4. INTERSYMBOL INTERFERENCE

Transcription

1 DATA COMMUNICATIONS INTERSYMBOL INTERFERENCE 4.1 OBJECT The effects of restricted badwidth i basebad data trasmissio will be studied. Measuremets relative to itersymbol iterferece, usig the eye patter ad the respose of the data trasmissio system to oe pulse, will be made. 4.2 THEORETICAL BACKGROUND The basebad data sigal d(t) is geerally composed of rectagular pulses with differet amplitudes a (Figure 4.1): d ( t) = ag( t T), g(t) beig a rectagular pulse with amplitude equal with uity (Figure 4.2). Fig. 4.1 Basebad data sigal. The amplitudes a ca take o a small set of discrete values. Usually, the umber M of these values is a power of 2, M=2 m, ad the spacig betwee levels is uiform: ±d; ±3d;...; ±(M 1)d. Each level ca represet m biary symbols. The frequecy spectrum of the rectagular data sigal d(t) is exteded over a ulimited frequecy bad. I most practical data trasmissio systems badwidth is expesive, ad tryig to preserve a rectagular sigal shape is ot ecoomical. Furthermore, it is preferable to exclude compoets of oise ad of other perturbatios, which have frequecies outside the bad cotaiig the most part of the sigal eergy. Eve if the data trasmissio system does t limit the Fig. 4.2 Rectagular pulse.

2 60 INTERSYMBOL INTERFERENCE sigal spectrum the trasmissio lie will limit it. The limitatio of frequecy spectrum will result i a differet shape of the received sigal compared to the trasmitted sigal d(t). But, i order to recover data from the received sigal, this oe is sampled at T itervals, so it is ot ecessary to preserve the trasmitted sigal shape. Now, we shall see which are the effects of spectrum limitatio of basebad data sigal for the process of data recoverig. From this poit of view, the simplified block diagram of a basebad data trasmissio system is show i Figure 4.3. Fig. 4.3 Basebad data trasmissio system. It is cosidered that the system employs trasmittig ad receivig filters, havig the trasfer fuctios G T (ω) ad G R (ω), ad a sampler ad threshold detector. Deotig by x(t) the system respose, at the output of the receivig filter, to a trasmitted pulse g(t), the system respose to a data sequece {a }, represeted by the data sigal d(t) is: y ( t) = ax( t T) + η ( t) (4.1) where η(t) is the additive oise. The shape of x(t) is determied by the chael (the trasfer fuctio C(ω)) ad by the trasmittig ad receivig filters. The effect of the restricted badwidth is a time extesio of the respose x(t) over may symbol itervals (Figure 4.4) so that the resposes to differet data symbols are overlappig. t 0 represets the delay ad x 0 represets the gai i sigal passig through the system.

3 DATA COMMUNICATIONS 61 Fig. 4.4 The respose x(t) to a pulse g(t). At time t 0 +kt the desired output voltage is a k ; however the actual value is y ( t0 + kt) = ax( kt T + t0) + η ( t0 + kt) (4.2) or i a cocise form yk = axk + η k (4.3) Isolatig the desired amplitude a k we have 1 ηk y k = x0( ak + axk + ) x x 0 0 (4.4) Takig ito accout the gai x 0, the threshold detector will compare y k /x 0 with the decisio thresholds 0, ±2d, ±4d,..., i order to determie which of the M possible values of a k is closest to the received, ormalized sample. A error occurs wheever k axk +ηk > x0d (4.5) The secod ad the third terms of equatio (4.4) represet itersymbol iterferece ad oise, respectively. The itersymbol iterferece arises from overlappig tails of other pulses addig to particular pulse a k x(t kt) which is examied at the k th samplig time. Notice from equatio (4.4) that itersymbol iterferece ca oly be elimiated by makig x =0 for all 0. I other words, the requiremet is that precursors ad tails of x(t) pass through 0 at regular T - sec. itervals. A example of such pulse is show i Figure 4.5. Obviously, these pulses could be amplitude

4 62 INTERSYMBOL INTERFERENCE modulated ad trasmitted at T - sec. itervals without overlap at the samplig istats. Fig. 4.5 Respose (pulse) correspodig to o itersymbol iterferece. I the desig ad aalysis of basebad trasmissio system it is coveiet, ad ofte ecessary, to be able to specify i the frequecy domai the requiremets for o itersymbol iterferece. So, the problem is how should X(ω), the Fourier trasform of x(t), be i order to have x =0 for 0. More geerally, our problem is to specify the Fourier trasform X(ω) of a time fuctio x(t) whe the samples x =x(t) are give. The fudametal aalytical tool, which we shall employ for this purpose, is the samplig theorem. The theorem eables us to express the time respose x(t) ad the frequecy respose X(ω) for a fuctio bad-limited to [ f Max, f Max ] i terms of samples take at 1/2f Max - sec itervals. The iterval 1/2f Max sec is kow as the Nyquist iterval or, alteratively, the frequecy 1/2T Hz for samplig iterval of T sec is kow as the Nyquist frequecy. The poit of the theorem is that a fuctio bad-limited to the frequecy rage [ f Max, f Max ] Hz has exactly 2f Max degrees of freedom per secod. Whe these are specified, the fuctio is kow exactly. I the basebad pulse trasmissio we are cocered with samples of x(t) take at T - sec itervals. If X(ω) is bad-limited to the Nyquist frequecy f N =1/2T Hz, these samples uiquely determie the fuctio x(t). If X(ω) is bad-limited to a frequecy smaller tha f N there is o fuctio x(t), ad implicitly a X(ω), correspodig to a imposed set of samples x(t). If X(ω) is bad-limited to some frequecy higher tha f N, a ifiity of fuctios x(t) exists, ad the correspodig X(ω), havig the same samples sequece {x }. All these characteristics X(ω),

5 DATA COMMUNICATIONS 63 correspodig to the same samples sequece {x }, are equivalet. The characteristic bad-limited to the Nyquist bad ad correspodig to the samples sequece {x } is called the equivalet Nyquist characteristic. For o iterferece, that meas to have x =0 for 0, the equivalet Nyquist characteristic (Figure 4.6) is x(t)=sic(πt/t); X(ω)=T for ω ω N ; X(ω)=0 for ω >ω N (4.6) The characteristic from equatio (4.6) is the oly meetig the coditios for o itersymbol iterferece because, i accordace to the samplig theorem, beig badlimited at the Nyquist frequecy (miimum badwidth) is uiquely determied by the samples {x }. Fig. 4.6 Equivalet Nyquist characteristic for o itersymbol iterferece. This characteristic is ideal i a double meaig. Oce, because it correspods to o symbol iterferece, ad twice, because it is ot physically realizable. x(t), the respose of the trasmissio system to a iput pulse g(t), starts to exist before applyig g(t). Practically, it is ecessary to approximate, as well as possible, the rectagular X(ω) whe realizig the trasmissio system, if we wat to use the miimum badwidth correspodig, theoretically, to o itersymbol iterferece. However, i early all practical cases of iterest the actual badwidth available is larger tha the miimum-required Nyquist badwidth for the desired symbol rate 1/T, but it does ot exceed twice this badwidth. If this restrictio is made, i.e., if X(ω)=0 for ω >2π/T (4.7)

6 64 INTERSYMBOL INTERFERENCE costructig X e (ω) is cosiderably simplified. This is illustrated i Figure 4.7 which shows a frequecy characteristic X(ω) cosidered to be a real fuctio of ω. Fig. 4.7 a) Equivalet Nyquist characteristic is superpositio of X -1, X 0, X 1 ; b) Foldig of the portio of characteristic i excess of Nyquist badwidth. The equivalet Nyquist badwidth characteristic is obtaied by superimposig the three bads labeled X 1, X 0, X 1. X 1 has o compoets for positive frequecies whe superimposed o X 0. Placig X 1 o X 0 is equivalet to foldig X(ω) back oto itself about the Nyquist frequecy π/t. I order to have o itersymbol iterferece, the equivalet Nyquist badwidth characteristic which ca be obtaied i this way must be rectagular. For this, the characteristic X(ω), whe it is a real oe, must have a odd symmetry about ω=ω N. Fig. 4.8 Raised cosie characteristics. Oe class of Nyquist characteristics which has bee extesively used ad studied is the so called raised cosie characteristic. A raised cosie characteristic

7 DATA COMMUNICATIONS 65 cosists of a flat amplitude portio ad a roll-off portio that has a siusoidal form (Figure 4.8): X(ω)=T for 0 ω ω N (1 α) T T X(ω)= 1 si ( ) ω ωn for ω N (1 α) ω ω N (1+α) (4.8) 2 2α The respose x(t) is give by: siπt T cosαπt T x( t) = (4.9) πt T 1 4α t T α is a parameter, called roll-off factor, which idicates the ratio betwee the supplemetary badwidth used i excess of the miimum Nyquist badwidth ad the Nyquist badwidth. It ca be oticed that x(t) decreases asymptotically as 1/t 3. Practically, it is ot possible to implemet a data trasmissio system with a ideal characteristic. Always, the itersymbol iterferece is preset ad, i this way, the received data sigal is distorted. Fig. 4.9 Biary sigals ad correspodig eye patters for udistorted (a) ad distorted (b) sigals.

8 66 INTERSYMBOL INTERFERENCE A useful method to appreciate the quality of a data trasmissio system ad to provide a great deal of iformatio about the performace characteristics is called eye patter. It shows us the distributio of itersymbol iterferece ad oise by usig a oscilloscope to sweep out the received voltage y(t) whe the horizotal sweep rate is 1/T or 1/T, with a iteger. The resultig image is kow as a eye patter from its resemblace to the huma eye for biary data. Figure 4.9 shows two biary sigals, a udistorted (a) ad a distorted (b) sigal ad the correspodig eye patters formed from superpositio of T duratio segmets. For the udistorted sigal (a) the eye patter is totally ope ad the sampled values, correspodig to the cetral vertical lie, are equal with dx 0 or dx 0. Because of the itersymbol iterferece the sampled values for the distorted sigal (b) are ot ±dx 0 ad the eye patter is partially closed. The distributio of itersymbol iterferece ca be observed alog the vertical lie correspodig to samplig istats. Eye patters provide useful iformatio cocerig the performace characteristics of a data trasmissio system. For a well-defied eye patter, schematized like i Figure 4.10, some performace parameters ca be determied. Fig Eye patter parameters. The vertical lie correspodig to the widest opeig of the eye patter idicates the best samplig istats. The sesitivity of the system to timig error is represeted by the rate of closure of the eye aroud the best samplig istats. The

9 DATA COMMUNICATIONS 67 maximum sigal distortio is represeted by width of the two braches of the patter at the samplig istats, ad the distace from the decisio threshold to the miimum sampled value, idicates the miimum margi agaist a oise error. The width of zero (or threshold) crossigs positios is a measure of zero-crossig distortio, importat for the systems usig these crossigs for timig adjustig. Such diagrams ca be obtaied for multilevel systems, too. A relative evaluatio of data systems ca be realized usig two criteria related to the eye patter: eye closure (peak distortio) ad mea square distortio. I the sequel, oly the peak distortio criterio is preseted. The maximum eye opeig i absece of oise shows the miimum margi agaist oise at the samplig istats. Normalizig the eye opeig i order to become equal to the uity i the ideal case, with o itersymbol iterferece, is preferable. The maximum value of the itersymbol iterferece, give by a x, is obtaied for that symbol sequece {a } for which to each symbol a the maximum level (M 1)d is used with the sig so as to have all the terms a x with the same sig. Deotig the measure of the itersymbol iterferece by ISI, ISI = 0 a x, a =±d; ±3d;..., ±(M 1)d, the the maximum value of ISI is: Max x 0 The peak eye closure (PEC), ormalized, is where 0 ( ISI ) = ( M 1) d (4.10) ( M 1) d 0 PEC = = ( M 1) D dx 0 x x D (4.11) 0 p = (4.12) x0 is called the peak distortio ad it depeds oly o the data system, x beig the samples of the system impulse respose. The eye opeig is EO=1-PEC. The eye opeig does t take ito accout the oise effect but it idicates the miimum margi agaist oise, determied for the data sequeces for which the itersymbol iterferece is maxim. p

10 68 INTERSYMBOL INTERFERENCE 4.3 EXPERIMENTAL DEVELOPMENT SIMULATION SCHEME PRESENTATION I this sectio, the performaces of a basebad data trasmissio scheme will be ivestigated usig the eye patter method. The block scheme of the data trasmissio system simulated usig Simulik is preseted i Figure Out1 NRZ Sigal Geerator +/-1V, vs = 1/T TRANSMITTER Square root Raised Cosie Trasmit Filter Eye Patter - TRANSMITTER AWGN AWGN Noise Adder FDATool Badwidth Limitatio with LPF - Digital Filter Desig CHANNEL MODEL Square root Raised Cosie Receive Filter I1 I2 I3 I4 Out1 Sigal Multiplexer -300 Z Iteger Delay Eye Patter - RECEIVER RECEIVER Fig Data trasmissio system performace evaluatio usig Simulik. The above scheme icludes three mai sectios: the trasmitter part (these blocks are marked with red), the chael model (the yellow blocks), ad the receiver part (the blue blocks). I the trasmitter, a double polarity (+1/-1 V) NRZ sigal is geerated. This NRZ sigal carries a sequece of L radom bits. The symbol iterval (equal to the bit iterval) is deoted by T. Therefore, the sigalig speed is give by v s = 1 / T. The, the NRZ sigal is filtered by a root raised cosie filter (this trasmit filter forms together with the receive couterpart the overall characteristic that miimizes the ISI). The roll-off factor of the RC filter is deoted as R. The output filtered sigal ca be aalyzed usig the eye patter trasmitter block. Cosiderig that Sigals Waveforms

11 DATA COMMUNICATIONS 69 this sigal is ot affected yet by the oise ad chael distortios the eye patter will show a wide ope eye. The chael model icludes a oise adder ad a LPF badwidth limiter. The additive white Gaussia oise (AWGN) block adds a radom oise with the power P N to the iput NRZ sigal with the power P S = 1W. The parameter that cotrols the oise power is the sigal-to-oise ratio deoted by SNRdB = 10 log 10 (P S / P N ) [db]. For example, a SNRdB = 0 db specifies a oise power of P N = P S = 1W. the LPF badwidth limiter restricts the badwidth of the trasmitted sigal betwee 0 Hz ad the cut-off frequecy f 1. The receiver part icludes a RC filter (idetical to the trasmitter oe) ad a eye-patter aalyzer. The eye-patter i the receiver will be compared with the trasmitter oe to otice how the chael affects the system performaces (the receiver eye closure) SIMULATION PROCEDURE Dowload the Matlab/Simulik files from the website cali.comm.pub.ro/didactice/dt/lab/dt_lab.htm. Copy the cotet of the dowloaded archive ito the folder MATLAB/Work (or the correspodig workig directory). Ope Matlab usig the ico placed o the desktop. Next, ope the iitializatio/cofiguratio file ISI_iit.m (you have to do this BEFORE opeig the simulatio model file RRC_filters_dec_2012.mdl). This iitializatio file fixes the scheme parameters, as followig: The data sequece legth (i umber of bits, or NRZ symbols): L = 1000; The data symbol iterval (secods) ad sigalig speed (Bauds): T = 1e-3; vs = 1/T; The root RC filter group delay (i o. of symbols; DO NOT CHANGE THIS VALUE!): G = 4; The roll-off factor (alfa = the badwidth excess factor for Raised-Cosie filters): R = 0.6; The filter up-samplig factor (DO NOT CHANGE THIS VALUE!):

12 70 INTERSYMBOL INTERFERENCE N = 20; The cut-off frequecy (f 1 ) of the chael LPF limittig the badwidth (the LPF trasitio bad is betwee f 1 ad f 2 ): f1 = 0.7 * vs; f2 = 1.1 * f1; The chael sigal-to-oise ratio (i decibells): SNRdB = 10; Every time whe you ru a ew simulatio (with ew parameters), you should start by ruig the ISI_iit.m file first. Next, ope the simulatio model file RRC_filters_dec_2012.mdl ad ru this oe, too. Aalyze the eye-patter diagrams from the trasmitter ad receiver. It is recommeded to chage oly oe parameter for each ew simulatio. Chage oe parameter ad aalyze the chages i the receiver eye-patter. IMPORTANT NOTE: Whe you chage the symbol iterval T or the cut-off frequecy f 1 you have to ope the FDA Tool block (used for Badwidth Limitatio with LPF Digital Filter Desig) ad press the Desig Filter butto. This butto ca be activated by emulatig the selectio of a differet value for ay parameter (for example you ca chage the Respose Type from Lowpass to Raised-cosie ad the, back to Lowpass). 4.4 QUESTIONS 1. Which are the mai causes of the errors i data trasmissio? 2. Does the filter respose preset itersymbol iterferece if the sigalig speed is half of that for which the filter was desiged? 3. Which are the data sequeces that will provide a maximum for the itersymbol iterferece i the respose of the aalyzed filter? 4. Why is it ot possible to realize a badlimited data trasmissio system without itersymbol iterferece? 5. Which are the advatages of the raised cosie characteristics?