EE359 Discussion Session 6 Midterm Review · EE359 Discussion 6 November 6, 2017 7 / 33. Fading Models attenuation due to EM waves combining with random phases due to multipath Recall:

EE359 Discussion Session 6Midterm Review

November 6, 2017

EE359 Discussion 6 November 6, 2017 1 / 33

Broad topics covered so far

Channel modelsI Path lossI ShadowingI Fading

Performance analysisI CapacityI Probability of outageI Probability of bit/symbol error

Combating fading (diversity)


Outline

1 ReviewChannel modelsPerformance analysisCombating fading


Path loss models

Models attenuation caused by “spread” of EM waves

Free space

2-ray and n-ray models

Simplified path loss models

Pr = PtK

(d0

d

)γValid in the far field, i.e. when d is large, γ is path loss exponent, K candepend on carrier frequency


Shadowing

Models attenuation caused by EM waves passing through randomlylocated buildings/obstacles

Log normal shadowing assumes

10 log10(Pr) = 10 log10(P̄r) + S,

where S ∼ N (0, σ2ψdB

) or equivalently

Pr(dB) = P̄r(dB) + S

S is associated with location, closely located points will havecorrelated S (can talk of decorrelation distance Xc)


Outage probability

Idea

Outage ≡ Received power γ is below threshold γ0

Reasons

Path loss (usually no randomness)

Shadowing (randomness if shadowing time scales are small)

Fading (randomness due to multipath combining)


Outage probability and cell coverage area

Outage probability

Defined for a particular location

Relates Pout, Pmin (dB), P̄r(d) (dB), σψdB at a location d via

Pout = Q

(P̄r(d)− Pmin

σψdB

)under log normal shadowing

Cell coverage area

Expected fraction of location within cell where received power isabove Pmin (dB) (averaged over both space and shadowing),

C = Q(a) + e2−2ab

b2 Q

(2− abb

)under simplified path loss and log normal shadowing, where

a = Pmin−P̄r(R)σψdB

,b = 10γ log10(e)σψdB


Fading

Models attenuation due to EM waves combining with random phases dueto multipath

Recall: Narrowband versus wideband

Received signal Re{∑N

n=1 an(t)e−jφn(t)u[τ − τn(t)]ej2πfct}Narrowband approximation u(t) ≈ u(t− τn(t)), i.e. received signal is

r(t) = Re{α(t)u(t)ej2πfct}

Time

Figure: Narrowband Tm � 1Bu

Time

Figure: Wideband Tm ≈,≥ 1Bu


Fading contd.

Narrowband fading

Effect of channel is just scalar multiplication by complex constant

α(t) = rI(t) + jrQ(t)

Specify distribution on envelope z(t) = |α(t)| =√rI(t)2 + rQ(t)2:

Rayleigh, Rician, Nakagami m, . . .

Wideband fading

Effect of channel no longer modeled by a single scalar multiplication

Characterized by multipath intensity profile, doppler power spectrum


On fading “types”

Depends on:

Signal Bandwidth Bu

Coherence Time Tc or Doppler Effects

Coherence Bandwidth Bc or Delay Spread

Tc high, slow fadingTc low, fast fading

Bc low, freq. sel. fading

Bc high, flat fading

Tc

Bc


Example Problems

Consider a 150m circular cell that follows a simplified pathloss model withK = 0.01, d = 1m, and pathloss exponent γ = 3. The transmitted poweris 14W. What percentage of locations have a received power greater than6.21× 10−8W?


Example Problems

Repeat the previous problem with σφdB = 4dB log-normal shadowingpresent.


Example Problems

What is Pout for a user that is 100m away from the center of the cell? UsePr < 6.21× 10−8W as the outage criteria


Outline



Capacity

Definition

Maximum data rate that can be supported by the channel with vanishingprobability of error

Capacity C under different models (γ is the instantaneous SNR at thereceiver, B is bandwidth)

Scheme Capacity ExpressionAWGN C = B log2(1 + γ)

Shannon capacity in fadingwith Rx CSI only

C =∫∞

0 B log2(1 + γ)p(γ)dγ

Shannon capacity with Tx, RxCSI (Waterfilling)

C =∫∞γ0B log2(γ/γ0)p(γ)dγ, where∫∞

γ0(1/γ0 − 1/γ)p(γ)dγ = 1


Capacity formulas continued ...

Capacity expressions C

Scheme Capacity Expression

Channel Inversion C = B log2

(1 + 1

E[1/γ]

)Truncated Channel Inversion C = B log2

(1 + 1

Eγ0 [1/γ]

)p(γ > γ0)

where Eγ0 [1/γ] =∫∞γ0

1γ p(γ)dγ


Example Problem

Discrete time-varying AWGN channel with 3 states: γ1 = 3dB, γ2 = 8dB,γ3 = 15dB, with p1 = 0.3, p2 = 0.5, p3 = 0.2. Assume average transmitpower P̄ and perfect CSI and TX and RX. Find optimal transmissionstrategy and capacity per unit bandwidth


Example Problem


Example Problem

What is the capacity of the previous example if TX power is fixed?


Average probability of bit/symbol error

Idea

Compute P̄s = Eγ [Ps(γ)]

May be simplified using alternate Q functions and MGFs of fadingdistributions

Regime of relevance

Metric Relevant regimeOutage probability Ts � TcAverage probability of error Ts ≈ TcAWGN probability of error Ts � Tc


Combined outage and average error probability

Setting

Shadowing time scales are small (e.g. moving receiver)

Idea

Three SNRs:

γs: Instantaneous (random)

γ̄s: Averaged over multipath fading (random)

¯̄γs: Averaged over multipath fading and shadowing (influenced by e.g.path loss)


Fading and fading/shadowing: Which formula to use ?

Question 1

Outage can be due to fading or shadowing, so which one to use?

Answer

In fading, use fading outage formula, relating Pout, γ0 or Ps(γ), γ̄

In combined fading and shadowing, outage is due to shadowing, so use

Pout = Q

(P̄r(d)− Pmin

σψdB


Fading outage formula

Pout = 1− e−γ0γ̄ in Rayleigh fading



Question 1


Answer



Pout = Q

(P̄r(d)− Pmin

σψdB






Question 1


Answer



Pout = Q

(P̄r(d)− Pmin

σψdB






Question 2

Given Ps or P̄s, what formula do we use to get target γ0?

Answer

In fading, you would want to use AWGN formulae relating γ to Ps

In combined fading and shadowing, you would want to use γ̄ versus P̄s

AWGN Ps formulae

Ps = Q(√

2γ)

for BPSK, Ps =1

2e−γ for DPSK

Average P̄s formulae

P̄s ≈1

4γ̄for BPSK, P̄s ≈

1

2γ̄for DPSK



Question 2

Given Ps or P̄s, what formula do we use to get target γ0?

Answer

In fading, you would want to use AWGN formulae relating γ to Ps

In combined fading and shadowing, you would want to use γ̄ versus P̄s

AWGN Ps formulae

Ps = Q(√

2γ)

for BPSK, Ps =1

2e−γ for DPSK

Average P̄s formulae

P̄s ≈1

4γ̄for BPSK, P̄s ≈

1

2γ̄for DPSK


Error floors

What is an error floor?

Error floor whenever Ps 9 0 as γ →∞

Summary of effects

Data rate cannot be too low with differential modulation schemesI Differential schemes assume channel is constant across subsequent

symbolsI Depends on Doppler or Tc

Data rate cannot be too high in any systemI Channel will “spread” symbols across time, causing self interference

(ISI - inter symbol interference)I Depends on Bc or bandwidth of channel


Example Problems

What SNR is required to acheive a BER of 10−3 given BPSK in AWGN?For DPSK in AWGN?


Example Problems

What SNR is required to acheive a BER of 10−3 given BPSK in RayleighFading? For DPSK in Rayleigh Fading?


Outline



Diversity

Idea

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

Some diversity combining schemes (with M i.i.d. realizations) withCSI

Selection Combining: γΣ = maxi γi, Pout,M = (Pout)M

Maximal Ratio Combining: γΣ =∑

i γi, P̄s,M ≈(P̄s,1

)M, can use

MGF expressions for P̄s,M

Benefits

Diversity gain (or diversity order)

SNR gain (or array gain)

Depending on available CSI, can employ these schemes at both receiverand transmitter


Diversity

Idea

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

Some diversity combining schemes (with M i.i.d. realizations) withCSI

Selection Combining: γΣ = maxi γi, Pout,M = (Pout)M

Maximal Ratio Combining: γΣ =∑

i γi, P̄s,M ≈(P̄s,1

)M, can use

MGF expressions for P̄s,M

Benefits

Diversity gain (or diversity order)

SNR gain (or array gain)

Depending on available CSI, can employ these schemes at both receiverand transmitter


Transmit diversity

Setup

Multiple antennas at the transmitter, single antenna at receiver

Observation

Can use the same SC and MRC techniques at the transmitter

Benefits

Simpler receiver processing with same diversity benefits

Needs CSIT (which has more overhead than CSIR, why?)


Example Problems

Consider a two branch MRC setup. The SNR on the first branch isuniformly distributed between 5 and 10 (linear units), and the SNR on thesecond branch is uniformly distributed between 5 and 20 units. What isthe outage probability for a target BER of 10−6?


Example Problems


Example Problems

Repeat the above problem for an SC diversity setup


Example Problems


Documents

EE359 Discussion Session 6 Midterm Review · EE359 Discussion 6 November 6, 2017 7 / 33. Fading Models attenuation due to EM waves combining with random phases due to multipath Recall: