principles of digital communications [1mm] part 1

Principles of Digital CommunicationsPart 1

Philippe Ciblat

Université Paris-Saclay & Télécom ParisTech

DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w. ISI

Outline

Section 1: What are Digital Communications ?Section 2: A toy exampleSection 3: Baseband and carrier signalsSection 4: Continuous-time propagation channelSection 5: Modulation (TX): from discrete-time tocontinuous-timeSection 6: Demodulation (RX): from continuous-time todiscrete-time− Matched filter + sampler− Equivalent discrete-time channel model

Section 7: Detection Theory− Optimal detector− General performances

Section 8: Application to Gaussian channelSection 9: Application to Frequency-Selective channel− Viterbi Algorithm− Linear equalization− OFDM

Philippe Ciblat Digital Communications 2 / 86


Section 1: What are Digital Communications?



Introduction

Except audio broadcasting (radio), current communicationsystems are digital− 2G, 3G, 4G, 5G, DVBT, Wifi, Free Space Optical Communications− ADSL, Optical fiber communications− MP3, DVD

Channels: copper twisted pair, powerline, wireless, optical fiber,...

Sources: analog (voice) or digital (data)



If analog source, ...

Sampling (no information loss)

Nyquist-Shannon Theorem

Let t 7→ x(t) be a continuous-time signal of bandwidth B. x(t) isperfectly characterized by the sequence {x(nT )}n where T is thesampling period satisfying 1/T ≥ B.

Quantization (information loss)

Example

Let us consider voice signal

Quality Bandwidth Sampling Quantization2G [300Hz, 3400 Hz] 8kHz 8 bitsHifi [20Hz, 20kHz] 44kHz 16 bits



What is digital?

Analog system: d(t) analog source

transmit signal : x(t) = f (d(t))

+ Pros: low complexity− Cons: data transmission, multiple access, performance, limited

information processing

Digital system: dn digital source (composed by 0 and 1)

transmit signal : x(t) = f (dn)



Design parameters

Data rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Db bits/sBandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .B HzError probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Pe

Transmit power (SNR) . . . . . . . . . . . . . . . . . . . . . . . . . . . .P mW or dBmLatency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L

Goal

max Db with min B,Pe,P,L

but− theoretical limits (information theory)− practical constraints (propagation, complexity)

Practical case: depends on Quality of Service (QoS)− 2G/3G/4G (voice): target L with fixed Db and variable Pe

− ADSL (data): max Db with target Pe and fixed B and P



A few systems

System Db B Pe Spectral efficiencyTV 10Mbits/s 8MHz 10−11 1,25 bits/s/Hz

voice 13kbits/s 25kHz 10−2 0,5 bits/s/HzData (ADSL) 500kbits/s 1MHz 10−7 0,5 bits/s/Hz



Transceiver/Receiver structure.

ModulationSource

Destination

Channelcoding

Channeldecoding

Demodulation

channel

propagation

x(t)

y(t)

an

an

dn

dn

.

Question ?How to design

Modulation/demodulation boxesCoding/decoding boxes

... depending on propagation channel



Section 2: A toy example



The “old” optical fiberGoal:

Sending a bit stream an ∈ {0,1} at data rate Db bits/sData an will be sent at time nTb with Tb = 1/Db s

How?x(t) = 0 if an = 0 within [nTb, (n + 1)Tb)⇒ No lightx(t) = A if an = 1 within [nTb, (n + 1)Tb)⇒ Light

.

��

1 0 0 1 1 0 1 00

A

Tb

t.

but

Light has a color(∼ wavelength)

xc(t) = x(t) cos(2πf0t)0 1 2 3 4 5 6 7 8

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t/Tb



Mathematical framework

Each data has a shape. Here, the rectangular functionEach shape is multiplied by an amplitude. Here, either A or 0Each data is shifted at the right time

x(t) =∑

n

sng(t − nTs)

withg(t) shaping filter. Here, g(t) rectangular functionsn symbol sequence. Here sn = Aan

Ts symbol period. Here, Ts = Tb

Finallyxc(t) = x(t) cos(2πf0t)



Degrees of freedom

carrier frequency f0− impact on propagation condition− impact on data rate (see later)

shaping filter g(t)− impact on bandwidth

Sx (f ) ∝ |G(f )|2

with G(f ) Fourier Transform of g(t)− impact on receiver complexity and performance (see later)

symbol sn

− impact on data rate: multi-level− impact on performance (see later)

symbol period Ts

− impact on data rate− impact on bandwidth (through the choice of g(t))



Section 3: Baseband/carrier signals



Questions

xc(t) = x(t) cos(2πf0t)

withxc(t): carrier signalx(t): baseband signal⇒ (complex) envelope

Q1: Is there another way to translate the signal? x(t)→ xc(t)YESI/Q modulatorComplex-valued signal

Q2: How retrieving x(t) from xc(t)?I/Q demodulator



Mathematical framework

Instead of using only cos, we can use simultaneously cos and sin

xc(t) = xp(t) cos(2πf0t)− xq(t) sin(2πf0t)

= <(

(xp(t) + ixq(t))e2iπf0t)

withxp(t) a baseband real-valued signal of bandwidth B: In-phasexq(t) another real-valued signal of bandwidth B: Quadrature

We may have two streams in baseband for one carrier signal!

Complex envelope

The baseband signal can be represented by the so-called complexenvelope

x(t) =1√2

(xp(t) + ixq(t))



Mathematical framework (cont’d)

Assuming B/2 < f0, we have

I/Q modulator I/Q demodulator.

+f0

π/2

xp(t)

xq(t)

xc(t)

x

x

.

.

f0

π/2

xq(t)

xc(t)

xp(t)

x

x

.

In practice, we work with complex envelopesmaller bandwidth B instead of 2f0 + Bno cos and sin disturbing terms

.

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�� ChannelI/Q mod I/Q demodan x(t) y(t) an

Supra-channelTX RX

xc(t) yc(t)

.



A few wireless systems

When f0 increasespropagation degrades (1/f 2)antenna size decreases (1/f )bandwidth B may increase

System f0 B Antenna sizeIntercont. ∼ 10MHz (HF) 100kHz 100m

DVBT 600MHz (UHF) ∼ 1 MHz 1m2G 900MHz ∼ 1 MHz 10cmWifi 5.4 GHz 10MHz ∼ 1cm

Satellite 11GHz 100MHz –Personal Network 60GHz – –



Section 4: Propagation channel



Multipath channel

typical wireless channelvalid also for ADSL and optical fiber (low SNR)

.

(ρ1, τ1)

(ρ2, τ2)

(ρ0, τ0)

.

y(t) =∑

k

ρk x(t − τk ) + w(t)

= c(t) ? x(t) + w(t)

with noise w(t)

Dispersion time: Td = maxk τk

Coherence bandwidth:Bc = minf arg maxδ {‖C(f )− C(f + δ)‖ < ε}

Bc = O(1/Td )



Noise property

Let wc(t) be the (random) noise at carrier levelwc(t) is zero-mean (real-valued) Gaussian variablewc(t) is stationary (E[wc(t)2] independent of t)wc(t) is almost white

.

f0

f

N0/2

B.

P =

∫Sw (f )df = N0B

What’s happened for complex envelope w(t)?



Noise property (cont’d)

w(t) =1√2

(wp(t) + iwq(t))

with1. wp(t) and wq(t) zero-mean (real-valued) stationary Gaussian

variable with the same spectrum.

fB

N0

.

2. wp(t) and wq(t) are independent



Model: Gaussian channel

Short multipaths (Td ) compared to symbol period (Ts)Holds for Hertzian beamsHolds for SatelliteHolds also for very low data rate transmission

y(t) = x(t) + w(t)



Model: Frequency-Selective channel

Holds for cellular systems (2G with Td = 4Ts)Holds for Local Area Network (Wifi with Td = 16Ts)Holds for ADSL (Td = 100Ts)Holds also for Optical fiber (the so-called chromatic dispersion)

y(t) = c(t) ? x(t) + w(t)

⇒ InterSymbol Interference (ISI)

RemarkChannel type (ISI?) is modified according to data rateThe higher the rate is, the stronger the ISI is (Td � Ts)



Section 5: Modulation (TX):from discrete-time to continuous-time



Question

.

x(t)Modulation

"Modulation"

an xc(t)I/Q modulator

.

How associating bits an with analog (baseband) signal x(t)?



Binary modulation

Waveform: x0(t) if bit ’0’ and x1(t) if bit ’1’Binary linear modulation

x0(t) = Ag(t) and x1(t) = −Ag(t)

with symbols −A and A, and the shaping filter g(t)

If the symbol period is Ts, then

x(t) =∑

k

sk g(t − kTs) with sk ∈ {−A,A}

Example (g(t) rectangular function).

t

Ts

A

−A10 1 0 0 1 0 1 0 0 .



Multi-level modulation

Bandwidth of x(t) (B) identical of that of g(t):- If B � 1/Ts, InterSymbol Interference (see rectangular case)- If B � 1/Ts, bandwidth is wasted (signal oscillates at 1/Ts)

B = O(1/Ts)

Spectral efficiency is 1bit/s/Hz in binary modulation

Multi-level modulation: one symbol contains more than one bit

Exemple (M = 4)′00′ 7→ A ′01′ 7→ −A ′10′ 7→ 3A ′11′ 7→ −3A

.

t

Ts

A

−A 0 1 1 0 0 1 0 1 0 0

3A

−3A.



Constellations

Constellation = set of possible symbols

Pulse Amplitude Modulation (PAM).

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Phase Shift Keying (PSK).

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Quadrature Amplitude Modulation (QAM).

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Section 6: Demodulation (RX):from continuous-time to discrete-time



Question

.

I/Q modulator Demodulation Detectoryc(t) y(t) anz(n)

.

Two main boxes:How coming back to discrete-time signal: demodulationHow detecting optimally the transmit bits (from z(n)): detector

GoalDescribing and justifying the demodulation



A mathematical tool: signal spaceLet L2 be the space of energy-bounded function

L2 =

{f s.t.

∫|f (t)|2dt < +∞

}L2 is an infinite-dimensional vectorial space

Properties

L2 has an inner product

< f1(t)|f2(t) >=

∫f1(t)f2(t)dt

− leads to “orthogonality” principle: < f1(t)|f2(t) >= 0− leads to a norm: ‖f (t)‖ =

√< f (t)|f (t) >

L2 has an infinite-dimensional orthonormal (otn) basis: {Ψm(t)}m

∀f ∈ L2,∃{βm}m, f (t) =∑

m

βmΨm(t) with βm =< f (t)|Ψm(t) >

Any function is described by complex-valued coefficientsPhilippe Ciblat Digital Communications 26 / 86


A signal subspaceLet E be a subspace of L2 generated by the functions {fm(t)}m=1,··· ,M

E = span({fm(t)}m=1,··· ,M) =

{M∑

m=1

αmfm(t) for any complex αm

}

Property

This subspace has a finite dimension and a finite otn basis

D = dimC E and E = span{Φ`(t)}`∈{1,··· ,D}

For instance, let f (t) be a function in E

f (t) =D∑`=1

s(`)Φ`(t) with s(`) =< f (t)|Φ` >∈ C

s = [s(1), · · · , s(D)]T corresponds to the analog signal f (t)Usually, we prefer to work with s (which will carry information)



Exhaustive demodulator

y(t) =∑

k

sk h(t − kTs) + w(t)

with any symbol sk and any filter h(t)

QuestionHow sampling without information loss?

Nyquist-Shannon Theorem: sampling at fe > B. Then y(n/fe)contains all the information on y(t)

Actually information ({sk}) is only a part of y(t)Exhaustive demodulator based on subspace principle

Information {sk} belongs to the subspace E

E = span({h(t − kTs)}k )

Noise w(t) belongs to E and E⊥ (orthogonal of E)

w(t) = wE (t) + wE⊥(t) (wE (t) and wE⊥(t) independent)

Consequently, projection on E contains any information on {sk} in y(t)Philippe Ciblat Digital Communications 28 / 86


Exhaustive demodulator (cont’d)

Projection on E

z(n) = < y(t)|h(t − nTs) >

=

∫y(τ)h(τ − nTs)dτ

= h(−t) ? y(t)|t=nTs

.

h(−t)y(t)

′nT ′s

z(n)

.

Projection = Matched filter + Sampling

Remark: Sampling at Ts and not at Te



Input/output discrete-time model

.

sn y(t)w(t)

z(n)

h(t)

h(t) h(−t)

.

z(n) =∑`

h(`Ts)sn−` + w(n)

withh(t) = h(−t) ? h(t)w(n) = h(−t) ? w(t)|t=nTs zero-mean complex-valued stationaryGaussian with spectrum

Sw (e2iπf ) = N0h(e2iπf ) = N0|h(e2iπf )|2



Orthogonal basis caseWhat’s happened when {h(t − kTs)}k is an otn basis⇔ No ISI

z(n) = sn + w(n)

Equivalent proposition

{h(t − kTs)}k otn basish(t) Nyquist filter

h(`Ts) = δ`,0 ⇔∑

k

H(

f − kTs

)= Ts

h(t) square-root Nyquist

h(t) = h(−t) ? h(t)⇔ H(f ) =

√H(f )

In practice, h(t) square-root Nyquist iffGaussian channel . . . . . . .no ISI provided by propagation channelg(t) square-root Nyquist . . . . . . . . no ISI provided by shaping filter



Nyquist filter

Main property

If h(t) square-root Nyquist, then

B >1Ts

Examples:h(t) rectangular⇔ h(t) triangularh(t) square-root raised cosine (srrc)⇔ h(t) raised cosine

-0.2

0

0.2

0.4

0.6

0.8

1

-10 -5 0 5 10

h(t)

t

Raised cosine with rho=0.5.

0

0.2

0.4

0.6

0.8

1

1.2

−1 −0.5 0 0.5 1

Ts = 1 et ρ = 0.5

H(f)

f

B

−1+ρ2Ts

−1−ρ2Ts

1−ρ2Ts

1+ρ2Ts

.

with roll-off ρ (ρ = 0.22 in 3G, ρ = 0.05 in DVB-S2, ρ = 0.01 inWDM-Nyquist)



Consequence on noise

If h(t) square-root Nyquist, then w(n) white noise

w(n) = wR(n) + iwI(n)

wR(n) and wI(n) independentE[wR(n)2] = E[wI(n)2] = N0/2E[|w(n)|2] = N0 and E[w(n)2] = 0

Probability density function (pdf)

pw (x) = pwR ,wI (xR , xI) = pwR (xR)pwI (xI)

=1√πN0

e−x2R

N0 × 1√πN0

e−x2I

N0 =1πN0

e−x2R+x2

IN0

=1πN0

e−|x|2N0



Non-orthogonal basis case

What’s happened when {h(t − kTs)}k is a non-otn basis

ISIColored noise

Equivalent model

By using whitening filter f , we have

y(n) = f ? z(n) =L∑`=0

h(`)sn−` + w(n)

with w(n) white Gaussian noise



General model

Any previous case can be written as follows

y = Hs + w

withy = [y(0), . . . , y(N)]T

s = [s−L, · · · , s−1︸︷︷︸known

, s0, . . . , sN ]T

Goal: decode the data s from the observations y

Matrix model valid for no-ISI case, ISI case, but alsowireless MIMO, multiple access, multi-core/multi-mode fibers

(with different matrix structures)



Section 7: Detection Theory



Main resultsHow to choose optimally s given y?

y = f (s,w, ...), with s ∈ {sm}m∈C

By optimality, we mean minimal error probability

min Pe, with Pe = Prob(s 6= s)

where s is the output of the detector.

Optimal detectors

If data are not equilikely, then the optimal receiver (minimizingthe error probability) is the Max A Posteriori (MAP),

s = arg maxs

p(s|y)

If data are equilikely, then the optimal receiver (minimizing theerror probability) is the Maximum Likelihood (ML),

sML = arg maxs

p(y|s)



Decision regionsIf y is seen as a point in a vector space, we can define a regiondecision for any sm as follows

Rm = {y | p(sm|y) ≥ p(sm′ |y),∀m′ 6= m}

.

Rm1

Rm2

sm2

sm1

.



Application to Gaussian additive noise’s model

We consider

y = d + w

withd = f (s) with s the transmit information vectordm = f (sm), and dm ∈ D (received alphabet)Example: f (s) = Hs

ML = Least Square (LS)

dML = arg mind‖y− d‖2

msML = arg min

s‖y− f (s)‖2



Application to Gaussian additive noise’s model (cont’d)

Decision regions are described through the medians of the receivedalphabet (if s, or equivalently d, are equilikely)

.

Rm1

Rm2

dm1

dm2

.



PerformancesOnly available for Gaussian additive noise’s model

.

dm1

dm2

dmin

.

dmin = minm 6=m′ ‖dm − dm′‖ (= minm 6=m′ ‖f (sm)− f (sm′)‖)Nmin = 1

M

∑Mm=1 Nmin,m with Nmin,m number of points at distance

dmin from dm

Pe ≈ NminQ(

dmin√2N0

)Philippe Ciblat Digital Communications 40 / 86


Section 8: Application to Gaussian Channel



Signal model

y = s + wwith w Gaussian white noise.

s = arg mins‖y− s‖2

with the Euclidian distance ‖.‖.Main result

If y = [y(1), . . . , y(N)]T and s = [s(1), . . . , s(N)]T, then

s = arg mins

N∑n=1

|y(n)− s(n)|2

s(n) = arg mins(n)|y(n)− s(n)|2

Separable functionComponent per component optimization is still optimal



Application to linear modulations

y = s + w (scalar)

with s belonging to PAM, or PSK, or QAM.

We look for the nearest constellation point from y (threshold detector,denoted by \: s = \(y))

.

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Application to linear modulations (cont’d)

Constellation Performance Q(√

γ EbN0

)M-PAM γ = 6 log2(M)/(M2 − 1)

M-PSK γ = log2(M)(1− cos( 2πM ))

M-QAM γ = 3 log2(M)/(M − 1)

with Eb the energy used per transmit information bit

RemarkQAM > PSK > PAM



Numerical illustrations

1e-05

0.0001

0.001

0.01

0.1

1

0 5 10 15 20

Ta

ux E

rre

ur

Sym

bo

le

Eb/No (en dB)

Taux Erreur Symbole en fonction du Eb/No

MDP-2MDP-4MDP-8

MDP-16

1e-05

0.0001

0.001

0.01

0.1

1

0 5 10 15 20

Ta

ux E

rre

ur

Sym

bo

leEb/No (en dB)

Taux Erreur Symbole en fonction du Eb/No

MDP-2MAQ-16MDP-16MDA-16

Pe for different M-PSK Pe for different constellations(with fixed M)


DC Ex BD/carrier Ch Mod Demod Detec Appli w/o ISI Appli w. ISI Optimal receiver Suboptimal receivers OFDM

Section 9: Application toFrequency-Selective Channel



Outline

y = Hs + w

with H a N ′H × NH (non-diagonal) Toeplitz-band matrix

Section 9.1: Optimal receiver− Some special cases− Viterbi algorithm

Section 9.2: Suboptimal receivers− ZF (with or without CSIT)− MMSE− DFE

Section 9.3: OFDM



Section 9.1 : Optimal receiver



Optimal detector

sML = arg mins‖y− Hs‖2

Simple optimization without constraint on s

Optimization problem consists in finding out the minimumof a (positive) quadratic form under (nonconvex) conditions

Exhaustive search− complexity in O(MNH ) with M-QAM− ex.: 4× 4 MIMO with 64-QAM leads to 16M s

Special simple cases: NH = 1 or unitary matrixTree approach− suitable for wireless MIMO with small NH ⇒ Sphere decoding

Using the structure of H− suitable for ISI since H Toeplitz⇒ Viterbi algorithm

Suboptimal detectors− Remove constraint and threshold (s = \(H−1y))⇒ ZF− MMSE, DFE



Optimal detector

sML = arg mins∈CNH

‖y− Hs‖2

Hard optimization due to constraint on s

Optimization problem consists in finding out the minimumof a (positive) quadratic form under (nonconvex) conditions

Exhaustive search− complexity in O(MNH ) with M-QAM− ex.: 4× 4 MIMO with 64-QAM leads to 16M s

Special simple cases: NH = 1 or unitary matrixTree approach− suitable for wireless MIMO with small NH ⇒ Sphere decoding

Using the structure of H− suitable for ISI since H Toeplitz⇒ Viterbi algorithm

Suboptimal detectors− Remove constraint and threshold (s = \(H−1y))⇒ ZF− MMSE, DFE



Special case: N ′H = NH = 1We consider scalar signal

y = hs + w

We have

sML = arg mins∈C|y − hs|

= arg mins∈C|h|.|h−1y − s|

= arg mins∈C|h−1y − s|

= \(h−1y)

.

h−1y sML

.

ZF=MLEasy to implement



Special case: SIMO

We consider a multivariate signal (typically N ′H antennas)

y(`) = h`s + w(`) with ` = 1, · · · ,N ′H ⇐⇒ y = hs + w

with h a N ′H × 1 column-vectorWe have

sML = arg mins∈C‖y− hs‖2

= arg mins∈C‖y‖2 + ‖h‖2|s|2 − yHhs − shHy

.

y sMLhH Normalization

1/‖h‖2.

ML=MRC=ZF (with pseudo-inverse h# = (hHh)−1hH = hH/‖h‖2)Easy to implement



Special case: SIMOWe consider a multivariate signal (typically N ′H antennas)

y(`) = h`s + w(`) with ` = 1, · · · ,N ′H ⇐⇒ y = hs + w



= arg mins∈C‖h‖2

∥∥∥∥ hHy‖h‖2

∥∥∥∥2

+ ‖h‖2(|s|2 − yHh

‖h‖2 s − shHy‖h‖2

)

.


1/‖h‖2.




Special case: SIMOWe consider a multivariate signal (typically N ′H antennas)

y(`) = h`s + w(`) with ` = 1, · · · ,N ′H ⇐⇒ y = hs + w



= arg mins∈C

∥∥∥∥ hHy‖h‖2 − s

∥∥∥∥2

.


1/‖h‖2.




Special case: unitary matrices

We consider HHH = INH , or equivalently, ‖Hx‖ = ‖x‖ for any xWe have

sML = arg mins∈CNH

‖y− Hs‖

= arg mins∈CNH

‖H(HHy− s

)‖

= arg mins∈CNH

‖H−1y− s‖

.

y sMLH−1 = HH

.

ZF=MLEasy to implementNot true in general since usually ‖AB‖ 6= ‖A‖.‖B‖



Go back to ISI problem

IdeaDue to filtering, H is ToeplitzSuch a structure enables a low-complex optimal detector

Dynamic programming⇒ Viterbi’s algorithm [Viterbi1973]

We remind that

‖y− Hs‖2 =N∑

n=0

∣∣∣∣∣y(n)−L∑`=0

h`sn−`

∣∣∣∣∣2

def .= JN([s0, · · · , sN ]︸︷︷︸

s(N)

)

Fundamental property

JN(s(N)) = JN−1(s(N−1)) + ∆J(sN ,EN) =N∑

n=0

∆J(sn,En)

with ∆J(sn,En) = |y(n)−∑L`=0 h`sn−`|2 and En = [sn−1, · · · , sn−L]T



Definition of Trellis

enables to view s(N) through states {En}n

corresponds to the possible transitions between states

Example : M = 2, L = 1⇒ two states E (0) = [−1] et E (1) = [1]

.

’E(0) = −1’

’E(1) = 1’

’received −1’

time 0 node

branch

path (one possible s )

’received1’

.

Branch:− only characterized by sn and En (at time n)− ∆J : Branch metric (cost to go through this branch)

Goal: find optimal path (with the lowest sum of branch metrics)



Algorithm principle

For sake of simplificationtwo states only E (0) = [−1] et E (1) = [1]

inspecting between time n − 1 and n

.sn = −1

sn =−1

’E(0) = −1’

’E(1) = 1’

s(n−1)

|E(0)

s(n−1)

|E(1)

time n− 1 time n.

Amongst paths arriving at E (0), only one minimizes Jn−1(.)

s(n−1)opt.|E (0) s.t. Jn−1(s(n−1)

opt.|E (0)) ≤ Jn−1(s(n−1)|E (0) )

s(n−1)opt.|E (1) s.t. Jn−1(s(n−1)

opt.|E (1)) ≤ Jn−1(s(n−1)|E (1) ) (similar result for E (1))



Algorithm principle (cont’d)

What’s happened at time n in state E (0) ?

In other words, what do we know about s(n)|E (0) and s(n)

opt.|E (0)

Jn−1(s(n−1)opt.|E (0)) ≤ Jn−1(s(n−1)

|E (0) )






opt.|E (0)

Jn−1(s(n−1)opt.|E (0)) + ∆J(−1,E (0)) ≤ Jn−1(s(n−1)

|E (0) ) + ∆J(−1,E (0))






opt.|E (0)

Jn([s(n−1)opt.|E (0) ,−1]) ≤ Jn([s(n−1)

|E (0) ,−1])

Any suboptimal path at time n − 1 remains suboptimal



Complexity analysis

Complexity in O((N + 1).ML+1)

− linear in N− polynomial in M− exponential in L

So M and L have to be small enough

Refresher: maxk τk = LTs, D = log2(M)/Ts and B = 1/Ts

Increasing data rate leads to− either increasing M ⇒ bad news for Viterbi’s algorithm− or increasing B and thus L⇒ bad news for Viterbi’s algorithm



Numerical example

Filter: h0 = 1 and h1 = 0.25 (L = 1)Noiseless case

Opening: ’−1’Data: ’1,−1,−1’Closing: ’−1’

Sought vector: [−1, s0?, s1?, s2?,−1]

Received vector: [y(−1)?,0.75,−0.75,−1.25,−1.25, y(4)?]

Branch metric (at time n)

∆J = |y(n)− sn − 0.25sn−1|2



Time 0

.

’1’

’−1’ ’−1’

.

.’−1’

’1

0 4 4

0

0

’−1’

.



Time 1.

’−1’

’1

0 4

0

0

4

0

4

4

02.25

6.25

0.254.25

’−1’

.

.’−1’

’1

0 0

4

4

0

0

0

’−1’

.Philippe Ciblat Digital Communications 57 / 86


Time 2.

’−1’

’1

0

0

0

0 4 4

0 00

4

0.5

6.25

10.25

4.5

4

’−1’

.

.

’−1’

’1

0

0

0

0

00

0

4

4

’−1’

.Philippe Ciblat Digital Communications 58 / 86


Time 3.

’−1’

’1

0

0

0

0

0 0 0

4

4’−1’0

0

0.25

4.25

.

.

’−1’

’1

0

0

0

0

0 0 0 ’−1’0

0

.



Section 9.3 : Suboptimal receivers



Principle

GoalViterbi’s algorithm not applicable due to high data rateDesign a simple receiver (but suboptimal)Idea: linear compensation for linear interference

⇒ linear receivers

.

≫

s H

w

y z s

ISIinput ISIoutput

Equalizer P

.

Question: how choosing matrix P ?



ZF equalizer

Principle

Forcing interference to be zero⇒ Zero-Forcing (ZF)Mathematically,

PH = I⇔ PZF = H−1

Soz = s + w′ with w′ = H−1w

Drawback : noise enhancement (we so derive the SNR percomponent)

SNRinput =Es

2N0

trace(HHH

)NH

SNRoutput =Es

2N0

1trace(H−1H−H)

NH

Thanks to convexity of x 7→ 1/x , one can prove

SNRoutput ≤ SNRinput

with equality when HHH = I, i.e., unitary matrix (cf. slide 49)Philippe Ciblat Digital Communications 61 / 86


Precompensation: ZF at the transmitter side!

If H known at the transmitter side, one can precompensate it!How? by sending x instead of s with

x = Ps

where P corresponds to a linear precoderHow choosing P ? e.g. ZF principle

Drawback:No noise enhancement anymore, butP =

√aH−1 with a = NH/trace

(H−1H−H

)(for energy purpose),

then

SNRw/o comp =Es

2N0

trace(HHH

)NH

SNRcomp =Es

2N0

1trace(H−1H−H)

NH

Same SNR issue (just not located at the same place!)

Solution suitable if CSIT and unitary transformation



MMSE equalizer

Principle

Choosing P s.t. Py close to s

PMMSE = arg minP

Es,w[‖Py− s‖2]

After algebraic manipulations,

PMMSE = Es(EsHHH + 2N0I

)−1 HH

Remarks :At high SNR: PMMSE ≈ PZF

At low SNR: PMMSE ∝ HH



DFE equalizer

Principle

Using previous decision on s for removing interferenceDecision Feedback Equalizer (DFE)Problem: causality principle in MIMO

Solution: the so-called QR decomposition

H = QR

with a unitary matrix Q and a upper triangular matrix R

z = QHy = Rs + w′

where w′ is still white Gaussian noise

.

s H

w

yQH z s

R

DFE equalizer

.

ZF-DFE: decode first sNH , thensNH−1 based on(zNH−1 − rNH−1,NH sNH ), and so on



Implementation issue

When NH is high, implementation issue for matrix inversion

When H corresponds to ISI,equivalence between Toeplitz matrices and filteringImplementation through filtering

Example: ZF

PZFH = I⇐⇒ pZF ? h = δ

sopZF(e2iπf ) =

1h(e2iπf )



Numerical illustrations

We assumeL = 1 with

h0 =1√

1 + ρ2and h1 =

ρ√1 + ρ2

Matrix implementation of introduced equalizers

The larger ρ is, the stronger ISI is



Viterbi’s algorithm performance

Eb/N

0 (dB)

0 1 2 3 4 5 6 7 8

BE

R

10 -4

10 -3

10 -2

10 -1

10 0

ρ=0ρ=0.2ρ=0.4ρ=0.6ρ=0.8ρ=1

ISI can not be totally removed



Performance of various equalizers for weak ISI

ρ = 0.2

Eb/N

0 (dB)

0 2 4 6 8 10 12 14

BE

R

10 -4

10 -3

10 -2

10 -1

10 0Performance with rho=0.2

Viterbi

No equalizer

ZF

MMSE

ZF-DFE

Close performance between any solution



Performance of various equalizers for strong ISI

ρ = 0.8

Eb/N

0 (dB)

0 2 4 6 8 10 12 14

BE

R

10 -4

10 -3

10 -2

10 -1

10 0Performance with rho=0.8

Viterbi

No equalizer

ZF

MMSE

ZF-DFE

DFE outperforms linear equalizers



Section 9.3 : OFDM



Refresher: Input/Ouput model

Letx(n) be the transmitted signal (may be different from symbols sn)y(n) be the received signal{h`}`=0,··· ,L be the filter associated with propagation channel

y(n) =L∑`=0

h`x(n − `)

Let us consider one block of size N→ y = [y(N − 1), · · · , y(0)]T

→ x = [x(N − 1), · · · , x(0)]T

→ x = [x(−1), · · · , x(−L)]T



Matrix model for Input/Output

y = T1x + T2xT1: N × N Toeplitz matrix whose the k -th row is given by− [0k−1, h0, h1, · · · , hL, 0N−L−k ] (if k ≤ N − L)− [0k−1, h0, h1, · · · , hN−k−1] (if k > N − L)

T2: N × L Toeplitz matrix whose the k -th row is given by− 0L (if k ≤ N − L)− [hL, hL−1, · · · , hN−k+1, 0N−k ] (if k > N − L)

In noisy case, we find again

y = Hx + wwhen null guard interval between blocks: H = T1 and x = 0when guard interval x is not empty and linearly depends oncurrent bloc, i.e., x = T3x: H = T1 + T2T3

Question: how obtaining an interference-free system at thereceiver side (by modifying the transmitter) ?



Lemma 1

Assumption:Perfect CSIT (H known at the transmitter side)Actually, extension of ZF principle described in slide 23

Let H = UHΛV be the singular value decomposition (svd) of H with U,V two unitary matrices, and Λ = diag(λ0, · · · , λN−1)

Instead of sending x = s, we send x = VHs (no energy purpose)Instead of detecting on y, we detect on z = Uy

⇒ z = Λs + w′

with w′ = Uw still white Gaussian noise

RemarksInformation located in eigenvectors (not interfer in-between!)Eigenvectors usually depend on HIssue: perfect CSIT assumption unrealistic in wireless



Lemma 2

Let C be a N × N circulant matrix associated with {h`}`=0,··· ,L

C =

h0 h1 · · · hL 0 · · · 00 h0 h1 · · · hL 0 0· · · · · · · · · · · · · · · · · · · · ·h1 · · · hL 0 · · · 0 h0

Properties of C

Eigenvectors: Fourier vectors (independent of the channel!)Eigenvalues: filter responses at Fourier frequencies

C = FHΛF

withF FFT matrix (so, FH = F−1)

λn = H(e2iπn/N) =∑L`=0 h`e−2iπ `n

N



OFDM principle

If x = [x(N − 1), · · · , x(N − L)]T, then

y = T1x + T2x⇔ y = Cx

Thus we havez = Λs

with z = Fy and x = F−1sFinally, for the k -th block and the n-th subcarrier, we get

z(k)n = H(e2iπn/N)s(k)

n ∀n, k

Remarkscyclic prefix transforms Toeplitz into Circulant (who diagonalizeswithin a basis independent of the channel, so no required CSIT !)OFDM: Orthogonal Frequency Division Multiplexing



Another way for introducing OFDM

The historical way

.

��

��

Bcoh

|C(f)|

f

.

y(t) = c(t) ? x(t)

with c(t) propagation channel

Bc : coherence bandwidth B: signal bandwidth

If B < Bc , almost no ISI.

��

��

|C(f)|

f

|X(f)|

f0

Y (f ) ≈ C(f0)X(f ) ⇒ y(t) ≈ C(f0)x(t) .

If B > Bc , ISI.

��

��

|C(f)|

f

|X(f)|

Y (f ) = C(f )X(f ) ⇒ y(t) = c(t) ⋆ x(t).

How being always in the configuration B < Bc?



Another way... naive ideaIdea: as B = 1/Ts (symbol period),splitting symbols sequence into Nsubsequence (with period T = NTs) s.t.

1T< Bc

Then each subsequence n transmitted todifferent subcarriers fn

.

��

��

|C(f)|

f

· · · · · ·

|X0(f)| |Xn(f)| |XN−1(f)|

fnf1f0 fN−1

.

On each subcarrier, no ISI

Let s(k)n = skN+n be a subsequence, the transmitted signal is

x(t) =N−1∑n=0

∑k∈Z

s(k)n g(t − kT )e2iπfnt

N: subcarriers number∑N−1n=0 s(k)

n e2iπfnt : OFDM symbol with period T = NTs



Another way... orthogonality principle

x(t) =∑k∈Z

N−1∑n=0

s(k)n Φn,k (t) with Φn,k (t) = g(t − kT )e2iπfnt

Orthogonal subcarriers⇔∫R

Φn,k (t)Φn′,k ′(t)dt = δn,n′δk,k ′

We force {Φn,k (t)}n,k to be orthonormal basis.

f f

(ii)(i)

Without overlap With overlap .

Case (ii) : OFDM (Orthogonal Frequency Division Multiplexing)fn equally spaced⇒ fn = n∆fg(t) rectangular function of support [0,T ]

orthogonality iff ∆f = 1/T = 1/NTs



Another way... transmitter description

Bandwidth: almost same spectral efficiency than single carrier

Btot = NBsc ≈ N1T

= N1

NTs=

1Ts

x (k)(m) = x(kT + mTs)

=1√N

N−1∑n=0

s(k)n e2iπ nm

N

︸︷︷︸inverse FFT

.

S/P

IFFT

Rate 1/Ts Rate 1/T Rate 1/T

x(t)ADC

s(k)N−1

s(k)0

sn

x(k)(0)

x(k)(N − 1)

.

Remarksx in time domain. k OFDM block number and m time index(within block)s in frequency domain. n frequency index



Another way... receiver description

z(k)(n) =

∫R

y(t)Φn,k (t)dt

=

√Ts√N

N−1∑m=0

y (k)(m)e−2iπ nmN

with y (k)(m) = y(kT + mTs)

.

FFTP/S

z(k)(0)

z(k)(N − 1)

s(k)n

y(k)(0)

y(k)(N − 1)

Ts

y(t) z(k)n

.

When channel present,

y(t) = c(t) ? x(t) =∑k∈Z

N−1∑n=0

s(k)n Ψn,k (t) with Ψn,k (t) = c(t) ? Φn,k (t)

Problem: Ψn,k (t) is not orthonormal anymore (actually, almost)Solution: go back to the previously-developed approach

⇒ Cyclic prefix



TX/RX Scheme

.

Add CPFFT−1 FFT

ConvolutionToeplitz matrix

RXTX

s y

Circular convolution / Circulant matrix

Remove CP∝ sFreq EQ.

(typ. ZF)

x zChannel h

.

Cyclic prefix is crucial!



Detection: OFDM-SISO

Formally, we havez = Hs + w

with H = diag(H(1), · · · ,H(e2iπn/N)︸︷︷︸H(n)

, · · · ,H(e2iπ(N−1)/N))

As H is diagonal (no interference occurs), one can work subcarrierper subcarrier, i.e.,

z(n) = H(n)sn + w(n)

with− z(n) received signal after FFT− sn transmitted symbol (before IFFT)− H(n) filter response at the n-th subcarrier

Optimal detector (application of Slide 47)

Threshold detector on H(n)−1z(n)



History

end-50: multicarriers concept (Case (i))

end-60: orthogonal multicarriers (Case (ii))⇒ OFDM

beginning-70: FFT

mid-80: European project “Eurêka” for DAB− cyclic prefix− coding and OFDM relationship in wireless context

beginning-90: First standard based on OFDM (DAB)

end-90: Very popular standards (ADSL, DVBT, Wifi, LTE, . . .)



OFDM design: two rules

N should be large enough

L� N ⇒ LTs � NTs ⇒ Td � NTs

N � B/Bc ⇔ ∆f � Bc

N should be small enough

(L + N)Ts � Tc ⇒ NTs � Tc

N � B/Bd ⇔ ∆f � Bd

but also− Mis-synchronization of VCO (a few ppm)− FFT complexity (O(N log(N)))− Latency



OFDM design: four examples

Wifi ADSL DVBT Optic 100GCarrier freq. 5.2GHz 0.6MHz 700MHz 1550nmBandwidth 20MHz 1.1MHz 9.15MHz 5GHzSampling period 50ns 0.9µs 0.11µs 0.2nsFilter length 800ns 135µs 224µs 0.69nsFilter degree 16 150 2036 4Cyclic prefix 16 32 2048 8Spect. eff. loss 20% 12.5 % 20% 3.125%# Subcarrier 64 256 8192 256OFDM duration 4µs 256µs 896µs 52.8nsSubcarrier spac. 312.5kHz 4.31kHz 1.11kHz 19.53 MHzCoherence band. 1.25MHz 7.4kHz 4.47kHz 1.44GHzDoppler band. 52Hz (3m/s) 0 52Hz (3m/s) 0

Remark: in ADSL, Time Equalizer added for channel shortening



Perspectives

MIMO : how to code, how to decode?

Multi-user interference: how to manage resource allocation

Artificial Intelligence tools : decoding and resource optimization



References

[Tse2005] D. Tse and P. Viswanath, Fundamentals of wireless communications, 2005

[Goldsmith2005] A. Goldsmith, Wireless Communications, 2005

[Proakis2000] J. Proakis, Digital Communications, 2000

[Benedetto1999] S. Benedetto and E. Biglieri, Principles of digital transmission with wireless applications, 1999

[Viterbi1979] A. Viterbi and J. Omura, Principles of digital communications and coding, 1979

[Gallager2008] R. Gallager, Principles of digital communications, 2008

[Wozencraft1965] J. Wozencraft and I. Jacob, Principles of communications engineering, 1965

[Barry2004] J. Barry, D. Messerschmitt and E. Lee, Digital communications, 2004

[Sklar2001] B. Sklar, Digital communications : fundamentals ans applications, 2001

[Ziemer2001] R. Ziemer and R. Peterson, Introduction to digital communication, 2001

[Viterbi1967] A. Viterbi, Error bounds for convolutional codes and an asymptotic optimum decoding algorithm, IEEE Trans. onInformation Theory, 1967

[VanTrees2013] H. VanTrees, K. Bell, Detection Estimation and Modulation Theory Part I, Wiley, 2013


principles of digital communications [1mm] part 1

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