ELEG 635 Digital Communication Theory

Lecture 9

10252012

Agenda● Optimal receiver for PSK

● Effects of fading on BER performance

– Two ray model

● Pulse shaping

– Rectangle

– Raised cosine

– Root raised cosine

– Receivers and pulse shaping

● FSK receiver

– Coherent

– Non-coherent

● CPM receiver

Simulations ● 4PAM % Program to calculate Pe for a PAM signal

N = 10000; %Number of samplesk = 2; %Bits per symboldmin = 1; %Minimum distancemodulation = 'PAM';SNR = 0:25; %SNR in dBM = 2.^k; %M

op = bin2gc(0:(M-1)); %Create Gray Code mapping

bits = Bit_Stream(k.*N);

[Ai, Aq, symbols, symbolsgc] = Mapper_gc(k, bits, modulation, 0);

%Add NoisePe = zeros(size(SNR));Pe_theory=zeros(size(SNR));for kk = 1:length(SNR); tmp = 10.^(SNR(kk)/10); sigma = sqrt(dmin.^2*(M.^2-1)/(6*k*tmp)); [gv1,gv2] = gngauss(N,0,sigma); r = Ai + gv1; s = PAM_Detector(r, M, dmin); Pe(kk) = sum(double(s ~= symbols)); Pe_theory(kk) = 2*(M-1)/M*Qfunc(sqrt(6*k/(M^2-1)*tmp));endsemilogy(SNR,Pe/N,'*')hold onsemilogy(SNR,Pe_theory)axis([min(SNR) max(SNR) 10^-8 10^0]);grid onxlabel('Ebavg/No [dB]');ylabel('Pe');hold off

Optimal Detection for PSK

I

Q

dmin

r=(r1,r

2)=(√(E)+n

1,n

2)

p (r1,r

2)=

1π N

o

e−

(√(E )+r1

)2

+r2

2

No

V =√r1

2

+r2

2

Θ=arctanr

2

r1

pV Θ

(v ,θ)=v

πNo

e

v2

+E−2√ Ecosθ

No

Optimal Detection for PSK

I

Q

dmin

pΘ(θ)=∫

0

∞

pV Θ

(v ,θ)dv=1

2πe

−γssin

2

θ

∫0

∞

v e−

(v−√2 γs

cosθ)2

2 dv

γs=

EN

o

pe=1− ∫

−π/M

π/ M

pΘ(θ)d θ

This can only be solved numerically

Frequency Selective Fading

● Consider a channel with a direct and single reflective fade path that inverts the signal. The difference in the time of arrival of these two paths in τ.

h (t )=1−δ(t−τ)

H ( f )=1−e j2π f τ

Note: when fτ is an integer there is complete cancellation

|H(f)|2

f

Frequency Selective Fading

● Consider 2PSK or 2PAM (These are both BPSK)

Flat Fadings(t )=A(t )sin(2π f

ot+ϕ(t ))

r (t)=∑k

s (t−τk)=∑

k

A(t−τk)sin (2π f

ot+ϕ(t−τ

k)−2π f

oτ

k)

The output of passing a signal through a delay line is

sd(t )=∫

−∞

∞

[ S ( f )e− j2 π f τ ]e j2 π f tdf

If the spectrum is confined to B << 1/τ

sd(t )=∫

−B

B

[S ( f )e− j2 π f τ ]e j2 π f t df =∫−B

B

S ( f )e j2π f tdf =s( t)

Flat Fading

r (t)=A(t)[sin(2π fot+ϕ)∑

k

αkcos(2π f

oτ

k)−cos(2π f

ot+ϕ)∑

k

αksin (2π f

oτ

k)]

r (t)=A(t)[a( t)sin (2π fot+ϕ(t))+b (t)cos(2π f

ot+ϕ(t ))]

r (t)=A(t)M (t )sin(2π fot+ϕ(t )+θ(t ))

M (t)=√a(t )2

+b( t)2

θ(t )=arctana (t )b(t)

Flat Fading

p (x , y)=1

2πσ2 e

−x

2

+ y2

2σ2

p (r ,θ)=1

2πσ2 e

−r

2

2σ2

p (r )=∫0

2π

p(r ,θ)r d θ=r

σ2 e

−r

2

2σ2

Rayleigh Distribution

Flat Fading

p (r )=r

σ2 e

−r

2

2σ2

Rayleigh Distribution

Pfade

=∫0

∞

P (e)P (A)dA=∫0

∞ A

A2e

−A

2

2 A2 1

2π∫

√ A2

T /No

∞

e−u2

/2du

For BPSK

Pfade

=0.5(1−[ SNR(1+SNR)]

.5

) SNR=A

2

T2 N

o

=

Eb

No

BER of BPSK w/ Fading Channel

Pe

SNR [dB]

Base Band Pulse Shaping● To constrain the bandwidth of the signals pulse

shaping is applied PAM, PSK, and QAM modulators

Xh(t)

cos(2πfct)

Ai(t)

X

sin(2πfct)

h(t)Aq(t)

h(t)Ai(t)

Base Band Pulse Shaping

● Several different pulse shapes– Brick wall filter

– Raised cosine

– Root raised cosine

– Rectangle

b = sinc(t/T) -cT < t < cT

b = sinc(t).*cos(pi*alpha*t)./(1-4*alpha^2*t.^2) -cT < t < cT

b = (4/pi*alpha*cos((1+alpha)*pi*t).+(1-alpha)*sinc((1-alpha)*t))./(1-(4*alpha*t).^2) -cT < t < cT

b = 1 0 < t < T = 0 otherwise

Base Band Pulse Shaping

rrc

brick

rect

rc

T

Spectrum

Brick RRC

RECT

Eye Diagram● BPSK with Raised Cosine

– Divide the modulator output into 2 * number of samples per symbol and then plot them on top of on another

Noise Margin

Distortion of zero crossing

Distortion of Peak

Sensitivity toTiming Error

Optimum Sampling Point

Matched Filter Receiver● BPSK with Raised Cosine

– The intersymbol inteference (ISI) from a raised cosine matched filter reduces the noise margin

Noise Margin

hrc(t)r(t)

Eye Diagram● BPSK with Root Raised Cosine

Noise Margin

Distortion of zero crossing

Distortion of Peak

Sensitivity toTiming Error

Optimum Sampling Point

Matched Filter Receiver● BPSK with Root Raised Cosine

– Root raised cosine introduces ISI which is removed at the receiver by using the same filter

Noise Margin

hrrc(t)r(t)

Eye Diagram for 4PAM w/RC

Constellation for QPSK w/RC

Spectrum of QPSK w/RC

● fc = 5 Hz;

The width would make it difficult to determine the exact center frequency

Raise signal to the 4th power

Spike at 20Hz = 4 * 5Hz

FSK Coherent Receiver%FSK ReceiverclearBits_per_Symbol = 1; %Bits per symbolN = 5000; %Number of Symbolshk = 1*ones(N,1); %Deviationsamples_per_symbol = 10; %samples per symbolmodulation = "LREC"; %Pulse shapeL = 1; %Pulse lengthTb = 1; %Bit durationfc = 1/Tb; %Center frequencySNR = 0:12; %SNR in dB

%SNR = 2E/sigma^2snr = 10.^(SNR/10);sigma = (2./(snr)).^.5;

%Generate Bitsbits = Bit_Stream(N*Bits_per_Symbol);

%Generate signal levels[Ik, Aq] = Mapper(Bits_per_Symbol, bits, 'PAM', 0);%Ik = 1-2*round(rand(N,1));

%Calculate the phase[phi,t] = CPM_Phase(Ik, hk, L, samples_per_symbol, modulation);sl = cos(phi + 2*pi*fc*t);for kk = 1:length(SNR) [gv1,gv2] = gngauss(N*samples_per_symbol,0,sigma(kk)); rl = sl + gv1; r11 = rl.*cos(2*pi*(fc+1/2)*t); r22 = rl.*cos(2*pi*(fc-1/2)*t); for kkk=1:N r111(kkk) = (sum(r11(((kkk-1)*samples_per_symbol+1):(kkk*samples_per_symbol)))/samples_per_symbol); r222(kkk) = (sum(r22(((kkk-1)*samples_per_symbol+1):(kkk*samples_per_symbol)))/samples_per_symbol); end bits_est = r111'>r222'; Pe(kk) = sum(double(bits ~= bits_est))/N; Pe_theory(kk) =Qfunc((snr(kk)).^.5);end

semilogy(SNR,Pe,'*')hold onsemilogy(SNR,Pe_theory)axis([min(SNR) max(SNR) 10^-8 10^0]);grid onxlabel('Ebavg/No [dB]');ylabel('Pe');hold off

FSK Coherent Receiver

Not right yet

Non-Coherent Detections(t )=√ 2 E

Tcos [2π f

ct+2πmΔ f t+ϕ

m]+n (t)

What if we do not know the phase?

X

cos(2πfct)

X

X

SelectLargest

r(t)

sin(2πfct)

X

cos(2πfct +2π∆ft)

sin(2πfct +2π∆ft)

rmc

=√Escosϕ

m+n

mc

rms

=√Essinϕ

m+n

mc

rm=√r

mc

2

+rms

2

Non-Coherent Detections(t )=√ 2 E

Tcos [2π f

ct+2πmΔ f t+ϕ

m]+n (t)

What if we do not know the phase?

PM=∑

n=1

M −1

(−1)n+1

(M−1n ) 1

n+1e

−nkEb/N

o(n+1)

Non-coherent detection of FSK – envelop detection

Pb=

2k−1

2k

−1P

M

Signaling Schemes with Memory

● Maximum Likelihood Sequence Detector (MLSD)

∫0

KTs

∣r (t)−s (t)∣2

dt=∑k=1

K

∫(k−1)T

s

k Ts

∣r (t )−s(t )∣2

dt

Optimal Detection Rule

(s(1)

, s( 2)

,...... , s(K )) =

argmin

(s(1)

, s(2)

, ...... , s(K )

)∈Υ∑k=1

K

∣∣r(k)

−s(k )∣∣2

dt

(s(1)

, s( 2)

,...... , s(K )) =

argmin

(s(1)

, s(2)

, ...... , s(K )

)∈Υ∑k=1

K

D(r(k)

, s(k ))

Trellis

Trellis for NRZI Signal● Assume we start with state s

o.

0/−√(Eb)

so

s1

1/√ (Eb

) 1/√ (Eb

)

1/√ (Eb

)

0/−√(Eb)

0/√(Eb)

0/−√(Eb)

0/√(Eb)

1/− √(

E b

)1/− √(

E b

)

t = T t = 2T t = 3T

D0(0,0)=(r 1

+√Eb)

2

+(r 2+√E

b)2

D0(1,1)=(r 1

−√Eb)

2

+(r 2−√E

b)2

● There are two paths to arrive at state so at t = 2T

● We can calculate the Euclidean distance using the output of the demodulator and discard the larger distance for state s

o

● There are also two paths to arrive at state s1 at t =

2T● We can calculate the Euclidean distance using the

output of the demodulator and discard the larger distance for state s

1

● This means we only need to keep two paths

D1(0,1)=(r 1

+√Eb)

2

+(r 2−√E

b)2

D1(1,0)=(r 1

−√Eb )

2

+(r 2+√E

b)2

Trellis for NRZI Signal● Assume we start with state s

o.

D0(0,0,0)=D

0(0,0)+(r 3

+√Eb)

2

D0(0,1,1)=D

1(0,1)+(r 3

+√Eb)

2

● Now consider t = 3T

D1(0,0,1)=D

0(0,0)+(r 3

−√Eb)

2

D1(0,1,0)=D

1(0,1)+(r 3

−√Eb)

2

0/−√(Eb)

so

s1

1/√ (Eb

) 1/√ (Eb

)

1/√ (Eb

)

0/−√(Eb)

0/√(Eb)

0/−√(Eb)

0/√(Eb)

1/− √(

E b

)1/− √(

E b

)

t = T t = 2T t = 3T