EE359 Discussion Session 5 Performance of Linear ... · MGF can be found from two-sided Laplace...

EE359 Discussion Session 5Performance of Linear Modulation in Fading, Diversity

November 1, 2017

EE359 Discussion 5 November 1, 2017 1 / 32

Outline

1 Recap

2 Performance analysis of linear modulation

3 Moment Generating Functions

4 Linear modulation in fading

5 Diversity


Announcement

Midterm review on Monday, November 6, 4-6 pm, in Packard 364

Midterm on Thursday, November 9, 6-8 pm, in Thornton 102

OH hour changed for next week — see calendar!


Last discussion session

Capacity formulas with CSIT and CSIR

Optimal power and rate adaptation policies

Suboptimal power adaptation policies

This session

Performance analysis of linear modulation in fading

Diversity


Outline

1 Recap




5 Diversity


What is linear modulation

Definition

Any modulation where the data is encoded in real or complex symbols (i.e.in amplitude or phase) is linear modulation

Observations

Performance dependent on constellation (not on baseband waveform)

Examples are BPSK, QPSK, MPAM, MQAM, etc.

FSK (frequency shift keying) is not linear, why ?


Linear modulation in AWGN

Idea

Noise is additive, hence

y[i] = γ ∗ x[i] + n[i]

Pb = Q(√

2γb) if n[i] ∼ N (0, 1), and x[i] ∈BPSK

In general, well approximated by Ps ≈ αMQ(√βMγ), where αM , βM

are constellation dependent

For differential PSK (DPSK) systems, Pb = 12e−γb


Outline

1 Recap




5 Diversity


Moments of a Random Variable

Given a random variable X, and its probability density function f(x), itsnth moment is given by:

µn = E[Xn] =

∫ ∞−∞

xnf(x)dx

Why do we care?

n = 1 is the mean

n = 2 is the variance

Higher order moments often interesting


Moments of a Random Variable

Given a random variable X, and its probability density function f(x), itsnth moment is given by:

µn = E[Xn] =

∫ ∞−∞

xnf(x)dx

Why do we care?

n = 1 is the mean

n = 2 is the variance

Higher order moments often interesting


Moment Generating Function

Many ways to compute the moment, one useful way is throught theMoment Generating Function (MGF):

MX(t) = E[etX]

Intuition:

MX(t) = E[etX]

= 1 + tE [X] +t2E

[X2]

2!+ · · ·

Differentiating and setting t = 0 gives moments!



Many ways to compute the moment, one useful way is throught theMoment Generating Function (MGF):

MX(t) = E[etX]

Intuition:

MX(t) = E[etX]

= 1 + tE [X] +t2E

[X2]

2!+ · · ·

Differentiating and setting t = 0 gives moments!



More concretely:

µn = E[Xn] =dnMX

dtn

∣∣∣∣t=0

Another useful observation:

L{fX} (s) =

∫ ∞−∞

e−sxfX(x)dx

MX(t) = L{fX} (−t) =

∫ ∞−∞

e−sxfX(x)dx

MGF can be found from two-sided Laplace tranform of PDF



More concretely:

µn = E[Xn] =dnMX

dtn

∣∣∣∣t=0

Another useful observation:

L{fX} (s) =

∫ ∞−∞

e−sxfX(x)dx

MX(t) = L{fX} (−t) =

∫ ∞−∞

e−sxfX(x)dx

MGF can be found from two-sided Laplace tranform of PDF


Sums of Random Variables

MGFs are also useful for dealing with linear combinations of randomvariables

Sn =

n∑i=1

aiXi,

ai arbitrary constants

Xi independant (not necessarily identical) random variables.

Moments of Sn are given by:

MSn(t) =

n∏i=1

MXi(ait)


Outline

1 Recap




5 Diversity


Performance in fading

System model

y[i] =√γ[i]x[i] + n[i]

Different metrics

Average probability of error: Relevant when channel is fast fading

Outage probability: Relevant when channel is slow fading

Combined Outage + Avg. probability of error: shadowing (slow) andfading (fast)


Homework 5

Problem 1

Outage probability relevant when symbol time is much less than coherencetime of channel


Computing the average probability of error

Idea

Integrate the Q function over fading distributions

Use change of integration order to try to get closed form expressions

Some useful relations

Pb for BPSK in Rayleigh ≈ 14γ̄ (Closed form also possible)

Pb for DPSK in Rayleigh ≈ 12γ̄ (Closed form possible)

Q(x) =∫∞x

1√2πe−z

2/2dz =∫ π/2

0 e−x2/(2 sin2 φ)dφ

Average BER computation is often MGF computation asQ(√γ) ≤ 1

2e−γ/2


P̄s using MGF (Mγ(s) =∫∞

0 esγp(γ)dγ)

Idea

Use fact that

Q(x) =

∫ ∞x

1√2πe−

y2

2 dy =1

π

∫ π/2

0e−x

2/2 sin2 φdφ

P̄s ≈∫ ∞

0

αMπQ(√βMγ)p(γ)dγ

=αMπ

∫ γ=∞

γ=0

∫ φ=π/2

φ=0e−βMγ/2 sin2 φp(γ)dφdγ

=αMπ

∫ φ=π/2

φ=0

∫ γ=∞

γ=0e−βMγ/2 sin2 φp(γ)dγdφ

=αMπ

∫ φ=π/2

φ=0Mγ(−βM/2 sin2 φ)dφ


P̄s using MGF (Mγ(s) =∫∞

0 esγp(γ)dγ)

Idea

Use fact that

Q(x) =

∫ ∞x

1√2πe−

y2

2 dy =1

π

∫ π/2

0e−x

2/2 sin2 φdφ

P̄s ≈∫ ∞

0

αMπQ(√βMγ)p(γ)dγ

=αMπ

∫ γ=∞

γ=0

∫ φ=π/2

φ=0e−βMγ/2 sin2 φp(γ)dφdγ

=αMπ

∫ φ=π/2

φ=0

∫ γ=∞

γ=0e−βMγ/2 sin2 φp(γ)dγdφ

=αMπ

∫ φ=π/2

φ=0Mγ(−βM/2 sin2 φ)dφ


Example: BPSK in Rayleigh Fading

BPSK in AWGN: Pb = Q(√

2γb)⇒ α = 1β = 2. Moment generatingfunction for Rayleigh Fading:

Mγb(s) = (1− sγ̄b)−1

Mγb

(− 1

sin2 φ

)=

(1 +

γ̄b

sin2 φ

)−1

Integral now becomes:

P̄b =1

π

∫ π/2

0

(1 +

γ̄b

sin2 φ

)−1

dφ





Mγb(s) = (1− sγ̄b)−1

Mγb

(− 1

sin2 φ

)=

(1 +

γ̄b

sin2 φ

)−1


P̄b =1

π

∫ π/2

0

(1 +

γ̄b

sin2 φ

)−1

dφ





Mγb(s) = (1− sγ̄b)−1

Mγb

(− 1

sin2 φ

)=

(1 +

γ̄b

sin2 φ

)−1


P̄b =1

π

∫ π/2

0

(1 +

γ̄b

sin2 φ

)−1

dφ


On error floors

Idea

As γs →∞, Perror → 0 usually. Not true if there is an error floor!

Some reasons

Differential modulation with large symbol times and/or fast fading(due to small Tc)

Due to intersymbol interference ISI (or small Bc) Pb ≈ ( σTs )2

Some factors

Correlation function of channel (channel coherence time Tc andbandwidth Bc)

Fading statistics, symbol time Ts

Question

What happens to error floors if Ts decreases or data rate increases?


Homework 5

Problem 2

Use average error probability requirement to get γ̄b

Use Pb requirement to set the target power in the presence ofshadowing and path loss

Use cell coverage area formula, if appropriate

Problem 3

Use formula for Rayleigh fading with DPSK, i.e. (6.90), with K = 0, withthe ρC function given by the Jakes’ formula (uniform scattering). Thisgives error floor due to doppler.


Outline

1 Recap




5 Diversity


Diversity

Idea

Use of independent fading realizations can reduce the probability oferror/outage events

Some observations

Diversity can be in time, space, frequency, polarization, . . .

Diversity order used as a measure of diversity, defined as

M = limγ̄→∞

− logPelog γ̄

, Pe = P̄s or Pout

Can also use array gain (or SNR gain) γ̄Σ/γ̄, where γ̄Σ is the averageSNR after “diversity combining”


Diversity order

Diversity order

Specifying diversity order M is roughly equivalent to saying that at high γ̄,

P̄e ≈ (γ̄)−M

Array gain

Array gain Ag is equivalent to ratio of average SNRs after diversitycombining

Ag =γ̄Σ

γ̄


Diversity order

Diversity order

Specifying diversity order M is roughly equivalent to saying that at high γ̄,

P̄e ≈ (γ̄)−M

Array gain

Array gain Ag is equivalent to ratio of average SNRs after diversitycombining

Ag =γ̄Σ

γ̄


Diversity combining techniques

In this class, we have talked about two schemes to exploit diversity, bothat the receiver

System model

r = γx+ n

Some receiver diversity combining schemes

Selection combining: Choose the largest SNR of the independentrealizations

Maximal ratio combining: Combine all the independent receivedSNRs to maximize SNR


Selection combining (SC)

Idea

Given M i.i.d. r.v., γ1, . . . , γM ≥ 0,

P (maxi

(γi) < c) = P (γi < c)M

Some observations

Define γΣ = maxi γi

In Rayleigh fading γ̄Σ = γ̄(∑M

i=1 1/i) (γ̄: average SNR at a branch)

P̄b in general difficult, but for DPSK and Rayleigh fading,

P̄b = M/2

M−1∑m=0

(−1)m(M−1m

)1 +m+ γ̄


Selection combining continued

Outage probability

Pout =(

1− e−γ0γ̄

)MQuestion (SC in Rayleigh fading)

What is the diversity gain?:

What is the SNR gain?:


Selection combining continued

Outage probability

Pout =(

1− e−γ0γ̄

)MQuestion (SC in Rayleigh fading)

What is the diversity gain?: M

What is the SNR gain?:∑M

i 1/i


Threshold Combining (SSC)

Idea

Use one RF chain with multiple antennas

PγΣ(γ) =

{Pγ1(γT )Pγ2(γ) γ < γT

p(γT ≤ γ2 ≤ γ) + Pγ1(γT )Pγ2(γ) γ ≥ γT


Maximal ratio combining (MRC)

Idea

Instead of discarding weaker branches, combine the SNRs of all branches,i.e.

γΣ =

M∑i

γi

Nuts and bolts

Need to make the received components of the same phase (not aproblem with modern DSP)

Maximal ratio combining maximizes received SNR, i.e. solves thefollowing problem

maxa:||a||2=1

E[|aHγx|2]

E[|aHn|2]

MGF of sums decompose into product of individual MGFs so easy toanalyse P̄s


MRC continued (Outage probability and P̄s)

Outage probability

Pout = 1− eγ0γ̄

(M−1∑i=0

(γ0

γ̄

)i/i!

)

Average probability of error P̄sThe MGF of sum decouples into product of MGFs

For DPSK and Rayleigh fading, average error probability is

1

2EγΣ [e−γΣ ] =

1

2

M∏i=1

Eγi [e−γi ] =

1

2

M∏i=1

M(−1)

For general constellations P̄s is approximately of the form

C

∫ φ=B

φ=A(M(−γ/2 sin2 φ))Mdφ


Questions

What is the diversity order for MRC?:

What is the SNR gain for MRC?:


Questions

What is the diversity order for MRC?: M

What is the SNR gain for MRC?: M


Homework 5

Problem 4

Concepts in SC (Selection combining) - probability of outage

Problem 5

DPSK under Rayleigh fading using SSC. Hint: Split integral into twoparts: 0→ γT and γT →∞.

Problem 6

Concepts in MRC - Deriving optimal weights, computing P̂s. Using MGFapproach in part (b).


Homework 5

Problem 4


Problem 5


Problem 6



Homework 5

Problem 4


Problem 5


Problem 6



Homework 5

Problem 7

Notions of combining - SC/MRC. For part (c), if A⇒ B or in other wordsA ⊆ B, then P (A) ≤ P (B).

Problem 8

Diversity gain in SC and MRC at high/low SNR.


Homework 5

Problem 7

Notions of combining - SC/MRC. For part (c), if A⇒ B or in other wordsA ⊆ B, then P (A) ≤ P (B).

Problem 8

Diversity gain in SC and MRC at high/low SNR.


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EE359 Discussion Session 5 Performance of Linear ... · MGF can be found from two-sided Laplace...

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