EECS 455 Solution to Problem Set 3

Transcription

1 EECS 455 Solution to Problem Set 3. (a) Is it possible to have reliably communication with a data rate of.5mbps using power P 3 Watts with a bandwidth of W MHz and a noise power spectral density of N 8 Watts/Hz? Soltuion: According to Shannon s theorem reliable communication is possible if R W log P N W log 3 6 log 3 6 log 4 6 Mbps Thus reliable communication is possible upto the rate of Mbps. So.5Mbps is not possible. (b) A communication system uses BPSK in a (null-to-null) bandwidth of MHz with power P 5 Watts in the presence of white Gaussian noise with two-side power spectral density N 8. An error probability of Q is desired. What data rate is possible? Soltuion: The error probability for BPSK is So the signal to noise ratio must satisfy P e Q E N PT N P R N E N P R N Thus there is enough power to provide reliable communication at a data rate of 5 kbps. In addition, at that data rate the (null-to-null) bandwidth is MHz so there is also enough bandwidth.

2 (c) For the same parameters as part (b) except the error probability requirement was Q what data rate would be possible? Solution: Since the energy requirement has been reduced by a factor of the data rate could potentially increase by a factor of. However, the bandwidth would not allow it. Thus the maximum data rate is still 5kbps.. (a) Simulate a communication system that uses rectangular pulses of amplitude to transmit data. The channel is an additive white Gaussian noise channel. The receiver uses a matched filter followed by a sampler and a decision device. For E b N 8 determine (and plot) the simulated bit error probability. Compare with the theoretical bit error probability. Solution The Matlab code is attached. Simulation P e,b 3 Theory E b /N (db) (b) Repeat the above experiment except with a RC filter with impulse response h t αe αt u t. Use the value α 56 T. Compare the result to part (a).

3 RC Filter P e,b 3 Matched Filter E b /N (db) % This program simulates a rectangular pulse shape signal % in the presence of noise clear; % Simulation Parameters % Tb=; % Bit duration of data Nbcount=input( Number of bit errors counted = ); Nbp=input( Number of bits in a packet= ); Nsb=input( Number of samples per bit= ); 3

4 fmax=nsb/(*tb) N=*Nbp*Nsb; N=N/; fs=*fmax; df=*fmax/n dt=./(df.*n) t=(:n)*dt-dt; Tmax=N*dt f=(:n)*df-df; f=f-n/*df; rb=/tb; fc= % Simulation bandwidth % Simulation samples % Half the number of samples % Sampling Frequency % Frequency spacing % Time spacing % Time samples % Simulation time % Frequency samples % Data rate % Center Frequency % Generate Pulse Shape % xf(:n)=ones(,n)*e-8; xf(:n/)=xf(:n/)+tb*sinc(f(:n/)*tb).*exp(-j*pi*f(:n/)*tb); xf(n/+:n)=conj(fliplr(xf(:n/))); xt=real(ifft(xf)./dt); Peb=zeros(,9); for ml=:9 EbNdB(ml)=ml- EbN=ˆ(EbNdB(ml)/); P=; Eb=P*Tb; N=Eb/EbN; % Received Power % Received Energy % Noise power Nberrors(ml)=; Np=; while Nberrors(ml)<Nbcount b=sign(rand(,nbp)-.5); zf=zeros(,n); for k=:nbp zf=zf+b(k)*exp(-j**pi*f*(k-)*tb); end xf=xf.*zf; 4

5 xt=real(ifft(xf)./dt); % Generate Noise % sigma=sqrt(n*fmax); nt=sigma*randn(,n); % Add signal to noise % rt=xt+nt; % Filter the signal and noise % rf=fft(rt)*dt; yf=rf.*xf; yt=real(ifft(yf)./dt); % Sample the filter output % sampling_time=(:nbp)*nsb+; bhat=sign(yt(sampling_time)); Np=Np+; Nberrors(ml)=Nberrors(ml)+sum(sign(abs(b-bhat))); if (mod(np,) == ) [Np, Nberrors(ml)] end; end Peb(ml)=Nberrors(ml)/(Np*Nbp) petheory(ml)=q(sqrt(*ebn)) 5

6 end hold off semilogy(ebndb,peb) hold on semilogy(ebndb,petheory, r ) 6

7 % This program simulates a rectangular pulse shape signal % in the presence of noise with an RC receiver filter clear; % Simulation Parameters % Tb=; % Bit duration of data alpha=input( RC Filter Parameter= ); Nbcount=input( Number of bit errors counted = ); Nbp=input( Number of bits in a packet= ); Nsb=input( Number of samples per bit= ); fmax=nsb/(*tb) N=*Nbp*Nsb; N=N/; fs=*fmax; df=*fmax/n dt=./(df.*n) t=(:n)*dt-dt; Tmax=N*dt f=(:n)*df-df; f=f-n/*df; rb=/tb; fc= % Simulation bandwidth % Simulation samples % Half the number of samples % Sampling Frequency % Frequency spacing % Time spacing % Time samples % Simulation time % Frequency samples % Data rate % Center Frequency % Generate Pulse Shape % xf(:n)=ones(,n)*e-8; xf(:n/)=xf(:n/)+tb*sinc(f(:n/)*tb).*exp(-j*pi*f(:n/)*tb); xf(n/+:n)=conj(fliplr(xf(:n/))); xt=real(ifft(xf)./dt); Peb=zeros(,9); for ml=:9 EbNdB(ml)=ml- EbN=ˆ(EbNdB(ml)/); 7

8 P=; Eb=P*Tb; N=Eb/EbN; % Received Power % Received Energy % Noise power Nberrors(ml)=; Np=; while Nberrors(ml)<Nbcount b=sign(rand(,nbp)-.5); zf=zeros(,n); for k=:nbp zf=zf+b(k)*exp(-j**pi*f*(k-)*tb); end xf=xf.*zf; xt=real(ifft(xf)./dt); % Generate Noise % sigma=sqrt(n*fmax); nt=sigma*randn(,n); % Add signal to noise % rt=xt+nt; % Filter the signal and noise % rf=fft(rt)*dt; yf=rf.*x3f; yt=real(ifft(yf)./dt); x3f(:n/)=alpha./(alpha+j**pi*f(:n/)); x3f(n/+:n)=conj(fliplr(x3f(:n/))); 8

9 % Sample the filter output % sampling_time=(:nbp)*nsb+; bhat=sign(yt(sampling_time)); Np=Np+; Nberrors(ml)=Nberrors(ml)+sum(sign(abs(b(9:Nbp)-bhat(9:Nbp)))); if (mod(np,) == ) [Np, Nberrors(ml)] end; end %loop for counting errors Peb(ml)=Nberrors(ml)/(Np*(Nbp-8)) end %loop for different SNR hold off semilogy(ebndb,peb) hold on 3. Simulate the communication system shown below. The system uses square-root raised cosine pulses with β 35 data amplitude to transmit data on the cosine and sine branch. Show the data waveform, the transmitted signal in the time domain s t and frequency domain S f. Plot the output of the mixers at the receiver in the frequency domain (show the double frequency terms). Plot the waveform at the output of the receiver filters for a sequence of 8 bits. Make an eye diagram for one of the outputs (use 5 bits for the eye diagram). Assume T and f c 4 for the simulation. Use a maximum frequency for the simulation of 6Hz. Solution: 9

10 lt lt sin sin b c t N l b c lδ t b s t N l b s lδ t H T f H T f cos π f c t s t cos π f c t H R f H R f ˆb c t ˆb s t π f c t π f c t.5 Pulse Shape.5 h(t) time.5 Spectrum H(f) frequency

11 Data Waveform x c (t) time Data Waveform x s (t) time Transmitted Signal s(t) time S(f) frequency

12 3 Output of Mixer at Receiver z c (t) time Z c (f) frequency Recevier Filter Output.5.5 y(t) time

13 Eye Diagram.5.5 y(t) time % This program simulates a raised cosine pulse shape signal % filtered by an raised cosine filter. clear; % Simulation Parameters % Tb=; % Bit duration of data %EbNdB=input( E_b/N_ (db)= ); %EbN=ˆ(EbNdB/); %P=; % Received Power %Eb=P*Tb; % Received Energy %N=Eb/EbN; % Noise power beta=input( Raised Cosine Filter Parameter= ); Nb=input( Number of bits simulated= ); fmax=input( Maximum frequency of simulation= ); N=input( Number of samples = ); N=N/; % Half the number of samples 3

14 fs=*fmax; df=*fmax/n dt=./(df.*n) Nsb=Tb/dt t=(:n)*dt-dt; Tmax=N*dt f=(:n)*df-df; f=f-n/*df; rb=/tb; fc=4 % Sampling Frequency % Frequency spacing % Time spacing % Time samples % Simulation time % Frequency samples % Data rate % Center Frequency % Generate Signals % N=round((-beta)/(*Tb)/df) N=round((+beta)/(*Tb)/df) hf(:n)=zeros(,n); hf(:n)=sqrt(tb*ones(,n)); hf(n:n)=sqrt(.5*tb*(+cos(pi*tb/beta*(f(n:n)-(-beta)/(*tb)))));; hf(n/+:n)=conj(fliplr(hf(:n/))); ht=real(ifft(hf)./dt); figure() subplot(,,), plot(t(:8*nsb)/tb,ht(:8*nsb)); grid axis([ 6*Tb -.5.5]) xlabel( time ); ylabel( h(t) ); title( Pulse Shape ) subplot(,,), plot(f(:n),fftshift(abs(hf(:n)))); axis([-fmax fmax.5]) grid xlabel( frequency ); ylabel( H(f) ); title( Spectrum ) % Generate data for cosine branch % bc=sign(rand(,nb)-.5); 4

15 bcf=zeros(,n); for k=:nb bcf=bcf+bc(k)*exp(-j**pi*f*k*tb); end xcf=hf.*bcf; xct=real(ifft(xcf)./dt); figure() subplot(,,), plot(t/tb,xct); grid axis([ 6 - ]) xlabel( time ); ylabel( x_c(t) ); title( Data Waveform ) % Generate data for sine branch % bs=sign(rand(,nb)-.5); bsf=zeros(,n); for k=:nb bsf=bsf+bs(k)*exp(-j**pi*f*k*tb); end xsf=hf.*bsf; xst=real(ifft(xsf)./dt); subplot(,,), plot(t/tb,xst); grid axis([ 6 - ]) xlabel( time ); ylabel( x_s(t) ); title( Data Waveform ) % Mix the signal to a carrier % st=xct.*cos(*pi*fc*t)-xst.*sin(*pi*fc*t); 5

16 sf=fft(st)*dt; figure(3) subplot(,,), plot(t/tb,st) xlabel( time ) ylabel( s(t) ) grid axis([ 6 - ]) subplot(,,), plot(f,*log(fftshift(abs(sf)))) grid xlabel( frequency ) ylabel( S(f) ) axis([ ]) % Mix the signal back to baseband % zct=*st.*cos(*pi*fc*t); zcf=fft(zct)*dt; zst=-*st.*sin(*pi*fc*t); zsf=fft(zst)*dt; figure(4) hold off subplot(,,), plot(t/tb,zct) axis([ 8-3 3]) xlabel( time ) ylabel( z_c(t) ) subplot(,,), plot(f,*log(fftshift(abs(zsf)))) grid axis([- -4 ]) xlabel( frequency ) ylabel( Z_c(f) ) title( Output of Mixer at Receiver ) % Filter the signal % yf=zcf.*hf; 6

17 yt=real(ifft(yf)./dt); figure(5) subplot(,,), plot(t/tb,yt) grid on axis([ 6 - ]) xlabel( time ) ylabel( y(t) ) title( Recevier Filter Output ) hold off figure(6) %subplot(,,), plot(t(:*nsb+)/tb,yt(:*nsb+)) %grid %yscale=*ceil(max(yt/)); %axis([ - ]) for k=:floor(nb/)- plot(t(:*nsb+)/tb,yt((k*)*nsb+:(k*+)*nsb+)) hold on end xlabel( time ); ylabel( y(t) ); title( Eye Diagram ) grid on hold off 7