EE359 Discussion Session 7 Adaptive Modulation, MIMO systems · Outline 1 Adaptive transmit schemes...

EE359 Discussion Session 7Adaptive Modulation, MIMO systems

November 15, 2017

EE359 Discussion 7 November 15, 2017 1 / 29

Outline

1 Adaptive transmit schemesVariable rate variable powerDiscrete rate variable powerChannel estimation errors

2 MIMO systemsRepresentationCapacity

Outline

Adapt to what?

Use the estimate of the instantaneous channel state fed back to thetransmitter

Adapt what ?

Modulation (rate)

Coding

QoS (quality of service or BER Pb)

Problems

Estimate of the channel may be:

Inaccurate (due to receiver measurement error)

Delayed (due to non zero delay of the feedback link)

Adapt to what?

Use the estimate of the instantaneous channel state fed back to thetransmitter

Adapt what ?

Modulation

Coding

QoS (quality of service or BER Pb)

Problems

Estimate of the channel may be:

Inaccurate (due to receiver measurement error)

Delayed (due to non zero delay of the feedback link)

Adaptive schemes used in many high performance systems

Lots of tunable parameters

Adaptive modulation: 802.11ac (latest commercial Wifi standard)uses BPSK, QPSK, 16 QAM, 64 QAM, 256 QAM

Adaptive power: Used more in multiuser systems (e.g. power controlin CDMA systems) (we are discussing point to point channels here)

Adaptive rate coding: Also used for QoS requirements, e.g. 802.11achas coding rates of 1/2, 2/3, 3/4, 5/6. LTE has 29 distinctmodulation and coding schemes!

Adaptive spatial streams: Different configurations for up to 8× 8MIMO in 802.11ac. 4× 4 for older 802.11 standards and LTE. LTEdoes not achieve more than 3 spatial streams

Outline

Spectral efficiency and target BER

Spectral efficiency

Simply defined as the number of bits sent on average per unit bandwidth(i.e. Ei[log2(Mi)] bits where an Mi-ary constellation is sent at time i)

Target BER

We can use a Chernoff bound to approximate the BER for MQAMconstellations:

Pb ≤ 0.2e−1.5γM−1

Spectral efficiency and target BER

Spectral efficiency

Simply defined as the number of bits sent on average per unit bandwidth(i.e. Ei[log2(Mi)] bits where an Mi-ary constellation is sent at time i)

Target BER

We can use a Chernoff bound to approximate the BER for MQAMconstellations:

Pb≈0.2e−1.5γM−1

An equivalent way of looking at Pb ≈ 0.2e−1.5γM−1

Given SNR γ, the maximum constellation size “supported” for BERPb is

M ≤ 1− 1.5

ln(5Pb)γ

Questions

What happens when K > 1?

Spectral efficiency greater than capacity

Is a bigger K better from a performance standpoint ?

Bigger K means more spectral efficiency for a given power

What does K depend on/how do we improve K ?

Coding, BER, blocklength/latency/delay

Does smaller BER mean smaller or larger K ?

Smaller K

M ≤ 1 +Kγ

Questions

What happens when K > 1?

Spectral efficiency greater than capacity

Is a bigger K better from a performance standpoint ?

Bigger K means more spectral efficiency for a given power

What does K depend on/how do we improve K ?

Coding, BER, blocklength/latency/delay

Does smaller BER mean smaller or larger K ?

Smaller K

M ≤ 1 +Kγ

Questions

What happens when K > 1?Spectral efficiency greater than capacity

Is a bigger K better from a performance standpoint ?Bigger K means more spectral efficiency for a given power

What does K depend on/how do we improve K ?Coding, BER, blocklength/latency/delay

Does smaller BER mean smaller or larger K ?Smaller K

Variable rate variable power MQAM

Maximize spectral efficiency (S.E.) subject to average power constraints

maximizeP (γ) E[log2(M)]s.t. E[P (γ)] = P

Solution

Optimal P (γ) given by

P (γ)/P =

{1/γ0 − 1/(γK) γ ≥ γ0/K0 γ < γ0/K

where γ0 satisfies E[(K/γ0 − 1/γ)1(γ > γ0/K)] = K. Optimal S.E. is

E[log2(Kγ/γ0)1(γ > γ0/K)]

Variable rate variable power MQAM

Maximize spectral efficiency (S.E.) subject to average power constraints

maximizeP (γ) E[log2(1 +KP (γ)/P )]s.t. E[P (γ)] = P

Solution

Optimal P (γ) given by

P (γ)/P =

{1/γ0 − 1/(γK) γ ≥ γ0/K0 γ < γ0/K

where γ0 satisfies E[(K/γ0 − 1/γ)1(γ > γ0/K)] = K. Optimal S.E. is

E[log2(Kγ/γ0)1(γ > γ0/K)]

More details about variable rate variable power MQAM

Comparison of solution with Shannon capacity expression

Same waterfilling ideas work in this case

“Rate” M is now the size of the constellation

Capacity expression if K = 1; does it imply BER for capacity is nonzero?

K represents “power loss” due to BER requirement

Extension of BER constrained schemes

Channel inversion: S.E. is log2(1 +Kσ), where σ = 1/E[1/γ]

Truncated channel inversion: S.E. is (1− P (γ < γ0)) log2(1 +Kσ),where σ = 1/E[1/γ1(γ > γ0)].

But....

Rate cannot be continuous in practice

Capacity expression if K = 1; does it imply BER for capacity is nonzero? No!

But....

Capacity expression if K = 1; does it imply BER for capacity is nonzero?

But....

Outline

Continuous rate continuous power MQAM

Problem

maxP (γ) E[log2(M(γ))]s.t. E[P (γ)] = P , Pb(γ) ≤ Pb

The solution

Use waterfilling

M(γ) = γ/γ∗K

γ1 γ2 γ3

Figure: M versus γ

Discrete rate variable power MQAM

Problem

maxP (γ),M(γ) E[log2(M(γ))]s.t. E[P (γ)] = P ,M(γ) ∈M, Pb(γ) ≤ Pb

Ideas to approximate solution

Given available constellations M = {M1, . . . ,Mn}, choose aconstellation which meets the Pb target for a given channel state γ

Reduce transmit power to keep BER constant if channel is good

M(γ) = γ/γ∗K

γ1 γ2 γ3

Figure: M versus γ

Discrete rate variable power MQAM

Problem

maxP (γ),M(γ) E[log2(M(γ))]s.t. E[P (γ)] = P ,M(γ) ∈M, Pb(γ) ≤ Pb

Ideas to approximate solution

Given available constellations M = {M1, . . . ,Mn}, choose aconstellation which meets the Pb target for a given channel state γ

Reduce transmit power to keep BER constant if channel is good

M(γ) = γ/γ∗K

γ1 γ2 γ3

Figure: M versus γ

Power adaptation

Assuming that Pb ≈ 0.2e−1.5γM−1 , for constant M , Pb ↓ as γ ↑

To conserve power simply reduce power if channel state is good!

(M1 − 1)/K

(M2 − 1)/K

(M3 − 1)/K

γ1 γ2 γ3

Figure: γP (γ)/P versus γ

Resultant spectral efficiency (S.E.)

R/B =∑

i log2(Mi)p(γi ≤ γ < γi+1)

Power adaptation

(M1 − 1)/K

(M2 − 1)/K

(M3 − 1)/K

γ1 γ2 γ3

Figure: γP (γ)/P versus γ

R/B =∑

Power adaptation

(M1 − 1)/K

(M2 − 1)/K

(M3 − 1)/K

γ1 γ2 γ3

Figure: P (γ)/P versus γ

R/B =∑

And so on . . .

What we have presented so far is achievable, but not necessarilyoptimal

Reduction in optimization complexity achieved by using insights fromwaterfilling solutions

Can also use insights from channel inversion/truncated channelinversion

Can extend ideas to discrete power, discrete rate

Homework 6

Problem 11 Use the adaptive rate and power adaptation assuming continuousM(γ) for first two parts

2 Use the truncated channel inversion formulation for the last part(again using continuous modulation assumption)

Problem 21 Use the fact that the number of constellation points is finite;

maximize spectral efficiency over all possible γ0 (dictated by theavailable constellations).

2 For variable rate-variable power schemes, need to compute regionwhen constellation is transmitted - gives spectral efficiency.

Outline

Using adaptive schemes

The channel cannot change too fast!

Valid when time that channel “stays” in a particular state is muchhigher than several symbol times

Calculation in terms of level crossing rates with Markov assumption(with jump only between adjacent regions)

R0 R1 . . .

Figure: Markov assumption (R0 = {γ : γ < γ0}, R1 = {γ : γ0 ≤ γ < γ1}, . . . )

Errors in channel estimation or due to delay

Basic effect

True value γ, but our estimate is γ; what is the error ?

Some expressions

Can be characterized by p(γ, γ)

Approximation for Pb gives

Pb(γ, γ) ≈ 0.2[5Pb]γ/γ

(depends only on ε = γ/γ!)

Question

What happens if γ > γ? (in terms of power and BER ?)

Homework 6

Problem 3 hint

The BER under the approximation we saw depends only on the statisticsof ε (not the joint statistics of γ and γ)

Problem 4

For part a, modulation used when Pb(γ) ≤ Pfloor.

For part b, yt, ht, ht+1 are jointly gaussian. yt = xtht + nt. Can beused to find MMSE estimate ht = E[ht|yt, xt]For part c, ht+1 = aht +

√1− a2nt,2 for appropriate a (Jakes’

scattering). Can be used to find MMSE estimate.

Pb(|ht+1|2, |ht+1|2) is the probability of error of modulation scheme(chosen by estimate |ht+1|2) with SNR |ht+1|2. ht+1 is a function ofyt. Need to integrate to find mean over joint distribution of (yt, ht+1).

Outline

Representing MIMO systems

Assumptions

Narrowband signals

Nt transmit antennas

Nr receive antennas

Noise n is zero mean with a covariance matrix of σ2INr

Represent gain from transmitter antenna j to receiver antenna i by hi,j

Tx antenna j Rx antenna ihi,j

System model

Model y1...yNr

︸︷︷︸

h1,1 . . . h1,Nt...

. . ....

hNr,1 . . . hNr,Nt

︸︷︷︸

x1...xNt

︸︷︷︸

n1...nNr

︸︷︷︸

where x is what transmitter sends and y is what receiver sees

Transmit power constraint

E[xx∗] =∑Nt

i=1E[|xi|2] ≤ ρ

Decomposition of H

Use the singular value decomposition (SVD) of channel matrix

H = UΣVH

Parallel channel decomposition

Transmitter sends x = Vx (transmit precoding)

Receiver obtains y = UHy (receiver shaping)

y = Σx

Figure: Equivalent parallel channels (no “crosstalk” or interchannel interference)

Question

How many such channels are there?

Outline

Channel capacity with Tx and Rx CSI

Expression

Under system model, can be shown to be

C = maxRx:Tr(Rx)≤ρ

B log2∣∣I + HRxHH

∣∣Equivalent expression

By using parallel decomposition, we get

C = maxρ:∑i ρi≤ρ

log2(1 + ρiσ2i )

Question

How do you acheive this?

Waterfill!

Channel capacity with Tx and Rx CSI

Expression

Under system model, can be shown to be

C = maxRx:Tr(Rx)≤ρ

By using parallel decomposition, we get

C = maxρ:∑i ρi≤ρ

log2(1 + ρiσ2i )

Question

How do you acheive this? Waterfill!

Channel capacity with Rx CSI only

Cannot use the channel realization at the transmitter; so spread energyequally at all the transmitters

Capacity expression

C = maxRx:Rx=ρ/NtINt

C =∑i

B log2(1 + ρσ2i /Nt)

Homework 6

Problem 6

For part a, use symmetry, Jensen’s inequality or AM-GM inequality

For part b, use Hadamard’s inequality

Problem 7

Use the law of large numbers and note that Mt is fixed

EE359 Discussion Session 7 Adaptive Modulation, MIMO systems · Outline 1 Adaptive transmit schemes...

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