RUHR-UNIVERSITY BOCHUM
Wireless CommunicationsLecture slides
Karlheinz Ochs
Chair of Digital Communication Systems
Communications Systems
Chair ofDigital
Faculty of
Electrical Engineering and
Information Technology
www.dks.rub.de WS 2017/18
Wireless Communications
Contents
1 Motivation
2 Linear Systems
3 Wireless Communication Channel
4 Single Input Single Output Systems
5 Multiple Input Multiple Output Systems
6 Optimal Transmission Strategies
7 Multiple Access Channel
8 X Channel
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Wireless Communications Motivation
Wireless CommunicationsMotivation
Karlheinz Ochs
Chair of Digital Communication Systems
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Motivation
Contents
1 Preliminaries
2 Transmission Scenario
3 Challenges
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Motivation Preliminaries
Contents
1 Preliminaries
2 Transmission Scenario
3 Challenges
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Preliminaries 1 / 152
Preliminaries
Trends in communication systems
mobile communication
high data rates
low latency
Constraints on mobile communication systems
expensive and limited bandwidth
limited transmitter signal power
time-variant transfer behavior
Problem-solving approaches
orthogonal frequency-division multiplexing (OFDM)
multiple input multiple output systems
multiple antenna systems
cooperative communication
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Motivation Transmission Scenario
Contents
1 Preliminaries
2 Transmission Scenario
3 Challenges
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Transmission Scenario 2 / 152
Transmission scenario
Multipath Propagation Channel
cellular phone
echos
noise
base station
Time-variancemultipath propagation due to mobile objectssample and hold devices, modulators, HF amplifiers, . . .
Transmission conditions are changing with time!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Motivation Challenges
Contents
1 Preliminaries
2 Transmission Scenario
3 Challenges
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Challenges 3 / 152
Challenges
Communication theorie
How to design and synthesize digital communication systems?
Information theorie
What is the maximum data rate of a reliable transmission?
Digital signal processing
What is the optimal processing strategy?
Programable hardware
How can a digital communication system be verified?
Determine and reach the limits of communications!
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Wireless Communications Linear Systems
Wireless CommunicationsLinear Systems
Karlheinz Ochs
Chair of Digital Communication Systems
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Linear Systems
Contents
1 Signals
2 Fourier Series and Transformation
3 Linear Systems
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Linear Systems Signals
Contents
1 Signals
2 Fourier Series and Transformation
3 Linear Systems
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Signals 4 / 152
Unit Step Function
Definition
u(ξ) =
1 for ξ > 00 for ξ < 0
Graph
u(ξ)
1
ξ
Remarks
argument can have a physical unit
at discontinuity we may define u(0) = 1/2
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Signals 5 / 152
Sign Function
Definition
sgn(ξ) =
1 for ξ > 0−1 for ξ < 0
Graph
sgn(ξ)
1
−1
ξ
Remarks
is a superposition of two unit step functions
argument can have a physical unit
at discontinuity we may define sign(0) = 0
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Signals 6 / 152
Rectangular Function
Definition
rect(ξ) =
1 for |ξ| < 10 for |ξ| > 1
Graph
rect(ξ)
−1 1
1
ξ
Remarks
is a superposition of two unit step functions
argument has no physical unit
at discontinuities we may define rect(±1) = 1/2
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Signals 7 / 152
Triangular Function
Definition
4(ξ) =
1− |ξ| for |ξ| ≤ 1
0 for |ξ| > 1
Graph
−1 1
1
(ξ)
ξ
Remarks
is the convolution of two rectangular functions
argument has no physical unit
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Signals 8 / 152
si-Function
Definition
si(ξ) =
1 for ξ = 0
sin(ξ)/ξ for ξ 6= 0
Graph
−4π −3π −2π −π π 2π 3π 4π
1
si(ξ)
ξ
Remarks
argument has no physical unit
decays reciprocal to its argument
zeros si(νπ) = 0 for ν ∈ Z\0
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Signals 9 / 152
Dirac Delta Function
Definition∫ t
−∞δ(ξ)dξ = u(t)
Especially ∫ ∞−∞
δ(ξ)dξ = 1
Properties
even
δ(t) = δ(−t)
scaling
δ(αt) =1|α|δ(t) for α ∈ R \ 0
sifting property
f (t)δ(t − t0) = f (t0)δ(t − t0)
Remarks
argument can have physical unit
physical unit is reciprocal to the physical unit of its argument
briefly called δ-function
Do not evaluate this function!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Linear Systems Fourier Series and Transformation
Contents
1 Signals
2 Fourier Series and TransformationFourier SeriesFourier Transformation
3 Linear Systems
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Fourier Series and Transformation Fourier Series 10 / 152
Fourier Series
Periodic Signal
s(t) = s(t − T)
fundamental periodsmallest possible positive T
Examples
sine
s(t) = sin(Ωt) , with ΩT = 2π
cosine
s(t) = cos(Ωt) , with ΩT = 2π
periodically repeated function
s(t) =∞∑
n=−∞
f (t − nT)
Fourier Series
s(t) =∞∑
n=−∞
Sn e jnΩt , with Sn =1T
∫ T
0s(t)e−jnΩtdt and ΩT = 2π
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Fourier Series and Transformation Fourier Series 11 / 152
Important Fourier Series
Cosine
s(t) = cos(Ωt) =12
[e jΩt + e−jΩt
]corresponds to real part
Sine
s(t) = sin(Ωt) =12j
[e jΩt − e−jΩt
]corresponds to imaginary part
Periodically Repeated Function
s(t) =∞∑
n=−∞
f (t − nT) =1T
∞∑n=−∞
F(jnΩ)e jnΩt ,
where f (t) −−•F(jω) and ΩT = 2πLehrstuhl fürDigitale Kommunikationssysteme
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Fourier Series and Transformation Fourier Series 12 / 152
Fourier Series of the Sampling Function
Definition
δT(t) =∞∑
k=−∞
δ(t − kT)
Graph
δT (t)
−3 −2 −1 0 1 2 3 t/T
Remarks
also called Dirac comb or impulse train
T-periodic function
δ(t) −−• 1
Fourier Series
δT(t) =1T
∞∑n=−∞
e jnΩt ,
where ΩT = 2π.Lehrstuhl fürDigitale Kommunikationssysteme
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Fourier Series and Transformation Fourier Transformation 13 / 152
Fourier Transformation
Correspondence
x(t) −−•X(jω)
Time Domain
x(t) = F−1X(jω) = 12π
∫ ∞−∞
X(jω)e jωtdω
Frequency Domain
X(jω) = Fx(t) =∫ ∞−∞
x(t)e−jωtdt
Remark
X(jω) is called Fourier transform or spectrum of x(t)
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Fourier Series and Transformation Fourier Transformation 14 / 152
Properties of the Fourier Transformation
Complex Conjugation
x∗(t) −−•X∗(−jω)
Time Shift
x(t − t0) −−• e−jωt0 X(jω) , with t0 ∈ R
Modulation
x(t)e jω0t −−•X(jω − jω0) , with ω0 ∈ R
Time Scaling
x(αt) −−• 1|α|X
(jωα
), with α ∈ R\0
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Fourier Series and Transformation Fourier Transformation 15 / 152
Properties of the Fourier Transformation
Differentiation
time domain
dx(t)dt−−• jωX(jω)
frequency domain
tx(t) −−• jdX(jω)
dω
Integration
time domain∫ t
−∞x(τ)dτ −−• X(jω)
jω+ πX(0)δ(ω) for X(0) <∞
frequency domain
jx(t)
t+ πx(0)δ(t) −−•
∫ ω
−∞X(jv)dv for x(0) <∞
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Fourier Series and Transformation Fourier Transformation 16 / 152
Properties of the Fourier Transformation
Convolution
time domain
x(t) ∗ y(t) =∫ ∞−∞
x(τ)y(t − τ)dτ −−•X(jω)Y(jω)
frequency domain
x(t)y(t) −−• 12π
X(jω) ∗ Y(jω) =1
2π
∫ ∞−∞
X(jv)X(jω − jv)dv
Parseval’s Theorem∫ ∞−∞
x(t)y∗(t)dt =1
2π
∫ ∞−∞
X(jω)Y∗(jω)dω
energy of x(t) if y(t) = x(t)
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Fourier Series and Transformation Fourier Transformation 17 / 152
Often Used Correspondences
Rectangular Function
rect( t
T
)−−• 2T si(ωT) ,
Ω
πsi(Ωt) −−• rect
( ωΩ
),
with T, Ω > 0
Triangular Function
4( t
T
)−−• T si2
(ωT2
),
Ω
πsi2(Ωt) −−•4
( ω
2Ω
),
with T, Ω > 0
Gaussian Function
e−αt2/2 −−•√
2πα
e−ω2/[2α] , with Reα > 0
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Fourier Series and Transformation Fourier Transformation 18 / 152
Often Used Correspondences
Unit Step Function
u(t) −−•πδ(ω) +[
1jω
], δ(t) +
[jπt
]−−• 2u(ω)
Signum Function
sgn(t) −−•[
2jω
],
[jπt
]−−• sgn(ω)
Delta Function
δ(t) −−• 1 , 1 −−• 2πδ(ω)
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Fourier Series and Transformation Fourier Transformation 19 / 152
Correspondence of the Sampling Function
Correspondence
δT(t) −−•∆T(jω)
Fourier Transform
∆T(jω) =∞∑
k=−∞
e−jωkT = Ω∞∑
n=−∞
δ(ω − nΩ)
Time Shift∞∑
k=−∞
δ(t − tk) =1T
∞∑n=−∞
e jnΩ[t−t0]
−−•
∞∑k=−∞
e−jωtk = Ω∞∑
n=−∞
δ(ω − nΩ)e−jnΩt0 ,
with tk = t0 + kT.Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Linear Systems Linear Systems
Contents
1 Signals
2 Fourier Series and Transformation
3 Linear SystemsTime-Invariant SystemsTime-Variant SystemsExamples
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Linear Systems Time-Invariant Systems 20 / 152
Linear Time-Invariant System
System
x(t) S y(t)
Reaction
x(t)→ y(t) = h(t) ∗ x(t) =∫ ∞−∞
h(t − t′)x(t′)dt′
Impulse Response
δ(t) S h(t)
Reaction
δ(t)→ y(t) = h(t) ∗ δ(t) = h(t)
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Linear Systems Time-Invariant Systems 21 / 152
Linear Time-Invariant System
Transfer Function
ejΩxt S H(jΩx)ejΩxt
Reaction
e jΩx t → y(t) = H(jΩx)e jΩx t
Proof
1 y(t) = e jΩx t ∗ h(t)
2 y(t) =∫∞−∞ e jΩx[t−t′]h(t′)dt′
3 y(t) =∫∞−∞ h(t′)e−jΩx t′dt′ e jΩx t
4 y(t) = H(jΩx)e jΩx t
with
h(t) −−•H(jω)
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Linear Systems Time-Invariant Systems 22 / 152
Linear Time-Invariant System
Periodic Excitation
∞∑n=−∞
XnejnΩxt S∞∑
n=−∞H(jnΩx)Xne
jnΩxt
Reaction
∞∑n=−∞
Xn e jnΩx t → y(t) =∞∑
n=−∞
H(jnΩx)Xn e jnΩx t
Proof
1 e jnΩx t → H(jnΩx)e jnΩx t
2 Xn e jnΩx t → H(jnΩx)Xn e jnΩx t
3∞∑
n=−∞Xn e jnΩx t → y(t) =
∞∑n=−∞
H(jnΩx)Xn e jnΩx t
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Linear Systems Time-Invariant Systems 23 / 152
Linear Time-Invariant System
Aperiodic Excitation
X(jω) S Y (jω)
Reaction
X(jω)→ Y(jω) = H(jω)X(jω)
Proof
1 x(t) =1
2π∫∞−∞ X(jω)e jωtdω = lim
Ω→0xΩ(t)
2 xΩ(t) =∞∑
n=−∞
Ω
2πX(jnΩ)e jnΩt → yΩ(t) =
∞∑n=−∞
Ω
2πH(jnΩ)X(jnΩ)e jnΩt
3 limΩ→0
xΩ(t)→ limΩ→0
yΩ(t)
4 x(t)→ y(t) = F−1H(jω)X(jω)Lehrstuhl fürDigitale Kommunikationssysteme
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Linear Systems Time-Variant Systems 24 / 152
Linear Time-Variant System
Transfer Function
ejΩxt St H(t, jΩx)ejΩxt
Reaction
e jΩx t → y(t) = H(t, jΩx)e jΩx t
Definition
H(t, jΩx) = y(t)e−jΩx t for all Ωx ∈ R
Time-variant Transfer Function
H(t, jω′)
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Linear Systems Time-Variant Systems 25 / 152
Linear Time-Variant System
Periodic Excitation
∞∑n=−∞
XnejnΩxt St
∞∑n=−∞
H(t, jnΩx)XnejnΩxt
Reaction
∞∑n=−∞
Xn e jnΩx t → y(t) =∞∑
n=−∞
H(t, jnΩx)Xn e jnΩx t
Proof
1 e jnΩx t → H(t, jnΩx)e jnΩx t
2 Xn e jnΩx t → H(t, jnΩx)Xn e jnΩx t
3∞∑
n=−∞Xn e jnΩx t → y(t) =
∞∑n=−∞
H(t, jnΩx)Xn e jnΩx t
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Linear Systems Time-Variant Systems 26 / 152
Linear Time-Variant System
Aperiodic Excitation
X(jω) St y(t)
Reaction
X(jω)→ y(t) =1
2π
∫ ∞
−∞H(t, jω′)X(jω′)e jω′tdω′
Proof
1 x(t) =1
2π∫∞−∞ X(jω)e jωtdω = lim
Ω→0xΩ(t)
2 xΩ(t) =∞∑
n=−∞
Ω
2πX(jnΩ)e jnΩt → yΩ(t) =
∞∑n=−∞
Ω
2πH(t, jnΩ)X(jnΩ)e jnΩt
3 limΩ→0
xΩ(t)→ limΩ→0
yΩ(t)
4 x(t)→ y(t) = F−1H(t, jω)X(jω)Lehrstuhl fürDigitale Kommunikationssysteme
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Linear Systems Time-Variant Systems 27 / 152
Linear Time-Variant System
Impulse Response
δ(t− Tx) St h(t, Tx)
Reaction
δ(t − Tx)→ h(t, Tx) with h(t, t′) =1
2π
∫ ∞
−∞H(t, jω′)e jω′[t−t′]dω′
Proof
1 x(t) = δ(t − Tx) −−•X(jω) = e−jωTx
2 y(t) =1
2π∫ ∞−∞ H(t, jω′)e−jω′Tx e jω′tdω′
3 h(t, Tx) =1
2π∫ ∞−∞ H(t, jω′)e jω′[t−Tx]dω′
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Linear Systems Time-Variant Systems 28 / 152
Linear Time-Variant System
System
x(t) St y(t)
Reactions
x(t)→ y(t) =∫ ∞
−∞h(t, t′)x(t′)dt′
X(jω)→ y(t) =1
2π
∫ ∞
−∞H(t, jω′)X(jω′)e jω′tdω′
Remark
Y(jω) 6= H(t, jω)X(jω)
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Linear Systems Examples 29 / 152
Examples of Linear Time-Variant Systems
Linear Time-invariant System
x(t) S y(t)
Special excitation
δ(t − Tx)→ h(t − Tx)
e jΩx t → H(jΩx)e jΩx t
System functions
impulse response h(t, t′)= h
(t − t′
)transfer function H
(t, jω′
)= H
(jω′)
Transfer function is constant with respect to time!
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Linear Systems Examples 30 / 152
Examples of Linear Time-Variant Systems
Linear Frequency-invariant System
x(t)
c(t)
y(t)
Special excitation
δ(t − Tx)→ c(Tx)δ(t − Tx)
e jΩx t → c(t)e jΩx t
System functions
impulse response h(t, t′) = c(t′)δ(t − t′)
transfer function H(t, jω′) = c(t)
Transfer function is constant with respect to frequency!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Wireless Communications Wireless Communication Channel
Wireless CommunicationsWireless Communication Channel
Karlheinz Ochs
Chair of Digital Communication Systems
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Wireless Communication Channel
Contents
1 Transmission Scenario
2 Passband Transmission
3 Baseband Transmission
4 Time-Discrete Transmission
Lehrstuhl fürDigitale Kommunikationssysteme
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Wireless Communication Channel Transmission Scenario
Contents
1 Transmission Scenario
2 Passband Transmission
3 Baseband Transmission
4 Time-Discrete Transmission
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Transmission Scenario 31 / 152
Wireless Communications
Transmission Scenario
cellular phone
echos
noise
base station
Time-variancemultipath propagation due to mobile objectssample and hold devices, modulators, HF amplifiers, . . .
Transmission conditions are changing with time!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Transmission Scenario 32 / 152
Time-Invariant Multipath Propagation
Input-Output Relation
y(t) =n∑ν=0
cνxν(t)
transmitted signal arrives at the receiver on different paths
xν(t) = x(t − Tν)
different durationsTν = T0 + νT
attenuation and change of phase
cν = |cν | e j arccν
x(t) T0 T T
c0 c1 cn−1 cn
y(t)
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Transmission Scenario 33 / 152
Time-Invariant Multipath Propagation
Impulse Response
definitionx(t) = δ(t − Tx) → y(t) = h(t − Tx)
impulse response of the multipath channel
h(t) =n∑ν=0
cνδ(t − Tν)
input-output relation
y(t) =∫ ∞−∞
h(t − t′)x(t′)dt′
x(t) T0 T T
c0 c1 cn−1 cn
y(t)
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Transmission Scenario 34 / 152
Time-Invariant Multipath Propagation
Transfer function
definition
x(t) = e jΩx t → y(t) = H(jΩx)e jΩx t
transfer function of the multipath channel
H(jω) =n∑ν=0
cν e−jωTν
input-output relation
Y(jω) = H(jω)X(jω)
x(t) T0 T T
c0 c1 cn−1 cn
y(t)
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Transmission Scenario 35 / 152
Time-Variant Multipath Propagation
Input-output Relation
y(t) =n∑ν=0
cν(t)x(t − Tν) , with cν(t) ∈ C
Transfer behavior
x(t) = e jΩx t → y(t) =
[n∑ν=0
cν(t)e−jΩxTν
]e jΩx t
x(t) T0 T T
c0(t) c1(t) cn−1(t) cn(t)
y(t)
Time-variant transfer behavior!Lehrstuhl fürDigitale Kommunikationssysteme
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Transmission Scenario 36 / 152
Time-Variant Multipath Propagation
Stochastic Modeling
y(t) = c(t)x(t) + w(t) , with c(t),w(t) ∈ C
c(t) associated with probability density function
Rice distributionRayleigh distributionNakagami distribution
additive white noise w(t)
Transmission Scheme
x(t)
c(t) w(t)
y(t)digitalsource
digitalmodulator
digitaldemodulator
digitalsink
baseband channel
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Wireless Communication Channel Passband Transmission
Contents
1 Transmission Scenario
2 Passband Transmission
3 Baseband Transmission
4 Time-Discrete Transmission
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Passband Transmission 37 / 152
Passband Transmission
Channel
real
center radian frequency ωc
bandwidth Bc
Bc
−ωc ωc ω
availablefrequency range
availablefrequency range
x0(t) y0(t)source transmitter
transmissionchannel
receiver sink
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Passband Transmission 38 / 152
Passband Transmission
Transmission Signal
real
center radian frequency ω0
bandwidth Bx
X0(jω)
Bx
−ω0 ω0 ω
x0(t) y0(t)source transmitter
transmissionchannel
receiver sink
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Passband Transmission 39 / 152
Passband Transmission
Transmission Signal
real
bandwidth Bx ≤ Bc
carrier radian frequency ω0
X0(jω) Bc
Bx
−ω0 ω0 ω
x0(t) y0(t)source transmitter
transmissionchannel
receiver sink
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Passband Transmission 40 / 152
Passband Transmission
Equivalent Baseband
channel is complex-valued
transmitter signal x(t) is complex-valued
receiver signal y(t) is complex-valued
X(jω)
Bc
Bxω
x0(t) y0(t)x(t) y(t)digitalsource
digitalmodulator
analogmodulator
transmissionchannel
analogdemodulator
digitaldemodulator
digitalesink
baseband channel
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Wireless Communication Channel Baseband Transmission
Contents
1 Transmission Scenario
2 Passband Transmission
3 Baseband Transmission
4 Time-Discrete Transmission
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Baseband Transmission 41 / 152
Baseband Transmission
Channel Resources
bandwidth Bc
dynamic Dc
duration Tc
x(t) y(t)
dynamic duration
bandwidth
Dc Tc
Bc
digitalsource
digitalmodulator
basebandchannel
digitaldemodulator
digitalsink
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Baseband Transmission 42 / 152
Baseband Transmission
Transmitter Signal
bandwidth Bx
dynamic Dx
duration Tx
x(t) y(t)
dynamic dynamicduration
duration
bandwidth bandwidth
Dx DxDcTx Tx Tc
Bx Bx
Bc
digitalsource
digitalmodulator
basebandchannel
digitaldemodulator
digitalsink
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Baseband Transmission
Limitations
bandwidth Bx ≤ Bc
dynamic Dx ≤ Dc
duration Tx ≤ Tc
x(t) y(t)
dynamic dynamic dynamicduration
duration
duration
bandwidth bandwidth bandwidth
Dx Dx DxDc DcTx Tx TxTc Tc
Bx Bx Bx
Bc Bc
digitalsource
digitalmodulator
basebandchannel
digitaldemodulator
digitalsink
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Baseband Transmission
Symbol Mapping
matching of the signal dynamic Dx ≤ Dc
finite alphabet A
information in symbols u(tk) ∈ A
u(t) x(t) y(t) v(t)
dynamic dynamic dynamicduration
duration
duration
bandwidth bandwidth bandwidth
Dx Dx DxDc DcTx Tx TxTc Tc
Bx Bx Bx
Bc Bc
digitalsource
impulseshaping
basebandchannel
symbolrecovery
digitalsink
symbolmapping
inversesymbolmapping
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Baseband Transmission 45 / 152
Baseband Transmission
Impulse Shaping
matching of the signal bandwidth Bx ≤ Bc
real pulse with finite energy q(t) ∈ R , Eq <∞crucial for symbol recovery
u(t) x(t) y(t) v(t)
dynamic dynamic dynamicduration
duration
duration
bandwidth bandwidth bandwidth
Dx Dx DxDc DcTx Tx TxTc Tc
Bx Bx Bx
Bc Bc
digitalsource
impulseshaping
basebandchannel
symbolrecovery
digitalsink
symbolmapping
inversesymbolmapping
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Baseband Transmission 46 / 152
Transmission Scheme
Transmission Scheme1 Nyquist pulse
z(tk) = u(tk)
2 separation into transmit and receive filter
S(jω) = R(jω)Q(jω)
3 channel with additive white noise
y(t) = x(t) + w(t)
4 optimal signal to noise ratio at decider
r(t) = q(−t)/Eq
u(t)
δT (t− t0)
s(t)z(t)
δT (t− t0)
PAM v(t)
transmitter receiver
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Baseband Transmission 46 / 152
Transmission Scheme
Transmission Scheme1 Nyquist pulse
z(tk) = u(tk)
2 separation into transmit and receive filter
S(jω) = R(jω)Q(jω)
3 channel with additive white noise
y(t) = x(t) + w(t)
4 optimal signal to noise ratio at decider
r(t) = q(−t)/Eq
u(t)
δT (t− t0)
q(t)x(t)
r(t)z(t)
δT (t− t0)
PAM v(t)
transmitter receiver
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Baseband Transmission 46 / 152
Transmission Scheme
Transmission Scheme1 Nyquist pulse
z(tk) = u(tk)
2 separation into transmit and receive filter
S(jω) = R(jω)Q(jω)
3 channel with additive white noise
y(t) = x(t) + w(t)
4 optimal signal to noise ratio at decider
r(t) = q(−t)/Eq
u(t)
δT (t− t0)
q(t)x(t)
w(t)
y(t)r(t)
z(t)
δT (t− t0)
PAM v(t)
transmitter additive noise receiver
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Baseband Transmission 46 / 152
Transmission Scheme
Transmission Scheme1 Nyquist pulse
z(tk) = u(tk)
2 separation into transmit and receive filter
S(jω) = R(jω)Q(jω)
3 channel with additive white noise
y(t) = x(t) + w(t)
4 optimal signal to noise ratio at decider
r(t) = q(−t)/Eq
u(t)
δT (t− t0)
q(t)x(t)
w(t)
y(t) q(−t)
Eq
z(t)
δT (t− t0)
PAM v(t)
transmitter additive noise receiver
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Baseband Transmission 47 / 152
Transmission Scheme
Transmission Scheme
5 low-pass band-limited pulse
Bx ≤ Bc
6 moderate timing jitter
|τ | T
u(t)
δT (t− t0)
q(t)x(t)
w(t)
y(t) q(−t)
Eq
z(t)
δT (t− t0)
PAM v(t)
transmitter additive noise receiver
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Baseband Transmission 47 / 152
Transmission Scheme
Transmission Scheme
5 low-pass band-limited pulse
Bx ≤ Bc
6 moderate timing jitter
|τ | T
u(t)
δT (t− t0)
q(t)x(t)
w(t)
y(t) q(−t)
Eq
z(t)
δT (t− t0 − τ)
PAM v(t)
transmitter additive noise receiver
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Baseband Transmission 48 / 152
Transmission Scheme
Transmission Scheme
7 stochastic model for flat fading and noise
8 minimizing decision error probability
transmitted u(tk) ∈ A
received z(tk − τ) ∈ C
decided v(tk) = Qz(tk − τ) ∈ A
u(t)
δT (t− t0)
q(t)x(t)
c(t) w(t)
y(t) q(−t)
Eq
z(t)
δT (t− t0 − τ)
PAM v(t)
transmitter basebandchannel
receiver
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Baseband Transmission 48 / 152
Transmission Scheme
Transmission Scheme
7 stochastic model for flat fading and noise8 minimizing decision error probability
transmitted u(tk) ∈ A
received z(tk − τ) ∈ C
decided v(tk) = Qz(tk − τ) ∈ A
u(t)
δT (t− t0)
q(t)x(t)
c(t) w(t)
y(t) q(−t)
Eq
z(t)
δT (t− t0 − τ)
PAM v(t)
transmitter basebandchannel
receiver
Lehrstuhl fürDigitale Kommunikationssysteme
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Wireless Communication Channel Time-Discrete Transmission
Contents
1 Transmission Scenario
2 Passband Transmission
3 Baseband Transmission
4 Time-Discrete Transmission
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Time-Discrete Transmission 49 / 152
Time-Discrete Channel
Preliminary Considerations
1 perfect timing synchronization
τ = 0
2 no receiver filter, transmitter filter is Nyquist filter
x(tk) = u(tk)
3 time-discrete channel
y(tk) = c(tk)x(tk) + w(tk)
4 decision
v(tk) = Qy(tk)
u(t)
δT (t− t0)
q(t)x(t)
c(t) w(t)
y(t) q(−t)
Eq
z(t)
δT (t− t0)
PAM v(t)
transmitter basebandchannel
receiver
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Time-Discrete Transmission 49 / 152
Time-Discrete Channel
Preliminary Considerations
1 perfect timing synchronization
τ = 0
2 no receiver filter, transmitter filter is Nyquist filter
x(tk) = u(tk)
3 time-discrete channel
y(tk) = c(tk)x(tk) + w(tk)
4 decision
v(tk) = Qy(tk)
u(t)
δT (t− t0)
s(t)x(t)
c(t) w(t)
y(t)
δT (t− t0)
PAM v(t)
transmitter basebandchannel
receiver
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Time-Discrete Transmission 49 / 152
Time-Discrete Channel
Preliminary Considerations
1 perfect timing synchronization
τ = 0
2 no receiver filter, transmitter filter is Nyquist filter
x(tk) = u(tk)
3 time-discrete channel
y(tk) = c(tk)x(tk) + w(tk)
4 decision
v(tk) = Qy(tk)
replacements
u(t)
δT (t− t0)
s(t)x(t)
c(t) w(t)
y(t)
δT (t− t0)
PAM v(t)
time-discrete channel
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Time-Discrete Transmission 49 / 152
Time-Discrete Channel
Preliminary Considerations
1 perfect timing synchronization
τ = 0
2 no receiver filter, transmitter filter is Nyquist filter
x(tk) = u(tk)
3 time-discrete channel
y(tk) = c(tk)x(tk) + w(tk)
4 decision
v(tk) = Qy(tk)
u(tk) = x(tk)
c(tk) w(tk)
y(tk)v(tk)
time-discrete channel
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Time-Discrete Transmission 50 / 152
Time-Discrete Channel
Time-Discrete Channel
y(tk) = c(tk)x(tk) + w(tk)
Communication scenariodynamic du
ration
bandwidth
DxDc Tx Tc
Bx
Bc
u(tk)x(tk)
c(tk)w(tk)
y(tk)v(tk)Tx Rx
Find an optimal strategy to exploit the communication resources!Lehrstuhl fürDigitale Kommunikationssysteme
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Time-Discrete Transmission 51 / 152
Time-Discrete Channel
Time-Discrete Channel
y(tk) = c(tk)x(tk) + w(tk)
Communication limits
U
equivo
cation
mutual information
irrelev
ance
V
u(tk)x(tk)
c(tk)w(tk)
y(tk)v(tk)Tx Rx
Use information theory to determine the communication limits!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Wireless Communications Single Input Single Output Systems
Wireless CommunicationsSingle Input Single Output Systems
Karlheinz Ochs
Chair of Digital Communication Systems
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Single Input Single Output Systems
Contents
1 Signal Space
2 AWGN Channel
3 Flat Fading Channel
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K. Ochs Wireless Communications WS 2017/18
Single Input Single Output Systems Signal Space
Contents
1 Signal Space
2 AWGN Channel
3 Flat Fading Channel
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Signal Space 52 / 152
Signal Space
Digital Modulator
u(t)digital
modulatorx(t)
1 Retrieve transmitted symbols
x(tk) = u(tk) , with tk = t0 + kT
2 Nyquist criterion
s(kT) =
1 for k = 00 for k ∈ Z \ 0
3 Minimal bandwidth
s(t) = si (Ωt/2) , with ΩT = 2π
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Signal Space 52 / 152
Signal Space
Digital Modulator
u(t)
δT (t− t0)
s(t) x(t)
1 Retrieve transmitted symbols
x(tk) = u(tk) , with tk = t0 + kT
2 Nyquist criterion
s(kT) =
1 for k = 00 for k ∈ Z \ 0
3 Minimal bandwidth
s(t) = si (Ωt/2) , with ΩT = 2π
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Signal Space 52 / 152
Signal Space
Digital Modulator
u(t)
δT (t− t0)
si(Ωt2
)x(t)
1 Retrieve transmitted symbols
x(tk) = u(tk) , with tk = t0 + kT
2 Nyquist criterion
s(kT) =
1 for k = 00 for k ∈ Z \ 0
3 Minimal bandwidth
s(t) = si (Ωt/2) , with ΩT = 2π
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Signal Space 52 / 152
Signal Space
Digital Modulator
u(t)
δT (t− t0)
si(Ωt2
)x(t)
1 Retrieve transmitted symbols
x(tk) = u(tk) , with tk = t0 + kT
2 Nyquist criterion
s(kT) =
1 for k = 00 for k ∈ Z \ 0
3 Minimal bandwidth
s(t) = si (Ωt/2) , with ΩT = 2π
Signaling with Nyquist rate!Lehrstuhl fürDigitale Kommunikationssysteme
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Signal Space 53 / 152
Signal Space
Digital modulator
u(t)
δT (t− t0)
si(Ωt2
)x(t)
Digitally modulated signal
x(t) =∞∑
k=−∞
u(tk) si
(Ω
2[t − tk]
)
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Signal Space 53 / 152
Signal Space
Digital modulator
u(t)
δT (t− t0)
si(Ωt2
)x(t)
Digitally modulated signal
x(t) =∞∑
k=−∞
xkϕk(t)
Definitions
1 samples
xk = u(tk)
2 base functions
ϕk(t) = si
(Ω
2[t − tk]
)−−• Φk(jω) = T rect
(2ωΩ
)e−jωtk
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Signal Space 54 / 152
Signal Space
Scalar Product
〈ϕk(t), ϕ`(t)〉 =∫ ∞
−∞ϕk(t)ϕ∗` (t)dt
Orthogonal base functions
〈ϕk(t), ϕ`(t)〉 = T
1 for k = `0 for k 6= `
Proof
1 〈ϕk(t), ϕ`(t)〉 =∫ ∞−∞ ϕk(t)ϕ∗` (t)dt
2 〈ϕk(t), ϕ`(t)〉 = 12π
∫ ∞−∞ Φk(jω)Φ∗` (jω)dω
3 〈ϕk(t), ϕ`(t)〉 = 12π
∫ ∞−∞ T2 rect
( 2ωΩ
)e−jω[tk−t`]dω
4 〈ϕk(t), ϕ`(t)〉 = T 1Ω
∫ Ω/2−Ω/2 e−jω[k−`]T dω
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Signal Space 55 / 152
Signal Space
Energy
Ex =
∫ ∞
−∞|x(t)|2dt = T
∞∑k=−∞
|xk|2
Proof
1 Ex = 〈x(t), x(t)〉 = ‖x(t)‖2
2 Ex = 〈∞∑
k=−∞xkϕk(t),
∞∑`=−∞
x`ϕ`(t)〉
3 Ex =∞∑
k=−∞
∞∑`=−∞
xkx∗` 〈ϕk(t), ϕ`(t)〉
4 Ex = T∞∑
k=−∞|xk|2
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Signal Space 56 / 152
Signal Space
Signal Vector (finite number of symbols)
x =[
x1, x2, . . . , xK]T
power
Px =Ex
T= ‖x‖2 , with ‖x‖2 = xHx =
K∑k=1
|xk|2
law of large numbers
1K
K∑k=1
|xk|2 ≈ E|X |2
relation to stochastic power
Px ≈ KPx , with Px = E|X |2Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Single Input Single Output Systems AWGN Channel
Contents
1 Signal Space
2 AWGN Channel
3 Flat Fading Channel
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AWGN Channel 57 / 152
Channel with Additive White Gaussian Noise
AWGN Channel
U
equivo
cation
mutual information
irrelev
ance
V
u(k)x(k)
1z(k)
y(k)v(k)Tx Rx
Remarkstransmitter sends message U to the receiver
K channel uses: Tx = KT ≤ Tc , k ∈ 1, . . . ,K
transmitter signal x(k) ∈ C with limited power Px ≤ P
additive noise z(k) ∈ C
independent and identically distributed
normal distribution, with zero mean and variance σ2z = Pz
receiver signal y(k) ∈ C
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AWGN Channel 57 / 152
Channel with Additive White Gaussian Noise
AWGN Channel
u(k)x(k)
1z(k)
y(k)v(k)Tx Rx
Remarks
transmitter sends message U to the receiver
K channel uses: Tx = KT ≤ Tc , k ∈ 1, . . . ,K
transmitter signal x(k) ∈ C with limited power Px ≤ P
additive noise z(k) ∈ Cindependent and identically distributed
normal distribution, with zero mean and variance σ2z = Pz
receiver signal y(k) ∈ C
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AWGN Channel 58 / 152
Channel with Additive White Gaussian Noise
AWGN Channel
Signal flow diagram
x(k)
z(k)
y(k)
Mathematical model
y(k) = x(k) + z(k) , 1 ≤ k ≤ K
Px =1K
K∑k=1
|x(k)|2 ≤ P
z(k) ∼ N (0,Pz)
Communication
encoding of message U
x(k) ∈ C is a symbol of a finite alphabet A
y(k) ∼ N (x(k),Pz)
Highest data rate for a reliable transmission?Lehrstuhl fürDigitale Kommunikationssysteme
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AWGN Channel 59 / 152
Channel with Additive White Gaussian Noise
AWGN Channel
Signal flow diagram
x
z
y
Mathematical model
y = x+ z
‖x‖ =√
KPx
‖z‖ =√
KPz
Remarks
transmitted signal vector x =[
x(1), . . . , x(K)]T
noise vector z =[
z(1), . . . , z(K)]T
received signal vector y =[
y(1), . . . , y(K)]T
Received vector y lies in a hypersphere with center x and radius√
KPz !Lehrstuhl fürDigitale Kommunikationssysteme
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AWGN Channel 60 / 152
Channel with Additive White Gaussian Noise
AWGN Channel
Illustration
xz
y0
noise hypersphere
Mathematical model
y = x+ z
E‖y‖2 = E‖x‖2+ E‖z‖2x and z are independent
Remarks
E‖y‖ ≤√E‖y‖2 , Jensen’s inequality
E‖y‖2 = E‖x‖2+ E‖z‖2 , independence
E‖x‖2 ≈ KPx , E‖z‖2 ≈ KPz , law of large numbers
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AWGN Channel 60 / 152
Channel with Additive White Gaussian Noise
AWGN Channel
Illustration
√ KPx
√KPz
E‖y‖≤√
K[Px +Pz]
noise hypersphere
Mathematical model
y = x+ z
E‖y‖2 = E‖x‖2+ E‖z‖2x and z are independent
Upper bound
E‖y‖ ≤√
K[Px + Pz]
All received signal vectors lie in a hypersphere of radius√
K[Px + Pz]!
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AWGN Channel 61 / 152
Channel with Additive White Gaussian Noise
AWGN Channel
Illustration
1
2
3
M
√KPz
√KPz
√KPz
√KPz
√K[Px + Pz ]
Decoding
transmitted vector x is center of thehypersphere
M nonoverlapping hypersheres
received vector y within hypershperebelongs to the center x
How many noise hyperspheres fit into the received vector hypersphere?
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AWGN Channel 62 / 152
Channel with Additive White Gaussian Noise
AWGN Channel
Illustration
1
2
3
M
√KPz
√KPz
√KPz
√KPz
√K[Px + Pz ]
Hypersphere
K (real) dimensions
radius r
volume VK(r) ∼ rK
Nonoverlapping hyperspheres
K complex dimensions
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AWGN Channel 62 / 152
Channel with Additive White Gaussian Noise
AWGN Channel
Illustration
1
2
3
M
√KPz
√KPz
√KPz
√KPz
√K[Px + Pz ]
Hypersphere
K (real) dimensions
radius r
volume VK(r) ∼ rK
Nonoverlapping hyperspheres
2K real dimensions
M ≤ V2K(√
2K[Px + Pz])
V2K(√
2KPz)
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AWGN Channel 62 / 152
Channel with Additive White Gaussian Noise
AWGN Channel
Illustration
1
2
3
M
√KPz
√KPz
√KPz
√KPz
√K[Px + Pz ]
Hypersphere
K (real) dimensions
radius r
volume VK(r) ∼ rK
Nonoverlapping hyperspheres
2K real dimensions
M ≤ VK(2K[Px + Pz])
VK(2KPz)
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AWGN Channel 62 / 152
Channel with Additive White Gaussian Noise
AWGN Channel
Illustration
1
2
3
M
√KPz
√KPz
√KPz
√KPz
√K[Px + Pz ]
Hypersphere
K (real) dimensions
radius r
volume VK(r) ∼ rK
Nonoverlapping hyperspheres
2K real dimensions
M ≤ [1 + Px/Pz]K
Lehrstuhl fürDigitale Kommunikationssysteme
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AWGN Channel 62 / 152
Channel with Additive White Gaussian Noise
AWGN Channel
Illustration
1
2
3
M
√KPz
√KPz
√KPz
√KPz
√K[Px + Pz ]
Hypersphere
K (real) dimensions
radius r
volume VK(r) ∼ rK
Nonoverlapping hyperspheres
2K real dimensions
M ≤ [1 + Px/Pz]K
Upper bound for nonoverlapping hyperspheres
M ≤ [1 + ΓSNR]K , with ΓSNR =
Px
Pz=E|x(k)|2E|z(k)|2
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AWGN Channel 63 / 152
Channel with Additive White Gaussian Noise
AWGN Channel Upper bound for the data rate
ld(M) ≤ K ld (1 + ΓSNR)
M different messages can be reliably distinguished
each message can be encoded with ld(M) bits
data rate R is the ratio of bits per channel use
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AWGN Channel 63 / 152
Channel with Additive White Gaussian Noise
AWGN Channel Upper bound for the data rate
R =ld(M)
K≤ ld (1 + ΓSNR)
M different messages can be reliably distinguished
each message can be encoded with ld(M) bits
data rate R is the ratio of bits per channel use
Maximum data rate for a reliable communication?
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AWGN Channel 63 / 152
Channel with Additive White Gaussian Noise
AWGN Channel Upper bound for the data rate
R =ld(M)
K≤ ld (1 + ΓSNR)
M different messages can be reliably distinguished
each message can be encoded with ld(M) bits
data rate R is the ratio of bits per channel use
Maximum data rate for a reliable communication?
AWGN channel capacity
C = ld(1 + ΓSNR) in bits/channel use
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AWGN Channel 63 / 152
Channel with Additive White Gaussian Noise
AWGN Channel Upper bound for the data rate
R =ld(M)
K≤ ld (1 + ΓSNR)
M different messages can be reliably distinguished
each message can be encoded with ld(M) bits
data rate R is the ratio of bits per channel use
Maximum data rate for a reliable communication?
AWGN channel capacity
C = ld(1 + ΓSNR) in bits/channel use
This is only a heuristic approach!Information theory provides a fundamental derivation!
Lehrstuhl fürDigitale Kommunikationssysteme
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Single Input Single Output Systems Flat Fading Channel
Contents
1 Signal Space
2 AWGN Channel
3 Flat Fading Channel
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Flat Fading Channel 64 / 152
Flat Fading Channel
Flat Fading Channel
u(k)x(k)
h(k)z(k)
y(k)v(k)Tx Rx
Remarks
transmitter signal x(k) ∈ C with limited power Px ≤ P
channel gain h(k) ∈ Csmall-scale fading caused by echoes of the transmitted signaltransmitted signal period is larger than multi-path delay spread
additive noise z(k) ∈ Cindependent and identically distributed (i.i.d.)
normal distribution, with zero mean and variance σ2z = Pz
receiver signal y(k) ∈ CLehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Flat Fading Channel 65 / 152
Flat Fading Channel
Flat Fading Channel
x(k)
h(k) z(k)
y(k)
Channel capacity?
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Flat Fading Channel 65 / 152
Flat Fading Channel
Flat Fading Channel
x(k)
h(k) z(k)z(k)
y(k)y(k) h(k)x(k)
Idea
equivalent to AWGN channel
surrogate input signal is product of input signal times channel gain
time-variant signal to noise ratio
γSNR(k) =E|h(k)x(k)|2E|z(k)|2 = ΓSNR|h(k)|2
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Flat Fading Channel 65 / 152
Flat Fading Channel
Flat Fading Channel
x(k)
h(k) z(k)
y(k)
Idea
equivalent to AWGN channel
surrogate input signal is product of input signal times channel gain
time-variant signal to noise ratio
γSNR(k) =E|h(k)x(k)|2E|z(k)|2 = ΓSNR|h(k)|2
Channel capacity
C(k) = ld(1 + ΓSNR|h(k)|2) , with ΓSNR =E|x(k)|2E|z(k)|2
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Flat Fading Channel 66 / 152
Flat Fading Channel
Rayleigh Fading Channel gain
distribution of real and imaginary part
Reh, Imh ∼ N (0, σ2)
transformation to polar coordinates h = re jϕ
magnitude r > 0 has Rayleigh distribution
fr(r) = u(r)rσ2 e−
r2
2σ2
phase ϕ ∈ (−π, π] has uniform distribution
fϕ(ϕ) =1
2πrect
(ϕπ
)
Real- and imaginary part have zero-mean, which indicates no line of sight!
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Flat Fading Channel 67 / 152
Flat fading Channel
Rayleigh Distribution (σ = 1)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
r
f r(r)
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Flat Fading Channel 68 / 152
Flat Fading Channel
Rician Fading Channel gain
distribution of real and imaginary part
Reh ∼ N (µRe, σ2) , Imh ∼ N (µIm, σ
2)
transformation to polar coordinates h = re jϕ
magnitude r > 0 has Rice distribution
fr(r) = u(r)rσ2 I0
( rµσ2
)e−
r2+µ2
2σ2 , with µ =√µ2
Re + µ2Im
and modified Bessel function of the first kind with order zero I0
Rician factor K =µ
2σ2
K = 0 yields Rayleigh fading
Line of sight if Rician factor is greater than zero!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Flat Fading Channel 69 / 152
Flat Fading Channel
Rician Distribution (σ = 1)
0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
r
f r(r)
µ = 0µ = 0.5µ = 1µ = 2µ = 4
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Flat Fading Channel 70 / 152
Flat Fading Channel
Nakagami Fading Channel gain
sum of multiple i.i.d. Rayleigh fading signals has Nakagami distributedmagnitude
magnitude r > 0 has Nakagami distribution
fr(r) = u(r)2
Γ (m)
[mΩ
]mr2m−1 e−
mr2Ω ,
with
gamma function Γ (m) =
∫ ∞0
e−rrm−1dr
received signal average power Ω = Er2
shape factor m =Ω2
E[r −Ω]2≥
12
m = 1 yields Rayleigh fading
Useful to model urban radio multipath channels!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Flat Fading Channel 71 / 152
Flat Fading Channel
Nakagami Distribution
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30
0.2
0.4
0.6
0.8
1
1.2
1.4
r
f r(r
)
Ω = 1,m = 0.5Ω = 1,m = 1Ω = 2,m = 1Ω = 3,m = 1Ω = 1,m = 2Ω = 2,m = 2Ω = 1,m = 3
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Wireless Communications Multiple Input Multiple Output Systems
Wireless CommunicationsMultiple Input Multiple Output Systems
Karlheinz Ochs
Chair of Digital Communication Systems
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Multiple Input Multiple Output Systems
Contents
1 Transmission Scenario
2 MIMO Detectors
3 Random Channels
4 Eigenmode Decomposition
5 Capacity and Degrees of Freedom
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Multiple Input Multiple Output Systems Transmission Scenario
Contents
1 Transmission Scenario
2 MIMO Detectors
3 Random Channels
4 Eigenmode Decomposition
5 Capacity and Degrees of Freedom
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Transmission Scenario 72 / 152
MIMO Passband Transmitter
MIMO Passband Transmitter
u(t)
S
P
symbolmapping
symbolmapping
δT (t− t0)
δT (t− t0)
q(t)
q(t)
ejω0t
ejω0t
Re
Re
u1(t)
um(t)
x1(t)
xm(t)
x10(t)
xm0(t)
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Transmission Scenario 73 / 152
MIMO Passband Receiver
MIMO Passband Receiver
PAM
PAM
e−jω0t
e−jω0t
LP
LP
r(t)
r(t)
δT (t− t0)
δT (t− t0)
detection
v(t)
w10(t)
wn0(t)
y10(t)
yn0(t)
y1(t)
yn(t)
v1(t)
vn(t)
Instead of single decisions a combined detection of transmitted symbols!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Transmission Scenario 74 / 152
MIMO passband transmission
MIMO Passband Transmitterm transmit antennas
symbol vector u(tk) ∈ Am , with u(t) =[
u1(t), . . . , um(t)]T
equivalent low-pass signal vector x(t) =∞∑
k=−∞
u(tk)q(t − tk)
baseband signal vector x0(t) = Rex(t)e jω0t
,
with x0(t) =[
x10(t), . . . , xm0(t)]T
MIMO Passband Receivern receive antennas
additive noise w0(t) =[
w10(t), . . . ,wn0(t)]T
baseband signal y0(t) =[
y10(t), . . . , yn0(t)]T
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Transmission Scenario 75 / 152
MIMO Baseband Transmission
MIMO Baseband Transmission
AM
AM
u(t)
S
P
P
P
symbolmapping
symbolmapping
δT (t− t0)
δT (t− t0)
δT (t− t0)
δT (t− t0)
q(t)
q(t)
u1(t)
um(t)
x1(t)
xm(t)
c11(t)
cn1(t)
c1m(t)
cnm(t)
r(t)
r(t)
detection
v(t)
w1(t)
wn(t)
y1(t)
yn(t)
v1(t)
vn(t)
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Transmission Scenario 76 / 152
MIMO Baseband Transmission (simplified)
MIMO Baseband Transmission (simplified)
AM
AM
u(t)
S
P
P
P
symbolmapping
symbolmapping
δT (t− t0)
δT (t− t0)
δT (t− t0)
δT (t− t0)
s(t)
s(t)
u1(t)
um(t)
x1(t)
xm(t)
h11(t)
hn1(t)
h1m(t)
hnm(t)
detection
v(t)
z1(t)
zn(t)
y1(t)
yn(t)
v1(t)
vn(t)
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Transmission Scenario 77 / 152
MIMO Digital Baseband Transmission
MIMO Digital Baseband Transmission
u(k)
x1(k)
xm(k)
h11(k)
hn1(k)
h1m(k)
hnm(k)
z1(k)
zn(k)
y1(k)
yn(k)
v(k)Tx Rx
Flat fading channel
y1(k)...
yn(k)
=
h11(k) · · · h1m(k)...
. . ....
hn1(k) · · · hnm(k)
x1(k)
...xm(k)
+
z1(k)...
zn(k)
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Transmission Scenario 78 / 152
MIMO Digital Baseband Transmission
MIMO Digital Baseband Transmission
Eu(k) Txx(k)
H(k)z(k)
y(k)v(k)
Transmitter
maps message U to signal x(k)
transmitter signal has limited power
Flat fading channel
y(k) =H(k)x(k) + z(k)
channel matrix has almost sure full rankReceiver
knows the channel state from estimationretrieves message V from signal y(k)
Channel capacity?Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Multiple Input Multiple Output Systems MIMO Detectors
Contents
1 Transmission Scenario
2 MIMO DetectorsZero-Forcing DetectorMinimum Mean-Squared Error DetectorMaximum Likelihood Detector
3 Random Channels
4 Eigenmode Decomposition
5 Capacity and Degrees of Freedom
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
MIMO Detectors 79 / 152
MIMO Detectors
MIMO Detector
x(k)
H(k) z(k)
y(k) MIMO-detector
x(k)
Remarks
m transmit and n receive antennas
flat fading channel
y(k) =H(k)x(k) + z(k)
channel matrix has (almost sure) full rank
H(k) ∈ Cn×m , with rankH(k) = minm, n
MIMO detector estimates transmitted signalLehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
MIMO Detectors Zero-Forcing Detector 80 / 152
Zero-Forcing Detector
Scenario 1same number of antennas at transmitter and receiverrank(H(k)) = m = n
equation system
x(k)H(k) z(k)y(k) +=
Zero-Forcing Detection
x =H−1y ⇒ x = x+H−1z
Remarkssimple case
multiplication with H−1 can significantly amplify the noiseLehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
MIMO Detectors Zero-Forcing Detector 81 / 152
Zero-Forcing Detector
Scenario 2
more transmit antennas than receive antennas
rank(H(k)) = n < m
m− n times underdetermined equation system
x(k)H(k) z(k)z(k)y(k) +=
Zero-Forcing Detection
x =H sy + [1−H sH]v , v arbitrary ⇒ Hx =Hx+HH sz
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
MIMO Detectors Zero-Forcing Detector 82 / 152
Zero-Forcing Detector
Remarks
semi-inverse H s , with HH sH =H and H sHH s =H s
Moore-Penrose right pseudoinverse can be used
H s =H+ =HH[HHH]−1
1−H sH is projection matrix to null space of H
Hx =HH sy
multiplication with HH s can significantly amplify the noise
Improper approach because of infinite many solutions!
Remedy
time variance of the channel is helpful
sent x again to increase number of linearly independent equationsLehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
MIMO Detectors Zero-Forcing Detector 83 / 152
Zero-Forcing Detector
Scenario 3
more receive antennas than transmit antennas
rank(H(k)) = m < n
n− m times overdetermined equation system
x(k)H(k) z(k)z(k)y(k) +=
Zero-Forcing Detection
x =H+y ⇒ x = x+H+z
Lehrstuhl fürDigitale Kommunikationssysteme
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MIMO Detectors Zero-Forcing Detector 84 / 152
Zero-Forcing Detector
Remarks
Moore-Penrose left pseudoinverse H+ = [HHH]−1HH
multiplication with H+ can significantly amplify the noise
Solution is an optimal approximation!
Optimization problem
x = argminxJ
, with J = ‖y −Hx‖2
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
MIMO Detectors Zero-Forcing Detector 85 / 152
Zero-Forcing Detector
Solution
necessary and sufficient conditions
∂J∂x
= 0T and∂J∂xH = 0
Wirtinger derivatives, x and xH independent
J = J∗ ⇒ ∂J∂x
=
[∂J∂xH
]H
,∂J∂x
= 0T ⇔ ∂J∂xH = 0
J = [yH − xHHH][y −Hx]
∂J∂xH = −HH[y −Hx]
HHy =HHHx
x =[HHH
]−1HHy
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
MIMO Detectors Minimum Mean-Squared Error Detector 86 / 152
Minimum Mean-Squared Error Detector
Detection
x =Dy , with y =Hx+ z
A minimum mean-squared error detector considers noise!
Optimization problem
D = argminDJ
, with J = E‖x− x‖2
Approach
J = J∗
necessary and sufficient conditions
∂J∂D
= 0T and∂J∂DH = 0
Wirtinger derivatives, D and DH independentLehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
MIMO Detectors Minimum Mean-Squared Error Detector 87 / 152
Minimum Mean-Squared Error Detector
Realness
J = J∗
necessary and sufficient condition
∂J∂DH = 0
Some basics
‖x‖2 = xHx = tr(xxH)
Kxy = ExyH
E‖x‖2
= tr (Kxx)
∂
∂Mtr (AMB) = ATBT
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
MIMO Detectors Minimum Mean-Squared Error Detector 88 / 152
Minimum Mean-Squared Error Detector
Error reformulation
x− x = x−Dy
[x−Dy][xH − yHDH] = xxH −DyxH +DyyHDH − xyHDH
J = tr(Kxx −DKyx +DKyyDH −KxyD
H)
J = tr(Kxx)− tr(DKyx) + tr(DKyyDH)− tr(KxyD
H)
Solution
∂J∂DH = [DKyy]
T −KTxy
DKyy =Kxy , regularity of Kyy is assumed!
Minimum Mean-Squared Error Detection
x = Dy , with D =KxyK−1yy
Lehrstuhl fürDigitale Kommunikationssysteme
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MIMO Detectors Minimum Mean-Squared Error Detector 89 / 152
Minimum Mean-Squared Error Detector
Uncorrelation
x,z uncorrelated
Kxy = Ex[xHHH + zH] =KxxH
H
Kyy = E[Hx+ z]
[xHHH + zH] =Kzz +HKxxH
H
Detection matrix
D =KxxHH [Kzz +HKxxH
H]−1
Minimum Mean-Squared Error Detection
x =KxxHH [Kzz +HKxxH
H]−1[Hx+ z]
Solution plausible?
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
MIMO Detectors Minimum Mean-Squared Error Detector 90 / 152
Minimum Mean-Squared Error Detector
Simplifying assumptions
1 Symbols and noise are spatially uncorrelated
Kxx = Px1 , Kzz = Pz1 , with Pz = σ2z
Detection matrix
D =HH[HHH +
1ΓSNR
1]−1
, with ΓSNR =Px
Pz
2 No noise
z = 0 , Pz = 0 ,1
ΓSNR= 0 , Kyy = PxHH
H
Detection matrix
D =H+ , with H+ =HH [HHH]−1
This MMSE detector is weak for high SNR and m < n!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
MIMO Detectors Maximum Likelihood Detector 91 / 152
Maximum Likelihood Detector
Stochastic Channel Model
y =Hx+ z , with known fy|x(y|x)
Detection
x = argmaxx∈Am fy|x(y|x)
Interpretation
fy|x(y|x) ≥ fy|x(y|x) for all x ∈ Am
A maximum likelihood detector chooses the most likely sent x!
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
MIMO Detectors Maximum Likelihood Detector 92 / 152
Maximum Likelihood Detector
Reformulation
z = y −Hx
fy|x(y|x) = fz(y −Hx)
Maximum Likelihood Detection
x = argmaxx∈Am fz(y −Hx)
Simplifying assumptions
1 circularly symmetric complex Gaussian random variables
fz =1
det(πKzz)exp
(−zHK−1
zzz)
2 spatially uncorrelated noise
Kzz = Pz1 , with Pz = σ2z
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
MIMO Detectors Maximum Likelihood Detector 93 / 152
Maximum Likelihood Detector
Intermediate results
fz =1
[πσ2z ]n
exp
(− 1σ2
z‖z‖2
)
x = argmaxx∈Am
1
[πσ2z ]n
exp
(− 1σ2
z‖y −Hx‖2
)Exponential function is strictly monotonically increasing!
Maximum Likelihood Detection
x = argminx∈Am
‖y −Hx‖
Remarksgeometrical task to find x ∈ Am, such that Hx has minimal distance to ythere are |A|m possible vectors x ∈ Am
Effort increases exponentially with the number of transmit antennas!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Multiple Input Multiple Output Systems Random Channels
Contents
1 Transmission Scenario
2 MIMO Detectors
3 Random ChannelsSpatially Uncorrelated ChannelSpatially Correlated Channel
4 Eigenmode Decomposition
5 Capacity and Degrees of Freedom
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Random Channels 94 / 152
Random MIMO Channel
Random MIMO Channel
x(k)
H(k) z(k)
y(k)
flat fading (frequency-non-selective)
channel matrix H(k) has (almost sure) full rank
elements hµν(k) of H(k) are random variables
Random channel matrix!
Example
no line of sight between transmit antenna µ and receive antenna ν
modeled e. g. with Rayleigh-distributed |hνµ(k)|Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Random Channels Spatially Uncorrelated Channel 95 / 152
Spatially Uncorrelated Channel
Independent and Identically Distributed Random MIMO Channel
Elements hµν(k) of channel matrix
independent and identically distributed random variableszero-mean
Ehµν(k) = 0
spatially uncorrelated
Ehνµ(k)hλκ(k) =σ2 for ν = λ and µ = κ0 otherwise
Channel Correlation Matrix
KHH = σ21 , with KHH = E
vec(H)vec(H)H =KHHH
Spatially uncorrelated MIMO channels are of minor practical importance!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Random Channels Spatially Correlated Channel 96 / 152
Spatially Correlated Channel
Dense antenna array
improved directional characteristiccorrelation between antenna signals
Spatially Correlated Channel
x(k)
√Kx(k) HIID(k)
√Ky(k)
Hz(k)
y(k)
at transmitter Kx =KHx ≥ 0 , with Kx =
√Kx
H√Kx
at receiver Ky =KHy ≥ 0 , with Ky =
√Ky
H√Ky
Random channel matrix
H =√Ky
HHIID√Kx
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Random Channels Spatially Correlated Channel 97 / 152
Spatially Correlated Channel
Intermediate calculations
vec(H) = vec(√
KyHHIID√Kx
)=[√Kx
T ⊗√Ky
H]
vec(HIID)
vec(H)H = vec(HIID)H[√Kx∗ ⊗
√Ky
]Derivation of the channel correlation matrix
KHH =[√Kx
T ⊗√Ky
H]E
vec(HIID)vec(HIID)H [√Kx
∗ ⊗√Ky
]= σ2
[√Kx
T ⊗√Ky
H] [√
Kx∗ ⊗
√Ky
]= σ2
[√Kx
T√Kx∗]⊗[√Ky
H√Ky
]= σ2
[√Kx
H√Kx
]∗⊗[√Ky
H√Ky
]Channel Correlation Matrix
KHH = σ2K∗x ⊗Ky
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Multiple Input Multiple Output Systems Eigenmode Decomposition
Contents
1 Transmission Scenario
2 MIMO Detectors
3 Random Channels
4 Eigenmode DecompositionSingular Value DecompositionEigenmodes of a MIMO Channel
5 Capacity and Degrees of Freedom
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Eigenmode Decomposition Singular Value Decomposition 98 / 152
Singular Value Decomposition
Singular Value Decomposition
H = UΣV H =[U1 U2
] [ Σr 00 0
] [V H
1
V H2
],
with
H ∈ Cn×m , r = rank(H) ≤ minm, n
U ∈ Cn×n , UHU = UUH = 1n , U1 ∈ Cn×r , U2 ∈ Cn×n−r
V ∈ Cm×m , V HV = V V H = 1m , V1 ∈ Cm×r , V2 ∈ Cm×m−r
Σ ∈ Cn×m , Σr = diag(σ1, . . . , σr) > 0
For every matrix H ∈ Cn×m with rank r there exists asingular value decomposition with positive singular values σ1, . . . , σr!
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Eigenmode Decomposition Eigenmodes of a MIMO Channel 99 / 152
Eigenmodes of a MIMO Channel
Scenario
PSfrag
x(k)Pre-
Encoder
x′(k)
H(k) z′(k)
y′(k)Post-
Encodery(k)
Transmitter has channel state information (CSIT)!
Channel
y′ =Hx′ + z′
Encoders
use encoders to decouple transmission paths
exploit singular value decomposition
H = UΣV H ⇔ UHHV = Σ
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Eigenmode Decomposition Eigenmodes of a MIMO Channel 100 / 152
Eigenmodes of a MIMO Channel
Scenario
PSfrag
x(k)
V (k)
x′(k)
H(k) z′(k)
y′(k)
UH(k)
y(k)
Encoders
pre-encoding
x′ = V x
post-encoding
y = UHy′
Encoded channel
y = UH [HV x+ z′]
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Eigenmode Decomposition Eigenmodes of a MIMO Channel 101 / 152
Eigenmodes of a MIMO Channel
Scenario
x(k)
V (k)
x′(k)
H(k) UH(k) UH(k)z′(k)
y(k)
Equivalent channel
y = UHHV x+UHz′
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Eigenmode Decomposition Eigenmodes of a MIMO Channel 101 / 152
Eigenmodes of a MIMO Channel
Scenario
x(k)
Σ(k) z(k)
y(k)
Equivalent channel
y = Σx+ z ,
with
encoded channel
Σ = UHHV
encoded noise
z = UHz′ , with ‖z‖2 =∥∥z′∥∥2
Noise power conserved after unitary transformation!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Eigenmode Decomposition Eigenmodes of a MIMO Channel 101 / 152
Eigenmodes of a MIMO Channel
Scenario
x(k)
Σ(k) z(k)
y(k)
Equivalent channel
y = Σx+ z
[yr
yn−r
]=
[Σr 00 0
] [xr
xm−r
]+
[zr
zn−r
]
yν = σνxν + zν for ν = 1, . . . , ryν = zν for ν = r + 1, . . . , n
Encoding yields r (relevant) parallel SISO channels!
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Eigenmode Decomposition Eigenmodes of a MIMO Channel 102 / 152
Eigenmodes of a MIMO Channel
Unitary transformation
Noise with circularly-symmetric and zero mean complex normal distribution
z = UHz′
z′ ∼ N (0,Kz′z′) i. e. f ′z(z) =exp
(−zHK−1
z′z′z)
|πKz′z′ |
z ∼ N (0,Kzz) , Kzz = UHKz′z′U
In addition, spatially uncorrelated with identical power
Kz′z′ = σ2z 1
z ∼ N(
0, σ2z 1)
, Kzz =Kz′z′
Stochastic properties conserved after unitary transformation!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Multiple Input Multiple Output Systems Capacity and Degrees of Freedom
Contents
1 Transmission Scenario
2 MIMO Detectors
3 Random Channels
4 Eigenmode Decomposition
5 Capacity and Degrees of FreedomCapacityDegrees of Freedom
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Capacity and Degrees of Freedom Capacity 103 / 152
Capacity of SISO Channels
Capacity of SISO Channels
AWGN channel
x(k)
z(k)
y(k)
C = ld(1 + ΓSNR) , with ΓSNR =Px
Pz=E|x(k)|2E|z(k)|2
Flat fading channel
x(k)
h(k) z(k)
y(k)
C(k) = ld(1 + ΓSNR|h(k)|2)
Capacity of a MIMO channel?Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Capacity and Degrees of Freedom Capacity 104 / 152
Capacity of MIMO Channels
Capacity of a Decomposed MIMO Channel
Parallel SISO channels
x (k)
σ (k) z(k)
y(k)
y%(k) = σ%(k) x%(k) + z%(k) for % = 1, . . . , r
Simplifying assumption
ΓSNR =E|x%(k)|2E|z%(k)|2
Capacity
C(k) =r∑
%=1
C%(k) , with C%(k) = ld(
1 + ΓSNRσ2% (k)
)Reformulation independent from singular value decomposition?
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Capacity and Degrees of Freedom Capacity 105 / 152
Capacity of MIMO Channels
Capacity of a Decomposed MIMO Channel
C(k) =r∑
%=1
ld(
1 + ΓSNRσ2% (k)
)
Refomulation
1r∑
%=1ld(ξ%) = ld
(r∏
%=1ξ%
)
C(k) = ld
(r∏
%=1
[1 + ΓSNRσ
2% (k)
])
2r∏
%=1ξ% = |diag(ξ%)|
C(k) = ld(∣∣∣diag
(1 + ΓSNRσ
2% (k)
)∣∣∣)Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Capacity and Degrees of Freedom Capacity 106 / 152
Capacity of MIMO Channels
Refomulation
3 diag(1 + ξ%) = 1 + diag(ξ%) , diag(αξ%) = α diag(ξ%) ,
C(k) = ld(∣∣∣1 + ΓSNR diag
(σ2% (k)
)∣∣∣)4 diag(ξ2
%) = diag(ξ%)2
C(k) = ld(∣∣∣1 + ΓSNRΣ
2r (k)
∣∣∣)
5
∣∣∣∣[ A 00 1
]∣∣∣∣ = |A|C(k) = ld
(∣∣1m + ΓSNRΣH(k)Σ(k)
∣∣)6 HHH = V ΣHΣV H i. e. ΣHΣ = V HHHHV
C(k) = ld(∣∣V H(k)
[1m + ΓSNRH
H(k)H(k)]V (k)
∣∣)Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Capacity and Degrees of Freedom Capacity 107 / 152
Capacity of MIMO Channels
Refomulation
7 |AB| = |A| |B| ,∣∣A−1
∣∣ = |A|−1
C(k) = ld(∣∣1m + ΓSNRH
H(k)H(k)∣∣)
8 |1n +AB| = |1m +BA| for A ∈ Cn×m , B ∈ Cm×n
C(k) = ld(∣∣1n + ΓSNRH(k)HH(k)
∣∣)Hint
m ≤ n
C(k) = ld(∣∣1m + ΓSNRH
H(k)H(k)∣∣)
n ≤ m
C(k) = ld(∣∣1n + ΓSNRH(k)HH(k)
∣∣)No need for singular value decomposition!
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Capacity and Degrees of Freedom Capacity 108 / 152
Capacity of MIMO Channels
Capacity of a MIMO Channel
C(k) = maxKxx
ld(∣∣Kzz +HKxxH
H∣∣)− ld (|Kzz|)
s. t. trace(Kxx) ≤ P
This is a (convex) optimization problem!
Special case
Kxx = Px1 , Kzz = σ2z 1 , ΓSNR =
Px
σ2z
C(k) = ld(∣∣1n + ΓSNRH(k)HH(k)
∣∣)Capacity of r parallel SISO channels with constant signal to noise ratio!
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Capacity and Degrees of Freedom Degrees of Freedom 109 / 152
Degrees of Freedom
Degrees of Freedom
η = limΓSNR→∞
Cld(ΓSNR)
average of symbols per channel use
synonymous DoF
closely related to multiplexing gain
SISO channel
η = 1
MIMO channel
η = r
The degrees of freedoms are equal to the rank of the channel matrix!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Capacity and Degrees of Freedom Degrees of Freedom 110 / 152
Degrees of Freedom
Proof
1 SISO is a special case of MIMO with r = 1
2 η = limΓSNR→∞
r∑%=1
ld(1+σ2%ΓSNR)
ld(ΓSNR)=
r∑%=1
limΓSNR→∞
ld(1+σ2%ΓSNR)
ld(ΓSNR)
3 limΓSNR→∞
ld(1+σ2%ΓSNR)
ld(ΓSNR)= limΓSNR→∞
ld(σ2%ΓSNR)
ld(ΓSNR)
4 limΓSNR→∞
ld(1+σ2%ΓSNR)
ld(ΓSNR)= limΓSNR→∞
[ld(σ2
%)
ld(ΓSNR)+ 1]
5 limΓSNR→∞
ld(1+σ2%ΓSNR)
ld(ΓSNR)= 1
6 η =r∑
%=11 = r
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Capacity and Degrees of Freedom Degrees of Freedom 111 / 152
Degrees of Freedom
Degrees of Freedom
η = limΓSNR→∞
Cld(ΓSNR)
Interpretation
η = limΓSNR→∞
CCSISO
, with CSISO = ld(1 + |σ|2ΓSNR)
Multiplexing Gain
C ≈ ηCSISO for ΓSNR →∞
Asymptotic measurement for the high signal to noise ratio regime!
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Wireless Communications Optimal Transmission Strategies
Wireless CommunicationsOptimal Transmission Strategies
Karlheinz Ochs
Chair of Digital Communication Systems
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Optimal Transmission Strategies
Contents
1 Maximum Ratio Combining
2 Maximum Ratio Transmission
3 Water-Filling
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Optimal Transmission Strategies Maximum Ratio Combining
Contents
1 Maximum Ratio Combining
2 Maximum Ratio Transmission
3 Water-Filling
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Maximum Ratio Combining 112 / 152
Maximum Ratio Combining
Scenario
w(k) Txx(k)
h1(k)
hn(k)
z1(k)
zn(k)
y1(k)
yn(k)
Rx w(k)
Transmitter has no channel state information (no CSIT)
Channel
y(k) = h(k)x(k) + z(k) , with Kzz = σ2z 1
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Maximum Ratio Combining 113 / 152
Maximum Ratio Combining
Strategy
x(k) = hH(k)y(k)
Channel with strategy
x(k)
‖h(k)‖2 hH(k)z(k)
x(k)
x(k) = ‖h(k)‖2x(k) + hH(k)z(k)
Maximum achievable data rate?
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Maximum Ratio Combining 114 / 152
Maximum Ratio Combining
Signal to noise ratio
γSNR = E|‖h‖2x|2E|hHz|2 = ‖h‖4E|x|2
EhHzzHh = ‖h‖4PxhHKzzh
= ‖h‖4Pxσ2
zhHh
= ‖h‖2ΓSNR
Achievable data rate
R(k) ≤ Rmax(k) = ld(
1 + ‖h(k)‖2ΓSNR
)Capacity of the (MIMO) channel
C(k) = ld(1 + hH(k)h(k)ΓSNR
)Performance of strategy
Rmax(k) = C(k)
Maximum ratio combining is optimal!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Optimal Transmission Strategies Maximum Ratio Transmission
Contents
1 Maximum Ratio Combining
2 Maximum Ratio Transmission
3 Water-Filling
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Maximum Ratio Transmission 115 / 152
Maximum Ratio Transmission
Scenario
w(k) Tx
x1(k)
xm(k)
h1(k)
hm(k)
z(k)
y(k)Rx w(k)
feed-back channel
Transmitter has perfect channel state information (CSIT)
Channel
y(k) = hH(k)x(k) + z(k) , with Pz = σ2z
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Maximum Ratio Transmission 116 / 152
Maximum Ratio Transmission
Strategy
x(k) =h(k)‖h(k)‖w(k)
Channel with strategy
w(k)
‖h(k)‖ z(k)
y(k)
y(k) = ‖h(k)‖w(k) + z(k)
Maximum achievable data rate?
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Maximum Ratio Transmission 117 / 152
Maximum Ratio Transmission
Signal to noise ratio
γSNR = E|‖h‖w|2E|z|2 = ‖h‖2E|w|2
σ2z
=‖h‖2E
∥∥∥ h‖h‖ w
∥∥∥2
σ2z
=‖h‖2E‖x‖2
σ2z
= ‖h‖2ΓSNR
Achievable data rate
R(k) ≤ Rmax(k) = ld(
1 + ‖h(k)‖2ΓSNR
)Capacity of the (MIMO) channel
C(k) = ld(1 + hH(k)h(k)ΓSNR
)Performance of strategy
Rmax(k) = C(k)
Maximum ratio transmission is optimal!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Optimal Transmission Strategies Water-Filling
Contents
1 Maximum Ratio Combining
2 Maximum Ratio Transmission
3 Water-Filling
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Water-Filling 118 / 152
Water-Filling
Scenario
w(k)
x1(k)
xm(k)
h11(k)
hn1(k)
h1m(k)
hnm(k)
z1(k)
zn(k)
y1(k)
yn(k)
w(k)Tx Rx
feed-back channel
Transmitter has perfect channel state information (CSIT)
Problem
Power allocation strategy to achieve maximum data rate?
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Water-Filling 119 / 152
Water-Filling
Prerequisite
CSIT allows for singular value decomposition
exploit SVD to reduce problem to parallel SISO channels
r = rank(H) , with H = UΣV H ,
same noise power at each receiver antenna
Pz% = Pz = σ2z for % = 1, . . . , r = rank(H)
variable transmit signal power
Px% = α%Px , with α% ≥ 0
limited total transmit power
E‖x(k)‖2 ≤ mPx , respectively Px
r∑%=1
α% ≤ mPx
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Water-Filling 120 / 152
Water-Filling
Equivalent Scenario
w(k)
α1
αr
x1(k)
xr(k)
σ1(k)
σr(k)
z1(k)
zr(k)
y1(k)
yr(k)
w(k)Tx Rx
feed-back channel
Transmitter has perfect channel state information (CSIT)
Reduced problem
Power allocation strategy to achieve maximum data rate?Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Water-Filling 121 / 152
Water-Filling
Equivalent parallel SISO channels
y%(k) = σ(k)√α%(k)x%(k) + z%(k) for % = 1, . . . , r = rank(H)
Capacity
C = maxα1,...,αr
r∑
%=1
ld(
1 + α%σ2%ΓSNR
)
r∑%=1
α% ≤ m
α% ≥ 0
How to compute this maximum?
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Water-Filling 122 / 152
Water-Filling
Optimization problem formulation
minα f (α) s. t. g(α) ≤ 0 , α 0
Objective function
f (α) = −r∑
%=1
ld(
1 + α%σ2%ΓSNR
)differentiable, convex
Inequality constraint function
g(α) =r∑
%=1
α% − m
differentiable, convex
Convex optimization problem!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Water-Filling 123 / 152
Water-Filling
Karush-Kuhn-Tucker conditions In particular , with % = 1, . . . , r
1 µ ≥ 0 1 µ = 0 or µ > 0
2 f ′(α) + µg′(α) = 0T 2 − σ2%ΓSNR
ln(2)[1 + α%σ2
%ΓSNR] + µ = 0
3 µg(α) = 0 3 µ
[r∑
%=1
α% − m
]= 0
4 g(α) ≤ 0 4
r∑%=1
α% − m ≤ 0
5 α ≥ 0 5 α% ≥ 0
For this convex optimization problem theKarush-Kuhn-Tucker conditions are necessary and sufficient!
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Water-Filling 124 / 152
Water-Filling
Consequences
2 ⇒ µ =σ2%ΓSNR
ln(2)[1 + α%σ2
%ΓSNR] > 0
1 is feasible
3 ⇒r∑
%=1
α% = m
4 is feasible
2 ⇒ α% =1
µ ln(2)− 1σ2%ΓSNR
Solution
5 ⇒ α% =
(1
µ ln(2)− 1σ2%ΓSNR
)+
for % = 1, . . . , r
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Water-Filling 125 / 152
Water-Filling
Water-Filling Algorithm σ1 ≥ σ2 ≥ · · · ≥ σr > 0
1 2 3 4 • • • r − 1 r
1σ21ΓSNR
1σ22ΓSNR
1σ23ΓSNR
1µ ln(2)
1σ24ΓSNR
1σ2r−1
ΓSNR
1σ2rΓSNR
α1 α
2
α3
α4=0
αr−1=0
αr=0
•••
Who has will be given more!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Wireless Communications Multiple Access Channel
Wireless CommunicationsMultiple Access Channel
Karlheinz Ochs
Chair of Digital Communication Systems
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Multiple Access Channel
Contents
1 Scenario
2 Time Division Multiple Access
3 Time Sharing
4 Successive Interference Cancelation
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Multiple Access Channel Scenario
Contents
1 Scenario
2 Time Division Multiple Access
3 Time Sharing
4 Successive Interference Cancelation
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Scenario 126 / 152
Scenario
Scenario
w1(k)
w2(k)
Tx1
Tx2
x1(k)
x2(k)
h∗1
h∗2
z(k)
y(k)Rx w1(k), w2(k)
Transmitters have no channel state information (no CSIT)
Channel
y(k) = hHx(k) + z(k) , with Kxx = diag(Px1 ,Px2) , Pz = σ2z
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Scenario 127 / 152
Scenario
Signal flow diagram
x1(k)
x2(k)
h∗1
h∗2
z(k)
y(k)
Channel
y(k) = h∗1 x1(k) + h∗2 x2(k) + z(k)
Objective
R1 + R2 → max
Optimal transmission strategy?
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Multiple Access Channel Time Division Multiple Access
Contents
1 Scenario
2 Time Division Multiple Access
3 Time Sharing
4 Successive Interference Cancelation
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Time Division Multiple Access 128 / 152
Time Division Multiple Access
Time Division Multiple Access Strategy
x1(k)
x2(k)
k ∈ K1
k ∈ K2
h∗1
h∗2
z(k)
y(k)
Transmitter Tx1
k ∈ K1 = 1, . . . ,κ
1K
κ∑k=1
P1 ≤ Px1
worst case κ = K, (K2 = ∅)P1 ≤ Px1
Transmitter Tx2
k ∈ K2 = κ + 1, . . . ,K
1K
K∑k=κ+1
P2 ≤ Px2
worst case κ = 0, (K1 = ∅)P2 ≤ Px2
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Time Division Multiple Access 129 / 152
Time Division Multiple Access
Channel usage proportions
Tx1: α =κK Tx2: 1− α
Maximum achievable data rates
Tx1: R1 ≤ αC1 ,
with C1 = ld
(1 + |h1|2 Px1
σ2z
) Tx2: R2 ≤ [1− α]C2 ,
with C2 = ld
(1 + |h2|2 Px2
σ2z
)
Rate region
R2 ≤ C2 − C2
C1R1
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Time Division Multiple Access 130 / 152
Time Division Multiple Access
TDMA Rate Region
0 C10
C2
TDMA
α = 0
α = 1
R1
R2
Optimal strategy?Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Multiple Access Channel Time Sharing
Contents
1 Scenario
2 Time Division Multiple Access
3 Time Sharing
4 Successive Interference Cancelation
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Time Sharing 131 / 152
Time Sharing
Time Sharing Strategy
x1(k)
x2(k)
k ∈ K1
k ∈ K2
1√α
1√1−α
h∗1
h∗2
z(k)
y(k)
Transmitter Tx1
k ∈ K1 = 1, . . . ,κ
1K
κ∑k=1
P1 ≤ Px1
average power constraintαP1 ≤ Px1 , α 6= 0
Transmitter Tx2
k ∈ K2 = κ + 1, . . . ,K
1K
K∑k=κ+1
P2 ≤ Px2
average power constraint[1− α]P2 ≤ Px2 , α 6= 1
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Time Sharing 132 / 152
Time Sharing
Maximum achievable data rates
Transmitter Tx1
R1 ≤ α ld
(1 +|h1|2α
Px1
σ2z
),
with C1 = R1|α=1
Transmitter Tx2
R2 ≤ [1− α] ld(
1 +|h2|2
1− αPx2
σ2z
),
with C2 = R2|α=0
Rate region
(R1(α),R2(α)) , with α ∈ [0, 1]
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Time Sharing 133 / 152
Time Sharing
Time Sharing Rate Region
0 C10
C2
TDMA
Time-Sharing
α = 0
α = 1
R1
R2
Optimal strategy?Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Multiple Access Channel Successive Interference Cancelation
Contents
1 Scenario
2 Time Division Multiple Access
3 Time Sharing
4 Successive Interference Cancelation
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Successive Interference Cancelation 134 / 152
Successive Interference Cancelation
Upper Bounds
1 Tx1 transmits only
R1 ≤ C1 , with C1 = ld
(1 + |h1|2 Px1
σ2z
)
2 Tx2 transmits only
R2 ≤ C2 , with C2 = ld
(1 + |h2|2 Px2
σ2z
)
3 Tx1, Tx2 are cooperating (MISO)
R2 ≤ −R1 + C , with C = ld
(1 + |h1|2 Px1
σ2z+ |h2|2 Px2
σ2z
)
Are the upper bounds achievable?
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Successive Interference Cancelation 135 / 152
Successive Interference Cancelation
Successive Interference Cancelation Rate Region
0 R12 C10
R21
C2Tx1 transmits only
Tx1 , Tx
2 are cooperating
Tx2
transmits
only
R1
R2
Upper bounds achievable?
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Successive Interference Cancelation 135 / 152
Successive Interference Cancelation
Successive Interference Cancelation Rate Region
0 R12 C10
R21
C2Tx1 transmits only
Tx1 , Tx
2 are cooperating
Tx2
transmits
only
TDMA
Time-Sharing
(0, C2)
(C1, 0)
(R12, C2)
(C1, R21)
R1
R2
Upper bounds achievable?Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Successive Interference Cancelation 136 / 152
Successive Interference Cancelation
Known upper bound
3 R1 ≤ ld
(1 + |h1|2 Px1
σ2z+ |h2|2 Px2
σ2z
)− R2
Achievability
(R1,R2) = (R12,C2)
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Successive Interference Cancelation 136 / 152
Successive Interference Cancelation
Known upper bound
3 R1 ≤ ld
(1 + |h1|2 Px1
σ2z+ |h2|2 Px2
σ2z
)− R2
Achievability
(R1,R2) = (R12,C2)
R12 ≤ ld
(1 + |h2|2 Px2
σ2z+ |h1|2 Px1
σ2z
)− ld
(1 + |h2|2 Px2
σ2z
)
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Successive Interference Cancelation 136 / 152
Successive Interference Cancelation
Known upper bound
3 R1 ≤ ld
(1 + |h1|2 Px1
σ2z+ |h2|2 Px2
σ2z
)− R2
Achievability
(R1,R2) = (R12,C2)
R12 ≤ ld
(1 +
|h1|2Px1
σ2z + |h2|2Px2
)
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Successive Interference Cancelation 136 / 152
Successive Interference Cancelation
Known upper bound
3 R1 ≤ ld
(1 + |h1|2 Px1
σ2z+ |h2|2 Px2
σ2z
)− R2
Achievability
(R1,R2) = (R12,C2)
R12 ≤ ld
(1 +
|h1|2Px1
σ2z + |h2|2Px2
)
Successive interference cancelation strategy
1 receiver decodes x1 treating x2 as noise : R1 = R12
2 receiver cancels x1 by decoding x2 from y− h∗1 x1 : R2 = C2
Point (R12,C2) is achievable!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Successive Interference Cancelation 137 / 152
Successive Interference Cancelation
Successive Interference Cancelation Rate Region
0 R12 C10
R21
C2Tx1 transmits only
Tx1 , Tx
2 are cooperating
Tx2
transmits
only
TDMA
Time-Sharing
(0, C2)
(C1, 0)
(R12, C2)
(C1, R21)
R1
R2
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Successive Interference Cancelation 138 / 152
Successive Interference Cancelation
Known upper bound
3 R2 ≤ ld
(1 + |h1|2 Px1
σ2z+ |h2|2 Px2
σ2z
)− R1
Achievability
(R1,R2) = (C1,R21)
R21 ≤ ld
(1 +
|h2|2Px2
σ2z + |h1|2Px1
)
Successive interference cancelation strategy
1 receiver decodes x2 treating x1 as noise : R2 = R21
2 receiver cancels x2 by decoding x1 from y− h∗2 x2 : R1 = C1
Point (C1,R21) is achievable!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Successive Interference Cancelation 139 / 152
Successive Interference Cancelation
Successive Interference Cancelation Rate Region
0 R12 C10
R21
C2Tx1 transmits only
Tx1 , Tx
2 are cooperating
Tx2
transmits
only
TDMA
Time-Sharing
(0, C2)
(C1, 0)
(R12, C2)
(C1, R21)
R1
R2
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Successive Interference Cancelation 140 / 152
Successive Interference Cancelation
Achievable Points
1 Tx1 transmits only
(R1,R2) = (C1, 0)
2 Tx2 transmits only
(R1,R2) = (0,C2)
3 receiver successively cancels interference
(R1,R2) = (R12,C2)
4 receiver successively cancels interference
(R1,R2) = (C1,R21)
Use TDMA of the particular strategiesto achieve points on the connecting line!
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Successive Interference Cancelation 141 / 152
Successive Interference Cancelation
Successive Interference Cancelation Rate Region
0 R12 C10
R21
C2
TDMA
Time-Sharing
(0, C2)
(C1, 0)
(R12, C2)
(C1, R21)
R1
R2
Successive interference cancelation is an optimal strategy!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Wireless Communications X Channel
Wireless CommunicationsX Channel
Karlheinz Ochs
Chair of Digital Communication Systems
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
X Channel
Contents
1 Transmission Scenario
2 Interference Alignment
3 Interference Cancelation
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
X Channel Transmission Scenario
Contents
1 Transmission Scenario
2 Interference Alignment
3 Interference Cancelation
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Transmission Scenario 142 / 152
X Channel
Transmission Scenario
u(t) h(t, t′) v(t)
Requirement specifications
k even [v1(kT)v2(kT)
]=
[u1(kT)u2(kT)
]
k odd [v1(kT)v2(kT)
]=
[u2(kT)u1(kT)
]
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Transmission Scenario 142 / 152
X Channel
Transmission Scenario
u(t) h(t, t′) v(t)
Requirement specifications
k even [v1(kT)v2(kT)
]=
[u1(kT)u2(kT)
]or H(kT, jω′) =
[1 00 1
]
k odd [v1(kT)v2(kT)
]=
[u2(kT)u1(kT)
]or H(kT, jω′) =
[0 11 0
]
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Transmission Scenario 142 / 152
X Channel
Transmission Scenario
u(t) h(t, t′) v(t) u(t) Encoderx(t)
Channely(t)
Decoder v(t)
Requirement specifications
k even [v1(kT)v2(kT)
]=
[u1(kT)u2(kT)
]or H(kT, jω′) =
[1 00 1
]
k odd [v1(kT)v2(kT)
]=
[u2(kT)u1(kT)
]or H(kT, jω′) =
[0 11 0
]
Optimal usage of channel resources?
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Transmission Scenario 143 / 152
X Channel
Transmission Scenario
u1(t)
u2(t)
x1(t)
x2(t)
h11(t)
h21(t)
h12(t)
h22(t)
z1(t)
z2(t)
y1(t)
y2(t)
v1(t)
v2(t)Encoder
Encoder
Decoder
Decoder
Frequency-invariant channel with additive noise
y1(t) = h11(t)x1(t) + h12(t)x2(t) + z1(t)
y2(t) = h21(t)x1(t) + h22(t)x2(t) + z2(t)
Optimal usage of channel resources?Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Transmission Scenario 144 / 152
X Channel
Simplified Transmission Scenario
u11, u21
u22, u12
x1(k)
x2(k)
h11(k)
h21(k)
h12(k)
h22(k)
y1(k)
y2(k)
u11, u12
u22, u21Encoder
Encoder
Decoder
Decoder
Frequency-invariant channel
y1(k) = h11(k)x1(k) + h12(k)x2(k)
y2(k) = h21(k)x1(k) + h22(k)x2(k)
Transmission of 4 symbols with 3 channel usages?Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Transmission Scenario 145 / 152
X Channel
Transmission Scenario for 3 Channel Usages
u11, u21
u22, u12
x1
x2
H11
H21
H12
H22
y1
y2
u11, u12
u22, u21Encoder
Encoder
Decoder
Decoder
Frequency-invariant channel µ, ν ∈ 1, 2
yν =
yν(1)yν(2)yν(3)
, Hνµ =
hνµ(1) 0 00 hνµ(2) 00 0 hνµ(3)
, xµ =
xµ(1)xµ(2)xµ(3)
hνµ(1) 6= hνµ(2) 6= hνµ(3) 6= hνµ(1)Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
X Channel Interference Alignment
Contents
1 Transmission Scenario
2 Interference Alignment
3 Interference Cancelation
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Interference Alignment 146 / 152
X Channel
Transmission Scenario for 3 Channel Usages
u11, u21
u22, u12
x1
x2
H11
H21
H12
H22
y1
y2
u11, u12
u22, u21Encoder
Encoder
Decoder
Decoder
Transmitted signals
x1 = q11u11 + q21u21
x2 = q12u12 + q22u22
Vectors qνµ are design parameters!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Interference Alignment 147 / 152
X Channel
Transmission Scenario for 3 Channel Usages
u11, u21
u22, u12
x1
x2
H11
H21
H12
H22
y1
y2
u11, u12
u22, u21Interference-Alignment
Interference-Alignment
Decoder
Decoder
Received signals
y1 =H11q11u11 +H12q12u12 +H11q21u21 +H12q22u22
y2 =H21q21u21 +H22q22u22 +H21q11u11 +H22q12u12
Reasonable choice of vectors qνµ?Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Interference Alignment 148 / 152
X Channel
Received signals
y1 =H11q11u11 +H12q12u12 +H11q21u21 +H12q22u22
y2 =H21q21u21 +H22q22u22 +H21q11u11 +H22q12u12
Interference Alignment
q21 =H−111 H12q22 , q12 =H−1
22 H21q11
Aligned received signals
y1 =H11q11u11 +H12H−122 H21q11u12 +H12q22[u21 + u22]
y2 =H21H−111 H12q22u21 +H22q22u22 +H21q11[u11 + u12]
Interferences lie in 1-dimensional subspaces!Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
X Channel Interference Cancelation
Contents
1 Transmission Scenario
2 Interference Alignment
3 Interference Cancelation
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Interference Cancelation 149 / 152
X Channel
Transmission Scenario for 3 Channel Usages
u11, u21
u22, u12
x1
x2
H11
H21
H12
H22
y1
y2
u11, u12
u22, u21Interference-Alignment
Interference-Alignment
Decoder
Decoder
Aligned received signals
y1 =H11q11u11 +H12H−122 H21q11u12 +H12q22[u21 + u22]
y2 =H21H−111 H12q22u21 +H22q22u22 +H21q11[u11 + u12]
Choice of q11 and q22, such that decoding is possible?Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Interference Cancelation 150 / 152
X Channel
Aligned received signals
y1 = R1
u11
u12
u21 + u22
and y2 = R2
u21
u22
u11 + u12
with matrices depending on vectors q11 and q22
R1 =[H11q11 H12H
−122 H21q11 H12q22
]R2 =
[H21H
−111 H12q22 H22q22 H21q11
]Decoding of transmitted symbols
Choice of q11 and q22, such that R1 and R2 are regular?
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Interference Cancelation 151 / 152
X Channel
Transmission Scenario for 3 Channel Usages
u11, u21
u22, u12
x1
x2
H11
H21
H12
H22
y1
y2
u11, u12
u22, u21Interference-Alignment
Interference-Alignment
Decoder
Decoder
Interference Cancelation u11
u12
u21 + u22
= R−11 y1 and
u22
u21
u11 + u12
= R−12 y2
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18
Interference Cancelation 152 / 152
X Channel
Transmission Scenario for 3 Channel Usages
u(t) Encoderx(t)
Channely(t)
Decoder v(t)
Remarks
encoder and decoder are time-variant transmission systems
channel state information at transmitter required
transmission of 4 symbols with 3 channel uses
strategy achieves DoF
DoF is a first order approximation of channel capacity
Information theory
Interference alignment and cancelation is a DoF-optimal strategy!
Lehrstuhl fürDigitale Kommunikationssysteme
K. Ochs Wireless Communications WS 2017/18