fund. of digital communications ch. 3: digital modulation · graz university of technology al...
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GRAZ UNIVERSITY OF TECHNOLOGY
al Processing and Speech Communications Lab
Fund. of Digital CommunicationsCh. 3: Digital Modulation
Klaus Witrisal
Signal Processing and Speech Communication Laboratory
www.spsc.tugraz.at
Graz University of Technology
November 26, 2015
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 1/39
GRAZ UNIVERSITY OF TECHNOLOGY
al Processing and Speech Communications Lab
Outline
� 3-1 Pulse Amplitude Modulation
� Baseband and Bandpass Signals
� One-, Two-, and Multidimensional Signals
� QAM and Complex Equivalent Baseband Signals
� 3-2 Pulse Shaping and ISI-free Transmission
� Signal Spectrum
� Nyquist Pulse Shaping
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 2/39
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References and Figures
� Figures refer to Chapter 7 of
� J. G. Proakis and M. Salehi, “CommunicationSystem Engineering,” 2nd Ed., Prentice Hall, 2002
� J. G. Proakis and M. Salehi, “Grundlagen derKommunikationstechnik,” 2. Aufl., Pearson, 2004 (inGerman)
� References to figures denoted as [7.x]
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 3/39
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3-1 Pulse Amplitude Modulation(PAM)
� In practice: TX signal is a stream of symbols
s(t) =
∞∑i=−∞
s[i](t− iT )
where s[i](t) ∈ {sm(t)}Mm=1 represents symbol s[i], taken
from an M -ary alphabet {sm(t)}� Basic assumption:
� Consecutive symbols do not interfere
� Thus we can concentrate on one single symbol
� Symbol index i is dropped “without loss of
generality”; the TX signal is s(t) ∈ {sm(t)}Mm=1
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 4/39
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Pulse Amplitude Modulation(PAM)
� Transmission of information through modulation ofsignal amplitude
� Signal shape gT (t) is tailored to channel
� Baseband signals – for baseband channels [7.4]
� Binary antipodal modulation;selects amplitude of a pulse waveform gT (t)
“1” =̂ A for s1(t) = AgT (t)
“0” =̂ −A for s2(t) = −AgT (t)� Bit rate Rb, bit interval Tb (= symbol interval)
Rb =1
TbFund. of Digital CommunicationsCh. 3: Digital Modulation – p. 5/39
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PAM, Baseband (cont’d)
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 6/39
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al Processing and Speech Communications Lab
PAM, Baseband (cont’d)
� M -ary PAM [7.5]
� Usually: M = 2k for integer k; k ... nb. bits/symbol
� Symbol interval: T = k/Rb = kTb [7.6]
� (Set of) M signal waveforms [7.7]
sm(t) = AmgT (t) , for m ∈ {1, 2, ...,M}, 0 ≤ t ≤ T
� Pulse shape gT (t) determines signal spectrum [7.9]
� Energy (can vary for m ∈ {1, 2, ...,M})
Em =
∫ T
0
s2m(t)dt = A2m
∫ T
0
g2T (t)dt = A2mEg
Eg ... energy of pulse gT (t)Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 7/39
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PAM, Baseband (cont’d)
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 8/39
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PAM, Baseband (cont’d)
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 9/39
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PAM, Bandpass (Passband)
� Bandpass signals – for bandpass channels
� Carrier modulation [7.8]
� Multiplication of sm(t) by carrier cos(2πfct)
um(t) = sm(t) cos(2πfct) = AmgT (t) cos(2πfct),
for m ∈ {1, 2, ...,M},fc ... carrier frequency (center frequency)
� in frequency domain [7.9]:
Um(f) =Am
2[GT (f − fc) +GT (f + fc)]
� DSB-SC-AM (Dual sideband, suppressed carr. AM)
� Channel bandwidth 2W (doubled w.r.t. baseband!)
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 10/39
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PAM, Baseband (cont’d)
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 11/39
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PAM, Baseband (cont’d)
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 12/39
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Definitions of Bandwidth
� Definitions
� Absolute bandwidth
� 3-dB bandwidth
� equivalent bandwidth (BW of block spectrum withequal energy and const. amplitude as at fc)
� first spectral zero (BW of main lobe)
� Time-bandwidth product is constant!
� e.g. first zero of rectangular pulse:
� Interval T vs. first zero Bz of its Fourier transform:
Bz = 1/T , hence TBz = 1
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 13/39
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Geometric Representation inSignal Space
� PAM Signals are one-dimensional
sm(t) = smψ(t)
� baseband:
ψ(t) =1√Eg
gT (t), 0 ≤ t ≤ T
sm =√
EgAm, m ∈ {1, 2, ...,M}� bandpass:
ψ(t) =
√2
Eg gT (t) cos 2πfct
sm =√
Eg/2Am, m ∈ {1, 2, ...,M}
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 14/39
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Geometric Representation(cont’d)
� Euclidean distance
dmn =√
|sm − sn|2
� Energy of PAM signals (baseband)
Em = s2m = EgA2m, m ∈ {1, 2, ...M}
� e.g.: symmetric PAM [7.11]
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 15/39
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Two- (and Multidimensional)Signals
� Simultaneous PAM of two (or more) basis functions
� yields additional points in N -dim. signal space;each representing a signal waveform
� orthogonal signals
� M -ary symbols are represented by N =Morthogonal waveforms
� (see [7.12]–[7.14] for M = 2)
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 16/39
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Two-dimensional Signals
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 17/39
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Two-dimensional Signals
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 18/39
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Two-dimensional Signals
� bi-orthogonal signals [7.15]
� binary antipodal PAM of the basis functions
� M = 4-ary signals with equal energies
� add signal vectors with inverted polarities:
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 19/39
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Two-dimensional Signals
� M = 8-ary signals with equal energies
� M = 8-ary signals with (two) different energies
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 20/39
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Two-dimensional BandpassSignals–QAM
� important example for 2D-signals: (digital) QAM
� PAM of the orthogonal carriers
um(t) = AmcgT (t) cos 2πfct− AmsgT (t) sin 2πfct
� in geometric representation:
um(t) = sm1ψ1(t) + sm2ψ2(t), with
ψ1(t) =√
2/Eg gT (t) cos 2πfctψ2(t) = −
√2/Eg gT (t) sin 2πfct
sm = [sm1, sm2]T = [
√EsAmc,
√EsAms]
T
� QAM signals have a complex-valued equivalentbaseband representation; no BW loss! (see Chapter 2)
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 21/39
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Two-dimensional QAM Signals
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 22/39
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Two-dimensional BandpassSignals–QAM (cont’d)
� Functional block diagram of a (digital) QAM modulator
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 23/39
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Multidimensional Signals
� Two-dimensional case: M = 2k signals have beenconstructed in 2D
� Multidimensional case: (OPAM)
� construct N orthogonal signals
� define signal points in these dimensions
sμ(t) =
N∑n=1
am,ngn(t)
{gn(t)}Nn=1 . . . N orthogonal waveforms (basis)
{am,n ∈ R(or C)} . . . PAM (or QAM) symbols (M -ary)
for n-th waveform; m ∈ 1, 2, . . . ,M
{sμ(t)}MNμ=1 . . . MN -ary set of waveforms
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 25/39
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Multidimensional Signals
� OFDM: parallel transmission to enlarge symbol durationagainst inter-symbol-interference (ISI)
gn(t) =1√Tej2πnt/Tw(t) . . . n-th subcarrier at f = n/T
sμ(t) =1√T
N/2−1∑n=−N/2
am,nej2πnt/Tw(t)
{gn(t)} . . . orthogonal subcarriers (Fourier basis)
w(t) . . . window function (e.g. rectangular)
{am,n ∈ C} . . . QAM symbols (e.g. QPSK, 16/64-QAM)
� the symbols sμ(t) are MN -ary (N subcarriers)
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 26/39
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Multidimensional Signals
� Spread Spectrum/CDMA: few points in N � 1dimensional space (large TB-product)
� divide T into N chip intervals Tc = T/N
� modulate chip waveform with “spreading code” {cn}
gT (t) =
N−1∑n=0
cngc(t− nTc); sm(t) = amgT (t) (PAM/QAM)
� chip wavef. gc(t) has N -fold bandwidth ∝ 1/Tc = N/T
� gT (t) is a broadband pulse of duration T , BW N/T ;i.e. it has N dimensions:
� N orthogonal sequences {cn} can be found formultiple access (CDMA); enhanced robustness
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 27/39
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Optimum Demodulation(preview)
� Intuitive introduction to the demodulator using thesignal-space concept
� a preview to Section 5-1
� complete treatment requires theory of randomprocesses
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 28/39
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Correlation-type demodulator
� Channel: Additive white Gaussian noise (AWGN) isadded
r(t) = sm(t) + n(t)
� transmitted signal {sm(t)},m = 1, 2, ...M is representedby N basis functions {ψk(t)}, k = 1, 2, ...N
� received signal r(t) is projected onto these basisfunctions {ψk(t)}∫ T
0
r(t)ψk(t)dt =
∫ T
0
[sm(t) + n(t)]ψk(t) dt
rk = smk + nk, k = 1, 2, ..., N
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 29/39
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Correlation demod. (cont’d)
[Proakis 2002]
� Output vector in signal space: r = sm + n
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 30/39
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Correlation demod. (cont’d)
� Received signal
r(t) =
N∑k=1
smkψk(t) +
N∑k=1
nkψk(t) + n′(t)
=
N∑k=1
rkψk(t) + n′(t)
� Correlator outputs r = [r1, r2, ...rN ]T are sufficientstatistik for the decision
� i.e.: there is no additional info in n′(t)
� n′(t) is part of n(t) that is not representable by{ψk(t)}
� Interpretation of r: noise cloud in signal spaceFund. of Digital CommunicationsCh. 3: Digital Modulation – p. 31/39
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3-2 Nyquist Pulse-Shaping
Filtering and pulse-shaping
� at transmitter:
� pulse-shaping to reduce signal bandwith
� at receiver
� filter out noise and interferences
→ hence filtering is applied at both sides
� Example: Low-pass (RX) filter: introducesinter-symbol interference (ISI)
� Objective is ISI-free transmission
� Achieved by Nyquist filtering (e.g. root-raised-cosfilter)
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 32/39
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Nyquist Pulse-Shaping (cont’d)
5 10 15 20
TX signal with rectangular pulse
5 10 15 20
RX signal corrupted by noise
5 10 15 20
RX signal after lowpass filter
time0 0.5 1 1.5 2
-1.5
-1
-0.5
0
0.5
1
1.5
time
RX signal after lowpass filter; eye-diagram
� Rectangular pulse at transmitter; noise added onchannel; lowpass filter at receiver for noise reduction
� Eye-diagram (right) shows RX signal qualityFund. of Digital CommunicationsCh. 3: Digital Modulation – p. 33/39
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Nyquist Pulse-Shaping (cont’d)
5 -4 -3 -2 -1 0 1 2 3 4 5
0
5
1
rectangular pulse used at TX
0 1 2 3 4 50
1
2 impulse response of lowpass filter used at RX
5 -4 -3 -2 -1 0 1 2 3 4 5
0
5
1 equivalent system impulse response
time
inter-symbolinterference (ISI)
0 0.5 1 1.5 2-1.5
-1
-0.5
0
0.5
1
1.5
time
eye-diagram without noise; LP-filtered rect. pulse
ISI reduceseye-opening
eye-opening
� Construction of the eye-diagram
� Lowpass filter introduces inter-symbol-interference(ISI)
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 34/39
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Nyquist Pulse-Shaping (cont’d)
� PAM signal after receiver filter
y(t) =
∞∑i=−∞
a[i]he(t− iT )
a[i] ∈ {am}Mm=1 ... PAM (or QAM) of symbol ii ... symbol (= time) indexhe(t) = gT (t) ∗ hc(t) ∗ h(t) ... cascade of TX pulse,channel IR, and RX filter
� Condition for ISI-free transmission
he(kT + τ) =
{C for k = 0 (constant)
0 for k = 0
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 35/39
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Nyquist Pulse-Shaping (cont’d)
Transmission at minimum bandwidth BN (Nyquist BW)
� Assume: sampling frequency equals symbol rate
fs = 1/T
sampled signal can represent signals up to
BN = fs/2 = 1/(2T )
� Consider a rectangular frequency response for He(f)
He(f) = rect(f,BN )F←→ he(t) =
1
Tsinc(t/T )
� fulfills condition for ISI-free transmission
� but cannot be realized (infinite extent; not causal)
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 36/39
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Nyquist Pulse-Shaping (cont’d)
2 -1.5 -1 -0.5 0 0.5 1 1.5 2frequency; normalized to Nyquist bandwidth B
N= 1/(2T)
r = 0r = 0.5r = 1
-5 -4 -3 -2 -1 0 1 2 3 4 5-0.5
0
0.5
1
sinc
-5 -4 -3 -2 -1 0 1 2 3 4 5-1
0
1
2
FT
ofco
s-pu
lse
-5 -4 -3 -2 -1 0 1 2 3 4 5-1
0
1
2
time; normalized to symbol period Tpr
oduc
t
r = 0r = 0.5r = 1
� Cosine roll-off: allow bandwidth extension by BN (1 + r)
� frequency: convolve rectangular with cos-pulse
� time: multiply sinc (upper) with Fourier transform ofcos-pulse (center); product
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 37/39
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Nyquist Pulse-Shaping (cont’d)
5 -4 -3 -2 -1 0 1 2 3 4 5time; normalized to symbol period T
r = 0r = 0.5r = 1
0 1 2 3 4 5 6 7 8 9 10-1.5
-1
-0.5
0
0.5
1
1.5
2
time; normalized to symbol period T
r = 0r = 0.5r = 1
� left-hand figure: equivalent system impulse responseshe(t) with cos-roll-off Nyquist filtering
� right-hand figure: received data sequences
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 38/39
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Nyquist Pulse-Shaping (cont’d)
1 2 3 4 5 6 7 8 9 10time; normalized to symbol period T
r = 0r = 0.5r = 1
0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
time; normalized to symbol period T
eye diagram (right-hand figure):
� (long) sequence of received, filtered data symbols
� superimposed in diagram over two symbol intervals
Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 39/39