Analyss and Optmzaton of the erformance of OFDM on Frequency- Selectve Tme-Selectve Fadng Channels Hed Steendam, Marc Moeneclaey Communcatons Engneerng Lab. Unversty of Ghent B-9 GET, BELGIUM Abstract In moble rado communcaton, the fadng channels generally exhbt both tme-selectvty and frequency-selectvty. Orthogonal frequency dvson multplexng (OFDM has been proposed to combat the frequency-selectvty, but ts performance s also affected by the tme-selectvty. In ths paper, we nvestgate how varous parameters, such as the number of carrers, the guard tme length and the samplng offset between recever and transmtter, affect the system performance. Further, we determne the optmum values of the above parameters, whch mnmze the degradaton of the sgnal-to-nose rato at the nput of the decson devce. I. Introducton Due to the enormous growth of wreless servces (cellular telephones, wreless LA s,... durng the last decade, the need of a modulaton technque that can transmt relably hgh data rates at a hgh bandwdth effcency arses []- [3]. In a moble rado channel, the sgnal s dsturbed by multpath fadng whch generally exhbts both tme-selectvty and frequency-selectvty. The sgnal power s carred by a large number of paths wth dfferent strengths and delays. For GSM, typcal multpath ntensty profles are defned for rural areas (RA, urban areas (TU and hlly terran areas (HT [4]-[5]. The nfluence of the ntersymbol nterference caused by the frequency-selectvty can be reduced by ncreasng the duraton of a transmtted symbol. Ths can be accomplshed by usng orthogonal frequency dvson multplexng (OFDM : the symbol sequence to be transmtted s splt nto a large number of lower speed symbol streams, whch each modulate a dfferent carrer; the carrer spacng s selected such that modulated carrers are orthogonal over a symbol nterval. In addton, a guard nterval (cyclc prefx s nserted n order to combat the frequency-selectvty of the channel [6]-[7]. The transmtter and recever for OFDM can be mplemented effcently by usng Fast Fourer Transform (FFT technques. OFDM has been proposed and/or accepted for varous applcatons, such as broadcastng of dgtal audo (DAB and dgtal televson (DTTB [8], moble rado []-[],[9]-[], and transmsson over twsted par cables (ADSL []-[3]. Increasng the duraton of a transmtted symbol however, makes the system more senstve to the tme-selectvty of the channel. As the tme-selectvty affects the orthogonalty of the carrers, a larger symbol duraton gves rse to more ntercarrer nterference (ICI. The lengthenng of the symbol duraton, ntroduced to combat the frequency-selectvty, therefore s lmted by the tme-selectvty. In [] and [], the number of subchannels of the OFDM system s optmzed n the case of a suffcent guard nterval,.e. the guard nterval s longer than the delay spread.. In ths contrbuton we nvestgate the effect of an nsuffcent guard nterval as well as the effect of the number of carrers and the samplng offset between recever and transmtter on the performance of the OFDM system. The OFDM system s descrbed n secton II. In secton III we consder the degradaton of the sgnal-to-nose rato at the nput of the decson devce as performance measure of the OFDM system. Secton IV focuses on two lmtng cases,.e. the tme-flat channel (for
a small number of carrers and the frequency-flat channel (for a large number of carrers, and analyses the resultng nterference powers. The problem of frame synchronsaton s consdered n secton V. umercal results, ncludng performance optmzatons, are presented n secton VI. Fnally, conclusons are drawn n secton VII. II. System Descrpton In orthogonal frequency dvson multplexng (OFDM, the avalable bandwdth s parttoned nto subchannels that are made orthogonal by usng carrers wth a spacng equal to the subchannel symbol rate. A bnary message s coded and mapped to a sequence of complex data symbols, whch are splt nto frames of symbols : a,n denotes the n-th symbol of the -th frame ( n -, - <<+. The n-th carrer s modulated by the symbols {a,n - <<+ }, and the modulated carrers are summed before transmsson. In a practcal mplementaton, the samples of the transmtted sgnal correspondng to the -th frame are generated by feedng {a,n n=,...,-} to an nverse dscrete Fourer transform (IDFT (see Fgure. The loss of orthogonalty between the carrers, caused by the dspersve channel, ntroduces ntercarrer and ntersymbol nterference (ISI and ICI at the recever. To combat ths nterference, each frame s preceded by a guard nterval of ν samples contanng a cyclc extenson of the transmtted tme doman samples (cyclc prefx. The -th transmtted frame (ncludng the prefx contans +ν tme doman samples, of whch the m-th sample s gven by : g = E + ν = jπ s ( m a e m = ν,, n nm, n Assumng the data symbols are statstcally ndependent and have a unt average energy,.e. E[a,n a * j,k]=δ,j δ n,k, the transmtted average energy per symbol equals E s. Many wreless communcaton channels can be modeled as multpath Raylegh fadng channels, havng an mpulse response h(k;, represented by a tapped delay lne where the k-th coeffcent s a Gaussan random process wth tme varable. Bello [4] ntroduced the wde-sense statonary uncorrelated scatterng (WSSUS model to easly descrbe fadng ( channels. Ths model, whch s vald for most rado channels, assumes that the sgnal varatons arrvng at dfferent delays are uncorrelated and that the correlaton propertes of the channel are statonary. The autocorrelaton functon, consderng these assumptons, yelds : [ ( ( ] * E h k, h k, = δ ( k k R( k ; ( The channel, havng an autocorrelaton functon R(k;, can be characterzed by a multpath ntensty profle R(k; and a = Doppler spectrum S D (e jπft, wth S D + + j ft ( e R( k π jπf T = k= ; e (3 Wthout loss of generalty, we assume that the largest value of the multpath ntensty profle occurs at k=,.e. R(; R(k;. In addton, the receved sgnal s corrupted by complex-valued addtve whte Gaussan nose (AWG wth a power spectral densty. For each transmtted frame of +ν samples, the recever selects consecutve samples to be processed further and drops the other ν samples (guard tme removal. The ndces of the remanng samples correspondng to the j-th frame are {k-k +j(+ν k=,...,-}. The samplng offset k s assumed to be provded by a frame synchronzaton algorthm, whch selects k such that the sgnal-to-nose rato at the nput of the decson devce s maxmum. The remanng samples r(k of the j-th frame, gven by r + = m= ν k = k + j ( k = g ( m h( k m ( + ν ; k + n( k ( + ν,, k + j( + ν + are demodulated usng a dscrete Fourer transform DFT. Each of the outputs of the DFT s scaled and rotated (sngle-tap equalzaton per DFT output and appled to the decson devce. III. System erformance In the followng, we concentrate on the detecton of the data symbols durng the frame j=. Due to the loss of orthogonalty caused by the fadng channel, the outputs of the dscrete Fourer transform are dsturbed by nterference, whch (4
3 adds to the channel nose. The power (n at the n-th output of the DFT can be decomposed as + ν ( n = Es ( U ( n + ICI ( n + ISI ( n + The useful power U denotes the contrbuton from the symbol a,n. The ntercarrer nterference (ICI power ICI contans the contrbutons from the other symbols transmtted n the consdered frame (=, whereas the ntersymbol nterference (ISI power ISI contans the contrbutons from all symbols transmtted n other frames (. Fnally, denotes the contrbuton from the addtve nose. Assumng all carrers are modulated, one obtans U ISI ( n = E γ ( k ICI n, n, = n ( n = E γ, n, ( k + ( n = E γ, n, ( k = = where γ,n,(k, gven by γ k nm, n, ; m= ν k= (5 (6 jπ ( k = e h( k k m ( + ν k k (7 denotes the sgnal component at the n-th DFT output durng the frame j=, caused by the symbol a, whch s transmtted on the -th carrer durng the -th frame. It can be verfed that, n the case of all carrers modulated, the useful power U (n and the nterference powers ICI (n and ISI (n are ndependent of the consdered carrer; n the followng we can therefore omt the carrer ndex n. The sgnal-to-nose rato (SR at the output of the DFT s defned as the rato of the power of the useful component to the power of the remanng contrbutons : SR = E s Es U + ν (8 ( ICI + ISI + + ν In the presence of the fadng channel, the SR s reduced as compared to the case of an AWG channel. The AWG channel yelds U =, ICI = ISI =, so that for ν= the SR equals E s /. Assumng the mpulse response h(k; has a unt k= R k average energy (.e. ( ; =, t can be verfed that the sum of the, the useful power and the nterference powers s gven by U + ICI + ISI = (see Appendx A. Under these consderatons, the degradaton of the SR expressed n db s gven by : U Deg = log + ν (9 Es + ( U + ν For large E s /, the SR (8 s lmted by U /(- U whch ndcates that the performance s lmted by the nterference. Hence, ncreasng E s / far beyond ((- U /(+ν - yelds only a margnal performance mprovement. In a further analyss of the powers n (6, takng nto account the above-mentoned consderatons, t can be verfed that (see Appendx A : + + U = w( k R( k k ICI = k= k= = k = ; ; ( R( k k ; U (a w k; (b ( w( k; R( k k ; = (c ISI where w(k; s a two-dmensonal weght functon (A and w(k; s shown n Fgure. Accordng to (, the ntersymbol nterference power ISI s ndependent of the tme correlaton propertes of the channel; takng nto account the weght functon w(k; from Fgure, t follows that ISI s determned only by the tals of the multpath ntensty profle of the fadng channel. The ntercarrer nterference conssts of two contrbutons: the frst contrbuton s ndependent of the tme correlaton propertes of the fadng channel and s manly determned by the central part (the body of the mult-path ntensty profle; the second contrbuton (- U depends on both the dspersve and tme-correlaton characterstcs of the fadng channel. Let us consder quanttatvely the effect of the system parameters (.e. the guard tme length ν and the number of carrers on the performance degradaton (9 For gven, ncreasng ν reduces the amount of channel dstorton on the samples that are kept by the recever for
4 further processng (see Fgure 3. Hence, ncreasng ν reduces ICI and ISI, so that U moves closer to. On the other hand, ncreasng ν reduces the power effcency through the factor /(+ν, because the recever keeps only of the +ν receved samples. ote that for an AWG channel wth ν> the degradaton (9 becomes log(/(+ν, whch reflects the power effcency loss caused by the guard nterval. For gven ν, the dspersve channel ntroduces a gven amount of lnear dstorton, whch s bascally confned to a few samples at the edges of the block of samples that are processed by the recever. Increasng reduces the relatve mportance of these dstorted samples and n addton, ncreases the power effcency. On the other hand, ncreasng makes the system more senstve to the tme-selectvty of the channel, because the transmtted frames get longer. The tme-selectvty affects the orthogonalty of the transmtted frames, and therefore ntroduces ICI at the DFT output. From the above consderatons, t follows that an optmum set (ν, exsts, whch mnmzes the degradaton (9 IV. Lmtng cases : Tme-Flat Channel and Frequency-Flat Channel The mpulse response of a moble rado channel generally exhbts a delay spreadng as well as a Doppler spreadng [4]. The delay spreadng orgnates from the multtude of propagaton paths taken by the transmtted sgnals. The coherence bandwdth,.e. the mnmum bandwdth for whch tmedsperson s observable, s nversely proportonal to the delay spreadng. The varaton of the moble rado channel n tme s expressed by the Doppler spreadng. The coherence tme, correspondng to the mnmum sgnal duraton for whch dstorton becomes notceable, s nversely proportonal to the Doppler spreadng. In ths secton, the system s evaluated for two lmtng cases : the tme-flat channel and the frequency-flat channel. When the duraton of a transmtted frame s small as compared to the coherence tme of the fadng channel, the varaton n tme of the channel durng a frame can be neglected : the channel can be approxmated by a tme-flat channel wth R(k; =R(k;. Consderng the relatonshp between the coherence tme and the Doppler spreadng, ths tme-flat channel approxmaton corresponds to a fadng channel wth a Doppler spreadng whch s small as compared to the carrer spacng. As the Doppler spreadng s proportonal to the velocty of the vehcle [4]-[5], the tme-flat channel can be used to model slowly movng vehcles. For a tme-flat channel, the general expresson (a of the power U smplfes to : k= ( k R( k k ; = w U ; ( where we have taken nto account that w( k; = w ( ;. Takng nto account that ISI + ICI =- = k U, t follows from ( that ( w ( k; R( k k ; k= + = ISI ICI ( Consderng the nature of w(k;, ( and ( ndcate that the useful power s manly determned by the body of the autocorrelaton functon, whle the total nterference power only nvolves the head and tal of R(k; (see Fgure. Assumng the tals of the multpath ntensty profle are much shorter than samples, ( s well approxmated by ISI + ICI k= k ( R( k k; + R( ν k + k; (3 whch ndcates that the total nterference s proportonal to /. In addton, t follows from (3 that the total nterference decreases wth ncreasng ν, n a way whch depends on the shape of the tal of the multpath ntensty profle. Under the same assumptons as above, the ISI power (c s well approxmated by ISI k( R( k k; + R( ν k + k; (4 k= o Comparng (3 and (4, we conclude that ISI and ICI equally contrbute to the nterference when the tme varatons of the channel can be gnored.
5 When ncreases for a gven ν and a gven channel autocorrelaton functon, the number of samples affected by the channel dsperson s small as compared to the total number ( of samples that are processed per frame : the effect of the channel dsperson becomes neglgble. When the frame duraton becomes comparable to the coherence tme of the channel, the effect of the tme varatons of the channel becomes notceable: the total nterference (ISI+ICI ncreases wth. The ISI power (c s ndependent of the coherence tme of the channel and s proportonal to / when s much larger that the duraton of the tals of the multpath ntensty profle (see (4. Hence the total nterference s manly ICI when the tme varatons of the channel become domnant. When the frame duraton s large as compared to the coherence tme and the delay spread, t s shown n Appendx B that the fadng channel can be approxmated by a frequencyflat channel wth ( ~ R k; = R( δ ( k, where ~ ( R = R( k; (5 k= Hence, the frequency-flat model corresponds to a sngle coeffcent h ( ; ~ wth the same Doppler spectrum as the actual channel. The resultng useful power U s gven by + U = ~ R( (6 = V. Frame Synchronzaton The recever makes a detecton of the symbols {a,n n=,...,-} by processng the samples {r(m-k m=,...,-}. The samplng offset k determnes how much these samples are affected by the channel dsperson, and should be selected such that the degradaton (9 s mnmal. As U + ISI + ICI =, mnmzng the degradaton (9 s equvalent to mnmzng the nterference ISI + ICI or maxmzng the useful power U. As the mpact of the channel dsperson ncreases when the frame gets shorter, we wll determne the optmum samplng offset k under the assumpton that the frame duraton s much less that the coherence tme of the channel; hence the tme-flat channel model apples. The resultng k mght be no longer optmum when the frame length s n the order of the coherence tme; however n ths case the value of k s less crtcal, because the effect of channel dsperson s less mportant than the effect of the tme varaton of the channel. When the tme-flat model apples, the optmum value k mnmzes the nterference power gven by (. In most cases of practcal nterest, s much larger than the tals of the multpath ntensty profle, so that mnmzng ( s essentally equvalent to mnmzng (3. Mnmzaton of (3 yelds an optmum value of k that does not depend on. otng that the maxmum of the ntensty profle R(k; occurs at k=, the optmum value of k s easly determned n the followng cases : For causal channels (.e. R(k;= for k<, the optmum value of k s k = For ant-causal channels (.e. R(k;= for k>, the optmum value of k s k =ν For symmetrc channels (.e. R(k;= R(-k;, the optmum value of k s k =ν/ In other cases, the optmum value of k has to be obtaned by numercally mnmzng ( or (3. In practce, a frame synchronzaton algorthm should provde the value of k to the recever. A sutable frame synchronzaton algorthm could perform measurements of the multpath ntensty profle R(k;, use these measurements n ( or (3 as f they were the correct values, and fnally mnmze the resultng ( or (3. For convenence of mplementaton, the functon to be mnmzed could be replaced be the area under the tals of R(k; : ( k = ( R( k k; + R( k + k; k= Area ν (7 A smlar frame synchronzaton mechansm has been proposed n [6], n the context of multcarrer transmsson over the twsted par cable. VI. umercal results In the computatons, a 5MHz channel bandwdth and a GHz carrer frequency have been assumed. The Doppler spreadng for a typcal outdoor rado channel can be calculated
6 straghtforwardly from the expresson f D =(v/cf C, v representng the velocty of the moble (35 km/hr, c the velocty of lght and f C the center frequency of the moble rado channel ( GHz. The resultng coherence tme T, accordng to the rule of thumb T =.5/f D [4]-[5], equals 4 ms. In the lterature [4]-[5], typcal channel mpulse responses for varous envronments are defned. For a typcal urban (TU area, a delay spread of 5 µs s taken. Consderng the 5MHz channel bandwdth, t follows that the duraton of a sample s. µs. The proposed autocorrelaton functon exhbts an exponentally decayng multpath ntensty profle and a Gaussan tme correlaton profle: k R k; = C exp( exp( k, < y σ (8 ( < + where C s a constant of normalzaton. Defnng the delay spread as the tme at whch the multpath ntensty profle falls db below the level of the strongest component, the parameter y s found to be about 5 samples. The coherence tme T s fxed to the duraton of twce the spreadng of the Gaussan tme correlaton profle yeldng σ = samples. In Fgures 4 and 5 we compare the total nterference power ISI + ICI for the followng cases : (a the frequency-selectve tme-selectve channel wth autocorrelaton functon R(k; from (8 (b the lmtng case of the tme-flat channel wth autocorrelaton functon R(k; (c the lmtng case of the frequency-flat channel wth auto- ~ R δ k R ~ gven by (5 correlaton functon ( (, wth ( (d the sum of the total nterference powers resultng from (b and (c Fgure 4 shows the total nterference power for ν=4 as functon of, whereas Fgure 5 shows the total nterference power for =56 as functon of ν. We observe from Fgure 4 that the total nterference power for the lmtng cases of the tme-flat channel (b and the frequency-flat channel (c converge to the total nterference power of the frequency-selectve tme-selectve channel (a, for small and large, respectvely. The total nterference power for the tme-flat channel s proportonal to /; ths agrees wth the result (3. ote that the total nterference power for the frequency-flat channel already approaches the total nterference power for the frequency-selectve tmeselectve channel, for values of the frame duraton that are consderably less that the coherence tme of the channel; the resultng total nterference power s proportonal to. Fnally, we have added the total nterference power resultng from the tme-flat channel (b and the frequency-flat channel (c. The resultng sum (d turns out to be an accurate approxmaton of the total nterference power for the actual frequencyselectve tme-selectve channel (a. ote that the approxmaton (d nvolves the computaton of only two sngle summatons (.e. ( and (6, whereas the correct result (a requres a double summaton (.e. (a. Hence, for large the computaton of the approxmaton s much more effcent than the computaton of the correct result. Fgure 5 shows that, for the tme-flat channel, the total nterference power decreases wth ncreasng ν; ths s because the effect of channel dsperson s reduced by ncreasng the guard nterval. Ths decrease wth ν s exponental, because of the exponentally decayng multpath ntensty profle. For the frequency-flat channel, the total nterference power does not depend on ν : the nterference s caused solely by the tme-varatons of the channel, whch cannot be counter-acted by a guard nterval. Agan, the sum of the total nterference powers resultng from the tme-flat and frequency-flat lmts of the channel s a very good approxmaton of the total nterference power for the frequency-selectve tme-selectve channel. Fgures 6 and 7 show the effect of the samplng offset k on the degradaton of the SR, for E s / =db, =56 and varous values of ν. In Fgure 6, the autocorrelaton functon (8 wth the causal exponentally decayng multpath ntensty profle has been taken; we observe that the mnmum degradaton occurs at k =. The degradaton shown n Fgure 7 corresponds to an autocorrelaton functon wth the same Gaussan tme correlaton profle as n (8, but wth a symmetrcal exponentally decayng multpath ntensty profle; the mnmum degradaton occurs for k ν/. ote that the optmum values
7 of k, found n Fgures 6-7, agree wth the optmum values for a tme-flat channel, determned n secton V. Fgures 8 and 9 show the results from the numercal mnmzaton of the degradaton (9 wth respect to the system parameters and ν; as we have assumed the causal multpath ntensty profle resultng from (8, the optmum tmng offset k = has been selected for all consdered cases. Fgure 8a dsplays the optmum values opt and ν opt as a functon of E s /, for y =5 and σ =. We observe that for ncreasng E s /, opt and ν opt are decreasng and ncreasng, respectvely. Ths behavor can be explaned as follows. For very large E s /, the degradaton (9 converges to log( U /(- U +log(e s /, n whch case the mnmzaton of the degradaton s equvalent to the maxmzaton of U ; let us denote by ( opt (,ν opt ( the correspondng optmum system parameters. For very small E s /, the degradaton (9 converges to log( U /(+ν, n whch case the mnmzaton of the degradaton s equvalent to the maxmzaton of U /(+ν; let us denote by ( opt (,ν opt ( the correspondng system parameters. As /(+ν s decreasng wth and decreasng wth ν, t follows that opt ( < opt ( and ν opt ( >ν opt (. The range of E s / dsplayed n Fgure 8a s an ntermedate range, n whch ( opt,ν opt has reached nether ts lmt ( opt (,ν opt ( for low E s / nor ts lmt ( opt (,ν opt ( for hgh E s /. Fgure 9a shows the mnmum degradaton, correspondng to the optmum values ( opt,ν opt from Fgure 8a. Ths mnmum degradaton ncreases wth E s /. As the consdered range of E s / s ntermedate, the above-mentoned lmts of the degradaton for very low and very hgh E s / are not reached n Fgure 9a. Fgure 8b shows the optmum system parameters ( opt,ν opt as functon of y (whch s proportonal to the delay spread, for E s / =3dB and σ =. When y ncreases, ν opt vares n proporton to y ; opt also ncreases wth y, n order to compensate for the power effcency reducton caused by the ncrease of ν opt. However, as ncreasng opt enhances the nterference caused by the tme varatons of the channel, the ncrease of opt s not lnear wth y. The mnmum degradaton, correspondng to ( opt,ν opt from Fgure 8b, ncreases wth y, as can be verfed from Fgure 9b. Fgure 8c shows the optmum system parameters ( opt,ν opt as a functon of σ (whch s proportonal to the coherence tme, for E s / =3dB and y =5. The optmum guard tme duraton ν opt does not depend on the coherence tme, because the guard nterval has no mpact on the nterference caused by the tme-selectvty. The optmum value opt s a compromse between the followng phenomena: (a Increasng reduces both the power effcency loss and the nterference caused by the frequency-selectvty (b Decreasng reduces the nterference caused by the tmeselectvty Hence, opt ncreases wth σ. Fgure 9c dsplays the mnmum degradaton correspondng to ( opt,ν opt from Fgure 8c; ths degradaton decreases wth σ. VII. Conclusons In ths paper, we have frst nvestgated the effect of the number of carrers and the guard tme duraton ν on the performance of an OFDM system operatng on a frequencyselectve tme-selectve fadng channel. Our man conclusons are the followng. For short frames, the tme-selectvty of the channel can be gnored. The frequency-selectvty of the channel yelds equal portons of ISI and ICI. The total nterference power decreases wth ν, and s proportonal to /. For long frames, the frequency-selectvty of the channel can be gnored. The tme-selectvty of the channel yelds ICI but no ISI. The ICI power does not depend on ν, and ncreases wth. The total nterference power for a channel wth both frequency-selectvty and tme-selectvty s well approxmated by addng the total nterference powers that result from the tme-flat lmt and the frequency-flat lmt of the consdered channel. The computaton of ths approxmated total nterference power s much faster that the computaton of the correct total nterference power.
& $ $ " Further, we have determned the optmum values of the tmng offset (k, the number of carrers ( and the guard tme duraton (ν, that mnmze the degradaton of the SR, caused by ISI and ICI. The optmum tmng offset s determned manly by the guard tme duraton ν and the multpath ntensty profle R(k;. Assumng R(; R(k;, the optmum tmng offsets are k = for a causal profle and k =ν/ for a symmetrc profle. The optmum number of carrers ncreases wth the delay spread and the coherence tme, but decreases wth E s /. The optmum guard tme duraton ncreases wth the delay spread and wth E s /, but s ndependent of the coherence tme. The resultng mnmum degradaton of the SR ncreases wth E s / and the delay spread, but decreases wth the coherence tme. Appendx A Let us consder the calculaton of the component E[ γ,n,(k ]. Substtutng n (7 the new summaton varable q=k-m and consderng the WSSUS channel (, we obtan : E γ, n, ( k ν + ν = + + q= k, k = q= ν k, k = q ν e ( k k ( n jπ R ( q k ( + ν ; k k + q q= ( k, k = (A After substtutng n (A the summaton varable r=k-k and defnng the weght functon w(q;r as : ( ; r w q r q + ν r = + q r the expresson (A yelds q ν, r ν q + ν, r q + ν q, r + q elsewhere (A E γ, n, ( k = w( q; r + + q= r= R e r( n jπ ( q k ( + ν ; r (A3 The sum of all powers U + ICI + ISI equals + = n= + ( k = R( q k; U + ICI + ISI = E γ, n, q= (A4 whch, consderng the normalzaton of the energy of the mpulse response of the channel, reduces to. Usng (A3, t can easly be verfed that the useful power, the power of the ntercarrer nterference and the power of the ntersymbol nterference are gven by (9. Appendx B When the frame length s large as compared to the coherence tme and the delay spread, the weght functon w(k;! (A can be approxmated by : " " ( ( w k; = w k; k, (B whch reduces the useful power U (9 to : + + + # # ( ( $ ~ = U R w k; R k k = ; = k= ~% defnng R ( as : & R ~ ( = R( m; m= ( ( 8 (B (B3 For large, the frst contrbuton of (B behaves nversely proportonal to, whle the second contrbuton behaves nversely proportonal to. In addton, f the guard nterval s of the order of the delay spread, the second contrbuton s neglgble as compared to the frst contrbuton. The useful power therefore can be obtaned usng expresson (9 where the autocorrelaton functon s substtuted by (B3, whch corresponds to the autocorrelaton functon of a frequency-flat fadng channel. In a smlar way, t can be found that the ntercarrer nterference power and the ntersymbol nterference ~'. power for large converge to values that correspond to a channel autocorrelaton functon R( δ ( k
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6 4 ν= ν=3 ν=4 ν=5 Degradaton(dB 8 6 4 Degradaton(dB 4 6 8. - - 3 4 5. E s/ o(db k o Fgure 7 : Degradaton as functon of k : symmetrc profle.4. 4 35 3 5 opt ν opt 3 4 5 6 7 E s / o (db opt 9 8 7 6 5 4 3 ν 8 7 6 5 Degradaton(dB Degradaton(dB.8.6.4..8.6.4..8.6.4. 4 y o 6 8 5 5 σ o 5 ν opt 4 ν 3 Fgure 9 : Optmal degradaton 5 4 6 8 y o 5 45 4 35 3 5 5 5 ν opt opt 5 σ o 5 4 35 3 5 5 5 ν Fgure 8 : Optmal system parameters