: Guided: Copper - wisted Pair, Coax Fiber (glass) Uguided (wireless) Fixed, Lie-of-sight (microwave, satellite) Mobile (cellular) 2
Notes: wo types of copper wirig are commoly used i etworks - ushielded twisted pair (UP) ad coxial cable (Coax). UP: wo isulated copper wires twisted to reduce atea effects. Very commo i curret telephoe system betwee home ad local office. Legacy phoe etworks limited useful badwidth of UP to 3 KHz. Actual copper wire has usable badwidth of -MHz Data rates decrease with distace: (usig etire badwidth) 5 Mbps over ~ m 5 Mbps over ~ km ADSL systems use UP to provide.5-6 Mbps to the home. Coaxial Cable (Coax) Costructio ad shieldig gives coax higher badwidth ad oise immuity tha twisted pair. (Also higher cost) wo commo types: 5 (Basebad) Coax (For km legth, data rate ~ -2 Gbps). Used to be commo i phoe etwork (replaced by fiber). Commo i LANs (e.g. some versios of Etheret) 75 (Broadbad) Coax: Most commoly foud i cable V etwork. Data rates up to Gbps possible depedig o distace. 2 Cable V Networks Program Provider Microwave or optical lik Head-ed Customer's home Set top Box V Splitter Amplifiers Drop poits 22
Notes: raditioal CAV etworks: Used aalog sigalig ad aalog amplifiers. ree topology made up of coaxial cable liks from the head ed to the customer's. Uses 5 MHz Badwidth divided ito 6 MHz chaels. Oe way trasmissio (used aalog amplifiers which oly amplified sigal i dowstream directio) I 999, CAV etwork reached over 5% of US households ad could reach over 9% with relatively mior additioal ifrastructure. he CAV Network has evolved to be able to support a mix of applicatio icludig iteret coectivity. Sigificat chages iclude: Digital trasmissio with advaced video compressio techiques. Replacig either the coax betwee the head-ed ad the drop poits with fiber, (this is called fiber to the curb (FC) alteratively some systems oly a portio of this coax is replaced resultig i a Hybrid Fiber Coax (HFC) system. Upgradig hardware to allow bi-directioal commuicatio -this is doe either by layig a secod cable for upstream commuicatio or by usig a "split system" which divides the available badwidth ito two portios - oe for upstream ad oe for dow stream 23 Optical Fiber Fiber has a much higher badwidth ad better oise immuity tha Coax. Curret systems at ~- Gbps Feasible limit i excess of 5 bps Used i telephoe etwork backboe ad most WANs Used i some high-speed LANs ad MANs. 2
Notes: A optical fiber is a very thi, flexible medium capable of guidig a optical sigal. his sigal is geerated by either a semicoductor laser or a Light Emittig Diode (LED) at the trasmitter. At the receiver, a optical detector, such as a photo diode is used. Optical fibers are typically made from very pure glass (fused silica). Plastic fibers are also used for short liks (such as i office buildigs); plastic fiber supports a lower data rate but is cheaper. he trasmitted sigal optical sigal is usually a o/off pulse at some wavelegth, e.g. the light source will be tured o for a legth of time to correspod to a ad tured off for the same legth of time to correspod to a. (For obvious reasos, this is sometimes called O/Off Keyig). A techique called Wavelegth Divisio Multiplexig (WDM) allows the sedig of idepedet sigals o differet wavelegths of the same fiber. (e.g. sigals at 2. Gbps each). ypically whe discussig optical systems, the wavelegth of the sigals is referred to istead of the frequecy. Of course the frequecy is simply the speed of light divided by the wavelegth. 25 Atteuatio of light through fiber i the ifrared regio. Shaded areas are commoly used bads. Some ewer "all-wave" fiber has o otch at. micros. 26
Compariso of Fiber ad Copper Fiber Advatages: Much higher badwidth Low atteuatio (fewer repeaters) Immue to power surges, e-mag iterferece hi ad lightweight Cheap core material (sad) Iheretly secure Copper Huge istalled base Cheaper iterfaces For ew istallatios over MAN or WAN distace, fiber is clear wier. 27 Wireless rasmissio Electromagetic waves i radio, microwave, ifrared or visible bads are used for wireless commuicatios. Fixed: Usually requires lie-of sight path. Ofte cheaper to deploy tha fiber. (but lower rates) Ex. s. Microwave - commoly used i telephoe ad V. Ca suffer iterferece due to atmospheric coditios. 2. Ifrared Used for idoor etwork applicatios, e.g., wireless LANs, PC to Palm Pilot. 3. Geosychroous Satellite basically a microwave repeater. Delays up to 3 msec. ypical satellites ca support 5 MHz of badwidth, divided ito multiple traspoders each with 36 MHz badwidth. Each ca hadle 5 Mbps. Note: A sigle fiber has greater badwidth tha all the commuicatios satellites ever lauched. 2
Mobile wireless (e.g. cellular systems) aywhere, aytime coectivity. ime-varyig chael Much higher error rates tha wired liks. Power limitatios Limited frequecy spectrum must be shared Mobility also has impact o higher layer fuctios (e.g. routig) 29