Consider the following passband digital communication system model. c t. modulator. t r a n s m i t t e r. signal decoder.

Transcription

1 PASSBAND DIGITAL MODULATION TECHNIQUES Consder the followng passband dgtal communcaton system model. cos( ω + φ ) c t message source m sgnal encoder s modulator s () t communcaton xt () channel t r a n s m t t e r xt () detector X sgnal decoder m ˆ recever Passband dgtal communcaton system

2 Assumptons about the channel:. The channel s lnear wth bandwdth wde enough to accommodate the transmsson bandwdth of s(t) to mnmze sgnal dstorton at the recever. N. The channel nose s of the AWGN type wth p.s.d. SW ( f) =, f. Although there s a wde varety of modulaton schemes, we shall manly be concerned wth Ampltude-Shft Keyng (ASK), Phase-Shft Keyng (PSK), Frequency-Shft Keyng (FSK), and ther varants. In the followng fgure, bnary OOK s a form of ASK where the absence of sgnal means the transmsson of a logcal.

3 Passband dgtal modulaton examples

4 At the recever end, we shall consder both coherent and noncoherent detectors. The former type needs to know both carrer frequency and phase nformaton of the modulated waveform s(t), n order to demodulate the message sgnal. The latter type does not need to know the exact carrer frequency and phase values to demodulate the sgnal. The penalty pad by the noncoherent detector s that ts performance s nferor than that of the coherent detector. Bandwdth Effcency Consder M-ary sgnalng, wth M = n. If we transmt blocks of n bts, then the symbol duraton s Ts = ntb and the requred bandwdth s proportonal to /( ntb). Ths mples that transmsson bandwdth can be reduced by the factor n. Def. Bandwdth effcency s the rato of the data rate n bts/s to the effectvely utlzed bandwdth. R B Example: ρ for BPSK, QPSK, 6-QAM, FSK, MSK s,, 4, <,.35, respectvely. b Let Rb be the bt rate and B be the effectve channel bandwdth, then ρ = [ bts / s / Hz ] 3

5 PULSE AMPLITUDE MODULATION (PAM) AWGN Channel: Consder the followng PAM communcaton system wth coherent demodulaton { b } n s () pb t T s { r m} Map Σ ( ) dt PAM o Decson { s ˆ } cos c T s ( ω t + θ) Wt () cos c T s ( ω t + θ) where {b m } s a stream of bnary symbols, s () t s the band pass PAM transmtted N sgnal, Wt () s WGN wth power spectral densty SW ( f) =, f, {r m } s a sequence of observed samples, and { s ˆ } s a sequence of estmated symbols. pb 4

6 Let s,,, M, be a PAM symbol that can take on M possble values, wth M an arbtrary postve nteger. Then the modulated waveform s descrbed by s () t = s cos t+ = s (), t () t = cos t+, ( ω θ) φ φ ( ω θ) pb c c Ts Ts where s s the baseband sgnal (symbol) that takes on the values ( ) M d s =, =,, M wth d the ntra-symbol dstance. Let M be an even postve nteger. Consder the followng sgnal constellaton for the PAM modulator 5

7 After down converson, the detector s descrbed by an M-level mdrse quantzer. The decson varable at the output of the detector s descrbed by r = s + W, m N where W G, mples fw ( w) e π N N = and w fr ( ) m s r m s e π N ( r s ) m N =. The quantzer recever s equvalent to the mnmum dstance recever when the ML decson crteron s appled. Therefore, for the nner ponts ( =,, M ), the probablty of a correct decson (cd) gven that s was transmtted s gven by d s + ( r ) m s N = R s m m= m R d π N s { } ( ) m P cd s f r s dr e dr d d N N u = u u e du e du e du π = + d π d N N d = Q, =,, M. N 6

8 Where t has been assumed that all symbols occur wth equal probablty,.e., P{ S} =, =,, M (the decson regons boundares are equdstant for the M- M nnermost constellaton ponts). For = : d s + d N r s N u d ( ) m P{ cd s } = fr ( ). m s rm s drm = e drm = e du Q πn = π N R Lkewse, for = M, d P{ cd sm } = P{ cd s} = Q. N Let P P{ cd}, then usng the total probablty theorem, we get cd M M M P = P cd, s = P cd s P S = P cd s. { } { } { } { } cd = = M = 7

9 The probablty of a correct decson s then gven by M { } { } P = P cd S P S cd = { } { } { } { } = P cd S P S + P cd S P S M = M d d = Q P{ S} + Q P{ S}. N = N Rewrtng the last equaton, we get M M d Pcd = P{ S} + P{ S} P{ S} P{ S} + Q. = = N 8

10 M d d Pcd = P{ S} + P{ S} P{ S} Q = Q. = N M N The symbol error probablty s therefore d M d Pse = Pcd = Q Q, M = N M N Now, let s compute the average symbol energy Es. M M M ( ) M d E = E S = s P S = s P S = d P S = M M { } { } { } { } ( ) s = = = 4 M = M Let K ( M), then ( ) M K = ( M + ) M M + + = 3, d M E symbol error probablty can be rewrtten as P = Q. se s M KN s = E and the K 9

11 For M = 6, K = 4.5 and 5 4E b E b Pse = Q.875Q.94 8 =. 4.5N N - - Symbol error probablty E b /N n db

12 QUADRATURE AMPLITUDE MODULATION (QAM) AWGN Channel: In quadrature ampltude modulaton both ampltude and phase are modulated. Hence, the resultng waveform wll have both an n-phase and a quadrature component. As before, ncreasng the dstance between constellaton ponts wll mprove system performance at the expense of hgher average transmtted power. Let the passband transmtted sgnal be descrbed by j( ωct+ θc) {( ) } ( ω θ ) ( ω θ ) s ( t) =R e a + jb e = a cos t + b sn t +, nt t ( n + ) T Pb c c c c s s where a and b are the n-phase and quadrature components, respectvely, of the th symbol, =,,M. An 8-QAM sgnal constellaton s shown n the next fgure.,

13 8-QAM sgnal constellaton

14 A sample 8-QAM waveform s shown next. It s clear from the fgure that the QAM modulated waveform does not have constant envelope. Hence, the transmtter needs to use a hghly lnear power amplfer and the recever needs an automatc gan control crcut to compensate for the ampltude varatons due to channel mperfectons. Unmodulated carrer and 8-QAM modulated sgnals 3

15 A 64-QAM sgnal constellaton s shown below. 64-QAM sgnal constellaton 4

16 Let the complex equvalent baseband transmtted sgnal be descrbed by ( ) s () t = a + jb, nt t n + T. bb s s Then, the complex equvalent baseband sgnal seen at the nput of the recever decson devce (after bandpass flterng and down converson) s descrbed by r () t = a + jb + n (), t ( ) = a + ni() t + j b + nq() t, nts t n + Ts. Performance wll depend on the type (shape) of constellaton that s used at the transmtter and on the detecton scheme used at the recever. Regardless of the constellaton shape, f the detector selects symbol sj when the observed measurement rt () les on regon R j of the observaton space (equvalent to mnmum Eucldean dstance detecton), then the system performance can be obtaned as follows: 5

17 Suppose that at the n th observaton nterval rt () falls on decson regon R. Then the probablty of a correct decson s descrbed by (gven that symbol s was transmtted) { = } = ( = ) P cd S s f r, r S s drdr, where R { } { } r R e r() t = a + n ( nt ) I s r Im r( t) = b + n ( nt ). Q s Let r r r and n ni ni( nts) n Q nq( nts) Wth E{ ni} = E{ nq} =, E{ ni } = E{ nq } = N and E{ nn I Q} =. Defne { } { } j j P cd S = s P r s r s S = s, j ;, j =,, M as the probablty that the observed vector r s closer to constellaton pont s than to s j, gven that symbol s was transmtted. 6

18 Conversely, defne the probablty of an ncorrect decson (d), gven that symbol s was transmtted as { } { } j j P d s P r s > r s s, j ;, j =,, M But, when symbol s s transmtted, r s = n = ni + nq and r s = ( a a ) + ( b b ) + ( a a ) n + ( b b ) n + n + n j j j j I j Q I Q Hence, { r s } > r sj s mples ( ) ( ) ( ) ( ) ni + nq > a aj + b bj + a aj ni b bj n + Q + ni + nq or ( aj a) ni + ( bj b) nq > ( a aj) + ( b bj) = ( aj a) + ( bj b). The left hand sde of the last equaton s a lnear combnaton of zero-mean Gaussan random varables. 7

19 Let ν j ( aj a ) ni + ( bj b ) nq and C j ( aj a ) + ( bj b ) ν (, σ ) j G ν j, where the varance ( ) ( ) σν = j j + j a a b b N = C N. j then σ ν j s gven by Hence, σ P{ d s } = P{ ν > C } = e dv j πσ and j j j ν C { } j j v j ( aj a) ( bj b) C j C + j = Q = Q = Q σ ν N 4N j P error s, j ;, j =,..., M j M { } { } P error s P error s, =,, M. j j= j j ν 8

20 Fnally, an upper bound of the average symbol error rate for QAM s gven by M M M C P = P error s P S Q P S j { } { } { } se = = j= σ ν j j Note that ths last equaton s an upper bound of the symbol error probablty when the decson metrc s the mnmum Eucldean dstance and the probablty of symbol transmsson s arbtrary. Also, M can be an arbtrary odd or even postve nteger. 9

21 Consder now a rectangular 6-QAM sgnal constellaton. Let the dstance to the nearest neghbor be d = a. Then the constellaton dagram s shown n the next fgure. 6-QAM rectangular constellaton

22 Suppose all 6 symbols are transmtted wth equal probablty,.e. { } P m =, =,,,6, then the average transmtted energy per symbol can be 6 computed as follows: E = E P{ m} 6 s, av =, where E = d s the energy assocated wth the th symbol and d s the dstance from the orgn to the th pont on the constellaton. From the fgure we can see that the 4 nnermost ponts have the same dstance d, the outer 8 rectangle ponts along the φ and φ axes have dstance d, and the outer 4 corner ponts have dstance d 3, where d a a a = ( ) + =, and ( ) ( ) d 3a a a = + =, d3 = 3a + 3a = 8a. The average transmtted energy per symbol s therefore Es, av = 4 ( a ) + 8 ( a ) + 4 ( 8a ) a a a a. = + + = 6

23 The nearest neghbor ntra-symbol dstance s d = a. Hence, Es, av.5 =. = d and d E, 5 s av System performance n AWGN can now be assessed. Assumng equal probablty of symbol transmsson, we can use the maxmum lkelhood detecton approach (mnmzaton of dstance). In ths case, the decson regon for one of the nner ponts (symbols) s shown n the next fgure. Decson regon for symbol m when all symbols occur wth equal probablty.

24 The probablty of a correct decson, gven that ths symbol was transmtted s descrbed by { correct decson gven } X( ) P m f x m dx dx = Let P c be the average probablty of a correct decson. Then 6 6 R { choose sent} { choose sent } { sent}. P = P m m = P m m P m c = = sent =,, then Pc = P choose m m sent = P X R m sent 6 6 If P{ m } { } { }. = = But, P{ X R m sent} P{ X R m sent} =. Therefore, 6 6 P = P X R m = P X R m ( { sent} ) { sent}. c 6 = 6 = 3

25 Let P e be the average probablty of symbol error (SER), then 6 SER = P = P{ symbol error} = P = P{ X R m sent}. e c 6 = For a suffcently hgh SNR, an error wll occur f m s confused wth one of ts nearest neghbors. Let R j be the decson regon of one of the nearest neghbors of symbol m. Then, for the 4 nnermost ponts of the constellaton (symbols) j =,, 3, 4; for the 8 edge ponts (excludng the corner ponts) j =,, 3; and for the 4 corner ponts j =,. 4

26 Consder now the 4 nnermost ponts. Then usng the unon bound, we get 4 4 P{ X R m sent } P X Rj m sent P{ X Rj m sent} j=. j= For the 8 edge ponts, 3 3 P{ X R m sent } P X Rj m sent P{ X Rj m sent} j=. j= For the 4 corner ponts, P{ X R m sent } P X Rj m sent P{ X Rj m sent} j=. j= In concluson, for any pont on the 6-QAM constellaton, a loose upper bound for the symbol error rate (SER) for hgh SNR s gven by 4 { sent } { j sent} P X R m P X R m. j= 5

27 Assumng Grey encodng of the QAM symbols, P{ X Rj m sent} s the same as the par-wse bt error probablty,.e. d d P{ X Rj m sent } = P X > = P X <, =, E d s, av 5 E s, av 4E b, av =Q = Q = Q = Q. N N 5N 5N and 4 4E b, av 4E b, av P{ X R m sent} Q = 4Q. j= 5N 5N Fnally, the average symbol error probablty (SER) s upper bounded by 4E 4E SER = Pe Q = Q b, av b, av 4 4. j= N N 6

28 In general, for M = k and a rectangular sgnal constellaton wth symbols transmtted wth equal probablty and Grey encodng, d SER = Pe 4 Q, N where d s the s the nearest neghbor ntra-symbol dstance, whch can be expressed n terms of the average symbol energy Es, whch must be computed for every specfc sgnal constellaton. Other tghter upper bounds can be found n the lterature. The followng fgure shows the SER performance of 6 QAM that uses a rectangular constellaton that has the same ntra-symbol dstance d. 7

29 - - SER SNR n db 6- QAM SER performance n AWGN 8