Fund. of Digital Communications Ch. 3: Digital Modulation · GRAZ UNIVERSITY OF TECHNOLOGY al...

GRAZ UNIVERSITY OF TECHNOLOGY

al Processing and Speech Communications Lab

Fund. of Digital CommunicationsCh. 3: Digital Modulation

Klaus Witrisal

[email protected]

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



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




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]




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




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



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











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!)




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




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}




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]




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)




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:





� M = 8-ary signals with equal energies

� M = 8-ary signals with (two) different energies




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)




Two-dimensional QAM Signals




Two-dimensional BandpassSignals–QAM (cont’d)

� Functional block diagram of a (digital) QAM modulator



al Processing and Speech Communications Lab Fund. of Digital CommunicationsCh. 3: Digital Modulation – p. 24/39



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





� 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)





� 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




Optimum Demodulation(preview)

� Intuitive introduction to the demodulator using thesignal-space concept

� a preview to Section 5-1

� complete treatment requires theory of randomprocesses




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




Correlation demod. (cont’d)

[Proakis 2002]

� Output vector in signal space: r = sm + n




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



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)




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




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)





� 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





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)





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





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





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


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