Lecture 8: MIMO Architectures (II) Receiver Theoretical Foundations - - PowerPoint PPT Presentation

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Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures D-BLAST

Lecture 8: MIMO Architectures (II) Theoretical Foundations of Wireless Communications1

Ragnar Thobaben CommTh/EES/KTH Wednesday, May 25, 2016 09:15-12:00, SIP

1Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication 1 / 20 Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures D-BLAST

Overview

Lecture 7: MIMO Architectures

• Generalized structure: V-BLAST.
• Fast fading channels: capacity with CSI-R.
• Slow fading channels: outage probability.

Lecture 8: MIMO Architectures (Ch. 8.3+5)

1 Receiver Architectures

Linear Decorrelator Successive Cancellation Linear MMSE Receiver MMSE-SIC

2 D-BLAST

Outage Probability Outage-Suboptimality of V-BLAST Coding Across the Antennas: D-BLAST

2 / 20

Notes Notes

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Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures Linear Decorrelator Successive Cancellation Linear MMSE Receiver MMSE-SIC D-BLAST

Receiver Architectures

– Linear Decorrelator

Motivation: achieving the capacity.

• With CSI-T: use SVD and transmit along the eigenmodes.
• With CSI-R and rich scattering: use the angular representation and

transmit along the angular windows.

• Goal: make sure that the receiver can separate the data streams

efficiently. Linear decorrelator

• Time-invariant channel model (with H = [h1 . . . hnt ]):

y[m] =

nt

• i=1

hixi[m] + w[m].

• Focusing on the k-th data stream (i.e., the k-th transmit antenna):

y[m] = hkxk[m] +

• i=k

hixi[m] + w[m]. → Interference from other streams.

3 / 20 Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures Linear Decorrelator Successive Cancellation Linear MMSE Receiver MMSE-SIC D-BLAST

Receiver Architectures

– Linear Decorrelator

• Goal: project y onto the subspace Vk which is orthogonal to the

space spanned by h1, . . . , hk−1, hk+1, . . . hnt .

• Assuming Vk is dk-dimensional, the projection

can be described by a matrix multiplication with a (dk×nr) matrix Qk: ˜ y[m] = Qky[m] = Qkhkxk[m] + ˜ w[m], with ˜ w[m] = Qkw[m].

(D. Tse and P. Viswanath, Fundamentals of Wireless Communications.)

→ Rows of Qk form an orthonormal basis of Vk. → hk has to be linearly independent of h1, . . . , hk−1, hk+1, . . . hnt . → The maximum number of data streams is nt ≤ nr; i.e., only subsets

• f antennas are used if nt > nr.

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Notes Notes

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Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures Linear Decorrelator Successive Cancellation Linear MMSE Receiver MMSE-SIC D-BLAST

Receiver Architectures

– Linear Decorrelator

• Optimal demodulation
• Matched filtering of ˜

y[m] with Qkhk, or equivalently filtering y[m] with a filter ck = (Q∗

k Qk)hk.

• SNR after matched filtering (k-th stream with power Pk):

PkQkhk2/N0.

• Decorrelating multiple streams

simultaneously

• Multiplying with the pseudoinverse:

H† = (H∗H)−1H∗.

• Bank of decorrelators.

(D. Tse and P. Viswanath, Fundamentals of Wireless Communications.) 5 / 20 Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures Linear Decorrelator Successive Cancellation Linear MMSE Receiver MMSE-SIC D-BLAST

Receiver Architectures

– Linear Decorrelator: Performance

Case 1: deterministic H.

• Maximum rate for the k − th stream and sum rate

Ck := log

• 1 + PkQkhk2

N0

• and

Rdecorr =

nt

• k=1

Ck.

• No inter-stream interference: SNR = Pkhk2/N0.
• Inter-stream interference reduces rate since Qkhk ≤ hk.
• Qkhk = hk if hk is orthogonal to the other spatial signatures

hi, with i = k.

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Notes Notes

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Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures Linear Decorrelator Successive Cancellation Linear MMSE Receiver MMSE-SIC D-BLAST

Receiver Architectures

– Linear Decorrelator: Performance

Case 2: fading channels.

• Fast fading, average over realizations of the channel process:

¯ Ck := E

• log
• 1 + PkQkhk2

N0

• and

Rdecorr =

nt

• k=1

¯ Ck → Generally less or equal to the capacity with CSI-R.

• High SNR , i.i.d. Rayleigh fading, nmin = nt:

Rdecorr ≈ nmin log SNR nt + E nt

• k=1

log(Qkhk2)

• =

nmin log SNR nt + ntE[log χ2

2(nr −nt+1)]

→ Decorrelator is able to fully exploit the degrees of freedom of the MIMO channel. → Second term shows the degradation in rate compared to the capacity.

7 / 20 Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures Linear Decorrelator Successive Cancellation Linear MMSE Receiver MMSE-SIC D-BLAST

Receiver Architectures

– Linear Decorrelator: Performance

• Example

(D. Tse and P. Viswanath, Fundamentals of Wireless Communications.) 8 / 20

Notes Notes

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Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures Linear Decorrelator Successive Cancellation Linear MMSE Receiver MMSE-SIC D-BLAST

Receiver Architectures

– Successive Cancellation

• Decorrelator: bank of separate filters for estimating the data

streams.

• But: the result from one of the filters can be used to improve the
• peration of the others; successive interference cancellation, SIC.
• Modified detector structure:
• For the k-th decorrelator, the

k − 1 previous streams have been removed. → Error propagation!

(D. Tse and P. Viswanath, Fundamentals of Wireless Communications.)

• hk is projected by ˜

Qk on a higher dimensional subspace orthogonal to that spanned by hk+1, . . . , hnt .

• Improved SNR on the k-th stream: SNRk = Pk˜

Qkhk2/N0

9 / 20 Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures Linear Decorrelator Successive Cancellation Linear MMSE Receiver MMSE-SIC D-BLAST

Receiver Architectures

– Successive Cancellation: Performance

• A similar derivation as above yields

Rdec-sic ≈ nmin log SNR nt + E nt

• k=1

log(˜ Qkhk2)

• =

nmin log SNR nt +

nt

• k=1

E[log χ2

2(nr −nt+k)]

• SIC does not gain additional degrees of freedom.
• Constant term is equal to that in the capacity expansion

(cf. (8.18)-(8.20) in the book) ⇒ Power gain by decoding and subtracting!

• Example

(D. Tse and P. Viswanath, Fundamentals of Wireless Communications.) 10 / 20

Notes Notes

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Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures Linear Decorrelator Successive Cancellation Linear MMSE Receiver MMSE-SIC D-BLAST

Receiver Architectures

– Linear MMSE Receiver

Comparison: decorrelator bank versus a bank of matched filters

(D. Tse and P. Viswanath, Fundamentals of Wireless Communications.)

• Matched filter: good at low SNR, bad at high SNR.

(Preserving the signal energy at the cost of interference.)

• Decorrelator: bad at low SNR, good at high SNR.

(Eliminating all interference at the cost of a low SNR.)

• Desirable receiver: maximize the signal-to-interference-plus-noise

ratio (SINR).

11 / 20 Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures Linear Decorrelator Successive Cancellation Linear MMSE Receiver MMSE-SIC D-BLAST

Receiver Architectures

– Linear MMSE Receiver

Derivation of a generic MMSE receiver

• Generic model: y = hx + z, with
• Complex circular symmetric colored noise z;
• An invertible covariance matrix Kz;
• A deterministic vector h;
• A scalar data symbol x.
• Apply a linear transform2 K−1/2

z

such that ˜ z = K−1/2

z

z is white, K−1/2

z

y = K−1/2

z

hx + ˜ z.

• Matched filtering with (K−1/2

z

h)∗: (K−1/2

z

h)∗K−1/2

z

y = h∗K−1

z y = h∗K−1 z hx + h∗K−1 z z

→ The linear receiver vmmse = K−1

z h maximizes the SNR.

• Achieved SINR: σ2

xh∗K−1 z h.

2Reminder: if Kz is invertible, then Kz = UΛU∗ and K1/2 z

= UΛ1/2U∗.

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Notes Notes

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Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures Linear Decorrelator Successive Cancellation Linear MMSE Receiver MMSE-SIC D-BLAST

Receiver Architectures

– Linear MMSE Receiver

Specialization for the MIMO case

• Channel model for the k-th stream,

y[m] = hkxk[m] + zk[m], with the noise plus interference term and its covariance matrix3 zk[m] =

• i=k

hixi[m] + w[m] and Kzk = N0Inr +

nt

• i=k

Pihih∗

i .

• Linear MMSE receiver:

vmmse =  N0Inr +

nt

• i=k

Pihih∗

i

 

−1

hk, achieving the output SINR Pkh∗

kK−1 zk hk = Pkh∗ k

 N0Inr +

nt

• i=k

Pihih∗

i

 

−1

hk.

3Note that Kzk is invertible. 13 / 20 Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures Linear Decorrelator Successive Cancellation Linear MMSE Receiver MMSE-SIC D-BLAST

Receiver Architectures

– Linear MMSE Receiver

Performance

• Low SNR: Kzk ≈ N0Inr , i.e., MMSE solution becomes matched filter.
• High SNR: MMSE receiver reduces to the decorrelator.
• Capacities for the k-th stream

Ck = log

• 1 + Pkh∗

kK−1 zk hk

• and

¯ Ck = E

• log
• 1 + Pkh∗

kK−1 zk hk

• .
• Example: i.i.d. Rayleigh fading and equal power allocation.

(D. Tse and P. Viswanath, Fundamentals of Wireless Communications.) 14 / 20

Notes Notes

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Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures Linear Decorrelator Successive Cancellation Linear MMSE Receiver MMSE-SIC D-BLAST

Receiver Architectures

– MMSE-SIC

(D. Tse and P. Viswanath, Fundamentals of Wireless Communications.)

• Successive interference cancellation using MMSE filters.
• MMSE-SIC achieves the highest possible rate for CSI-R

log det

• Inr + 1

N0 HKxH∗

• 15 / 20

Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures Linear Decorrelator Successive Cancellation Linear MMSE Receiver MMSE-SIC D-BLAST

Receiver Architectures

– MMSE-SIC: Information Theoretic Optimality

• Consider again the generic model y = hx + z, with additive colored

noise and Gaussian x and z.

• MMSE receiver is information lossless; i.e., it provides a sufficient

statistic to detect x such that I(x; y) = I(x; v∗

mmsey).

• Consider now the MIMO model y[m] = Hx[m] + w[m], with

x ∼ CN(0, diag{P1, . . . , Pnt }).

• Using the chain rule of mutual information, we get

I(x; y) = I(x1, . . . , xnt ; y) = I(x1; y) + I(x2; y|x1) + . . . + I(xnt ; y|x1, . . . , xnt−1).

• Considering the k-th term in the chain rule, we get

I(xk; y|x1, . . . , xk−1) = I(xk; y′) = I(xk; v∗

mmseky′)

using y′ = y −

k−1

• i=1

hixi = hkxk +

• i>k

hixi + w → The rate achieved in the k-th stage is precisely I(xk; y|x1, . . . , xk−1).

16 / 20

Notes Notes

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Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures D-BLAST Outage Probability Suboptimality of V-BLAST Coding Across the Antennas

D-BLAST

– Outage Probability

• Reliable communication at rate R is possible as long as

log det

• Inr + 1

N0 HKxH∗

• > R

subject to Tr[Kx] ≤ P.

→ Information theory guarantees the existence of a channel-state independent coding scheme that achieves reliable communications whenever this condition is met. → Universal code: works for all channels that satisfy the above condition. → If the condition is not satisfied, we are in outage.

• Outage probability

pMIMO

• ut

(R) = min

Kx :Tr[Kx ]≤P Pr

• log det
• Inr + 1

N0 HKxH∗

• < R
• .

→ Choose the transmit strategy (i.e., Kx) to minimize the outage probability. → Solution depends on statistics of H.

17 / 20 Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures D-BLAST Outage Probability Suboptimality of V-BLAST Coding Across the Antennas

D-BLAST

– Outage-Suboptimality of V-BLAST (with MMSE-SIC)

V-BLAST: capacity achieving for the fast fading MIMO channel

• Independent data streams are multiplexed and transmitted over the

antenna; stream k with power Pk and rate Rk.

• MMSE-SIC receiver.

V-BLAST: diversity

• Each stream has at most diversity order nr while the MIMO channel

provides diversity gain nr × nt. → V-BLAST does not exploit full diversity and cannot be outage

• ptimal.
• Due to interference, the diversity can be lower than nr.

(Example: for SIC with decorrelator the diversity loss equals the number of uncanceled interferers.)

18 / 20

Notes Notes

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Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures D-BLAST Outage Probability Suboptimality of V-BLAST Coding Across the Antennas

D-BLAST

– Outage-Suboptimality of V-BLAST (with MMSE-SIC)

• For a given H, V-BLAST achieves

log det

• Inr + 1

N0 HKxH∗

• =

nt

• k=1

log(1 + SINRk). (SINRk is random since it is a function of H.)

• Assume that the outage-optimal transmit strategy Kx is employed

and that the target rate R is split into rates R1, . . . , Rnt .

• The channel is in outage if

nt

• k=1

log(1 + SINRk) = log det

• Inr + 1

N0 HKxH∗

• < R =

nt

• k=1

Rk → V-BLAST is in outage whenever any stream is in outage (i.e., log(1 + SINRk) < Rk for some k), and this can occur even though the channel is not in outage. Summary: Problem with V-BLAST

• Each stream sees only one efficient channel with SINRk; there is no

coding across the channels.

• But each stream should see all channels.

19 / 20 Lecture 8 MIMO Architectures (II) Ragnar Thobaben CommTh/EES/KTH Receiver Architectures D-BLAST Outage Probability Suboptimality of V-BLAST Coding Across the Antennas

D-BLAST

– Coding Across the Antennas: D-BLAST

Two-antenna example

• The i-th codeword x(i) is made up of two

codewords x(i)

A and x(i) B .

• First time slot: antenna-1 is silent; antenna-2

transmits x(1)

A ; receiver performs MRC of the

received antennas to estimate x(1)

A .

→ x(1)

A

sees effectively SINR2.

• Second time slot: antenna-1 transmits x(1)

B ;

antenna-2 transmits x(2)

A ; receiver performs linear

MMSE treating x(2)

A

as noise and estimates x(1)

B .

→ x(1)

B

sees effectively SINR1.

• The codeword x(i) can be decoded if

log(1 + SINR1) + log(1 + SINR2) > R and x(1)

B

can be subtracted such that x(2)

A

sees again an interference-free channel.

(D. Tse and P. Viswanath, Fundamentals of Wireless Communications.)

Drawbacks: Rate-loss due to initialization phase and error propagation.

20 / 20

Notes Notes