Lecture 8: Digital Modulation in Wireless Communication Systems

Transcription

1 Lecture 8: Digital Modulation in Wirele Communication Sytem 8.. Digital Signal Modulation: Main Definition A wa mentioned in Lecture 7, modulation i the proce where the aeand meage information i added to the andpa carrier. In digital modulation the digital eam tream i tranmitted a a meage, and then i converted into the analog ignal of type a( t)co( ω t + θ ) that modulate the digital it tream into a RF carrier. he analog ignal ha amplitude a(t), frequency f = ω / π, and phae θ = ωt. Changing thee three characteritic, we can formulate three kind of digital modulation. hey are: Amplitude hift keying (ASK) for phae and frequency kept contant; Frequency hift keying (FSK) for amplitude and phae kept contant; Phae hift keying (PSK) for amplitude and frequency kept contant. In o-called hyrid modulation method we can ue comination of thee three kind of modulation. Namely, for frequency i contant, ut amplitude and phae are not contat, we have quadrarure amplitude modulation (QAM). Some modulation method are linear, a inary phae hift keying (BPSK), quadrarure phae hift keying (QPSK), including π / 4 QPSK, DQPSK, including π / 4 DQPSK, and o on. At the ame time, FSK, a well a minimum hift keying (MSK) and Gauian minimum hift keying (GMSK) are non-linear modulation method. We will give ome example later (full information you can find in [, 4, ]) ut now will put a quetion what afvantage of digital modulation method compared with analog modulation method. Becaue digital modulation offer many advantage over analog modulation, it often ued in modern wirele communication ytem. Some advantage include greater noie immunity and routne to channel impairment, eaier multiplexing of variou form of information (uch a voice, data, and video), and greater ecurity. Moreover, digital tranmiion accommodate digital error-control code, which detect and correct tranmiion error, and upport complex ignal proceing technique uch a ource coding, encryption etc. In digital wirele communication ytem, the modulating ignal, i.e., the meage, can e repreented a a time equence of ymol or pule, where each ymol ha m finite tate. Each ymol repreent n it of information, where n = log m it/ymol. Many digital modulation cheme are ued in modern wirele communication ytem, and many are in a tage to e introduced. Depite difference etween technique, each technique elond to a family of related modulation method. Advantage to ue digital modulation: Cot Effective - VLSI and DSP technologie Greater noie immunity Routne to channel impairment Eaier multiplexing of information (voice, data, video)

2 Error detection and correction code Signal encryption Equalization to improve overall performance. What influence the choice of digital modulation? Requirement: low BER at low received SNR good performance in multipath and fading condition (ee elow) low andwidth (BW) implementation cot effectivene. he power efficiency and andwidth efficiency. he efficiency of each digital modulation technique depend on many factor. he main goal of uch a modulation i to otain in variou multipath and fading condition low it error rate (BER) at low received SNR, minimum of andwidth occupation and o on. he performance of a modulation cheme i often meaured in term of it power efficiency and andwidth efficiency. Firt term decrie the aility of a modulation technique to preerve the fidelity of the digital meage at low power level. In a digital communication ytem, in order to increae noie immunity, it i neceary to increae the ignal power. However, the amount y which the ignal power hould e increaed to otain a certain level of fidelity, i.e., an acceptale it error proaility, depend on the particular type of modulation employed. he power efficiency, η p, of a digital modulation cheme i a meaure of how favoraly thi tradeoff etween fidelity and ignal power i made, and i often expreed a a ratio of the ignal energy per it to noie power pectral denity, E / N, required at the receiver input for a certain proaility of error (ay 5 ). he andwidth efficiency, η B, decrie the aility of a modulation cheme to accomodate data within a limited andwidth. In general, increaing the data rate implie decreaing the pule width of a digital ymol, which increae the andwidth of the ignal. Bandwidth efficiency reflect how efficiently the allocated andwidth i utilized and i defined a the ratio of the throughput data rate (it per econd, p) per Hertz in given andwidth. If R i the data rate in it per econd, and B i the andwidth occupied y the modulated RF ignal, then andwidth efficiency i expreed a R η B = p / Hz (8.) B he ytem capacity of a digital communication ytem i directly related to the andwidth efficiency of the modulation cheme, ince a modulation with a greater value of η B will tranmit more data in a given pectrum allocation. here i a fundamental upper ound on achievale andwidth efficiency, which can e defined from well-known Shannon channel coding theorem []. hi theorem tate that for an aritrary nall proaility of error, the maximum poile andwidth efficiency i limited y the noie in the channel, and i given y the channel capacity formula:

3 3 C S S η = = + = + B max log log (8.) B N NB where C i the channel capacity (in p), B i the RF andwidth, and S/N i the SNR. Bandwidth and power pectral denity of digital ignal. he definition of ignal andwidth varie with context, and there i no ingle definition which cover all application []. All definition, however, are aed on ome meaure of the power pectral denity (PSD) of the ignal. he PSD of a random ignal x(t) i defined a [3] X f Px () t = lim ( ) (8.3) where denote an enemle average, X ( f ) i the Fourier tranform of x (), t which i the truncated verion of the ignal x(t), defined a xt () / < t< / x () t = t elewhere (8.4) he power pectral denity of a modulated (andpa) ignal i related to the power pectral denity of it aeand complex envelope. If a andpa ignal (t) i repreented a t () = Re[ gt ()exp( jπ ft)] (8.5) where g(t) i the complex aeand envelope. hen the PSD of the andpa ignal i given y P( f ) = [ Pg( f + fc) + Pg( f fc) ] (8.6) 4 where Pg ( f ) i the PSD of g(t). he aolute andwidth of a ignal i defined a the range of frequencie over which the ignal ha non-zero power pectral denity. For ymol repreented a rectangular aeand pule, the PSD ha a (in f ) / f profile which extend over an infinite range of frequencie, and ha an aolute andwidth of infinity. A impler and more widely accepted meaure of andwidth i the firt null-to-null andwidth. he null-to-null andwidth i equal to the width of the main pectral loe. A very often ued meaure of andwidth which meaure the diperion of the power pectrum i the half-power andwidth, which i defined a the interval etween frequencie at which the PSD ha dropped to half power, or 3 db elow the peak value. Half-power andwidth i alo called the 3 db andwidth. According to definition adopted y the Federal Communication Committee, the occupied andwidth i the and which conit 99% of the ignal power:.5 percent of the ignal power leave aove the upper and limit and.5 percent of the ignal power leave elow the lower and limit. Line coding. Digital aeand ignal often ue line code. he mot common code for wirele communication are return-to-zero (RZ), non-return-to-zero (NRZ), and o-called the Mancheter code (ee Fig. 8.). All of thee code may either e c

4 4 unipolar (with voltage level eing either or V) or ipolar (with voltage level eing either -V or V) (ee Fig. 8.). RZ code implie that the pule return to zero within every it period. hi lead to pectral widening, ut improve timing ynchronization. NRZ code do not return to zero during a it period; the ignal tay at contant level throughout a it period. NRZ code are more pectrally efficient than RZ code, ut offer poorer ynchronization capailitie. he Mancheter code i a pecial type of NRZ line code that i ideally uited for ignaling that mut pa through phone line and other DC locking circuit, a it ha no dc component and offer imple ynchronization. hi code ue two pule to repreent each inary ymol, and therey provide eay clock recoverty ince zero-croing are guaranteed in every it period. Example: he RF andwidth of the wirele network i B = 3 khz and SNR within the link i db. Find: he maximum theoretical data rate C that can e tranmitted through uch a link. Compare it with USA Digital Cellular (USDC) tandard of 48.6 kp. Solution: According to (8.) and taking into account that S/N= db=, we get: S C = B log + = 3 Hz log ( + ) = N kp hi rate i four time fater than in tandard USDC for the given limit of S/N= db. 8.. Pule Shaping echnique During their pa through a and-limited communication channel, rectangular pule, which are preented in form of the meage of ymol or it, will pread in time, and the pule of each ymol will pread into the time interval of neighour ymol. hi effect caue o-called the interymol interference (ISI) and finally lead to an increaed proaility of the receiver making an error in detecting a ymol. One way to minimize the ISI i to increae the channel andwid. However, wirele communication ytem often operate with minimal andwidth, and technique that reduce the modulation andwidth and uppre out-of-and radiation, while reducing ISI, are highly deirale. Since it i difficult to directly control the tranmitter pectrum at RF frequencie, pectral haping i done through aeand or intermediate frequency (IF) proceing. here are ome effective pule-haping technique are ued to imultaneouly reduce the ISI effect and the pectral width of a modulated digital ignal Nyquit criterion for ISI cancellation Nyquit wa the firt who oerved that the effect of ISI could e completely cancelled if the overall repone of the communication ytem, including tranmitter, receiver, and channel, i deigned o that at every ampling intant at the receiver, the repone due to all ymol except the current ymol i equal to zero []. If heff ()i t the impule repone of the ytem, the Nyquit condition can e tate a

5 5 h eff K n = ( n ) = n (8.7) where i the ymol period, n i an integer, and K i a non-zero contant. he effective tranfer function of the ytem can e repreented y the δ -function, the pule hape of a ymol, p(t), the channel impule repone, hc ( t), and the receiver impule repone, hr (), t a h () t = δ () t p() t h () t h () t (8.8) eff c r Nyquit derived tranfer function Heff ( f ) which atify the condition of (8.7). He aed on the two aumption: firt, that heff ( t) hould have a fat decay with a mall magnitude near the ample value for n ; econd, the channel i ideal, hc () t =δ () t, and it i poile to realize haping filter at oth the tranmitter and receiver to produce the deired H f ). Conidering the impule repone a eff ( h eff () t = in( πt / ) ( πt)/ (8.9) it i clear een that thi repone atifie the Nyquit condition (8.7) for ISI cancelation. Nyquit ideal pule hape for zero ISI i hown in Fig h NYQ (t).5?4?3?? 3 4 Fig. 8.3 Hence, if the overall communication ytem can e modeled a a filter with the impule repone decried y (8.9), it i poile to completely eliminate the effect of ISI. he tranfer function for uch a filter can e otained from (8.9) y taking the Fourier tranform of the impule repone, that i, H eff ( f ) = f Π (8.) f f hi tranfer function correpond to a rectangular rick-wall filter with aolute andwidth f /, wheref i the ymol rate. While thi tranfer function atifie the zero ISI criterion with a minimum of andwidth, there are practical difficultie in

6 6 implementing it, ince it correpond to a noncaual ytem ( heff ( t) exit for t<) and i thu difficult to approximate. Nyquit alo provide that any filter with a tranfer function having a rectangular filter of andwidth f /, convolved with any aritrary even function Z(f) with zero magnitude outide the paand of the rectangular filter, atifie the zero ISI condition (8.9). Such a filter with zero ISI condition can e expreed a H eff f ( f ) = Z( f ) Π (8.) f where Z(f)=Z(-f), and Z(f)= for f f /. Expreed in term of the impule repone, the Nyquit criterion tate that any filter with an impule repone h eff () t = in( πt / ) zt () πt / (8.) can achieve ISI cancellation. Filter which atifie the Nyquit criterion i called Nyquit filter, the tranfer function of which i hown in Fig H NYQ (f) Fig. 8.4 frequency 8... Raied coine filter he often ued pule haping filter, which atifie the Nyquit criterion, i the raied coine filter, the tranfer function of which i given a [] H RC f ( α)/ f + ( f ) = + co ( )/ < f ( + )/ π α α α α f > ( + α)/ (8.3) where α i the rolloff factior which range etween and. When α =, the raied coine rolloff filter correpond to a rectangular filter of minimum andwidth (ee Fig. 8.5). he correponding impule repone of the filter can e otained y taking the invere Fourier tranform of the tranfer function, that i,

7 7 h RC () t = in( πt / ) coπαt ( πt / ) ( αt) (8.4) 4 A follow from formula preented, a the rolloff factor α increae, the andwidth of the filter alo increae, and the time ideloe level decreae in adjacent ymol lot. hi implie that increaing α increae the occupated andwidth from R/ to R for α = to α = (ee Fig. 8.6). he ymol rate that can paed through a aeand raied coine rolloff filter i given y [] R B = = + α (8.5) where B i the aolute filter andwidth. For RF ytem, for example, the RF paand andwidth doule and B R = (8.5a) + α he coine rolloff tranfer function can e achieved y uing identical H f RC ( ) filter at the tranmitter and receiver, while providing a matched filter for optimum performance in a flat fading channel. o implement the filter repone, a a rule, pule haping filter are ued on the aeand data. Moreover, thee filter are typically implemented for ±6 aout the t= point for each ymol. A an example, aume inary aeand pule tranmitted uing a raied coine rolloff filter with α =/. If the modulator tore 3 it at a time, then there are 8 poile waveform tate that may e produced at random for the group (ee Fig. 8.7). If ±6 i ued to repreent time pan for each ymol (a ymol i the ame a a it in thi cae), then the time pan of the dicrete-time waveform will e 4.If, for example, the RF time waveform correpond to the data equence,,, then the optimal it deciion point occur at 4, 5, and 6, and the time diperive nature of pule haping can e een Gauian pule-haping filter It i alo poile to ue non-nyquit technique for pule haping. A Gauian pulehaping filter i often ued for uch technique for pecial type of digital modulation, known a minimum hift keying modulation (MSK) and i applicale for power nonlinear amplifier. hi i very important quetion, ecaue there i a dilemma for deigner of digital communication ytem that reducing the andwidth of pule haping filter according to Nyquit require linear amplifier which are not power efficient. Gauian pule-haping filter are well uited for power nonlinear amplifier and reolve thi prolem. Unlike Nyquit filter which have zero-croing at adjacent ymol peak and a truncated tranfer function, the Gauian filter ha a mooth tranfer function with no zero-croing. he Gauian lowpa filter ha a tranfer function given y HG ( f ) = exp α f (8.6) ( )

8 8 he parameter i related to B, the 3-dB andwidth of the aeand haping filter (the impulee repone of Gauian filter give rie to a tranfer function which i highly dependent upon the 3-dB andwidth), ln 774. α= = B B (8.7) A α increae, the pectral accupancy of the Gauian filter decreae and time diperion of the applied ignal increae (ee Fig. 8.8). he impule reponce of the Gauian filter i given y hg () t = exp π π t (8.8) α α he Gauian filter ha a narrow aolute andwidth (although not a narrow a a raied coine rolloff filter), and ha harp cut-off, low overhoot, and pule area preervation propertie which make it very attractive for ue in modulation techniwue that ue nonlinear RF amplifier and do not accurately preerve the tranmitted pule hape Geometric Repreentation of Digital Modulation Signal Vector-pace preentation of the digital modulated ignal For digital modulation cheme, the complex envelope r(t) during the kth aud k t ( k + ) elond to a finite et of waveform ri (), t i=,,..., M. In other word, the procedure of digital modulation involve chooing a particular ignal waveform ri ( t) from a finite et of poile ignal waveform (or ymol) aed on the information it applied to the modulator. If there are a total of M poile waveform, the modulation ignal et can e repreented in the vector form, that i a the M point in a vector pace: { r r r r } r i = i i i 3 in,,,...,, i=,,..., M (8.9) M Conequently, the waveform { r i } i = can e repreented in term of a et of ai M function { φ i } i = which are defined on the interval [, ] and are orthogonal to one another, that i, φ () t φ () t dt = δ (8.) i j ij where δ ij i the Dirac delta function, δ ij =, i j; δ =, i = j. he parameter N ij i the dimenionality of the vector-pace that i needed to repreent the finite et of waveform. hi finite et of dicrete ignal repreent the modulation ignal ri () t M on a vector pace a a linear comination of the ai ignal { φ i } i = uch that N i ij j= j r () t = r φ (), t i =,,..., M (8.)

9 9 where * r = r () t φ () t dt (8.) ij i j Here φ * j () t i complex conjugate to φ j ( t). Each of the ai ignal i normalized to have unit energy, that i, W = φ () t dt = (8.3) i he ai ignal form a coordinate ytem for the vector pace. A wa hown in [, 4, 8-], the Gram-chmidt orthonormalization procedure provide a ytematic way of otaining the ai ignal for a given et of waveform. Let u preent, firt of all, an example of linear modulation. Linear Modulation i modulation where amplitude of the tranmitted ignal varie linearly with the modulating digital ignal m(t) according to the following low: ( t) = Re = A m m( t) = m [ Am( t)exp( jπf ct) ] [ ( t)co(πf t) m ( t)in(πf t) ] R R ( t) + jm ( t) hi kind of modulation ha a good pectral efficiency, ut linear amplifier have poor power efficiency. At the ame time, nonlinear amplifier (which are ued) have good power efficiency. Sideloe are generated, increaing adjacent channel interference and cancel the enefit of linear modulation. a) Firt, let u conider the et of Amplitude Shift Keying (ASK) ignal, where keying (or witching) the carrier inuoid on if the input it i and off if (o-called On-Off-Keying-OOK). hi kind of modulation i hown in Fig. 8.9: I c I c Fig. 8.9 ) Secondly, let u conider the et of Binary Phae Shift Keying (BPSK) ignal (the meaning of thi areviation ee later) r( t) and r ( t) given y and W r () t = co( π ft c ), t (8.4a)

10 W ( t) = co(π fct), t (8.4) r where W i the energy per it, i the it period, and a rectangular pule hape pt () = Π( ( t / )/ ) i aumed. Bai ignal φ i for thi ignal et in Dvector-pace imply conit of a ingle wave form φ where φ () t = co( πft c ), t (8.5) Reult of uch kind of modulation i preented in Fig. 8.. Fig. 8. Uing thi ai ignal, the BPSK ignal et can e repreented a { W φ t W φ t } r ibpsk = ( ), ( ) (8.6) Such a repreentation i called a contellation diagram which provide a graphical repreentation of the complex envelope of each poile ymol tate. he x-axi of thi diagram repreent the in-phae component I of the complex envelope, and the y- axi repreent the quadrature component Q of the complex envelope. he ditance etween ignal on a contellation diagram relate to how different the modulation waveform are, and how well a receiver can differentiate etween all poile ymol when random noie i preent. It hould e noted that the numer of ai ignal will alway e le than or equal to the numer of ignal in the et. he numer of ai ignal required to repreent the complete modulation ignal et i called the dimenion of the vector pace (in our example aove it i two-dimenional (D) vector pace). If there are many ai ignal in the modulation ignal et, then all of them mut e orthogonal according to (8.). c) hirdly, let u conider Differential Phae Shift Keying (DPSK) modulated ignal: d k = mk d k ale 8.

11 {mk} {dk-} {dk} he proaility of DPSK modulated it with energy E i E P e, DPSK = exp N Noncoherent PSK (no need of reference ignal), eay to uild, cheap, energy efficiency i inferior to PSK (3 db le). d) Forth example i Quadrature Phae Shift Keying (QPSK) ignal. It advantage i that it ha twice the andwidth efficiency or two it at a time: QPSK ( t) = = E E π co πf ct + i π co i co(πf ct) t π in i in(πf ct) hi ignal et i hown geometrically in Fig. 8., where left diagram i for pure QPSK and the right one for π / 4 QPSK modulation: E Q Q i =,,,3 E I I E Fig. 8. he proaility of QPSK modulated it with energy E i P e, QPSK = Q E N e) Fifth example i a et of Frequency Shift Keying (FSK) ignal, where keying (witching) the carrier inuoid frequency into if the input it i and into if input it i. Reult of modulation are preented in Fig. 8..

12 Fig. 8. Now we will conider a quetion on how different kind of digital modulation affect different communication channel Digital modulation in AWGN channel he implet practical cae of the wirele communication channel i an additive white Gauian noie (AWGN) channel. When the modulated ignal i tranmitted over uch a channel, the ignal arriving at the demodulator i pertured only y the addition of ome noie. hi channel applie only in the tatic cae, where the terminal antenna and the otruction are not in motion. Such a noie i white, that i, with a contant power pectral denity (PSD) and Gauian, i.e., ha a normal ditriution. he received ignal in time t, (t), i then given y t () = Ag() t + n() t (8.7) where n(t) i the noie waveform, g(t) i the modulated ignal and A i overall path lo, aumed not to vary in time. Equation (8.7) i a complex aeand ignal (oth modulated ignal and noie) repreentation. If o, oth the real and imaginary part of noie n(t) are zero mean, independent, real Gauian procee, each of with a tandard deviation of σ n [9]. For digital ignal, which conit of ymol with an individual energyw and a finite duration. hen W = A /. Similarly, if the noie i contain within a andwidth B = /, and ha power pectral denity PSD N, then the mean noie power [, 4, 8-] * Pn = n() t n () t σ n = BN = N / (8.8) he ignal-to-noie ratio (SNR) at the input of the demodulator i then (ee aove all definition): γ Ag() t A g () t W A SNR = = = = P σ N N n n (8.9) It i uual to expre the error rate performance of a digital ytem in term of thi parameter or in term of the correponding SNR per it: γ γ = = m W N (8.3)

13 3 where m i the numer of it per ymol. he SNR i the key parameter in calculating the digital modulation ytem performance in the AWGN channel. Let u, a aove in an example preented aove, calculate the it nerror rate (BER) performance of inary phae hift keying (BPSK) ignal in AWGN channel. We, firt of all will conider the two-dimenional cae of BPSK ignal, that i, two ignal which correpond to a inary and. heir complex aeand preentation i g W W =, g = (8.3) where the duration of each ymol i, the energy of of each ymol i W, and A=. hee ignal conit of egment of carrier of duration and a phae difference of 8, and can e repreented in the BPSK contellation diagram (ee Fig. 8.4). For D-vector pace preentation it can e hown according to [, 8-] that the error rate performance of digital modulation cheme in AWGN channel with PSD of N depend on the Euclidean ditance d etween the tranmitted waveform, correponding to different tranmitted it according to [, 8-], and i determined y the proaility of error P = P ( γ) p( γ) dγ (8.3) e e where P e ( γ) i the proaility of error for a digital modulation at a pecific value of SNR, γ, where γ=aw / N and p( γ ) i a proaility denity function (PDF) due to AWGN channel. In [, 4, 8-] thi proaility of error wa etimated through the Q- function: P Q Ad d e = Q N N (8.33) It i clear from Fig. 8.4 that in our example d = W and P r = Q d N W Q Q = 4 = N ( ) γ (8.34) hi reult i illutrated in Fig he rapid decreae in it error rate a SNR increae i main pecific feature of an AWGN channel. hi decreae i the fatet, which could take place for digital modulation channel, o the AWGN channel i a et cae channel. For high SNR the it error rate decreae y a factor of approximately, that i, in decade for every db increae in SNR. In the M-dimenional vector pace the proaility of it error i alo proportional to the Euclidean ditance etween the cloet point in the contellation diagram. he ame upper ound for the proaility od ymol error, a in D-cae for AWGN channel, can e otained for the M-dimenional cae following to derivation carried out in [, 4, 8]. According to [, 4, 8], the average proaility of error for a particular modulation ignal, P ( ε r ) r i

14 4 P r Q d ij r( ε i) N j= j i (8.35) where d ij i the Euclidean ditance etween the ith and jth ignal point in the contellation diagram, Q-function, we will repeat it definition once more x Qx ( ) = exp dx π x (8.36) If all the M modulation waveform are equally likely to e tranmitted, then the average denity proaility of error for a modulation can e preented a M P P r P r M P r r( ε) = r( ε i) r( i) = r( ε i) i= (8.37) For ymmetric cantellation, the ditance etween all contellation point are equivalent, and the conditional error proaility Pr( ε ri) i the ame for all i. Hence (8.35) give the average proaility of ymol error for a particular contellation et Performance of digital modulation in flat fading channel A wa dicued [, ], flat fading channel caue a multiplicative (gain) variation in the tranmitted ignal (t) or in it envelope r(t). he AWGN digital channel decried aove, i a typical low flat fading channel, for which the proaility of error i decried y Q-function in (8.34) veru SNR, γ. Let u now conider the cae of narroand flat digital channel in which fading affect all frequencie in the modulated ignal equally, o it can e modeled a a ingle multiplicative proce. In thi channel the fading proce differ from AWGN conidered aove. We will how thi for Rayleigh fading channel and will compare oth kind of channel. Since the fading varie with time, the SNR at the demodulator input alo varie with time. It i neceary, in contrat with AWGN cae, to ditinguih etween the intantaneou SNR, γ, and the mean SNR, Γ. Since fading in the channel change much low than the applied modulation meage m(t), it can e aumed that the attenuation and phae hift of the ignal i contant over at leat one ymol interval. herefore the received ignal r(t) may e expreed a r() t = Aα () t g() t + n() t (8.38) where α( t ) i the complex fading coefficient at time t. Equation (8.38) mathematically decrie the narrowand fading channel which i preented in Fig. 8.6 y the correponding lock-cheme. If the fading i aumed contant over the tranmitted ymol duration, then i alo contant over a ymol and i given and A α( t) g( t) A α( t) γ() t = = P P n n (8.39)

15 5 Γ = γ( t ) (8.4) Uually we conider the flat fading, having unit variance, and lump any change in mean ignal power into the path lo, o Γ= γ( t) = A P n (8.4) wo thing are therefore needed in order to find the performance of a ytem in the narrowand low fading channel: the mean SNR and a decription of the way the fading caue the intantaneou SNR to vary relative to thi mean a hown in Fig Let u now conider the ditriution of SNR for a Rayleigh fading flat narrowand channel. We will ue now expreion (8.3) for the proaility of error in a low, flat fading channel, where now γ = α W / N (all parameter for digital modulated ignal (ymol) are defined aove). for Rayleigh ditriution the PDF of error i now [, 4, 8, ] γ p( γ) = exp (8.4) Γ Γ where Γ=( W / N ) α i the average value of the ignal-to-noie ratio. If o, for the comulative denity function CDF we have γ γ CDF( γ) Pr( γ < γ ) = p( γ) dγ = exp Γ (8.43) Here γ i the contant threhold of SNR in the Rayleigh fading channel. he reult of (3.43) i illutrated in Fig hi reult can e ued to calculate the mean SNR required to otain an SNR aove ome threhold for an acceptale percentage of the time. But the ame PDF from (8.4) can e uccefully ued to predict the error rate performance of digital modulation cheme in a Rayleigh channel y auming that the SNR i contant over one ymol duration. In thi cae the Rayleigh error rate performance can e predicted directly from the AWGN channel cae. hu for inary phae hift keying (BPSK) ignal, a aove in previou ection, uing AWGN error rate performance from (8.34) and the Rayleigh tatitic according to (8.4) for the intantaneou SNR, γ, according to illutration in Fig Following (8.4) and thi illutration, the average it error proaility P e can e defined a P = P ( γ) = P ( γ) p( γ) dγ ebpsk e e ( γ ) = Q γ exp dγ = Γ Γ Γ + Γ (8.44)

16 6 hi reult, a i an approximation, which lead from the cae of large Γ, P e /( 4Γ ) (ee Fig. 8.9). hi invere proportionality i characteritic of decoded modulation in a Rayleigh channel leading to a reduction of it-error-rate (BER) y one decade for every db increae in SNR. hi contrat harply with ~ db per decade variation in the AWGN channel (ee Fig. 8.5). he performance of other digital modulation cheme in Rayleigh fading channel can e analyzed in a imilar way following to [, 4, 8, ]. Becaue BPSK i an example of linear digital modulation technique, where the amplitude of the tranmitting ignal (t) varie linearly with the modulating ignal m(t), let u preent the ame average it error proaility P e for coherent inary frequency hift keying (FPSK) ignal a an example of nonlinear modulation method following to [, ] P efpsk = Γ + Γ (8.45) It alo can e hown according to [, 4, ] that the average error proaility for differential BPSK, a linear modulation method, and for incoherent inary FPSK, a nonlinear modulation method, in a low, flat Rayleigh fading channel are given, repectively, y P edpsk = (8.46) ( + Γ) and P eicfpsk = ( + Γ) (8.47) Figure 8. illutrate how the BER for variou modulation change a a function of W / N with repect to that for AWGN low, flat channel. he ame low rate of reduction in BER otained for BPSK ignal a compared to a typical performance curve in AWGN i oerved. he rate of reduction in BER may e increaed with increae of Ricean factor, K, introduced in [-4], a a ratio etween LOS component and NLOS component of the total multipath ignal. he Ricean flat low-fading channel i uually ued whenever one path field component exceed or at the ame level to the other multipath component due to multi-cattering, multi-reflection and multi-diffraction. he it error rate (BER) for BPSK modulated ignal and for a Ricean fading channel, depending on K value, i intermediate etween the AWGN and Rayleigh channel cae, a hown in Fig. 8.. It i clear that the Ricean channel ehave like AWGN channel in the limit a k without any multipath or non-line-of-ight component. Biliography [] Rappaport,. S., Wirele Communication, New York: Prentice Hall PR, 996. [] Amorozo, F., he andwidth of digital data ignal, IEEE Communication Magazine, Nov. 98, pp. 3-4.

17 7 [3] Couch, L. W., Digital and Analog Communication Sytem, Macmillan, New York, 993. [4] Stuer, G. L., Principle of Moile Communication, Kluwert Academic Puliher, Boton-London, 996. [5] Goling, W., J. P. McGeehan, and P. G. Holland, Receiver for the Wolfon SSB/VHF land moile radio ytem, Proc. IEEE Conf. on Radio Receiver and Aociated Sytem, Southampton, England, pp , July 978. [6] Luignan, B. B., Single-ideand tranmiion for land moile radio, IEEE Spectrum, July 978, pp [7] Well, R., SSB for VHF moile radio at 5 khz channel pacing, Proc. IERE Conf. on Radio Receiver and Aociated Sytem, Southampton, England, pp , July 978. [8] Ziemer, R. E., and R. L. Peteron, Introduction to Digital Communication, Macmillan Pulihing Co., 99. [9] Saunder, S. R., Antenna and Propagation for Wirele Communication Sytem, John Wiley & Son, New York, 999. [] Proaki, J. G., Digital Communication, McGraw-Hill, New York, 989. [] Blauntein, N., and J. B. Anderen, Multipath Phenomena in Cellular Network, Artech Houe, Boton-London,.