MIMO Fundamentals and Signal Processing Course · Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing MIMO Design Space Fast fading:

MIMO Fundamentals and Signal Processing Course

Erik G. LarssonLinkoping University (LiU), SwedenDept. of Electrical Engineering (ISY)Division of Communication Systemswww.commsys.isy.liu.se

slides version: September 25, 2009

Linkoping 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 andsignal 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.

1


Organization

➮ 4 lectures:Le 1: MIMO fundamentalsLe 2: Low-complexity MIMOLe 3: MIMO receiversLe 4: MIMO in 3G-LTE (by P. Frenger, Ericsson Research)

➮ Reading, in addition to course notes:

➠ Le 1: D. Tse & P. Viswanath, Fundamentals of WirelessCommunications, Cambridge Univ. Press 2005, Chs. 7–8

➠ Le 2: E. G. Larsson & P. Stoica, Space-time block coding for wirelesscommunications, 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): TBD2


Le 1: MIMO fundamentals

3


Basic channel model

➮ Flat fading; linear, time-invariant channel

TX RX

11

nrnt

➮ Complex data {x1, . . . , xnt} are transmitted via the nt antennas

➮ Received data: ym =

nt∑

n=1

hm,nxn + em

4


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

......


]

︸︷︷︸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.

5


MIMO Design Space

➮ Fast fading: codeword spans ∞ number of channel realizationsChannel 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 fadingChannel V-BLAST optimal V-BLAST suboptimalunknown at TX no coding across antennas coding across antennas

D-BLAST optimalChannel waterfilling waterfillingknown at TX over space& time over space

6


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, UHU = I, V HV = I

➮ Dimensions: U ∈ Cnr×nr, Λ ∈ R

nr×nt, V ∈ Cnt×nt

7


➮ Introduce transform: y = UΛV H︸︷︷︸

H

x + e

y , UHy = ΛV Hx︸︷︷︸

,x

+UHe︸︷︷︸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 = λ1x1 + e1

...

yn = λnxn + en

8


➮ 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

9


Architecture

x1

x2

xnt

x1

x2

xn

e1

enr

y1

y2

ynr

y1

y2

yn

HV :,1:n UH:,1:n

➮ xk are independent streams with powers Pk and rates Rk

10


Optimal power allocation over n subchannels

➮ Each subchannel can offer the rate

Ck = log2

(

1 +Pk

N0λ2

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

maxPk,Pn

k=1 Pk≤P

n∑

k=1

Ck ⇔ maxPk,Pn

k=1 Pk≤P

n∑

k=1

log2

(

1 +Pk

N0λ2

k

)

11


Waterfilling solution

➮ Problem: maxPk,Pn

k=1 Pk≤P

n∑

k=1

log2

(

1 +Pk

N0λ2

k

)

➮ Solution: (need solve for µ)

C =

n∑

k=1

log2

(

1 +P ∗

k λ2k

N0

)

P ∗k =

(

µ − N0

λ2k

)+

,

n∑

k=1

P ∗k = P

➮ Special case: H = λuvH. Transmit one stream

C = log2

(

1 +P

N0λ2

)

= log2

(

1 +P

N0‖H‖2

)

12


Waterfilling solution

1 2 3... n k

N0

λ2k

µ

P∗ 1

=0

P∗ 2

P∗ 3

=0

P∗ 4

P∗ 5

=0

13


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

λ2k

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 streams14


Waterfilling at high SNR

��

��

��

��

��

��

��

��

��

��

��

��

1 2 3... n k

N0

λ2k

P ∗k ≈ P

n

15


Waterfilling at low SNR

➮ At low SNR, the water is shallow. Then

P ∗k =

{

P, k = argmax λ2k

0, else

C = log2

(

1 +P

N0λ2

max

)

≈(

P

N0λ2

max

)

· log2(e)

➮ MIMO provides an array gain (power gain of λ2max) 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)

16


Waterfilling at low SNR

1 2 3... n k

N0

λ2k

µ

P∗ 4

=P

17


In practice

➮ Feedback of channel state information, requires quantization

➮ Potentially, by scheduling only “good” users, one may always operate athigh 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!

18


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 MIMOpropagation and how they connect to the DoF concept.

19


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−j2πλ ∆ cos φ

...

e−j(m−1)2πλ ∆ cos φ

➮ Signal from point source impinging on RX array (large TX-RX distance):y = αu(φ) · s + e, (α ∈ C, dep. on distance) 20


Line-of-sight MIMO channel

TX array

RX arrayφr

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! 21


Lobes and resolvability

➮ Consider unit-power point sources at φ1, φ2 with sign. u(φ1), u(φ2).How similar do these signatures look?

1

m‖s1u(φ1) − s2u(φ2)‖2

= 2 − 2Re

s∗1s2 ·1

muH(φ1)u(φ2)

︸︷︷︸

|·|=f(·)

where the lobe pattern f(cos(φ1) − cos(φ2)) , 1m|uH(φ1)u(φ2)|.

➮ If f(·) < 1, then φ1, φ2 resolvable.

➮ Resolvability criterion: | cos φ1 − cos φ2| ≥ 2πA , A , (m − 1)∆

➮ Grating lobes avoided if ∆ ≤ λ

2⇒ A ≤ (m − 1)

λ

2.

22


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(· · · ) 6= 1 ⇔ φ1, φ2 resolvable ⇔ | cos φ1−cos φ2| > 2πA23


MIMO with two plane scatters

TX array

RX array

A

B

➮ Here, HTX-RX = HAB-RX · HTX-AB

➮ We have rank(HTX-RX) = 2only 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

24


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 UHU = I.

➮ Let U r and U t be the U matrices associated with the TX and RXarrays. Note that UH

r U r = I and UHt 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 ∆ 6= λ/2 is more involved. 25


Angular decomposition, cont.

1

2

3

4

5

1

2

3

4

5

TX

RX

26


Angular decomposition, cont.

➮ Now defineHa , UH

r HU t

⇒ Ha,(k,l) = uH(φrk)Hu(φt

l)

uH(φrk)

[∑

i

αiu(φ′ri )uH(φ′t

i )

]

︸︷︷︸

physical model

u(φtl)

=∑

i

αi

uH(φr

k)u(φ′ri )

︸︷︷︸

=0 unless φ′ri falls in lobe φr

k

·

uH(φ′ti )u(φt

l)︸︷︷︸

=0 unless φ′ti 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 k27


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.

28


Fast fading, no CSI at TX

➮ Each codeword spans ∞ number of H

➮ The V-BLAST architecture is optimal hereNote: Reminiscent of architecture for slow fading and full CSI@TX

➮ Transmit vectorsx = Qx

where x1, ..., xn are independent streams with powers Pk and rates Rk

29


V-BLAST architecture

x1

x2

xnt

x1

x2

xn

e1

enr

y1

y2

ynr

HQ optimalreceiver

30


➮ Transmit covariance: Kx , cov(x) = Q

P1 0 · · · 00 P2

. . . ...... . . . . . . 00 · · · 0 Pn

QH

➮ Achievable rate, for fixed H:

R = log2

∣∣∣∣I +

1

N0HKxHH

∣∣∣∣

➮ Intuition: Volume of noise ball is |N0I|N .Volume of signal ball is |HKxHH + NoI|N .

31


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

C = E

[

log2

∣∣∣∣I +

1

N0HKxHH

∣∣∣∣

]

➮ Choose Q and Pk to

maxKx,Tr(Kx)≤P

E

[

log2

∣∣∣∣I +

1

N0HKxHH

∣∣∣∣

]

➮ Optimal Kx depends on the statistics of H

32


➮ In i.i.d. Rayleigh fading, K∗x = P

ntI (i.i.d. streams) and

C = E

[

log2

∣∣∣∣I +

P

N0

1

ntHHH

∣∣∣∣

]

=n∑

k=1

E

[

log2

(

1 +SNR

ntλ2

k

)]

wheren = 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.

33


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)]

34


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

0

(

log2(1 + t · SNR)

√

1

t− 1

4

)

dt

)

35


Fast fading, no CSI at TX, high SNR

➮ Here

C =

n∑

k=1

E

[

log2

(

1 +SNR

ntλ2

k

)]

≈ n log2(SNR) + const

➮ Both nr and nt must be large to provide DoF gain

➮ “Capacity grows as min(nr, nt)”

36


Fast fading, no CSI at TX, low SNR

➮ Here

C =

n∑

k=1

E

[

log2

(

1 +SNR

ntλ2

k

)]

≈ log2(e) ·SNR

nt·

n∑

k=1

E[λ2k]

= 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)37


V-BLAST in practice

➮ Transmitter architecture “simple” but the receiver must separate thestreams ➠ major challenge

➮ Problems are conceptually similar to uplink MUD in CDMA and toequalization 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 338


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)λ

2k

N0

)]

39


➮ High SNR: P ∗(λk) ≈ Pn (equal power)

C ≈n∑

k=1

E

[

log2

(

1 +SNR

nλ2

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!

40


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≤PP

[

log2

∣∣∣∣I +

1

N0HKxHH

∣∣∣∣< 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)41


➮ 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

42


Example: Outage probability at rate R = 2 bpcu

−5 0 5 10 15 20 25 30 35 4010

−4

10−3

10−2

10−1

SNR

FE

R

nt=1, n

r=1 (SISO)

nt=2, n

r=1

nt=1, n

r=2

nt=2, n

r=2

43


D-BLAST architecturereplacements

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 MMSE3. 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

44


Le 2: Low-complexity MIMO

45


Antenna diversity basics

➮ Recall transmission model (single time interval):

y1...

ynr

︸︷︷︸y (RX data)

=

h1,1 · · · h1,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

......


]

︸︷︷︸H

[x1,1 · · · x1,N

......

xnt,1· · · xnt,N

]

︸︷︷︸X∈X

“code matrix”

+

[e1,1 · · · e1,N

......

enr,1 · · · enr,N

]

︸︷︷︸E

46


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-timeblocks (small N) with good error probability performance. Outer FECcan 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) 47


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 − hs‖2 = ... = ‖h‖2 ·∣∣∣s − s

∣∣∣

2

+ const., where s ,hHy

‖h‖2

➮ MRC+scalar detection problem!

➮ Distribution of s determines performance:

s ,hHy

‖h‖2∼ N

(

s,N0

‖h‖2

)

, SNR|h =‖h‖2

N0· E[|s|2]

︸︷︷︸=P

= ‖h‖2 · P

N0︸︷︷︸

SNR48


Diversity order (n × 1 fading vector h)

➮ P (e|h) = Q

(√

SNR · ‖h‖2

)

and h ∼ N(0,Q) (SNR up to a constant)

➮ Then P (e) = E[P (e|h)] ≤∣∣∣∣I +

SNR

2Q

∣∣∣∣

−1

=

n∏

k=1

(

1 +SNR

2λk(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[‖h‖2

]

with eq. if λ1 = · · · = λn so Q ∝ I minimizes the bound on P (e) 49


Diversity order

10−1

10−2

10−3

10−4

10−5

0 10 20 30 40 50

d = 1

d = 2

d = 3

P (e)

SNR 50


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 − Hws‖2 = ... = ‖Hw‖2 · |s − s|2 + const.

where s ,wHHHy

‖Hw‖2∼ N

(

s,N0

‖Hw‖2

)

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

➮ Resulting SNR|H = 1N0

λmax(HHH)

︸︷︷︸

≥‖H‖2/nt

·E[|s|2]≥ 1

N0

‖H‖2

nt· E[|s|2].

➮ Diversity order: d = nrnt

➮ For nr = 1 take wopt = h∗‖h‖ so SNR|h = ‖h‖2

N0E[|s|2]

(same as for RX-d)51


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):

maxX

P (X|Y ,H) ⇔ minX

‖Y − HX‖2

➮ 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

︸︷︷︸

∼“

1N0

”−d∼SNR−d

d=“diversity order”. Note: d ≤ nrnt and d = nrnt if X0 − X full rank52


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 + isnBn)

where {An, Bn} are fixed matrices

➮ Typically N small. Need N ≥ nt for max diversity (why?)

➮ Rate: R ,N

nsbits/channel use

53


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) ≤ |WW H|−nr|s − s0|−2nrnt

( 1

4N0

)−nrnt

➮ What is the optimum W ? Try to maximize:

maxW

|WW H|

s.t. Tr{WW H

}= ‖W ‖2 ≤ 1 (power constraint)

➮ Solution: WW H = 1nt

I, (antenna cycling). Diversity but rate 1/N ! 54


Alamouti scheme for nt = 2

➮ X = 1√2

[s1 s∗2s2 −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

mins1,s2

∥∥∥∥∥∥∥∥∥

[y1

y∗2

]

︸︷︷︸y

− 1√2

[h1 h2

−h∗2 h∗

1

]

︸︷︷︸

,G

[s1

s2

]

︸︷︷︸s

∥∥∥∥∥∥∥∥∥

2

55


➮ Observation: GHG = 12

[hH

1 −hT2

hH2 hT

1

] [h1 h2

−h∗2 h∗

1

]

= ‖h1‖2+‖h2‖2

2 I

➮ Hence min ‖y − Gs‖2 ⇔ min ‖s − s‖2, s = 2

GHy

‖h1‖2+ ‖h2‖2

➮ Distribution of s:

s = 2GHy

‖h1‖2+ ‖h2‖2 = 2

GH(Gs + e)

‖h1‖2+ ‖h2‖2 ∼ N

(

s,2N0

‖H‖2I

)

➮ SNR|H =‖H‖2

2N0. For 2 × 1 system, 3 dB less than 1 × 2 with MRC

➮ Diversity order: 2nr

56


Overview of 2-antenna systems

Method SNR rate TX knows h1, h2

1 TX, 2 RX, MRC|h1|2 + |h2|2

N01 no

2 TX, 1 RX, BF|h1|2 + |h2|2

N01 yes

2 TX, 1 RX, ant. cycl.|h1|2 + |h2|2

2N01/2 no

2 TX, 1 RX, Alamouti|h1|2 + |h2|2

2N01 no

2 TX, 1 RX, Ant. sel. ≥ |h1|2 + |h2|22N0

1 partly

➮ For antenna selection, note that

max |hn|2N0

≥ 1

2

|h1|2 + |h2|2N0

57


2-antenna systems, cont

10−1

10−2

10−3

10−4

10−5

0 10 20 30 40 50

1 RX, 1 TX

2 RX, 1 TX, MRC

or 1 TX, 2 RX, BF

1 RX, 2 TX, antenna selection

1 RX, 2 TX, Alamouti

P (e)

SNR58


Orthogonal STBC (OSTBC)

➮ Important special case of linear STBC:

X =

ns∑

n=1

(snAn+isnBn) for which XXH =

ns∑

n=1

|sn|2 · I = ‖s‖2 · I

Notation: (·)=real part, (·)=imaginary part

➮ This is equivalent to requiring for n = 1, . . . , ns, p = 1, . . . , ns

AnAHn = I,BnBH

n = I

AnAHp = −ApA

Hn , BnBH

p = −BpBHn , n 6= p

AnBHp = BpA

Hn

59


Proof

➮ To prove ⇒), expand:

XXH =

ns∑

n=1

ns∑

p=1

(snAn + isnBn)(spAp + ispBp)H

=

ns∑

n=1

(s2nAnAH

n + s2nBnBH

n )

+

ns∑

n=1

ns∑

p=1,p>n

(

snsp(AnAHp + ApA

Hn ) + snsp(BnBH

p + BpBHn ))

+ i

ns∑

n=1

ns∑

p=1

snsp(BnAHp − ApB

Hn )

➮ Proof of ⇐), see e.g., EL&PS book.60


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 = 161


Diversity order of OSTBC

➮ Suppose {s0n}ns

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

X − X0 =

ns∑

n=1

(

(sn − s0n)An + i(sn − s0

n)Bn

)

⇒ (X − X0)(X − X0)H =

ns∑

n=1

|sn − s0n|2 · I

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

62


Derivation of decoupled detection

➮ Write the ML metric as

‖Y − HX‖2

=‖Y ‖2 − 2ReTr{Y HHX

}+ ‖HX‖2

=‖Y ‖2 − 2

ns∑

n=1

ReTr{Y HHAn

}sn + 2

ns∑

n=1

Im Tr{Y HHBn

}sn

+ ‖H‖2 · ‖s‖2

=

ns∑

n=1

(

− 2ReTr{Y HHAn

}sn + 2 Im Tr

{Y HHBn

}sn + |sn|2‖H‖2

)

+ const.

=‖H‖2 ·ns∑

n=1

∣∣∣∣∣sn − ReTr

{Y HHAn

}− i Im Tr

{Y HHBn

}

‖H‖2

∣∣∣∣∣

2

+ const.63


Decoupled detection, again

➮ Linearity: Y = HX + E ⇔ y = Fs′ + e, s′ , [sT sT ]T

➮ Theorem: X is an OSTBC if and only if

Re{F HF

}= ‖H‖2 · I ∀ H

➮ ML metric:

‖y − Fs′‖2 = ‖y‖2 − 2Re{yHFs′}+ Re

{s′TF HFs′}

= ‖y‖2 − 2Re{yHFs′}+ ‖H‖2 · ‖s′‖2

= ‖H‖2 · ‖s′ − s′‖2 + const.

where s′,

[ˆsˆs

]

=Re{F Hy

}

‖H‖2∼ N

([ss

]

,N0/2

‖H‖2I

)

➮ F is a “Spatial/temporal (code) matched filter”64


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· ‖H‖2

nt· P

N065


Example: Alamouti’s code is an OSTBC

➮ Consider the Alamouti code (re-normalized):

X =

[s1 s∗2s2 −s∗1

]

, XXH = (|s1|2 + |s2|2)I

➮ Identification of An and Bn gives

A1 =

[1 00 −1

]

A2 =

[0 11 0

]

B1 =

[1 00 1

]

B2 =

[0 −11 0

]

66


Examples of OSTBC

➮ Best known OSTBC for nt = 3, N = 4, ns = 3:

X =

s1 0 s2 −s3

0 s1 s∗3 s∗2−s∗2 −s3 s∗1 0

Code rate: 3/4 bpcu

➮ For nt = 4, N = 4, ns = 3:

X =

s1 0 s2 −s3

0 s1 s∗3 s∗2−s∗2 −s3 s∗1 0s∗3 −s2 0 s∗1

Rate is 3/4 bpcu.67


Summary of OSTBC Relations

XXH =

ns∑

n=1

|sn|2 · I = ‖s‖2 · I

m

AnAHn = I , BnBH

n = I

AnAHp = −ApA

Hn , BnBH

p = −BpBHn , n 6= p

AnBHp = BpA

Hn

m

Re{F HF

}= ‖H‖2 · I where F is such that

vec(Y ) = F ·[s

s

]

+ vec(E)68


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

Nlog

(

1 +N

ns

‖H‖2

ntN0

)

➮ Theorem:

CMIMO ≥ COSTBC, equality only for nt = 2, nr = 169


Capacity comparison (1% outage)

0 5 10 15 20 25 30 35 4010

−1

100

101

SNR

Cap

acity

[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 − OSTBCOutage capacity, 2 TX, 2 RX Outage capacity, 2 TX, 2 RX − OSTBC

70


Non-orthogonal linear STBC

➮ Also called linear dispersion codes

➮ Different approaches:

➠ Optimization of mutual information between the TX & RX:

max1

2EH

[

log2

∣∣∣I +

2

N0Re{F HF

}∣∣∣

]

(no explicit guarantee for full diversity here)

➠ Quasi-orthogonal codes

➠ Codes based on linear constellation (complex-field) precoding

s′ = Φs

71


Example: A non-OSTBC

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

X =

[s1 00 s2

]

➮ Then

A1 =

[1 00 0

]

, A2 =

[0 00 1

]

, B1 =

[1 00 0

]

B2 =

[0 00 1

]

➮ ML metric for symbol detection:

‖Y − HX‖2 =‖Y ‖2 − 2ReTr{XHHHY

}+ ‖H‖2

= − 2Re{[HHY ]1,1 · s1

}− 2Re

{[HHY ]2,2 · s2

}+ const.

➮ Decoupled detection, but not OSTBC, and not full diversity72


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 =

[10

]

,A2 =

[01

]

B1 =

[10

]

,B2 =

[01

]

73


Linearly precoded STBC

➮ Transmit WX where W ∈ {W 1, . . . ,W K}. Data model:

Y = HWX + E

➮ Fact: If rank{W k} = nt then same diversity order as but without W

➮ Consider correlated fading: R = E[hhH] = RTt ⊗ 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, minW

∣∣∣I +

ns

N0W HRtW

∣∣∣

74


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

➮ One bit used to choose between

W 1 =

[|a| 0

0√

1 − |a|2]

︸︷︷︸

if ‖h1‖>‖h2‖

, W 2 =

[√

1 − |a|2 00 |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)

75


4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2010

−5

10−4

10−3

10−2

10−1

SNR

BE

RUnweighted OSTBC. Optimal weighting (No feedback error). Optimal weighting with feedback error. Error tolerant weighting (No feedback error),Error tolerant weighting with feedback error

76


MIMO with feedback - optimized transmission

ssxH

ny

I I(H)

UW (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 techniques77


Frequency-selective channels

➮ Maximum diversity order (with ML detection) will be nrntL whereL=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

78


Le 3: MIMO receivers

79


Summary of MIMO receivers

➮ Optimal architectures (from Le 1):

➠ CSI@TX (any fading): linear processing, y = UHy, 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 − s1|2 + |s2 − s2|2

)

➠ With non-OSTBC, min ‖Y − HX‖ does not decouple- problem similar to for V-BLAST 80


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.81


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 MMSE3. 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

82


Receivers for linear STBC architectures

Training Data 2Data 1

➮ Received block: Y = HX + E.

➮ X linear in {s1, ..., sns} so with appropriate F , the ML metric is

‖Y − HX‖2=

∥∥∥∥

[vec(Y )

vec(Y )

]

− F

[s

s

]∥∥∥∥

2

➮ For OSTBC, F TF = ‖H‖2I so detection decouples

➮ Spatial multiplexing (V-BLAST) can be seen a degenerated special caseof this architecture, with

X =

s1...

snt

83


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)84


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.

85


The problem

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

mins∈Sn

‖y − Gs‖2, y ∈ R

m, G ∈ Rm×n

➮ Let G = QL where

{

Q ∈ Rm×n is orthonormal (QTQ = I)

L ∈ Rn×n is lower triangular

Then ‖y − Gs‖2=∥∥∥QQT (y − Gs)

∥∥∥

2

+∥∥∥(I − QQT )(y − Gs)

∥∥∥

2

=∥∥∥Q

Ty − Ls

∥∥∥

2

+∥∥∥(I − QQT )y

∥∥∥

2

so mins∈Sn

‖y − Gs‖2 ⇔ mins∈Sn

‖y − Ls‖ where y , QTy86


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 thecorrect solution quickly, with high probability

87


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

88


Zero-Forcing

➮ Lets , arg min

s∈Rn‖y − Gs‖ = arg min

s∈Rn‖y − Ls‖ = L−1y

E.g., Gaussian elimination: s1 = y1/L1,1

s2 = (y2 − s1L2,1)/L2,2

...

➮ Then project onto S: sk = [sk] , arg minsk∈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 89


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 = −1s2 = −1 s2 = +1s2 = +1

s3 = −1s3 = −1s3 = −1s3 = −1 s3 = +1s3 = +1s3 = +1s3 = +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

90


Zero-Forcing with Decision Feedback (ZF-DF)

➮ Consider the following improvement

i) Detect s1 via: s1 =

[y1

L1,1

]

= arg mins1∈S

f1(s1)

ii) Consider s1 known and set s2 =

[y2 − s1L2,1

L2,2

]

= arg mins2∈S

f2(s1, s2)

iii) Continue for k = 3, ..., n:

sk =

[

yk −∑k−1l=1 Lk,lsl

Lk,k

]

= arg minsk∈S

fk(s1, ..., 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.

91


Zero-Forcing with Decision Feedback (ZF-DF)

r1

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

92


Sphere decoding (SD)

➮ Select a sphere radius, R. Then traverse the tree, but once encounteringa node with cumulative metric > R, do not follow it down

➮ Enumerates all leaf nodes which lie inside the sphere ‖y − Ls‖2 ≤ 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

93


SD, without pruning, R = 6

r2

r2a 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

94


SD, with pruning, R = ∞

r3

r3a 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

95


“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 structure96


FCSD, r = 1

r4

r4a 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

97


Semidefinite relaxation (for sk ∈ {±1})

➮ Let s ,

[s

1

]

, S , ssT =

[s

1

][sT 1

], Ψ ,

[LTL −LT y

−yTL 0

]

Then‖y − Ls‖2

= sTΨs + ‖y‖2

= Trace{ΨS} + ‖y‖2

so the problem is to

mindiag{S}={1,...,1}

rank{S}=1sn+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 Sn98


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 Ts ∈ Sn ∀s ∈ Sn

➮ Find one such T for which LT ∝ I

➮ Then solve s′, arg min

s′∈Sn‖y − (LT )s′‖2, and set s = T−1s

′

➮ Critical steps:➠ Find suitable T (computationally costly, but amortize over many y)➠ s ∈ Sn, but s /∈ Sn in general, so clipping is necessary

99


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)

)

100


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

0

@

P

s:bi(s)=1 p(y|s)“

Qnp

i′=1P (bi′ = bi′(s))

”

P

s:bi(s)=0 p(y|s)“

Qnp

i′=1P (bi′ = bi′(s))

”

1

A

= log

0

B

@

P

s:bi(s)=1 p(y|s)“

Qnp

i′=1,i′6=iP (bi′ = bi′(s))

”

· P (bi = 1)

P

s:bi(s)=0 p(y|s)“

Qnp

i′=1,i′6=iP (bi′ = bi′(s))

”

· P (bi = 0)

1

C

A

= log

0

B

@

P

s:bi(s)=1 p(y|s)“

Qnp

i′=1,i′6=iP (bi′ = bi′(s))

”

P

s:bi(s)=0 p(y|s)“

Qnp

i′=1,i′6=iP (bi′ = bi′(s))

”

1

C

A+ L(bi)

In Gaussian noise p(y|s) = 1(2πσ)m/2 exp

(− 1

2σ‖y − Gs‖2)

so

L(bi|y) = log

0

B

@

P

s:bi(s)=1 exp“

− 12σ‖y − Gs‖2

”“

Qnp

i′=1,i′6=iP (bi′ = bi′(s))

”

P

s:bi(s)=0 exp“

− 12σ‖y − Gs‖2

”“

Qnp

i′=1,i′6=iP (bi′ = bi′(s))

”

1

C

A+ L(bi)

101


➮ With a priori equiprobable bits

L(bi|y) = log

(∑

s:bi(s)=1 exp(− 1

2σ‖y − Gs‖2)

∑

s:bi(s)=0 exp(− 1

2σ‖y − Gs‖2)

)

➮∑

can be relatively well approximated by its largest term

That gives problems of the type

mins∈Sn,bi(s)=β

‖y − Gs‖2

This is called “max-log” approximation

➮ If many candidates s are examined, then use these terms in∑

➠ List-decoding algorithm

102


Incorporating a priori probabilities (BPSK/dim case)

➮ Consider sk ∈ {±1}, and let

sk = 2bk − 1

γk ,1

2log (P (sk = −1)P (sk = 1)) =

1

2log (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

103


➮ Write

L(sk|y) = log

∑

s:sk=1 exp(

−1σ‖y − Gs‖2 + 1

2

∑ni=1,i 6=k (γi + λisi)

)

∑

s:sk=0 exp(

−1σ‖y − Gs‖2 + 1

2

∑ni=1,i 6=k (γi + λisi)

)

+λk

➮ Define y , [yT 1 · · · 1]T and G ,

[G

Λk

]

where

Λk , diag{

σ4λ1, · · · , σ

4λk−1,σ4λk+1, . . . ,

σ4λn

}

➮ Then

L(sk|y) = log

∑

s:sk=1 exp(

−1σ‖y − Gs‖2 +

∑ni=1,i 6=k

(σλ2

i16 + γi

2

))

∑

s:sk=0 exp(

−1σ‖y − Gs‖2 +

∑ni=1,i 6=k

(σλ2

i16 + γi

2

))

+λk

➮ A priori information on sk ➠ “virtual antennas”104


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

0.01

0.1

−4 −3.5 −3 −2.5 −2

Fra

me-e

rror-

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

105


Channel (H) estimation and associated receivers

➮ Very often pilots are used to form a channel estimate.

ConsiderY t = HXt + Et

➮ Estimate H via training:

➠ Maximum likelihood (in Gaussian noise):

H = argminH

‖Y t − HXt‖2 = Y tXHt (XtX

Ht )−1

➠ Can show, estimate is the same also in colored noise (e.g. co-channelinterference in multiuser system)

➠ Can be somewhat improved by using MMSE estimation106


➮ Estimate noise (co)variance:

Λ = 1Nt

Y tΠ⊥XH

tY H

t , for colored noise

N0 = 1Ntnr

Tr{

Y tΠ⊥XH

tY H

t

}

, for white noise

➠ This estimate can be used to prewhiten the received signal to suppressco-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.

107


More on training

➮ Training-based detector uses H, N0 and Λ in the coherent detector

➮ Can be improved, e.g., cyclic detection

➮ How should training be designed?

Optimum pilots (in many respects) satisfy XtXHt ∝ I

➮ Inserting pilot-based estimate in LF is not optimal

Cf. the use ofp(X|Y , H) (coherent)p(X|Y , H)|

H:=H(training-based)

p(X|Y , H) (best possible given H)p(X|Y , Y t) (best possible given training data)

Later, we will give p(X|Y , H) on explicit form 108


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

109


Optimal training

➮ Recall: ML channel estimate H = Y tXHt (XtX

Ht )−1

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

E[h] = h

Σ , E[(h − h)(h − h)H] = . . . = N0

“

(XtXHt )

−T ⊗ I”

Tr {Σ} = nrTr˘

(XtXHt )

−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:

XtXHt ∝ I 110


Channel estimation for frequency-selective channels

➮ MIMO channel as matrix-valued FIR filter:

H(z−1) =

L∑

l=0

H lz−l

➮ L is the length of the channel, L = 0 for ISI-free channel

➮ Transfer function:

H(ω) =

L∑

l=0

H le−iωl

➮ Transmission methods:

➠ Single-carrier (block-based)

➠ Multicarrier (e.g. OFDM)111


Training for frequency selective channels

➮ Two basic approaches:

➠ Frequency-domain estimation (estimate H(ω))➠ Time-domain estimation (estimate H l, then compute H(ω) via FT)

➮ Frequency-domain estimation of H(ω) is straightforward:

➠ Estimate in the frequency domain (via ML):

H(ω) = Y t(ω)XHt (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

H le−iωl

112


Time-domain estimation of H(ω)

➮ Let x(n) be time-domain training, y(n) received (t-d) training and define

Xt =

xTt (0) · · · · · · · · · xT

t (−L)

xTt (1) . . . xT

t (1 − L)... . . . ...

xTt (N − 1) · · · · · · xT

t (0) · · · xTt (N − 1 − L)

H =

HT0

...

HTL

, Y t =

yTt (0)...

yTt (N − 1)

➮ Then the ML estimate of H(ω) is:

H = (XHt Xt)

−1XHt Y t, H(ω) =

L∑

l=0

H le−iωl

➮ Exploits structure (L unknowns but N equations) 113


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

➮ G not known perfectly ➠ replacing G with G in p(y|s,G) not optimal!

➮ Instead, need to work with p(y|s, G). Write

y = Gs + e ⇔ y = (sT ⊗ I)h + e, h = vec(G), e = vec(E)

Suppose ‖s‖2 = n and

{

h ∼ N(0, ρI), e ∼ N(0, σI)

h = h + δ, δ ∼ N(0, ǫI)

➮ Then

[y

h

]

∼ N

(

0,

[(nρ + σ)I ρ(sT ⊗ I)ρ(s∗ ⊗ I) (ρ + ǫ)I

])

so

p(y|h, s) =1

πn

1nǫ

1+ǫ/ρ + σexp

(

− 1nǫ

1+ǫ/ρ + σ

∥∥∥∥y −

(ρ

ρ + ǫ

)

Gs

∥∥∥∥

2)

114


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

0.01

0.1

1

−5 −4 −3 −2 −1 0 1 2 3

Fra

me-e

rror-

rate

(FER)


Practical detector

Perfect CSI

Imperfect CSI

Brute-force

Practical detector, mismatched metric

Practical detector, optimal metric

Brute-force, optimal metric

Brute-force, mism.

115


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

0.01

0.1

1

−5 −4 −3 −2 −1 0 1 2 3

Fra

me-e

rror-

rate

(FER)


Practical detector

Perfect CSI

Imperfect CSI

Brute-force

Practical detector, mismatched metricPractical detector, optimal metric

Brute-force, optimal metric

Brute-force, mism.

116


Last slide

117

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