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)
EE359 Discussion 6 November 6, 2017 2 / 33
Outline
1 ReviewChannel modelsPerformance analysisCombating fading
EE359 Discussion 6 November 6, 2017 3 / 33
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
EE359 Discussion 6 November 6, 2017 4 / 33
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)
EE359 Discussion 6 November 6, 2017 5 / 33
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)
EE359 Discussion 6 November 6, 2017 6 / 33
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
EE359 Discussion 6 November 6, 2017 7 / 33
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
EE359 Discussion 6 November 6, 2017 8 / 33
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
EE359 Discussion 6 November 6, 2017 9 / 33
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
EE359 Discussion 6 November 6, 2017 10 / 33
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?
EE359 Discussion 6 November 6, 2017 11 / 33
Example Problems
Repeat the previous problem with σφdB = 4dB log-normal shadowingpresent.
EE359 Discussion 6 November 6, 2017 12 / 33
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
EE359 Discussion 6 November 6, 2017 13 / 33
Outline
1 ReviewChannel modelsPerformance analysisCombating fading
EE359 Discussion 6 November 6, 2017 14 / 33
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
EE359 Discussion 6 November 6, 2017 15 / 33
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
EE359 Discussion 6 November 6, 2017 16 / 33
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
EE359 Discussion 6 November 6, 2017 17 / 33
Example Problem
EE359 Discussion 6 November 6, 2017 18 / 33
Example Problem
What is the capacity of the previous example if TX power is fixed?
EE359 Discussion 6 November 6, 2017 19 / 33
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
EE359 Discussion 6 November 6, 2017 20 / 33
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)
EE359 Discussion 6 November 6, 2017 21 / 33
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
)under log normal shadowing
Fading outage formula
Pout = 1− e−γ0γ̄ in Rayleigh fading
EE359 Discussion 6 November 6, 2017 22 / 33
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
)under log normal shadowing
Fading outage formula
Pout = 1− e−γ0γ̄ in Rayleigh fading
EE359 Discussion 6 November 6, 2017 22 / 33
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
)under log normal shadowing
Fading outage formula
Pout = 1− e−γ0γ̄ in Rayleigh fading
EE359 Discussion 6 November 6, 2017 22 / 33
Fading and fading/shadowing: Which formula to use ?
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
EE359 Discussion 6 November 6, 2017 23 / 33
Fading and fading/shadowing: Which formula to use ?
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
EE359 Discussion 6 November 6, 2017 23 / 33
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
EE359 Discussion 6 November 6, 2017 24 / 33
Example Problems
What SNR is required to acheive a BER of 10−3 given BPSK in AWGN?For DPSK in AWGN?
EE359 Discussion 6 November 6, 2017 25 / 33
Example Problems
What SNR is required to acheive a BER of 10−3 given BPSK in RayleighFading? For DPSK in Rayleigh Fading?
EE359 Discussion 6 November 6, 2017 26 / 33
Outline
1 ReviewChannel modelsPerformance analysisCombating fading
EE359 Discussion 6 November 6, 2017 27 / 33
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
EE359 Discussion 6 November 6, 2017 28 / 33
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
EE359 Discussion 6 November 6, 2017 28 / 33
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?)
EE359 Discussion 6 November 6, 2017 29 / 33
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?
EE359 Discussion 6 November 6, 2017 30 / 33
Example Problems
EE359 Discussion 6 November 6, 2017 31 / 33
Example Problems
Repeat the above problem for an SC diversity setup
EE359 Discussion 6 November 6, 2017 32 / 33
Example Problems
EE359 Discussion 6 November 6, 2017 33 / 33