Lecture 7: MIMO Capacity and Multiplexing Architectures - - PowerPoint PPT Presentation

Lecture ¡7: ¡MIMO ¡Capacity ¡and ¡ Multiplexing ¡Architectures

I-Hsiang Wang ihwang@ntu.edu.tw 5/15, 2014

Design ¡of ¡MIMO ¡Systems

• Regarding MIMO, what we have done so far:
• Established solid foundation on the statistical channel modeling
• Analyzed AWGN (no fading) MIMO capacity
• Indeed, MIMO is capable of the following:
• Multiplex multiple data streams simultaneously
• Provide spatial diversity
• Increase power gain
• What’s next:
• Derive MIMO capacity under fading
• Design transceiver architectures to extract multiplexing gain,

diversity gain, and power gain

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Plot

• Derive capacity of fading MIMO channel
• Fast fading: CSIR only and full CSI
• Slow fading: outage probability
• Discuss the nature of performance gains
• Introduce transceiver architectures for fast fading
• The V-BLAST family
• Introduce a transceiver architecture for slow fading
• D-BLAST

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Outline

• Capacity of fast fading MIMO
• V-BLAST
• Receiver architectures:
• Linear filters: decorrelator, matched filter, MMSE
• Successive interference cancellation (SIC)
• Outage probability of slow fading MIMO
• D-BLAST

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5

Fast ¡Fading ¡MIMO ¡Channel

Fast ¡Fading ¡MIMO ¡with ¡Full ¡CSI

• Channel model: y[m] = H[m]x[m] + w[m]
• {H[m]}: random fading process which is stationary and ergodic
• With full CSI, Tx and Rx can perform pre- and post-

processing based on the SVD of H[m] at each time:

• H[m] = U[m]Λ[m]V[m]*
• Convert the fading MIMO into a fading parallel channel:

(nmin := min{nt,nr})

• Water-filling to find the optimal power allocation

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e yi[m] = λi[m]e xi[m] + e wi[m], i = 1, 2, . . . , nmin

Ergidic ¡Capacity ¡with ¡Full ¡CSI

• Capacity via water-filling:

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V*[m]

...

U[m] V[m] U*[m] λ1[m] e w1[m] λnmin[m] e wnmin[m] x[m] y[m] e x[m] e y[m] CMIMO =

nmin

X

i=1

E  log ✓ 1 + λ2

i P ∗(λi)

σ2 ◆ P ∗(λ) = ✓ ν − σ2 λ2 ◆+ , ν satisfies

nmin

X

i=1

E "✓ ν − σ2 λ2

i

◆+# = P

Transceiver ¡Architecture ¡with ¡Full ¡CSI

8 + AWGN coder AWGN coder {x1[m]} ~ {y1 [m]} ~ {xnmin[m]} ~ {ynmin[m]} ~ . . . . . . . . . n min information streams {0} {0} {w[m]} U* H V Decoder Decoder

pre-processing post-processing

Receiver ¡CSI ¡Only: ¡V-­‑BLAST

• Tx cannot apply the pre-processing matrix V
• V-BLAST architecture:
• Tx prepares nt data streams, each encoded with a rate Ri coder
• Generate x by multiplying them with a unitary matrix Q
• Rx carries out joint decoding of the streams (eg., ML)
• We will discuss Rx architecture later

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+ Pnt P1 Q x[m] H[m] w[m] y[m] Joint decoder AWGN coder rate R1 AWGN coder rate Rnt

· · · · · · · · · · · ·

Capacity ¡of ¡Fast ¡Fading ¡MIMO ¡with ¡CSIR

• Using information theoretic arguments (or a sphere

packing argument), one can show that V-BLAST can achieve the capacity, which is given by the following:

• Kx := the covariance matrix of transmit signal vector x
• With V-BLAST,
• The issue boils down to finding the optimizing Kx for a

given stationary distribution of H

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Kx = Q diag (P1, . . . , Pnt) Q∗ C = max

Kx:Tr(Kx)≤P E

 log det ✓ Inr + HKxH∗ σ2 ◆

Multiplexing ¡in ¡the ¡Angular ¡Domain

• In V-BLAST, Q can be thought of as the coordinate

system onto which Tx modulates its data streams

• The question is, which coordinate system Q should be used?
• The choice of Q (and the power allocation {P1, … ,Pn}) depends on

the statistical property of H, so let’s focus on the angular domain representation: Ha = Ur*HUt

• Under rich scattering, entries of Ha are statistically

independent and zero-mean ⟹ it is reasonable to multiplex data on the coordinate system (indeed, optimal. HW.)

• Choose Q = Ut and hence Kx = Ut ΛpUt*.
• Still need to determine the power allocation Λp

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Uniform ¡Power ¡Allocation

• If further symmetry is present the random Ha, uniform

power allocation (

• ) turns out to be optimal
• Sufficient condition: the nt column vectors of Ha are i.i.d.
• For example, i.i.d. Rayleigh faded Ha.
• Hence the covariance matrix Kx = Ut ΛpUt* =
• This gives us the capacity formula: (SNR := P/σ2)
• In this case, Q can be any unitary matrix; in particular, it

suffices to choose Q = Int

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Λp = P

nt Int P nt Int

C = E  log det ✓ Inr + SNR nt HH∗ ◆

V-­‑BLAST ¡under ¡i.i.d. ¡Rayleigh

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+ Pnt P1 Q x[m] H[m] w[m] y[m] Joint decoder AWGN coder rate R1 AWGN coder rate Rnt

· · · · · · · · · · · ·

I

• Effectively, each of the Tx antennas, say, antenna i,

transmits an independent data stream of rate Ri

• How to determine Ri?
• For joint ML, it does not matter as long as ΣRi = the total capacity
• For other Rx architectures (later), the individual rate depends on

the effective channel it faces with, after the MIMO detector

P1 = P2 = · · · = Pnt = P

nt

Receiver ¡CSI ¡vs. ¡Full ¡CSI

• Capacity formula can be rewritten using singular values
• f the random matrix H:
• No CSIT ⟹ water-filling is not possible
• Recall the capacity with full CSI:

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λ1 ≥ λ2 ≥ · · · ≥ λnmin ≥ 0 CCSIR =

nmin

P

i=1

E h log ⇣ 1 + SNR

nt λ2 i

⌘i CFull CSI =

nmin

P

i=1

E ⇥ log

• 1 + SNR∗(λi)λ2

i

⇤

SNR∗(λ) =

• ν −

1 λ2

+, ν satisfies

nmin

P

i=1

E ⇣ ν −

1 λ2

i

⌘+ = SNR

DoF ¡Gain ¡at ¡High ¡SNR

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nt = nr = 1 nt = nr = 4 nt = 1 nr = 4 C (bits /s / Hz) 35 30 25 20 15 10 5 –10 10 20 30 SNR (dB) nt = nr = 1 nt = nr = 8 nt = 1 nr = 8 C (bits /s / Hz) 70 60 50 40 30 20 10 –10 10 20 30 SNR (dB)

• High SNR regime: nmin-fold DoF gain
• nmin := min{nt,nr} determines the high-SNR slope

CCSIR ≈ nmin log ⇣

SNR nt

⌘ +

nmin

P

i=1

E ⇥ log

• λ2

i

⇤ CFull CSI ≈ nmin log ⇣

SNR nmin

⌘ +

nmin

P

i=1

E ⇥ log

• λ2

i

⇤

Power ¡Gain ¡at ¡Low ¡SNR

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• nr determines the power gain under CSIR
• If CSIT is available, power gain is larger due to both

beamforming and dynamic power allocation

C C1,1 (bits / s / Hz) 4 3.5 2.5 3 10 –10 –20 –30 nt = 1 nr = 4 nt = nr = 4 SNR (dB) C C1,1 (bits / s / Hz) 8 7 6 5 4 3 SNR (dB) 10 –10 –20 –30 nt = 1 nr = 8 nt = nr = 8

= nrSNR log2 e CCSIR ≈

nmin

P

i=1 SNR nt E

⇥ λ2

i

⇤ log2 e = SNR

nt E [Tr (HH∗)] log2 e

Nature ¡of ¡Peformance ¡Gain ¡(CSIR ¡only)

• High SNR (DoF-limited):
• min{nt,nr}-fold DoF gain
• capacity scaling linearly with nmin := min{nt,nr}
• MIMO is crucial
• Low SNR (Power-limited):
• nr-fold power gain
• capacity scaling linearly with nr
• Only need multiple Rx antennas
• At moderate SNR
• min{nt,nr}-fold gain
• Due to a combination of both effects

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Receiver ¡Architectures

Decoding ¡at ¡the ¡Receiver

• In the previous V-BLAST architecture, Rx uses ML:
• ML is optimal
• But the complexity grows exponentially with the # of data streams
• A natural approach:
• First separate the signal of each data stream from others with

certain linear operations

• Then decode each data stream using single-user decoder
• In the following we focus on Rx architectures that use

linear operations in the first step

• Assuming V-BLAST with Q = Int , that is, each Tx antenna sends

an independent data stream

• If not, we can just lump Q into the channel matrix H

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Decorrelator: ¡Interference ¡Nulling

• Rewrite the received signal vector y as follows:
• xi denotes the signal sent from the i-th Tx antenna
• hi denotes the i-th column of channel matrix H, representing the

signal direction of xi .

• To decode xi , a simple idea is to use a decorrelator:
• First null out interference by projecting y onto the null space of

the directions of all interfering vectors {hj | j≠i}

• Then apply matched filter to the projected signal
• The Rx architecture consists of a bank of decorrelators.
• Also called interference nulling, zero forcing, etc.

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y = hixi + X

j6=i

hjxj + w

Bank ¡of ¡Decorrelators

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Decorrelator for stream nt Decorrelator for stream 2 Decorrelator for stream 1 y[m]

e y1 := (Q1h1)∗ (Q1y) e y2 := (Q2h2)∗ (Q2y) Rows of Qk form a

• rthonormal basis
• f the null space
• f {hj | j≠k}.

nulling matched filter after projection Note: for successful decorrelation, nt ≤ nr and hence nmin = nt Rk = E ⇥ log

• 1 + Pk

σ2 ||Qkhk||2⇤

Performance ¡in ¡i.i.d. ¡Rayleigh

• Uniform power allocation ⟹ the overall achievable rate:
• At high SNR:
• At low SNR: lose the power gain (homework)
• The decorrelator fully extracts the spatial multiplexing

gain of V-BLAST, but not the power gain

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Rdecorr =

nmin

P

k=1

E h log ⇣ 1 + SNR

nt ||Qkhk||2⌘i

Rdecorr ≈ nmin log ⇣

SNR nt

⌘ +

nmin

P

k=1

E ⇥ log

• ||Qkhk||2⇤

CCSIR ≈ nmin log ⇣

SNR nt

⌘ +

nmin

P

i=1

E ⇥ log

• λ2

i

⇤

Performance ¡Gap ¡from ¡Capacity

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Decorrelator Capacity 70 60 50 40 30 20 10 –10 –5 5 10 15 20 25 30 bits / s / Hz SNR (dB)

Constant gap due to loss of power gain; quite substantial

Removing ¡the ¡Gap: ¡SIC

• Recall in Lecture 5, to achieve the uplink capacity, we

use successive interference cancellation (SIC) at Rx

• Similar idea can be applied here to remove this gap
• The only difference from the previous derivation
• The nulling projection Qk is replaced by
• The rows of

now form an orthonormal basis of the null space

• f {hj | j > k}.
• Indeed this removes the gap at high SNR

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e Qk e Qk

Successive ¡Interference ¡Cancellation

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Decode stream nt Decode stream 3 Decode stream 2 Decode stream 1 Decorrelator 2 Decorrelator 3 Decorrelator nt Decorrelator 1 Stream nt Stream 1 Subtract stream 1, 2, ..., nt –1 y[m] Stream 3 Subtract stream1, 2 Subtract stream1 Stream 2

Zero ¡Forcing ¡vs. ¡Matched ¡Filtering

• Decorrelator (zero forcing): remove all interference at the

expense of reducing the received SNR

• Matched filter: projecting onto hi to maximize the SNR

but SINR may be bad

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y = hixi + X

j6=i

hjxj + w

Zero ¡Forcing ¡vs. ¡Matched ¡Filtering

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Decorrelator Matched fillter SNR (dB) 20 30 0.1 0.8 0.9 0.7 0.6 0.5 0.4 0.3 0.2 –30 –20 –10 10 1

• Low SNR: power-limited, interference not significant
• High SNR: interference-limited, power not significant

R C

Optimal ¡Linear ¡Filter: ¡MMSE

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y = hixi + X

j6=i

hjxj + w

• The optimal linear filter should maximize SINR at all SNR
• It offers the optimal compromise between ZF and MF
• Procedure:
• First whiten the aggregate interference + noise
• The apply MF since matched filter is optimal for white noise
• MMSE filter:

zi := P

j6=i

hjxj + w e y1 := ⇣ K

− 1

2

z1 h1

⌘∗ ⇣ K

− 1

2

z1 y

⌘ whitening matched filter after whitening =

• K−1

z1 h

∗ y

Geometric ¡Picture

29 Interference subspace Decorrelator Optimal filter Signal direction (matched filter)

• Low SNR: MMSE → ZF
• High SNR: MMSE → MF

Linear ¡MMSE: ¡Performance

• For the k-th stream, its data rate is
• Since
• , we have
• Hence the achievable rate can be computed
• We can further improve the performance of the MMSE

receiver by using SIC

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Rk = E h log ⇣ 1 + Pk||K

− 1

2

zk hk||2⌘i

= E ⇥ log

• 1 + Pkh∗

kK−1 zk hk

⇤ zk := P

j6=k

hjxj + w Kzk := σ2Inr + P

j6=k

Pjhjh⇤

j

Linear ¡MMSE: ¡Performance

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Decorrelator 20 10 –10 –20 –30 30 SNR (dB) MMSE Matched filter 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

R C

• Low SNR: MMSE → ZF
• High SNR: MMSE → MF

MMSE-­‑SIC

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Subtract stream 1, 2, ... , nt –1 Stream 2 Decode stream nt Stream nt Subtract stream 1 Stream 1 Decode stream 1 Decode stream 2 Decode stream 3 Subtract stream 1, 2 MMSE receiver 1 MMSE receiver nt MMSE receiver 3 MMSE receiver 2 y[m] Stream 3

SINRk,MMSE = Pkh∗

kK−1 zk hk

Kzk := σ2Inr + P

j>k

Pjhjh∗

j

MMSE-­‑SIC ¡Achieves ¡MIMO ¡Capacity

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R C

–30 10 –10 –20 20 Decorrelator 30 SNR (dB) MMSE–SIC MMSE Matched filter 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

• In fact for any channel matrix H and Kx, we have

nt

P

k=1

log (1 + SINRk,MMSE) = log det (Inr + HKxH∗)

Optimality ¡of ¡MMSE-­‑SIC

• Sending independent streams at individual Tx antennas

⟹ can be viewed as a multi-user uplink channel

• Hence, the optimality of SIC is straightforward
• At each stage, MMSE filter gives a sufficient statistics for

decoding the data from the vector signal

• This is because the capacity achieving code for each data stream

looks like Gaussian

• Hence the aggregate noise + interference is still Gaussian
• Hence, MMSE is information lossless at each stage

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