MIMO Fundamentals and Signal Processing Course Erik G. Larsson Link - - PowerPoint PPT Presentation

MIMO Fundamentals and Signal Processing Course

Erik G. Larsson Link¨

• ping University (LiU), Sweden
• Dept. of Electrical Engineering (ISY)

Division of Communication Systems www.commsys.isy.liu.se

slides version: September 25, 2009

Link¨

• ping University, ISY, Communication Systems, E. G. Larsson

MIMO Fundamentals and Signal Processing

Objectives and Intended Audience

➮ This short course will give an introduction to the basic principles and signal processing for MIMO wireless links. ➮ Intended audience are graduate students and industry researchers ➮ Prerequisites: General mathematical maturity. Solid knowledge of linear algebra and probability theory. Good understanding of digital and wireless communications. Some basic coding/information theory.

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MIMO Fundamentals and Signal Processing

Organization

➮ 4 lectures: Le 1: MIMO fundamentals Le 2: Low-complexity MIMO Le 3: MIMO receivers Le 4: MIMO in 3G-LTE (by P. Frenger, Ericsson Research) ➮ Reading, in addition to course notes: ➠ Le 1: D. Tse & P. Viswanath, Fundamentals

• f

Wireless Communications, Cambridge Univ. Press 2005, Chs. 7–8 ➠ Le 2: E. G. Larsson & P. Stoica, Space-time block coding for wireless communications, Cambridge Univ. Press 2003, Chs. 2, 4–8 ➠ Le 3: E. G. Larsson, ”MIMO detection methods: How they work”, IEEE SP Magazine, pp. 91–95, May 2009 ➮ Examination (for Ph.D. students): TBD

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MIMO Fundamentals and Signal Processing

Le 1: MIMO fundamentals

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MIMO Fundamentals and Signal Processing

Basic channel model

➮ Flat fading; linear, time-invariant channel

TX RX

1 1 nr nt ➮ Complex data {x1, . . . , xnt} are transmitted via the nt antennas ➮ Received data: ym =

nt

• n=1

hm,nxn + em

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MIMO Fundamentals and Signal Processing

Basic MIMO Input-Output Relation

➮ Transmission model (single time interval):   y1 . . . ynr  

y (RX data) =

  h1,1 · · · h1,nt . . . . . . hnr,1 · · · hnr,nt  

• H (channel)

  x1 . . . xnt  

x (TX data) +

  e1 . . . enr  

e (noise)

➮ Transmission model (N time intervals): y1,1

· · · y1,N . . . . . . ynr,1 · · · ynr,N

• Y

= h1,1

· · · h1,nt . . . . . . hnr,1 · · · hnr,nt

• H

x1,1

· · · x1,N . . . . . . xnt,1 · · · xnt,N

• X∈X

“code matrix”

+ e1,1

· · · e1,N . . . . . . enr,1 · · · enr,N

• E

➮ AWGN: ek,l ∼ N(0, N0), i.i.d.

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MIMO Fundamentals and Signal Processing

MIMO Design Space

➮ Fast fading: codeword spans ∞ number of channel realizations Channel can be time- or frequency-variant (e.g., MIMO-OFDM), or both ➮ Slow fading: codeword spans one channel realization ➮ For point-to-point MIMO, four basic cases (reality inbetween) Fast fading Slow fading Channel V-BLAST optimal V-BLAST suboptimal unknown at TX no coding across antennas coding across antennas D-BLAST optimal Channel waterfilling waterfilling known at TX

• ver space& time
• ver space

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MIMO Fundamentals and Signal Processing

Slow fading, full CSI at TX

➮ Deterministic channel, known at TX and RX y = Hx + e H ∈ Cnr×nt ➮ Power constraint: E[||x||2] ≤ P ➮ e is white noise, N(0, N0I) ➮ SVD: H = UΛV H, U HU = I, V HV = I ➮ Dimensions: U ∈ Cnr×nr, Λ ∈ Rnr×nt, V ∈ Cnt×nt

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MIMO Fundamentals and Signal Processing

➮ Introduce transform: y = UΛV H

H

x + e ˜ y U Hy = Λ V Hx

˜ x

+ U He

˜ e

➮ ˜ e is white noise, N(0, N0I) ➮ ˜ x is precoded TX data. Note: E[||˜ x||2] = E[||x||2] ➮ Equivalent model with parallel channels: (n = rank(H)) ˜ y1 = λ1˜ x1 + ˜ e1 ... ˜ yn = λn˜ xn + ˜ en

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MIMO Fundamentals and Signal Processing

➮ No gain by coding across streams ➠ ˜ xk independent ➮ Operational meaning is multistream beamforming: x = V ˜ x =

n

• k=1

˜ xkvk ➠ n indep. streams {˜ xk} are transmitted over orthogonal beams {vk} ➠ We’ll assume that each stream ˜ xk has power Pk ➠ In the special case of n = 1 (rank one channel), then we have MRT: H = λuvH ⇒ x = ˜ xv

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MIMO Fundamentals and Signal Processing

Architecture

x1 x2 xnt ˜ x1 ˜ x2 ˜ xn e1 enr y1 y2 ynr ˜ y1 ˜ y2 ˜ yn

H V :,1:n UH

:,1:n

➮ ˜ xk are independent streams with powers Pk and rates Rk

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MIMO Fundamentals and Signal Processing

Optimal power allocation over n subchannels

➮ Each subchannel can offer the rate Ck = log2

• 1 + Pk

N0 λ2

k

• (bits/cu)

➮ The power constraint is E[||˜ x||2] = E[||x||2] =

n

• k=1

Pk ≤ P ➮ Optimal power allocation P ∗

1 , ..., P ∗ n

max

Pk,Pn

k=1 Pk≤P

n

• k=1

Ck ⇔ max

Pk,Pn

k=1 Pk≤P

n

• k=1

log2

• 1 + Pk

N0 λ2

k

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MIMO Fundamentals and Signal Processing

Waterfilling solution

➮ Problem: max

Pk,Pn

k=1 Pk≤P

n

• k=1

log2

• 1 + Pk

N0 λ2

k

• ➮ Solution: (need solve for µ)

C =

n

• k=1

log2

• 1 + P ∗

kλ2 k

N0

• P ∗

k =

• µ − N0

λ2

k

+ ,

n

• k=1

P ∗

k = P

➮ Special case: H = λuvH. Transmit one stream C = log2

• 1 + P

N0 λ2

• = log2
• 1 + P

N0 H2

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MIMO Fundamentals and Signal Processing

Waterfilling solution

1 2 3 ... n k

N0 λ2

k

µ P ∗

1 = 0

P ∗

2

P ∗

3 = 0

P ∗

4

P ∗

5 = 0

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MIMO Fundamentals and Signal Processing

Waterfilling at high SNR

➮ At high SNR, the water is deep, so P ∗

k ≈ P n and

C =

n

• k=1

Ck ≈

n

• k=1

log2

• 1 + P

N0 λ2

k

n

• ≈ n · log2

P N0

• +

n

• k=1

log2 λ2

k

n

• ➮ With SNR P/N0, we have C ∼ n log2(SNR). We say that

The channel offers n = rank(H) degrees of freedom (DoF) ➮ Best capacity for well conditioned H (all λk’s equal) ➮ Transmit n equipowered data streams spread on orthogonal beams ➮ Channel knowledge ∼unimportant ➮ No coding across streams

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MIMO Fundamentals and Signal Processing

Waterfilling at high SNR

• 1

2 3 ... n k

N0 λ2

k

P ∗

k ≈ P n

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MIMO Fundamentals and Signal Processing

Waterfilling at low SNR

➮ At low SNR, the water is shallow. Then P ∗

k =

• P,

k = argmax λ2

k

0, else C = log2

• 1 + P

N0 λ2

max

• ≈

P N0 λ2

max

• · log2(e)

➮ MIMO provides an array gain (power gain of λ2

max) but no DoF gains.

➮ Channel rank does not matter, only power matters. ➮ Transmit one beam in the direction associated with largest λk ➮ Knowing H is very important! (to select what beam to use)

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MIMO Fundamentals and Signal Processing

Waterfilling at low SNR

1 2 3 ... n k

N0 λ2

k

µ P ∗

4 = P

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MIMO Fundamentals and Signal Processing

In practice

➮ Feedback of channel state information, requires quantization ➮ Potentially, by scheduling only “good” users, one may always operate at high SNR ➮ Selection of modulation scheme — e.g., M-QAM per subchannel, different M — better channel, larger constellation — should be done with outer code in mind ➮ Imperfect CSI ➠ cross-talk!

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MIMO Fundamentals and Signal Processing

MIMO channel models

➮ MIMO channel modeling is a rich research field, with both empirical (measurement) work and theoretical models. ➮ We will explore the main underlying physical phenomena of MIMO propagation and how they connect to the DoF concept.

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MIMO Fundamentals and Signal Processing

Line-of-sight SIMO channel

➮ Consider m-ULA at TX and RX, wavelength λ = c/fc, ant. spacing ∆

TX RX array ∆ φ

➮ Let u(φ)      1 e−j 2π

λ ∆ cos φ

. . . e−j(m−1)2π

λ ∆ cos φ

     ➮ Signal from point source impinging on RX array (large TX-RX distance): y = αu(φ) · s + e, (α ∈ C,

• dep. on distance)

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MIMO Fundamentals and Signal Processing

Line-of-sight MIMO channel

TX array RX array φr φt

➮ MIMO channel: y = α · u(φr)

nr×1

·

H

u(φt)

nt×1

• H

·x + e, n = rank(H) = 1 ➮ The LoS-MIMO channel has rank one, so no DoF gain!

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MIMO Fundamentals and Signal Processing

Lobes and resolvability

➮ Consider unit-power point sources at φ1, φ2 with sign. u(φ1), u(φ2). How similar do these signatures look? 1 ms1u(φ1) − s2u(φ2)2 = 2 − 2Re    s∗

1s2 · 1

muH(φ1)u(φ2)

• |·|=f(·)

    where the lobe pattern f(cos(φ1) − cos(φ2)) 1

m|uH(φ1)u(φ2)|.

➮ If f(·) < 1, then φ1, φ2 resolvable. ➮ Resolvability criterion: | cos φ1 − cos φ2| ≥ 2π

A ,

A (m − 1)∆ ➮ Grating lobes avoided if ∆ ≤ λ 2 ⇒ A ≤ (m − 1)λ 2.

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MIMO Fundamentals and Signal Processing

Two separated point sources and an m-receive-array

TX1 TX2 RX array φ1 φ2

➮ Define H = [h1 h2], hi = αiu(φi) ➮ Condition number κ(H) = λmax(H) λmin(H) =

• 1 + f(cos(φ1) − cos(φ2))

1 − f(cos(φ1) − cos(φ2)) ➮ κ(H) is small if f(· · · ) = 1 ⇔ φ1, φ2 resolvable ⇔ | cos φ1−cos φ2| > 2π

A

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MIMO Fundamentals and Signal Processing

MIMO with two plane scatters

TX array RX array A B

➮ Here, HTX-RX = HAB-RX · HTX-AB ➮ We have rank(HTX-RX) = 2

• nly if rank(HAB-RX) = 2

and rank(HTX-AB) = 2 ➮ For H to offer 2 DoF, A and B must be sufficiently separated in angle, as seen both from TX and RX

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MIMO Fundamentals and Signal Processing

Angular decomposition of MIMO channel

➮ For φ1, φ2, ..., φm define U 1 √m[u(φ1) · · · u(φm)] ➮ Can show: With cos(φk) = k/m, {u(φk)} forms ON-basis. Then U HU = I. ➮ Let U r and U t be the U matrices associated with the TX and RX

• arrays. Note that U H

r U r = I and U H t U t = I

➮ If ∆ = λ/2, then u(φi) correspond to simple, perfectly resolvable beams, with a single mainlobe. ➮ We assume ∆ = λ/2 from now on. The case of ∆ = λ/2 is more involved.

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MIMO Fundamentals and Signal Processing

Angular decomposition, cont.

1 2 3 4 5 1 2 3 4 5 TX RX

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MIMO Fundamentals and Signal Processing

Angular decomposition, cont.

➮ Now define Ha U H

r HU t

⇒ Ha,(k,l) = uH(φr

k)Hu(φt l)

uH(φr

k)

• i

αiu(φ′r

i )uH(φ′t i )

• physical model

u(φt

l)

=

• i

αi    uH(φr

k)u(φ′r i )

• =0 unless φ′r

i falls in lobe φr k

  ·     uH(φ′t

i )u(φt l)

• =0 unless φ′t

i falls in lobe φt l

    ➮ Elements of Ha correspond to different propagation paths ➮ Ha,(k,l)=gain of ray going out in TX lobe l and arriving in RX lobe k

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MIMO Fundamentals and Signal Processing

Angular decomposition, key points

➮ “Rich scattering” if all angular bins filled (Ha has “no zeros”) ➮ “Diversity order” = measure of error resilience = number of propagation paths = number of nonzero elements in Ha ➮ Number of DoF = rank(H) = rank(Ha) ➮ If Ha,(k,l) are i.i.d. then Hk,l are i.i.d. ➮ With i.i.d. Ha and many terms in , then we get i.i.d. Rayleigh fading.

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MIMO Fundamentals and Signal Processing

Fast fading, no CSI at TX

➮ Each codeword spans ∞ number of H ➮ The V-BLAST architecture is optimal here Note: Reminiscent of architecture for slow fading and full CSI@TX ➮ Transmit vectors x = Q˜ x where ˜ x1, ..., ˜ xn are independent streams with powers Pk and rates Rk

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MIMO Fundamentals and Signal Processing

V-BLAST architecture

x1 x2 xnt ˜ x1 ˜ x2 ˜ xn e1 enr y1 y2 ynr

H Q

• ptimal

receiver

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MIMO Fundamentals and Signal Processing

➮ Transmit covariance: Kx cov(x) = Q     P1 · · · P2 ... . . . . . . ... ... · · · Pn    QH ➮ Achievable rate, for fixed H: R = log2

• I + 1

N0 HKxHH

• ➮ Intuition: Volume of noise ball is |N0I|N.

Volume of signal ball is |HKxHH + NoI|N.

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MIMO Fundamentals and Signal Processing

➮ Fast fading, coding over ∞ number of H matrices gives ergodic capacity C = E

• log2
• I + 1

N0 HKxHH

• ➮ Choose Q and Pk to

max

Kx,Tr(Kx)≤P

E

• log2
• I + 1

N0 HKxHH

• ➮ Optimal Kx depends on the statistics of H

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MIMO Fundamentals and Signal Processing

➮ In i.i.d. Rayleigh fading, K∗

x = P ntI (i.i.d. streams) and

C = E

• log2
• I + P

N0 1 nt HHH

• =

n

• k=1

E

• log2
• 1 + SNR

nt λ2

k

• where

n = rank(H) = min(nr, nt) SNR P N0 {λk} are the singular values of H ➠ Antennas then transmit separate streams. ➠ Coding across antennas is unimportant.

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MIMO Fundamentals and Signal Processing

Some special cases

➮ SISO: nt = nr = 1 C = E

• log2(1 + SNR|h|2)
• At high SNR, the loss is -0.83 bpcu relative to AWGN channel

➮ SIMO: nt = 1 (power gain relative to SISO) C = E

• log2
• 1 + SNR

nr

• k=1

|hk|2

• ➮ MISO: nr = 1 (no power gain relative to SISO)

C = E

• log2
• 1 + SNR

nt

nt

• k=1

|hk|2

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MIMO Fundamentals and Signal Processing

Large arrays (infinite apertures)

➮ Large MISO (nt TX, 1 RX) becomes AWGN channel: C = E

• log2
• 1 + SNR

nt

nt

• k=1

|hk|2

• → log2(1 + SNR)

➮ Large SIMO (1 TX, nr RX) C = E

• log2
• 1 + SNR

nr

• k=1

|hk|2

• ≈ log2(nrSNR) = log2(nr)+log2(SNR)

➮ Large square MIMO (nt TX, nr RX, nr = nt = n): Linear incr. with n: C ≈ n ·

• 1

π 4

• log2(1 + t · SNR)
• 1

t − 1 4

• dt
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MIMO Fundamentals and Signal Processing

Fast fading, no CSI at TX, high SNR

➮ Here C =

n

• k=1

E

• log2
• 1 + SNR

nt λ2

k

• ≈ n log2(SNR) + const

➮ Both nr and nt must be large to provide DoF gain ➮ “Capacity grows as min(nr, nt)”

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MIMO Fundamentals and Signal Processing

Fast fading, no CSI at TX, low SNR

➮ Here C =

n

• k=1

E

• log2
• 1 + SNR

nt λ2

k

• ≈ log2(e) · SNR

nt ·

n

• k=1

E[λ2

k]

= log2(e) · SNR nt · E[||H||2

• =nrnt

] = log2(e) · nr · SNR ➮ Capacity independent of nt! ➮ No DoF gain. All what matters here is power ➮ Relative to SISO, a power gain of nr (array/beamforming gain) ➮ Multiple TX antennas do not help here (but with CSI at TX, things are very different)

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MIMO Fundamentals and Signal Processing

V-BLAST in practice

➮ Transmitter architecture “simple” but the receiver must separate the streams ➠ major challenge ➮ Problems are conceptually similar to uplink MUD in CDMA and to equalization for ISI channels ➮ Stream-by-stream receivers: Successive-interference-cancellation ➠ MMSE-SIC is theoretically optimal but suffers from error propagation ➠ Rate allocation necessary ➮ Iterative architectures ➠ Iteration between outer code and demodulator ➠ Demodulator design is major problem ➮ Receivers for MIMO to be discussed more in lecture 3

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MIMO Fundamentals and Signal Processing

Fast fading, full CSI at TX

➮ The transmitter can do waterfilling over both space and time ➮ Parallel channels: ˜ yk[m] = λk[m]˜ xk[m] + ˜ ek[m] ➠ Waterfilling over space (k) and time (m). ➠ Optimal powers P ∗

k[m]

➠ Capacity C =

n

• k=1

E

• log2
• 1 + P ∗(λk)λ2

k

N0

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MIMO Fundamentals and Signal Processing

➮ High SNR: P ∗(λk) ≈ P

n (equal power)

C ≈

n

• k=1

E

• log2
• 1 + SNR

n λ2

k

• ,

n D.o.F. An SNR gain (compared to no CSI) of nt n = nt min(nt, nr) = nt nr , if nt ≥ nr ➮ Low SNR: Even larger gain, so here multiple antennas do help!

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MIMO Fundamentals and Signal Processing

Slow fading, no CSI at TX

➮ Reliable communication for fixed H if log2

• I +

1 N0HKxHH

• > R

➮ Outage probability, for fixed R: Pout = P

• log2
• I +

1 N0HKxHH

• < R
• ➮ Optimal Kx as function of H’s statistics:

K∗

x =

argmin

Kx,Tr Kx≤P

P

• log2
• I + 1

N0 HKxHH

• < R
• For H i.i.d. Rayleigh fading:

➠ K∗

x = P ntI optimal at large SNR

➠ K∗

x = P n′ diag{1, ..., 1, 0, ..., 0} at low SNR (n′ < nt)

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MIMO Fundamentals and Signal Processing

➮ Notion of diversity: Pout behaves as SNR−d where d=diversity order ➮ Maximal diversity: d = nrnt ➮ To achieve diversity, we need coding across streams ➮ V-BLAST does not work here. Each stream has diversity at most nr, while the channel offers nrnt ➮ Architectures for slow fading: ➠ Theoretically, D-BLAST is optimal ➠ Pragmatic approaches include STBC combined with FEC

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MIMO Fundamentals and Signal Processing

Example: Outage probability at rate R = 2 bpcu

−5 5 10 15 20 25 30 35 40 10

−4

10

−3

10

−2

10

−1

SNR FER

nt=1, nr=1 (SISO) nt=2, nr=1 nt=1, nr=2 nt=2, nr=2

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MIMO Fundamentals and Signal Processing

D-BLAST architecture

epl xA(1) xB(1) xA(2) xB(2) xA(3)

➮ Decoding in steps:

• 1. Decode xA(1)
• 2. Decode xB(1), suppressing xA(2) via MMSE
• 3. Strip off xB(1), and decode xA(2)
• 4. Decode xB(2), suppressing xA(3) via MMSE

➮ One codeword: x(i) = [xA(i) xB(i)] ➮ Requires appropriate rate allocation among xA(i), xB(i) ➮ In practice, error propagation and rate loss due to initialization

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MIMO Fundamentals and Signal Processing

Le 2: Low-complexity MIMO

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MIMO Fundamentals and Signal Processing

Antenna diversity basics

➮ Recall transmission model (single time interval):   y1 . . . ynr  

y (RX data) =

  h1,1 · · · h1,nt . . . . . . hnr,1 · · · hnr,nt  

• H (channel)

  x1 . . . xnt  

x (TX data) +

  e1 . . . enr  

e (noise)

➮ Transmission model (N time intervals): y1,1

· · · y1,N . . . . . . ynr,1 · · · ynr,N

• Y

= h1,1

· · · h1,nt . . . . . . hnr,1 · · · hnr,nt

• H

x1,1

· · · x1,N . . . . . . xnt,1 · · · xnt,N

• X∈X

“code matrix”

+ e1,1

· · · e1,N . . . . . . enr,1 · · · enr,N

• E

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MIMO Fundamentals and Signal Processing

Introduction and preliminaries

➮ Transmitting with low error probability at fixed rate requires N large. ➮ For practical systems, it is often of interest to design short space-time blocks (small N) with good error probability performance. Outer FEC can then be used over these blocks. ➮ Throughout, we will assume Gaussian noise, e ∼ N(0, N0I) Usually, we assume i.i.d. Rayleigh fading, Hi,j i.i.d. N(0, 1) Sometimes, for SIMO/MISO, we take h ∼ N(0, Q)

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MIMO Fundamentals and Signal Processing

Receive diversity (nt = 1)

➮ Suppose s transmitted, and h known at RX. ➮ Receive: y = hs + e ➮ Detection of s via maximum-likelihood (in AWGN): y − hs2 = ... = h2 ·

• s − ˆ

s

• 2

+ const., where ˆ s hHy h2 ➮ MRC+scalar detection problem! ➮ Distribution of ˆ s determines performance: ˆ s hHy h2 ∼ N

• s, N0

h2

• ,

SNR|h = h2 N0 · E

• |s|2

=P

= h2 · P N0

• SNR

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MIMO Fundamentals and Signal Processing

Diversity order (n × 1 fading vector h)

➮ P(e|h) = Q

• SNR · h2
• and h ∼ N(0, Q) (SNR up to a constant)

➮ Then P(e) = E[P(e|h)] ≤

• I + SNR

2 Q

• −1

=

n

• k=1
• 1 + SNR

2 λk(Q) −1 ➮ As SNR → ∞, P(e) ≤ SNR 2 − rank(Q) · 1 rank(Q)

k=1

λk(Q) ➮ Diversity order d −log P(e) log SNR = rank(Q) ➮ Note that n

• k=1

λk 1/n ≤ 1 n

n

• k=1

λk = 1 nTr{Q} = 1 nE[h2] with eq. if λ1 = · · · = λn so Q ∝ I minimizes the bound on P(e)

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MIMO Fundamentals and Signal Processing

Diversity order

10−1 10−2 10−3 10−4 10−5 10 20 30 40 50 d = 1 d = 2 d = 3 P(e) SNR

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Transmit diversity, H known at transmitter

➮ Try transmit w · s where w is function of H! (as we did in Le 1) ➮ RX data is y = Hws+e and optimal decision minimizes the ML metric: y − Hws2 = ... = Hw2 · |s − ˆ s|2 + const. where ˆ s wHHHy Hw2 ∼ N

• s,

N0 Hw2

• ➮ The SNR|H in ˆ

s is max for w=normalized dominant RSV of H ➮ Resulting SNR|H =

1 N0 λmax(HHH)

• ≥H2/nt

·E

• |s|2

≥

1 N0 H2 nt

· E

• |s|2

. ➮ Diversity order: d = nrnt ➮ For nr = 1 take wopt =

h∗ h so SNR|h = h2 N0 E

• |s|2

(same as for RX-d)

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MIMO Fundamentals and Signal Processing

Transmit diversity, H unknown at transmitter

➮ From now on, TX does not know H! ➮ Consider Y = HX + E. Optimal receiver in AWGN (H known at RX): max

X P(X|Y , H) ⇔ min X Y − HX2

➮ Pairwise error probability P(X0 → X|H) = Q

• H(X0−X)2

2N0

• ➮ Consider P(X0 → X) = E[P(X0 → X|H)]. For i.i.d. Rayleigh fading,

P(X0 → X) ≤

• I +

1 4N0 (X0 − X)(X0 − X)H

• −nr
• ∼

“

1 N0

”−d ∼SNR

−d

d=“diversity order”. Note: d ≤ nrnt and d = nrnt if X0 − X full rank

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MIMO Fundamentals and Signal Processing

Linear space-time block codes (STBC)

➮ STBC maps ns complex symbols onto nt × N matrix X: {s1, . . . , sns} → X ➮ Linear STBC: X =

ns

• n=1

(¯ snAn + i˜ snBn) where {An, Bn} are fixed matrices ➮ Typically N small. Need N ≥ nt for max diversity (why?) ➮ Rate: R N ns bits/channel use

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MIMO Fundamentals and Signal Processing

STBC with a single symbol

➮ Transmit one symbol s during N time intervals, weighted by W : X = W · s, Y = HX + E = HW s + E ➮ Average error probability in Rayleigh fading: P(s0 → s) ≤ |W W H|−nr|s − s0|−2nrnt 1 4N0 −nrnt ➮ What is the optimum W ? Try to maximize: max

W

|W W H| s.t. Tr

• W W H

= W 2 ≤ 1 (power constraint) ➮ Solution: W W H = 1

ntI, (antenna cycling). Diversity but rate 1/N!

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MIMO Fundamentals and Signal Processing

Alamouti scheme for nt = 2

➮ X =

1 √ 2

• s1

s∗

2

s2 −s∗

1

• . That is:

Time 1 Time 2 Ant 1 s1/ √ 2 s∗

2/

√ 2 Ant 2 s2 √ 2 −s∗

1/

√ 2 ➮ RX data: y1 y2

• =

1 √ 2

h1s1 + h2s2 h1s∗

2 − h2s∗ 1

• +

e1 e2

• ➮ Consider
• y1

y∗

2

• =

1 √ 2

• h1s1 + h2s2

h∗

1s2 − h∗ 2s1

• +
• e1

e∗

2

• =

1 √ 2

• h1

h2 −h∗

2

h∗

1

s1 s2

• +
• e1

e∗

2

• ➮ ML detector

min

s1,s2

• y1

y∗

2

• y

− 1 √ 2

• h1

h2 −h∗

2

h∗

1

• G
• s1

s2

• s
• 2

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MIMO Fundamentals and Signal Processing

➮ Observation: GHG = 1

2

• hH

1

−hT

2

hH

2

hT

1

h1 h2 −h∗

2

h∗

1

• = h12+h22

2

I ➮ Hence min y − Gs2 ⇔ min ˆ s − s2, ˆ s = 2 GHy h12 + h22 ➮ Distribution of ˆ s: ˆ s = 2 GHy h12 + h22 = 2 GH(Gs + e) h12 + h22 ∼ N

• s, 2N0

H2I

• ➮ SNR|H = H2

2N0 . For 2 × 1 system, 3 dB less than 1 × 2 with MRC ➮ Diversity order: 2nr

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MIMO Fundamentals and Signal Processing

Overview of 2-antenna systems

Method SNR rate TX knows h1, h2 1 TX, 2 RX, MRC |h1|2 + |h2|2 N0 1 no 2 TX, 1 RX, BF |h1|2 + |h2|2 N0 1 yes 2 TX, 1 RX, ant. cycl. |h1|2 + |h2|2 2N0 1/2 no 2 TX, 1 RX, Alamouti |h1|2 + |h2|2 2N0 1 no 2 TX, 1 RX, Ant. sel. ≥ |h1|2 + |h2|2 2N0 1 partly ➮ For antenna selection, note that max |hn|2 N0 ≥ 1 2 |h1|2 + |h2|2 N0

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MIMO Fundamentals and Signal Processing

2-antenna systems, cont

10−1 10−2 10−3 10−4 10−5 10 20 30 40 50 1 RX, 1 TX 2 RX, 1 TX, MRC

• r 1 TX, 2 RX, BF

1 RX, 2 TX, antenna selection 1 RX, 2 TX, Alamouti P(e) SNR

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MIMO Fundamentals and Signal Processing

Orthogonal STBC (OSTBC)

➮ Important special case of linear STBC: X =

ns

• n=1

(¯ snAn+i˜ snBn) for which XXH =

ns

• n=1

|sn|2 · I = s2 · I Notation: ¯ (·)=real part, ˜ (·)=imaginary part ➮ This is equivalent to requiring for n = 1, . . . , ns, p = 1, . . . , ns AnAH

n = I, BnBH n = I

AnAH

p = −ApAH n ,

BnBH

p = −BpBH n ,

n = p AnBH

p = BpAH n

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MIMO Fundamentals and Signal Processing

Proof

➮ To prove ⇒), expand: XXH =

ns

• n=1

ns

• p=1

(¯ snAn + i˜ snBn)(¯ spAp + i˜ spBp)H =

ns

• n=1

(¯ s2

nAnAH n + ˜

s2

nBnBH n )

+

ns

• n=1

ns

• p=1,p>n
• ¯

sn¯ sp(AnAH

p + ApAH n ) + ˜

sn˜ sp(BnBH

p + BpBH n )

• + i

ns

• n=1

ns

• p=1

˜ sn¯ sp(BnAH

p − ApBH n )

➮ Proof of ⇐), see e.g., EL&PS book.

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MIMO Fundamentals and Signal Processing

Some properties of OSTBC

➮ Manifests the intuition that unitary matrices are good ➮ Alamouti code is an OSTBC (up to 1/ √ 2 normalization) ➮ Pros ➠ Diversity of order nrnt ➠ Detection of {sn} is decoupled ➠ Converts space-time channel into ns AWGN channels ➠ Combination with outer coding is straightforward ➮ Cons ➠ Rate loss for nt > 2, i.e., nt > 2 ⇒ R = ns N < 1 ➠ Information loss except for when nt = 2, nr = 1

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MIMO Fundamentals and Signal Processing

Diversity order of OSTBC

➮ Suppose {s0

n}ns n=1 are true symbols and {sn} are any other symbols.

Then X − X0 =

ns

• n=1
• (¯

sn − ¯ s0

n)An + i(˜

sn − ˜ s0

n)Bn

• ⇒

(X − X0)(X − X0)H =

ns

• n=1

|sn − s0

n|2 · I

➮ Full rank ➠ full diversity for i.i.d. Gaussian channel

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MIMO Fundamentals and Signal Processing

Derivation of decoupled detection

➮ Write the ML metric as Y − HX2 =Y 2 − 2ReTr

• Y HHX
• + HX2

=Y 2 − 2

ns

• n=1

ReTr

• Y HHAn
• ¯

sn + 2

ns

• n=1

Im Tr

• Y HHBn
• ˜

sn + H2 · s2 =

ns

• n=1
• − 2ReTr
• Y HHAn
• ¯

sn + 2 Im Tr

• Y HHBn
• ˜

sn + |sn|2H2 + const. =H2 ·

ns

• n=1
• sn − ReTr
• Y HHAn
• − i Im Tr
• Y HHBn
• H2
• 2

+ const.

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MIMO Fundamentals and Signal Processing

Decoupled detection, again

➮ Linearity: Y = HX + E ⇔ y = F s′ + e, s′ [¯ sT ˜ sT]T ➮ Theorem: X is an OSTBC if and only if Re

• F HF
• = H2 · I

∀ H ➮ ML metric: y − F s′2 = y2 − 2Re

• yHF s′

+ Re

• s′TF HF s′

= y2 − 2Re

• yHF s′

+ H2 · s′2 = H2 · s′ − ˆ s′2 + const. where ˆ s′ ˆ ¯ s ˆ ˜ s

• = Re
• F Hy
• H2

∼ N

• ¯

s ˜ s

• , N0/2

H2I

• ➮ F is a “Spatial/temporal (code) matched filter”

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MIMO Fundamentals and Signal Processing

Interpretation of decoupled detection

➮ Space-time channel decouples into ns AWGN channels

(a) nt × nr space-time channel

signal 2 signal 1 AWGN AWGN AWGN signal ns

(b) ns independent AWGN channels

➮ SNR per subchannel: SNR|H = N ns · H2 nt · P N0

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MIMO Fundamentals and Signal Processing

Example: Alamouti’s code is an OSTBC

➮ Consider the Alamouti code (re-normalized): X =

• s1

s∗

2

s2 −s∗

1

• ,

XXH = (|s1|2 + |s2|2)I ➮ Identification of An and Bn gives A1 =

• 1

−1

• A2 =
• 1

1

• B1 =
• 1

1

• B2 =
• −1

1

• 66

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MIMO Fundamentals and Signal Processing

Examples of OSTBC

➮ Best known OSTBC for nt = 3, N = 4, ns = 3: X =   s1 s2 −s3 s1 s∗

3

s∗

2

−s∗

2

−s3 s∗

1

  Code rate: 3/4 bpcu ➮ For nt = 4, N = 4, ns = 3: X =     s1 s2 −s3 s1 s∗

3

s∗

2

−s∗

2

−s3 s∗

1

s∗

3

−s2 s∗

1

    Rate is 3/4 bpcu.

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MIMO Fundamentals and Signal Processing

Summary of OSTBC Relations

XXH =

ns

• n=1

|sn|2 · I = s2 · I

• AnAH

n

= I , BnBH

n

= I AnAH

p

= −ApAH

n

, BnBH

p

= −BpBH

n ,

n = p AnBH

p

= BpAH

n

• Re
• F HF
• = H2 · I

where F is such that vec(Y ) = F · ¯ s ˜ s

• + vec(E)

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MIMO Fundamentals and Signal Processing

Mutual Information Properties of OSTBC

➮ Average transmitted energy per antenna and time interval = 1/nt ➮ Channel mutual information, with i.i.d. streams of power 1/nt: CMIMO(H) = log

• I + 1

nt HHH N0

• ➮ Mutual information of OSTBC coded system:

COSTBC(H) = ns N log

• 1 + N

ns H2 ntN0

• ➮ Theorem:

CMIMO ≥ COSTBC, equality only for nt = 2, nr = 1

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MIMO Fundamentals and Signal Processing

Capacity comparison (1% outage)

5 10 15 20 25 30 35 40 10

−1

10 10

1

SNR Capacity [bits/sec/Hz]

Average capacity, 1 TX, 1 RX Outage capacity, 1 TX, 1 RX Average capacity, 2 TX, 2 RX Average capacity, 2 TX, 2 RX − OSTBC Outage capacity, 2 TX, 2 RX Outage capacity, 2 TX, 2 RX − OSTBC

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MIMO Fundamentals and Signal Processing

Non-orthogonal linear STBC

➮ Also called linear dispersion codes ➮ Different approaches: ➠ Optimization of mutual information between the TX & RX: max 1 2EH

• log2
• I + 2

N0 Re

• F HF
• (no explicit guarantee for full diversity here)

➠ Quasi-orthogonal codes ➠ Codes based on linear constellation (complex-field) precoding s′ = Φs

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MIMO Fundamentals and Signal Processing

Example: A non-OSTBC

➮ Consider the following diagonal code, where |sn| = 1: X =

• s1

s2

• ➮ Then

A1 =

• 1
• ,

A2 =

• 1
• ,

B1 =

• 1
• B2 =
• 1
• ➮ ML metric for symbol detection:

Y − HX2 =Y 2 − 2ReTr

• XHHHY
• + H2

= − 2Re

• [HHY ]1,1 · s1
• − 2Re
• [HHY ]2,2 · s2
• + const.

➮ Decoupled detection, but not OSTBC, and not full diversity

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MIMO Fundamentals and Signal Processing

More examples of linear but not orthogonal STBC

➮ Alamouti code with forgotten conjugates X =

• s1

s2 s2 −s1

• ➮ “Spatial multiplexing” (R = nt, N = 1, ns = nt, d = nr).

For nt = 2: A1 =

• 1
• , A2 =
• 1
• B1 =
• 1
• , B2 =
• 1
• 73

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MIMO Fundamentals and Signal Processing

Linearly precoded STBC

➮ Transmit W X where W ∈ {W 1, . . . , W K}. Data model: Y = HW X + E ➮ Fact: If rank{W k} = nt then same diversity order as but without W ➮ Consider correlated fading: R = E[hhH] = RT

t ⊗ Rr,

h = vec(H) ➮ Error probability:

EH[P (X0 → X)] ≤ const. · ˛ ˛ ˛I + 1 N0 (X0 − X)(X0 − X)H · W HRtW ˛ ˛ ˛

−nr

➮ For OSTBC, (X0−X)(X0−X)H ∝ I. Hence, min

W

• I + ns

N0 W HRtW

• 74

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MIMO Fundamentals and Signal Processing

• Ex. OSTBC with One-Bit Feedback for nt = 2

➮ One bit used to choose between W 1 = |a|

• 1 − |a|2
• if h1>h2

, W 2 =

• 1 − |a|2

|a|

• if h2>h1

➮ Let Pc be the probability that the feedback bit is correct ➮ For Pc = 1 (reliable feedback), a = 1 is optimal ➠ antenna selection W may be multiplied with fixed unitary matrix ➠ grid of beams ➮ For Pc < 1 (erroneous feedback), EH[P(X0 → X)] ≤ 2 SNR2 · Pc |a|2 + 1 − Pc 1 − |a|2

• (for nr = 1)

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MIMO Fundamentals and Signal Processing

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 10

−5

10

−4

10

−3

10

−2

10

−1

SNR BER Unweighted OSTBC. Optimal weighting (No feedback error). Optimal weighting with feedback error. Error tolerant weighting (No feedback error), Error tolerant weighting with feedback error

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MIMO Fundamentals and Signal Processing

MIMO with feedback - optimized transmission

s ˜ s x H n y I I( ˜ H) U W(I)

Encoder Decoder

˜ H

➮ Here ➠ U depends on long-term feedback ➠ I depends on short-term (few bits) feedback ➮ State-of-the art designs rely on vector quantization techniques

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MIMO Fundamentals and Signal Processing

Frequency-selective channels

➮ Maximum diversity order (with ML detection) will be nrntL where L=length of CIR ➮ Variety of techniques to achieve maximum diversity ➮ Most widely used transmission technique is MIMO-OFDM ➠ coding across multiple OFDM symbols ➠ coding across subcarriers within one OFDM symbol ➮ Basic model per subcarrier is Y n = HnXn + En

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MIMO Fundamentals and Signal Processing

Le 3: MIMO receivers

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MIMO Fundamentals and Signal Processing

Summary of MIMO receivers

➮ Optimal architectures (from Le 1): ➠ CSI@TX (any fading): linear processing, ˜ y = U Hy, separates streams ➠ no CSI@TX, fast fading: V-BLAST, optimal receiver is more involved

• linear receiver (channel inversion) is grossly suboptimal
• successive interference cancellation (SIC)
• using soft MIMO demodulator + decoder, possibly iterative

➠ no CSI@TX, slow fading: D-BLAST ➮ Architectures with STBC+outer FEC (from Le 2) ➠ With OSTBC, decoupled detection and thing are simple: min Y − HX ∼ min

• |s1 − ˆ

s1|2 + |s2 − ˆ s2|2 ➠ With non-OSTBC, min Y − HX does not decouple

• problem similar to for V-BLAST

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MIMO Fundamentals and Signal Processing

Theoretically optimal V-BLAST receiver based on SIC

decoder 1 decoder 2 decoder 3 decoder nt MMSE 1 MMSE 2 MMSE 3

(MMSE nt)

y = Hs + e

decoded stream 1 decoded stream 2 decoded stream 3 decoded stream nt

➮ Optimality only for fast fading. Requires rate allocation on streams. ➮ Major drawback: Requires long codewords. Prone to error propagation.

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MIMO Fundamentals and Signal Processing

Theoretically optimal D-BLAST receiver based on SIC

xA(1) xB(1) xA(2) xB(2) xA(3)

➮ One codeword split as x(i) = [xA(i) xB(i)], with rate allocation ➮ Decoding in steps:

• 1. Decode xA(1)
• 2. Decode xB(1), suppressing xA(2) via MMSE
• 3. Strip off xB(1), and decode xA(2)
• 4. Decode xB(2), suppressing xA(3) via MMSE

➮ Drawbacks: ➠ error propagation ➠ rate loss due to initialization ➠ requires long codewords

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MIMO Fundamentals and Signal Processing

Receivers for linear STBC architectures

Training Data 2 Data 1 ➮ Received block: Y = HX + E. ➮ X linear in {s1, ..., sns} so with appropriate F , the ML metric is Y − HX2 =

• vec( ¯

Y ) vec( ˜ Y )

• − F

¯ s ˜ s

• 2

➮ For OSTBC, F TF = H2I so detection decouples ➮ Spatial multiplexing (V-BLAST) can be seen a degenerated special case

• f this architecture, with

X =   s1 . . . snt  

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MIMO Fundamentals and Signal Processing

Demodulator+decoder architectures

MIMO demodulator channel decoder a priori information P(bi) soft output P(bi|y) y = Hs + e

➮ Demodulator computes P(bi|y) given a priori information P(bi) ➮ Decoder adds knowledge of what codewords are valid ➮ Added knowledge in decoder is fed back to demodulator as a priori ➮ Iteration until convergence (a few iterations, normally)

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MIMO Fundamentals and Signal Processing

MIMO demodulation (hard)

➮ General transmission model, with Gi,j ∈ R y

• m×1

= G

• m×n

· s

• n×1

+ e

• m×1

, sk ∈ S ➮ Models V-BLAST architectures, and (non-O)STBC architectures ➮ Other applications: multiuser detection, ISI, crosstalk in cables, ... ➮ Typically, m ≥ n and G is full rank and has no structure.

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MIMO Fundamentals and Signal Processing

The problem

➮ If e ∼ N(0, σI) then the problem is to detect s from y min

s∈Sn y − Gs2,

y ∈ Rm, G ∈ Rm×n ➮ Let G = QL where

• Q ∈ Rm×n

is orthonormal (QTQ = I) L ∈ Rn×n is lower triangular Then y − Gs2 =

• QQT(y − Gs)
• 2

+

• (I − QQT)(y − Gs)
• 2

=

• QTy − Ls
• 2

+

• (I − QQT)y
• 2

so min

s∈Sn y − Gs2

⇔ min

s∈Sn ˜

y − Ls where ˜ y QTy

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Some remarks

➮ Integer-constrained least-squares problem, known to be NP hard ➮ Brute force complexity O(|Sn|) ➮ Typical dimension of problem: n ∼ 8−16, so |S| ∼ 2–8, |Sn| ∼ 256–1014 ➮ Needs be solved ➠ in real time ➠ once per received vector y ➠ in power-efficient hardware (beware of heavy matrix algebra) ➠ possibly fixed-point arithmetics ➠ preferably, in a parallel architecture ➮ In communications, we can accept a suboptimal algorithm that finds the correct solution quickly, with high probability

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MIMO Fundamentals and Signal Processing

Some remarks, cont.

➮ For G ∝ orthogonal (OSTBC), the problem is trivial. ➮ Our focus is on unstructured G ➮ If G has structure (e.g., Toeplitz) then use algorithm that exploits this ➮ Generally, slow fading (no time diversity) is the hard case

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Zero-Forcing

➮ Let ˜ s arg min

s∈Rn y − Gs = arg min s∈Rn ˜

y − Ls = L−1˜ y E.g., Gaussian elimination: ˜ s1 = ˜ y1/L1,1 ˜ s2 = (˜ y2 − ˜ s1L2,1)/L2,2 . . . ➮ Then project onto S: ˆ sk = [˜ sk] arg min

sk∈S |sk − ˜

sk| ➮ This works very poorly. Why? Note that ˜ s = s + L−1QTe = s + ˜ e, where cov(˜ e)=σ · (LTL)−1 ZF neglects the correlation between the elements of ˜ e

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MIMO Fundamentals and Signal Processing

Decision tree view

min

{s1,...,sn}

sk∈S

{f1(s1) + f2(s1, s2) + · · · + fn(s1, . . . , sn)} where fk(s1, ..., sk)

• ˜

yk −

k

• l=1

Lk,lsl 2

{−1, −1, −1} {−1, −1, 1} {−1, 1, −1} {−1, 1, 1} {1, −1, −1} {1, −1, 1} {1, 1, −1} {1, 1, 1} s1 = −1 s1 = +1 s2 = −1 s2 = −1 s2 = +1 s2 = +1 s3 = −1 s3 = −1 s3 = −1 s3 = −1 s3 = +1 s3 = +1 s3 = +1 s3 = +1 f1(−1) = 1 f1(1) = 5 f2(−1, −1) = 2 f2(−1, 1) = 1 f2(1, −1) = 2 f2(1, 1) = 3 f3(· · · ) = 4 f3(· · · ) = 1 3 4 3 1 1 9

1 5 3 2 7 8 7 4 5 6 10 8 9 17 root node leaves

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Zero-Forcing with Decision Feedback (ZF-DF)

➮ Consider the following improvement i) Detect s1 via: ˆ s1 = ˜ y1 L1,1

• = arg min

s1∈S f1(s1)

ii) Consider s1 known and set ˆ s2 = ˜ y2 − ˆ s1L2,1 L2,2

• = arg min

s2∈S f2(ˆ

s1, s2) iii) Continue for k = 3, ..., n: ˆ sk =

• ˜

yk − k−1

l=1 Lk,lˆ

sl Lk,k

• = arg min

sk∈S fk(ˆ

s1, ..., ˆ sk−1, sk) ➮ This also works poorly. Why? Error propagation. Incorrect decision on si ➠ most of the following sk wrong as well. ➮ Optimized detection order (start with the best) does not help much.

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Zero-Forcing with Decision Feedback (ZF-DF)

1 5 2 1 2 3 4 1 3 4 3 1 1 9 1 5 3 2 7 8 7 4 5 6 10 8 9 17

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Sphere decoding (SD)

➮ Select a sphere radius, R. Then traverse the tree, but once encountering a node with cumulative metric > R, do not follow it down ➮ Enumerates all leaf nodes which lie inside the sphere ˜ y − Ls2 ≤ R ➮ Improvements: ➠ Pruning: At each leaf, update R according to R := min(R, M) ➠ Improvements: optimal ordering of sk ➠ Branch enumeration (e.g., sk = {−5, −3, −1, −1, 3, 5} vs. sk = {−1, 1, −3, 3, −5, 5}) ➮ Known facts: ➠ The algorithm solves the problem, if allowed to finish ➠ Runtime is random and algorithm cannot be parallelized ➠ Under relevant circumstances, average runtime is O(2αn) for α > 0

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SD, without pruning, R = 6

1 5 2 1 2 3 4 1 3 4 3 1 1 9 1 5 3 2 7 8 7 4 5 6 10 8 9 17

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SD, with pruning, R = ∞

1 5 2 1 2 3 4 1 3 4 3 1 1 9 1 5 3 2 7 8 7 4 5 6 10 8 9 17

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“Fixed complexity” sphere decoding (FCSD)

➮ Select a user parameter r, 0 ≤ r ≤ n ➮ For each node on layer r, consider {s1, ..., sr} fixed and solve (∗) min

{sr+1,...,sn}

sk∈S

{fr+1(s1, ..., sr+1) + · · · + fn(s1, ..., sn)} ➮ Subproblem (*) solved using |S|r times ➮ Low-complexity approximation (e.g. ZF-DF) can be used. Why? (*) is overdetermined (equivalent G is tall) ➮ Order can be optimized: start with the “worst” ➮ Fixed runtime, fully parallel structure

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FCSD, r = 1

1 5 2 1 2 3 4 1 3 4 3 1 1 9 1 5 3 2 7 8 7 4 5 6 10 8 9 17

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Semidefinite relaxation (for sk ∈ {±1})

➮ Let ¯ s

• s

1

• ,

S ¯ s¯ sT =

• s

1 sT 1

• ,

Ψ

• LTL

−LT ˜ y −˜ yTL

• Then

˜ y − Ls2 = ¯ sTΨ¯ s + ˜ y2 = Trace{ΨS} + ˜ y2 so the problem is to min diag{S}={1,...,1} rank{S}=1

¯ sn+1=1

Trace{ΨS} ➮ SDR proceeds by relaxing rank{S} = 1 to S positive semidefinite ➮ Interior point methods used to find S ➮ s recovered, e.g., by taking dominant eigenvector and project onto Sn

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Lattice reduction

➮ Extend Sn to lattice. For example, if S = {−3, −1, 1, 3}, then ¯ Sn = {. . . , −3, −1, 1, 3, . . .} × · · · × {. . . , −3, −1, 1, 3, . . .}. ➮ Decide on orthogonal integer matrix T ∈ Rn×n that maps ¯ Sn onto itself: Tk,l ∈ Z, |T | = 1, and T s ∈ ¯ Sn ∀s ∈ ¯ Sn ➮ Find one such T for which LT ∝ I ➮ Then solve ˆ s′ arg min

s′∈ ¯ Sn ˜

y − (LT )s′2, and set ˆ s = T −1ˆ s′ ➮ Critical steps: ➠ Find suitable T (computationally costly, but amortize over many y) ➠ ˆ s ∈ ¯ Sn, but ˆ s / ∈ Sn in general, so clipping is necessary

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MIMO demodulation (soft)

➮ Data model y

• m×1

= G

• m×n

· s

• n×1

+ e

• m×1

, sk = S(b1, ..., bp) ∈ S, |S| = 2p ➮ Bits bi a priori indep. with L(bi) = log P(bi = 1) P(bi = 0)

• ,

i = 1, ..., np ➮ Objective: Determine L(bi|y) = log P(bi = 1|y) P(bi = 0|y)

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Posterior bit probabilities

L(bi|y) = log „P (bi = 1|y) P (bi = 0|y) «

(a)

= log P

s:bi(s)=1 P (s|y)

P

s:bi(s)=0 P (s|y)

!

(b)

= log P

s:bi(s)=1 p(y|s)P (s)

P

s:bi(s)=0 p(y|s)P (s)

!

(c)

= log @ P

s:bi(s)=1 p(y|s)

“Qnp

i′=1 P (bi′ = bi′(s))

” P

s:bi(s)=0 p(y|s)

“Qnp

i′=1 P (bi′ = bi′(s))

” 1 A = log B @ P

s:bi(s)=1 p(y|s)

“Qnp

i′=1,i′=i P (bi′ = bi′(s))

” · P (bi = 1) P

s:bi(s)=0 p(y|s)

“Qnp

i′=1,i′=i P (bi′ = bi′(s))

” · P (bi = 0) 1 C A = log B @ P

s:bi(s)=1 p(y|s)

“Qnp

i′=1,i′=i P (bi′ = bi′(s))

” P

s:bi(s)=0 p(y|s)

“Qnp

i′=1,i′=i P (bi′ = bi′(s))

” 1 C A + L(bi)

In Gaussian noise p(y|s) =

1 (2πσ)m/2 exp

• − 1

2σy − Gs2

so

L(bi|y) = log B @ P

s:bi(s)=1 exp

“ − 1

2σy − Gs2”“Qnp i′=1,i′=i P (bi′ = bi′(s))

” P

s:bi(s)=0 exp

“ − 1

2σy − Gs2

”“Qnp

i′=1,i′=i P (bi′ = bi′(s))

” 1 C A + L(bi)

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➮ With a priori equiprobable bits L(bi|y) = log

• s:bi(s)=1 exp
• − 1

2σy − Gs2

• s:bi(s)=0 exp
• − 1

2σy − Gs2

• ➮ can be relatively well approximated by its largest term

That gives problems of the type min

s∈Sn,bi(s)=β y − Gs2

This is called “max-log” approximation ➮ If many candidates s are examined, then use these terms in ➠ List-decoding algorithm

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Incorporating a priori probabilities (BPSK/dim case)

➮ Consider sk ∈ {±1}, and let sk = 2bk − 1 γk 1 2 log (P(sk = −1)P(sk = 1)) = 1 2 log (P(bk = 0)P(bk = 1)) λk log P(sk = 1) P(sk = −1)

• = log

P(bk = 1) P(bk = 0)

• = L(bk)

➮ The prior is linear in sk: log(P(sk = s)) = 1 2[(1 + s) log(P(sk = 1)) + (1 − s) log(P(sk = −1))] = 1 2γk + 1 2λksk

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➮ Write L(sk|y) = log   

• s:sk=1 exp
• −1

σy − Gs2 + 1 2

n

i=1,i=k (γi + λisi)

• s:sk=0 exp
• −1

σy − Gs2 + 1 2

n

i=1,i=k (γi + λisi)

• 

 +λk ➮ Define ˜ y [yT 1 · · · 1]T and ˜ G

• G

Λk

• where

Λk diag σ

4λ1, · · · , σ 4λk−1, σ 4λk+1, . . . , σ 4λn

• ➮ Then

L(sk|y) = log   

• s:sk=1 exp
• −1

σ˜

y − ˜ Gs2 + n

i=1,i=k

• σλ2

i

16 + γi 2

• s:sk=0 exp
• −1

σ˜

y − ˜ Gs2 + n

i=1,i=k

σλ2

i

16 + γi 2

• 

 +λk ➮ A priori information on sk ➠ “virtual antennas”

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Example w. iter. decod. 4×4, r = 1/2-LDPC, 1000 bits

0.01 0.1 −4 −3.5 −3 −2.5 −2

Frame-error-rate (FER) Normalized signal-to-noise-ratio (SNR) [dB] Practical method, r = 3, no iteration Practical method, r = 3, 1 iteration Practical method, r = 3, 2 iterations Brute-force, no iteration Brute-force, 1 iteration Brute-force, 2 iterations

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Channel (H) estimation and associated receivers

➮ Very often pilots are used to form a channel estimate. Consider Y t = HXt + Et ➮ Estimate H via training: ➠ Maximum likelihood (in Gaussian noise): ˆ H = argmin

H

Y t − HXt2 = Y tXH

t (XtXH t )−1

➠ Can show, estimate is the same also in colored noise (e.g. co-channel interference in multiuser system) ➠ Can be somewhat improved by using MMSE estimation

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➮ Estimate noise (co)variance: ˆ Λ =

1 NtY tΠ⊥ XH

t Y H

t ,

for colored noise ˆ N0 =

1 NtnrTr

• Y tΠ⊥

XH

t Y H

t

• ,

for white noise ➠ This estimate can be used to prewhiten the received signal to suppress co-channel interference. E.g. ˜ Y = Λ−1/2Y ➠ At most nr − 1 rank-1 interferers can be suppressed. The receive array has nr degrees of freedom. At high SNR, Λ−1 becomes a projection matrix that projects out interference.

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More on training

➮ Training-based detector uses ˆ H, ˆ N0 and ˆ Λ in the coherent detector ➮ Can be improved, e.g., cyclic detection ➮ How should training be designed? Optimum pilots (in many respects) satisfy XtXH

t ∝ I

➮ Inserting pilot-based estimate in LF is not optimal

• Cf. the use of

p(X|Y , H) (coherent) p(X|Y , H)|H:= ˆ

H (training-based)

p(X|Y , ˆ H) (best possible given ˆ H) p(X|Y , Y t) (best possible given training data) Later, we will give p(X|Y , ˆ H) on explicit form

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Example, joint detection and estimation schemes

Re−estimate No Yes Initialize Detect Symbols Convergence? Channel

➮ Re-estimation step may make use of soft decoder output

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Optimal training

➮ Recall: ML channel estimate ˆ H = Y tXH

t (XtXH t )−1

➮ Let h = vec(H), ˆ h = vec( ˆ H). Can show:

E[ˆ h] = h Σ E[(ˆ h − h)(ˆ h − h)H] = . . . = N0 “ (XtXH

t )−T ⊗ I

” Tr {Σ} = nrTr ˘ (XtXH

t )−1¯

N0

➮ Lemma: Suppose Tr

• XXH

≤ nt. Then Tr

• (XXH)−1

≥ nt with equality if and only if XXH = I ➮ Application of lemma ➠ optimal training block is (semi-)unitary: XtXH

t ∝ I

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Channel estimation for frequency-selective channels

➮ MIMO channel as matrix-valued FIR filter: H(z−1) =

L

• l=0

Hlz−l ➮ L is the length of the channel, L = 0 for ISI-free channel ➮ Transfer function: H(ω) =

L

• l=0

Hle−iωl ➮ Transmission methods: ➠ Single-carrier (block-based) ➠ Multicarrier (e.g. OFDM)

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Training for frequency selective channels

➮ Two basic approaches: ➠ Frequency-domain estimation (estimate H(ω)) ➠ Time-domain estimation (estimate Hl, then compute H(ω) via FT) ➮ Frequency-domain estimation of H(ω) is straightforward: ➠ Estimate in the frequency domain (via ML): ˆ H(ω) = Y t(ω)XH

t (ω)(Xt(ω)XH t (ω))−1

➠ Problem: suboptimal because parameterizations is not parsimonious. It does not exploit structure that H(ω) =

L

• l=0

Hle−iωl

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Time-domain estimation of H(ω)

➮ Let x(n) be time-domain training, y(n) received (t-d) training and define Xt =   

xT

t (0)

· · · · · · · · · xT

t (−L)

xT

t (1)

... xT

t (1 − L)

. . . ... . . . xT

t (N − 1)

· · · · · · xT

t (0)

· · · xT

t (N − 1 − L)

   H =   HT . . . HT

L

 , Y t =   yT

t (0)

. . . yT

t (N − 1)

  ➮ Then the ML estimate of H(ω) is: ˆ H = (XH

t Xt)−1XH t Y t,

ˆ H(ω) =

L

• l=0

ˆ Hle−iωl ➮ Exploits structure (L unknowns but N equations)

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Metrics with imperfect CSI (complex y, s, G, e)

➮ G not known perfectly ➠ replacing G with ˆ G in p(y|s, G) not optimal! ➮ Instead, need to work with p(y|s, ˆ G). Write y = Gs + e ⇔ y = (sT ⊗ I)h + e, h = vec(G), e = vec(E) Suppose s2 = n and

• h ∼ N(0, ρI),

e ∼ N(0, σI) ˆ h = h + δ, δ ∼ N(0, ǫI) ➮ Then y ˆ h

• ∼ N
• 0,
• (nρ + σ)I

ρ(sT ⊗ I) ρ(s∗ ⊗ I) (ρ + ǫ)I

• so

p(y|ˆ h, s) = 1 πn 1

nǫ 1+ǫ/ρ + σ exp

• −

1

nǫ 1+ǫ/ρ + σ

• y −
• ρ

ρ + ǫ

• ˆ

Gs

• 2

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Example: 4 × 4 slow Rayl. fading MIMO, QPSK, est. G

0.01 0.1 1 −5 −4 −3 −2 −1 1 2 3

Frame-error-rate (FER) Normalized signal-to-noise-ratio (SNR) [dB] Practical detector Perfect CSI Imperfect CSI Brute-force Practical detector, mismatched metric Practical detector, optimal metric Brute-force, optimal metric Brute-force, mism.

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Example: 4 × 4 slow Rayl. fad., QPSK, outdat. G

0.01 0.1 1 −5 −4 −3 −2 −1 1 2 3

Frame-error-rate (FER) Normalized signal-to-noise-ratio (SNR) [dB] Practical detector Perfect CSI Imperfect CSI Brute-force Practical detector, mismatched metric Practical detector, optimal metric Brute-force, optimal metric Brute-force, mism.

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Last slide

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