Lecture 25,26,27: Digital communication

Aliazam Abbasfar

OutlineDigital communication

Baseband systems

Optimum receiver

Digital communication Transfer of digital messages from source to destination reliably

Sometimes called signaling

Digital message Sequence of symbols (digits) Symbols are chosen from an alphabet (M symbols)

Binary symbols : bits : alphabet {0,1}

Data rate Symbol/Baud/Signaling rate (symbols per second) (r) bit rate (bits per second) (rb)

Reliability is measured by probability of error Symbol/Bit error rate (BER) Packet error rate (PER)

BER targets Voice : 10-5 Data : 10-6 Video : 10-7

Digital systems

Digital source Digitized voice/images Data

Source encoder and decoder Data compression

Encryption

Channel encoder and decoder Error detection/correction Example : repetition code

Modulation/demodulation Digital Baseband/bandpass

SourcedecoderChannelSourc

eencoder

messagex(t) y(t)

Digital

Source

Pulse Amplitude Modulation (PAM) A sequence of pulses with varying amplitudes

y(t) = ak p(t- kT) + n(t) T : symbol time

Inter-symbol interference (ISI) y(kT) = ak p(0) + am p(mT) + n(kT) p(0) = 1; p(mT) = 0; for all m<>0

Rectangular pulse Sinc pulse

Symbols are mapped into pulse amplitudes (ak) M-PAM has M levels unipolar 2-PAM levels: {0, A} Alphabet {0,1} bipolar 2-PAM levels: {-A, A} Alphabet {0,1} bipolar 2-PAM levels: {-A, A} Alphabet {0,1,2} bipolar 3-PAM levels: {-A, 0, A} Alphabet {0,1,2,3} bipolar 4-PAM levels: {-3A, -A, A, 3A}

Data rate Symbol rate : r= 1/T Bit rate : rb = log2(M)/T

Example: binary signaling with rectangular pulse Bipolar 2-PAM RZ and NRZ

T

y(t)

Performance with noise AWGN with power 2

E[n2(t)] = 2

Sampled signal distribution No ISI and p(0)=1 z = y(kT) = ak + n(kT)

Symbol detection Compare with thresholds Slicer or A/D

Probability of error Pe = Pi Pe|i Pe|i : probability of error for ith symbol

Unipolar binary : Pe = Q(A/2) Bipolar binary : Pe = Q(A/) Bipolar M-PAM : Pe = 2(1-1/M) Q(A/)

= 2(1-1/M) Q(Amax/(M-1))

Analog vs Digital repeaterDigital (regenerative) repeater detects the

symbols and regenerate them againPem = 1-(1-Pe)m m Pe Accumulate errors

Analog repeater amplifies signal + noise

Accumulate noisem

2 = m 2 Pem = 2(1-1/M) Q(A/m)

Hybrid repeater : A digital repeater after every m analog repeaterPemxk = k Pem

Pulse detector x(t) = {0 or p(t)} + n(t)

p(t) is time-limited pulse p(t) = 0; t<0 or t> T

AWGN with power spectral density of N0/2 Rn() = N0/2 () Gn(f) = N0/2

Filter x(t) with H(f) and sample at time T Signal amplitude : Noise power :

Maximize A/2 Matched filter

H(f) = P(f)* e-j2fT h(t) = p(T-t)

Amax = Ep = EpN0/2

Probability of error

-

fT j2π dfP(f)H(f)e A

-

202 dfH(f)2

N

0

pmaxe 2N

EQ )

2

AQ( p

Correlator Matched filter output is the correlation of the signal and the pulse

Detecting one out of two different pulses y(t) = {p0(t) or p1(t)} + n(t) y(t)-p0(t) = {0 or p1(t)-p0(t)} + n(t) Correlate y(t) with p1(t)-p0(t) Decision level : corr( [p1(t)+p0(t)]/2, p(t) )

Error probability

Correlator receiver Correlate y(t) with all pi(t) Detected symbol based on the output of the correlators

If we have a series of pulses, each pulse is detected by correlation

y(t) = ak p(t- kT) + n(t) Correlate y(t) with p(t-kT) ak

T

0 p E dt p(t) x(t) z(T)

0

p0-p1maxe 2N

EQ )

2

AQ( p

ISI free matched filteringISI free : Matched filter output due to other

pulses = 0 Shifted versions of the pulse are orthogonalcombT(Rp())= Ep() rep1/T(|P(f)|2) = Cte

Folded spectrum is flat

Band-limited pulsesSinc pulseRoot raised cosine

δ[k] E dt kT)-p(t p(t) p

Power spectrumx(t) = ak p(t- kT) = [ ak (t- kT)] p(t)

Gx(f) = Ga(f) |P(f)|2

Bipolar PAM : Ga(f) = E[ak

2]/T

Gx(f) = E[ak2]/T |P(f)|2

Px = E[ak2] Ep/T = Es/T

Bandpass modulationsAmplitude shift keying (ASK)

x(t) = ak p(t- kT)

p(t) = cos(wct)

ak = 0 or A

Coherent detectionDown convert unipolar 2-PAM

Envelope detectorSimilar to AM (a strong carrier)

0

b

0

pe N

EQ

2N

EQ p

0

be 2N

Eexp

2

1 p

PSKPhase shift keying (PSK)

x(t) = p(t- kT) p(t) = cos(wct + k)

BPSK Modulated bipolar 2-PAM x(t) = ak p(t- kT)

ak = -A or A p(t) = cos(wct)

QPSK x(t) = ak p1(t- kT) + bk p2(t- kT)

ak = -A or A p1(t) = cos(wct) p2(t) = sin(wct)

0

be N

2EQ p

QAMQuadrature amplitude modulation(QAM)

Amplitude and phase modulations

x(t) = ak p1(t- kT) + bk p2(t- kT)

p1(t) = cos(wct)

p2(t) = sin(wct)

2 independent PAM

FSKFrequency shift keying (FSK)

Two different frequencies fc1 and fc2

x(t) = {A cos(c1t) or A cos(c2t)}

Coherent detectionEp1-p2 = 2K Eb

K=1 when orthogonal pulses

Non-coherent detectionUse frequency detectors

0

b

0

p2-p1e N

EK Q

2N

EQ p

0

be 2N

Eexp

2

1 p

ReadingCarlson Ch. 11.1, 11.2, 11.3