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