ELEG 635 Digital Communication Theory. Lecture 9

Transcription

1 ELEG 635 Digital Cmmunicatin Thery Lecture

2 Agenda Optimal receiver fr PSK Effects f fading n BER perfrmance Tw ray mdel Pulse shaping Rectangle Raised csine Rt raised csine Receivers and pulse shaping FSK receiver Cherent Nn-cherent CPM receiver

3 Simulatins 4PAM % Prgram t calculate Pe fr a PAM signal N = 10000; %Number f samples k = ; %Bits per symbl dmin = 1; %Minimum distance mdulatin = 'PAM'; SNR = 0:5; %SNR in db M =.^k; %M p = bingc(0:(m-1)); %Create Gray Cde mapping bits = Bit_Stream(k.*N); [Ai, Aq, symbls, symblsgc] = Mapper_gc(k, bits, mdulatin, 0); %Add Nise Pe = zers(size(snr)); Pe_thery=zers(size(SNR)); fr kk = 1:length(SNR); tmp = 10.^(SNR(kk)/10); sigma = sqrt(dmin.^*(m.^-1)/(6*k*tmp)); [gv1,gv] = gngauss(n,0,sigma); r = Ai + gv1; s = PAM_Detectr(r, M, dmin); Pe(kk) = sum(duble(s ~= symbls)); Pe_thery(kk) = *(M-1)/M*Qfunc(sqrt(6*k/(M^-1)*tmp)); end semilgy(snr,pe/n,'*') hld n semilgy(snr,pe_thery) axis([min(snr) max(snr) 10^-8 10^0]); grid n xlabel('ebavg/n [db]'); ylabel('pe'); hld ff

4 Optimal Detectin fr PSK Q r=(r 1, r )= ( ( E)+n n 1, ) d min I p (r 1, r V = r 1 )= 1 π N +r e ( ( E)+r N ) +r 1 Θ=arctan r r 1 p (v,θ)= v V Θ π N e v + E Ecs θ N

5 Optimal Detectin fr PSK p Θ (θ)= 0 p (v,θ)dv= 1 sin θ V Θ π e γ s v e 0 (v γ s cs θ) dv p e γ π/ M =1 π/ M s = E N p (θ)d θ Θ Q d min This can nly be slved numerically I

6 Frequency Selective Fading Cnsider a channel with a direct and single reflective fade path that inverts the signal. The difference in the time f arrival f these tw paths in τ. h (t)=1 δ(t τ) H ( f )=1 e j π f τ Nte: when fτ is an integer there is cmplete cancellatin H(f) f

7 Frequency Selective Fading Cnsider PSK r PAM (These are bth BPSK)

8 Flat Fading s(t )= A(t)sin( π f t+ϕ(t)) r (t)= k s(t τ k )= k A(t τ k )sin ( π f t+ϕ(t τ k ) π f τ k ) The utput f passing a signal thrugh a delay line is s d (t )= [ S ( f )e j π f τ ]e j π f t df If the spectrum is cnfined t B << 1/τ s d B B (t )= [S ( f )e j π f τ ]e j π f t df = B B S ( f )e j π f t df =s(t)

9 Flat Fading r (t)= A(t) [sin(π f t+ϕ) k α k cs(π f τ ) cs( π f k t+ϕ) k α k sin (π f τ k) ] r (t)= A(t) [ a(t)sin (π f t+ϕ(t))+b(t)cs(π f t+ϕ(t )) ] r (t)= A(t) M (t)sin(π f M (t)= a(t ) +b(t) t+ϕ(t )+θ(t )) θ(t )=arctan a(t ) b(t)

10 Flat Fading p(x, y)= 1 π σ p (r,θ)= 1 π p (r)= 0 π σ e e x r + y σ σ p(r,θ)r d θ= r σ e r σ Rayleigh Distributin

11 Flat Fading p(r)= r Fr BPSK P fade σ = 0 e r σ P (e) P (A)dA= 0 Rayleigh Distributin A A e A A 1 π A T / N e u / du P fade=0.5( 1 [ (1+SNR)].5 ) SNR= A T N E = N b

12 BER f BPSK w/ Fading Channel Pe SNR [db]

13 Base Band Pulse Shaping T cnstrain the bandwidth f the signals pulse shaping is applied PAM, PSK, and QAM mdulatrs A i (t) h(t) X cs(πf c t) sin(πf c t) A q (t) h(t) X

14 Base Band Pulse Shaping Several different pulse shapes Brick wall filter b = sinc(t/t) -ct < t < ct Raised csine b = sinc(t).*cs(pi*alpha*t)./(1-4*alpha^*t.^) -ct < t < ct Rt raised csine b = (4/pi*alpha*cs((1+alpha)*pi*t).+(1-alpha)*sinc((1-alpha)*t))./(1-(4*alpha*t).^) -ct < t < ct Rectangle b = 1 0 < t < T = 0 therwise

15 Base Band Pulse Shaping rrc rect brick rc T

16 Spectrum RECT Brick RRC

17 Eye Diagram BPSK with Raised Csine Divide the mdulatr utput int * number f samples per symbl and then plt them n tp f n anther Distrtin f Peak Optimum Sampling Pint Sensitivity t Timing Errr Nise Margin Distrtin f zer crssing

18 Matched Filter Receiver BPSK with Raised Csine The intersymbl inteference (ISI) frm a raised csine matched filter reduces the nise margin r(t) hrc(t) Nise Margin

19 Eye Diagram BPSK with Rt Raised Csine Optimum Sampling Pint Distrtin f Peak Sensitivity t Timing Errr Nise Margin Distrtin f zer crssing

20 Matched Filter Receiver BPSK with Rt Raised Csine Rt raised csine intrduces ISI which is remved at the receiver by using the same filter r(t) hrrc(t) Nise Margin

21 Eye Diagram fr 4PAM w/rc

22 Cnstellatin fr QPSK w/rc

23 Spectrum f QPSK w/rc fc = 5 Hz; Spike at 0Hz = 4 * 5Hz The width wuld make it difficult t determine the exact center frequency Raise signal t the 4 th pwer

24 FSK Cherent Receiver %FSK Receiver clear Bits_per_Symbl = 1; %Bits per symbl N = 5000; %Number f Symbls hk = 1*nes(N,1); %Deviatin samples_per_symbl = 10; %samples per symbl mdulatin = "LREC"; %Pulse shape L = 1; %Pulse length Tb = 1; %Bit duratin fc = 1/Tb; %Center frequency SNR = 0:1; %SNR in db %SNR = E/sigma^ snr = 10.^(SNR/10); sigma = (./(snr)).^.5; %Generate Bits bits = Bit_Stream(N*Bits_per_Symbl); %Generate signal levels [Ik, Aq] = Mapper(Bits_per_Symbl, bits, 'PAM', 0); %Ik = 1-*rund(rand(N,1)); %Calculate the phase [phi,t] = CPM_Phase(Ik, hk, L, samples_per_symbl, mdulatin); sl = cs(phi + *pi*fc*t); fr kk = 1:length(SNR) [gv1,gv] = gngauss(n*samples_per_symbl,0,sigma(kk)); rl = sl + gv1; r11 = rl.*cs(*pi*(fc+1/)*t); r = rl.*cs(*pi*(fc-1/)*t); fr kkk=1:n r111(kkk) = (sum(r11(((kkk-1)*samples_per_symbl+1): (kkk*samples_per_symbl)))/samples_per_symbl); r(kkk) = (sum(r(((kkk-1)*samples_per_symbl+1): (kkk*samples_per_symbl)))/samples_per_symbl); end bits_est = r111'>r'; Pe(kk) = sum(duble(bits ~= bits_est))/n; Pe_thery(kk) =Qfunc((snr(kk)).^.5); end semilgy(snr,pe,'*') hld n semilgy(snr,pe_thery) axis([min(snr) max(snr) 10^-8 10^0]); grid n xlabel('ebavg/n [db]'); ylabel('pe'); hld ff

25 FSK Cherent Receiver Nt right yet

26 Nn-Cherent Detectin s(t )= E T cs[π f t+ π m Δ f t +ϕ ]+n (t) c m cs(πf c t) What if we d nt knw the phase? r(t) X sin(πf c t) X X cs(πf c t +π ft) Select Largest r r r mc= E s ms= E s m= r mc csϕ sin ϕ +r ms +n m mc +n m mc sin(πf c t +π ft) X

27 Nn-Cherent Detectin s(t )= E T cs[π f t+ π m Δ f t +ϕ ]+n (t) c m What if we d nt knw the phase? Nn-cherent detectin f FSK envelp detectin P P M b M 1 = ( 1) n=1 n+1( M 1 ) 1 e nke / N (n+1) b n n+1 = k 1 k 1 P M

28 Signaling Schemes with Memry Maximum Likelihd Sequence Detectr (MLSD) K T 0 s r (t) s(t) Optimal Detectin Rule (ŝ (1),ŝ ( ),..., ŝ K dt= k=1 ( K ) ) = k T s (k 1)T (s (1) s r(t) s(t ),s () argmin,..., s dt (K ) ) Υ K k=1 Trellis r (k) s (k ) dt (ŝ (1),ŝ ( ),..., ŝ ( K ) ) = (s (1),s () argmin,..., s K (K ) ) Υ k=1 D(r (k),s (k ) )

29 Trellis fr NRZI Signal Assume we start with state s. 0/ (E b ) 0/ (E b ) 0/ (E b ) s s 1 1/ (E b) 1/ ( E b) 1/ (E b) 1/ ( E b) 1/ (E b) 0/ (E b) 0/ (E b) t = T t = T t = 3T D D 0(0,0)= ( r 1+ E b) + ( r + E b) 0(1,1)= ( r 1 E b) + ( r E b) D (0,1)= 1 ( r 1+ E b) + ( r E b) D (1,0)= 1 ( r 1 E b) + ( r + E b) There are tw paths t arrive at state s at t = T We can calculate the Euclidean distance using the utput f the demdulatr and discard the larger distance fr state s There are als tw paths t arrive at state s 1 at t = T We can calculate the Euclidean distance using the utput f the demdulatr and discard the larger distance fr state s 1 This means we nly need t keep tw paths

30 Trellis fr NRZI Signal Assume we start with state s. 0/ (E b ) 0/ (E b ) 0/ (E b ) s s 1 1/ (E b) 1/ ( E b) 1/ (E b) 1/ ( E b) 1/ (E b) 0/ (E b) 0/ (E b) t = T t = T t = 3T D D (0,0,0)= D (0,0)+ 0 0 ( r + E 3 b) (0,1,1)= D (0,1)+ 0 1 ( r + E 3 b) D (0,0,1)= D (0,0)+ 1 0 ( r E 3 b) D (0,1,0)= D (0,1)+ 1 1 ( r E 3 b) Nw cnsider t = 3T