Lecure 4 Goals: Be able o deermine bandwidh o digial signals Be able o conver a signal rom baseband o passband and back IV-1
Bandwidh o Digial Daa Signals A digial daa signal is modeled as a random process Y (wide sense) version o a process X ; which is a saionary Y X U where U is a random variable needed inorder o make Y saionary. In many digial communicaion sysems X sequence j pulses or waveorms i.e. X b x is an ininie wide sense In his case i U is uniormly disribued beween 0 and hen Y is a wide snese saionary random process. We desire hen o compue he auo correlaion o Y and also he specrum o Y. Assume ha b is a sequences o i.i.d. random variables wih zero mean and variance σ 2 (e.g. IV-2
are Y P b 1 1 independen. 2 P b 1 1 2). Also assume U and b Claim: RY SY ω τ σ 2 1 F ω ω d x x τ 2 where F F x Derivaion: For any and τ E Y τ E m m b σ 2 E δ x m x E bm b U σ 2 m 0m E U x x τ bmx U τm U x U τm U IV-3
1 σ 2 σ 2 1 σ2 σ 2 w dw x x u 0 x v du x 1 x w τ x τ u v x x v dv σ 2 τ w d x τ u v dv x v v τ hus Y Now le 1 is wide sense saionary wih 1 x 2 τ 2 RY τ x 1 σ 2 hen 2 d τ d x x τ d x x τ u dv du IV-4
So RY SY F1 F2 SY ω ω ω ω τ d x τ x σ 2 1 2 τ F F F σ2 1 σ 2 F ω ω ω x x 2 F 2 F τ σ 2 F 1 ω F2 ω IV-5
x b p 1 σ 2 x x d Example 1 0 0 elsewhere 1. τ p 0 d τ p τ τ τ 0 τ τ 0 elsewhere 0 1 0 τ 0 elsewhere 1 τ 0 elsewhere τ IV-6
F ω ω RY τ SY F 2 SY SY ω ω 1 F 1 p F p τ τ 0 elsewhere 2 sin2 ω ω 2 1 F ω 2 2 2 2 sinω ω 2 sin2 ω ω 2 2 2 2 e jω 2 sinc2 sin2 π π 2 sinc2 IV-7
Specrum or Recangular Pulses 1 0.9 0.8 0.7 0.6 Specrum 0.5 0.4 0.3 0.2 0.1 0 4 3 2 1 0 1 2 3 4 requency IV-8
Specrum or Recangular Pulses 0 5 10 S X () (db) 15 20 25 30 2 1.5 1 0.5 0 0.5 1 1.5 2 requency IV-9
Specrum or Recangular Pulses 0 5 10 S X () (db) 15 20 25 30 35 40 8 6 4 2 0 2 4 6 8 requency IV-10
Deiniion o Bandwidh or Digial Signals 1. Null-o-Null bandwidh bandwidh o main lobe o power specral densiy 2. 99% power bandwidh conainmen bandwidh such ha 1/2% o power lies above upper band limi and 1/2% lies below lower band limi 3. x db bandwidh bandwidh such ha specrum is xdb below specrum a cener o band (e.g. 3dB bandwidh) 4. Noise bandwidh value o specrum a WN P c. S c wherepis oal power and S c is IV-11
σ where σ 2 WA (Area under curve = P) S WN U c 5. Gabor bandwidh c S 2 S d d 6. Absolue bandwidh min W : S 0 IV-12
Recangular Pulse Example S Y sin2 π π 2 sinc2 sinc π c n c n 0aπ 1 2 c hπ h 1 2 3 Null o null bandwidh widh o main lobe o specral densiy. For PSK null o null bandwidh = 2 Fracional Power conainmen Bandwidh widh o requency band which leaves 1/2% o singal power above upper band limi and 1/2% o signal power below band limi For PSK 99% energy bandwidh = 20 56 We would like o ind modulaion schemes which decrease he bandwidh while reaining accepable perormance IV-13
Modulaion 1 2 3 35dB 4 5 3 3dB BPSK 2.0 20.56 35.12 1.00 0.88 QPSK 1.0 10.28 17.56 0.50 0.44 MSK 1.5 1.18 3.24 0.62 (0.5) 1 2 0.59 hese hree modulaion schemes all have same error probabiliy. MSK has minimum possible Gabor bandwidh over all modulaion schemes whose basic pulse is limied o 2 seconds. All o hese have ininie absolue bandwidh. IV-14
Shannon s heorem revisied heorem: (Shannon) For a whie Gaussian noise channel here exis signals 1 wih absolue bandwidh W conveying R bis o inormaion per second wih arbirarily small error probabiliy provided where E b R 1 W log 2 1 E b WN 0 P is he energy per daa bi and P is he power o he signal P R lim 1 2 W log 1 s 2 R W E b N 0 d P N 0 ln2 as W IV-15
R W log 1 R W E b N 0 E b N0 Eb ln N0 2 2 R W R Eb N0 1 W db ln2 as R W 10log10 0 ln 2 1 6dB IV-16
2π c 2π c Up and Down Conversion In communicaion sysems ypically he signal are generaed a baseband and hen up convered o he desired carrier requency. A he receiver his process is reversed. x c y c cos y c xc cos IV-17
2π c Y c 1 2 Xc c Xc c hus muliplicaion in he ime domain by cos and down by c and reduces each par by 1/2. shis he specrum up IV-18
Re Xc, ImX c W W IV-19
Im Xc W W IV-20
Re Yc, Im Yc c c IV-21
Im Yc c W c c W c W c c W IV-22
2π c 2π c 2π c Now consider muliplicaion by sin. x s y s sin y s x s sin Y c j 2 Xc c Xc hus muliplicaion by sin shis he specrum up and down also excep ha he real par becomes he imaginary par and he imaginary par is invered and becomes he real par in addiion o a reducion by 1/2. 2π c c IV-23
Re Xs W W IV-24
Im Xs W W IV-25
Re Ys c W c c W c W c c W IV-26
Im Ys c W c c W c W cc W IV-27
Now consider adding hese wo uncions ogeher. IV-28
xc xs cos sin 2π c 2π c yc ys y IV-29
2π c y yc y s xc cos 2π c x s sin xe cos 2π c θ he signal x e is called he envelope and θ is called he phase. Y Yc Ys θ 1 an x s x c x e x2c x 2 s 1 2 y Re x c jx s e j2π c IV-30
Re Y c W c c W c W c c W IV-31
Im Y c W c c W c c W W c IV-32
Signal Decomposiion he signals x c and x s can be recovered rom y baseband and ilering ou he double requency erms. Noe ha we need he exac phase o he local oscillaors o do his perecly. by mixing down o IV-33
2π c 2π c 2cos z c G LP x c y z s G LP x s 2sin IV-34
G LP is an ideal low pass iler wih ranser uncion G LP and G LP 0 oherwise. 1 W IV-35
G LP IV-36
Consider he specrum o z c. his is given by Z c Y c Y c Similarly he specrum o z s is Z s j Y c Y c IV-37
Re Zc 2 c W W 2 c IV-38
Im Zc 2 c W 2 c W IV-39
Re Zs 2 c W W 2 c IV-40
Im Zs 2 c W W 2 c IV-41
2π c 2π 1 2π 1 2π 2 2π 1 2π 1 2π 2 Example x c ac 1 cos a c 2 sin a c 3 cos x s as 1 cos a s 2 sin a s 3 sin where a c 1 a s 1 1 0 25 1 1 0 2 ac 2 ac 2 2 0 5 0 25 ac 3 ac 3 1 1 he signals are upconvered wih a quadraure modulaor o produce xc cos 2π c x s sin xe cos 2π c θ where c 16. IV-42
3 2 1 xc() 0-1 -2-3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ime 3 2 1 xs() 0-1 -2-3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ime Figure 28: Example o Up and Down Conversion o Signals IV-43
xe() 2.5 2 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ime 4 2 phi.() 0-2 -4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Figure 29: Example o Up and Down Conversion o Signals IV-44
2 1.5 1 0.5 xs() 0-0.5-1 -1.5-2 -2-1.5-1 -0.5 0 0.5 1 1.5 2 xc() Figure 30: Real and Imaginary Par o Complex Signal IV-45
2 1.5 1 0.5 y() 0-0.5-1 -1.5-2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ime Figure 31: Example o Up and Down Conversion o Signals IV-46
3 2 1 zc() 0-1 -2-3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ime 3 2 1 zs() 0-1 -2-3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ime Figure 32: Example o Up and Down Conversion o Signals IV-47
Now consider wha happens i here is an imperec local oscillaor IV-48
2cos 2π c φ z c G LP w c y z s G LP w s 2sin 2π c φ IV-49
w c xc cos φ φ φ φ x s sin w s x c sin x s cos w c jw s xc jx s e jφ Noe ha his is equivalen o a phase roaion by angle φ. IV-50
We can recover he orignial signal by roaing he signal. x c jx s wc jw s ejφ IV-51
wc ws cos cos φ φ sin φ xc xs IV-52
Binary Phase Shi Keying (BPSK) b 2Pcos Modulaor 2π c s r n b l b l p l bl p 1 1 p 1 0 0 oherwise. 1 IV-53
2π c 2π c s 2P l b l cos p l 2P b cos 2Pcos 2π c φ where φ is he phase waveorm. he signal has power P. he energy o he ransmied bi is EP. he phase o a BPSK signal can ake on one o wo values as shown below. IV-54
1 b() 0-1 0 1 2 3 4 5 6 4 ime phi() 2 0 0 1 2 3 4 5 6 ime 1 s() 0-1 0 1 2 3 4 5 6 ime Figure 33: Signals or BPSK Modulaion IV-55
2π c τ 2π c τ 2π c Demodulaor r LPF X i i 0decbi 1 0 dec b i 1 1 1 2 cos he low pass iler (LPF) is a iler mached o he baseband signal being ransmied. For BPSK his is jus a recangular pulse o duraion. he impulse response is h p X X i 2 2 cos cos h p i τ r τ τ r dτ τ dτ IV-56
2π c τ i i 1 2 cos n τ 2π c τ 2Pb dτ τ cos i i 1 2 P bi 1 cos 2π c τ 2π c τ cos dτ ηi η i is Gaussian random variable, mean 0 variance N 0 2π c 2πn 2. Assuming X i P bi 1 ηi E bi 1 ηi IV-57
P e,-1 P e,+1 E 0 E X(i) Figure 34: Probabiliy Densiy o Decision Saisic or Binary Phase Shi Keying IV-58
Bi Error Probabiliy o BPSK P e b Q 2E N 0 Q 2E b N 0 where Q x x 1 2π e u2 2 du For binary signals his is he smalles bi error probabiliy, i.e. BPSK are opimal signals and he receiver shown above is opimum (in addiive whie Gaussian noise). For binary signals he energy ransmied per inormaion bi E b is equal o he energy per signal E. For P e b 10 5 we need a bi-energy, E b o noise densiy N 0 raio o E b 9 6dB. Noe: Q x is a decreasing uncion which is 1/2 a x0. here are eicien algorihms (based on aylor series expansions) o calculae Q x. Since Q x x2 2 2 he error N0 e IV-59
probabiliy can be upper bounded by P e b 1 2 e which decreases exponenially wih signal-o-noise raio. E b N0 IV-60
Error Probabiliy o BPSK P e,b 1 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 0 2 4 6 8 10 12 14 E b /N 0 (db) 16 Figure 35: Error Probabiliy o BPSK. IV-61
Bandwidh o BPSK he power specral densiy is a measure o he disribuion o power wih respec o requency. he power specral densiy or BPSK has he orm where Noice ha S P 2 sinc 2 sinc x S c sin πx d P sinc 2 he power specrum has zeros or nulls a c i 0; ha is here is a null a c 1called he irs null; a null a c 2 called he second null; ec. he bandwidh beween he irs nulls is called he null-o-null bandwidh. For BPSK he null-o-null bandwidh is 2 ha he specrum alls o as 2 as moves away rom c. (he c c πx excep or i. Noice IV-62
specrum o MSK alls o as he ourh power, versus he second power or BPSK). I is possible o reduce he bandwidh o a BPSK signal by ilering. I he ilering is done properly he (absolue) bandwidh o he signal can be reduced o 1 power is concenraed in he requency range1 2 c 2. he drawbacks are ha he signal loses is consan envelope propery (useul or nonlinear ampliiers) and he sensiiviy o iming errors is grealy increased. he iming sensiiviy problem can be grealy alleviaed by ilering o a slighly larger bandwidh 1α2 α 2. wihou causing any inersymbol inererence; ha is all he c 1 1 IV-63
S() 0.50 0.40 0.30 0.20 0.10 0.00-4 -3-2 -1 0 1 2 3 4 (-c) Figure 36: Specrum o BPSK IV-64
S() (db) 0-20 -40-60 -80-100 -5-4 -3-2 -1 0 1 2 3 4 5 (- c ) Figure 37: Specrum o BPSK IV-65
S() (db) 0-20 -40-60 -80-100 -10-8 -6-4 -2 0 2 4 6 8 10 (- c ) Figure 38: Specrum o BPSK IV-66
Example Given: Noise power specral densiy o N 0 2 150 dbm/hz =10 18 Was/Hz. P r 3 10 10 Was Desired P e 10 7. Find: he daa rae ha can be used and he bandwidh ha is needed. Soluion: Need Q 2Eb 10 7 or E b 11 3dBorEb Bu E b Pr 13 52. hus he daa bi mus be a leas 9 0 10 8 seconds long, i.e. he daa rae 1 Mbis/second. Clearly we also need a (null-o-null) bandwidh o 22 MHz. N0 N0 N0 N0 N0 mus be less han 11 13 52. IV-67
An alernaive view o BPSK is ha o wo anipodal signals; ha is s 0 Eψ 0 and s 1 Eψ where ψ 2 0 above describes he signals ransmied only during he inerval. Obviously his is repeaed or oher inervals. he receiver correlaes wih ψ 0 and compares wih a hreshold (usually 0) o make a decision. he correlaion receiver is shown below. over he inerval cos 2π c 0 is a uni energy waveorm. he 0 r 0 γ dec s0 γ dec s1 ψ IV-68
his is called he Correlaion Receiver. Noe ha synchronizaion o he symbol iming and oscillaor phase are required. IV-69
Now Le Z Y o Y. hen I Z Y1 cos independen hen Specrum or Passband Pulses R Z S Z ωc cos τ R Z Applicaion: or PSK x ω ωc θ 1 2 R Y 1 4 θ1 Y2 τ 1 2 Ap wih θ uniorm on [0,2π] and independen S Y RY1 τ cos cosω c τ ωc τ RY2 ωωc S Y ω ωc θ2 wih Y1 τ cosω c τ. he specrum is and Y 2 S Z A 2 4 sinc 2 c sinc2 c IV-70