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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
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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.
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
Le 1: MIMO fundamentals
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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Linkoping University, ISY, Communication Systems, E. G. Larsson 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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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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Linkoping University, ISY, Communication Systems, E. G. Larsson 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, UHU = I, V HV = I
➮ Dimensions: U ∈ Cnr×nr, Λ ∈ R
nr×nt, V ∈ Cnt×nt
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
➮ 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
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Linkoping University, ISY, Communication Systems, E. G. Larsson 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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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
)
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
)
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
Waterfilling solution
1 2 3... n k
N0
λ2k
µ
P∗ 1
=0
P∗ 2
P∗ 3
=0
P∗ 4
P∗ 5
=0
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Linkoping University, ISY, Communication Systems, E. G. Larsson 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
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
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Waterfilling at high SNR
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1 2 3... n k
N0
λ2k
P ∗k ≈ P
n
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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)
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
Waterfilling at low SNR
1 2 3... n k
N0
λ2k
µ
P∗ 4
=P
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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!
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Linkoping University, ISY, Communication Systems, E. G. Larsson 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 MIMOpropagation and how they connect to the DoF concept.
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Linkoping University, ISY, Communication Systems, E. G. Larsson 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−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
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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Linkoping University, ISY, Communication Systems, E. G. Larsson 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
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.
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Linkoping University, ISY, Communication Systems, E. G. Larsson 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(· · · ) 6= 1 ⇔ φ1, φ2 resolvable ⇔ | cos φ1−cos φ2| > 2πA23
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Linkoping University, ISY, Communication Systems, E. G. Larsson 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) = 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
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Linkoping University, ISY, Communication Systems, E. G. Larsson 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 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
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
Angular decomposition, cont.
1
2
3
4
5
1
2
3
4
5
TX
RX
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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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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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
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V-BLAST architecture
x1
x2
xnt
x1
x2
xn
e1
enr
y1
y2
ynr
HQ optimalreceiver
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➮ 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 .
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➮ 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
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➮ 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.
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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
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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
)
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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)”
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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
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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
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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
)]
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➮ 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!
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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
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➮ 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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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
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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
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Le 2: Low-complexity MIMO
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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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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
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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
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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
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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
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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 − 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
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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
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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
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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
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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
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➮ 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
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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
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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
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
]
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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
∣∣∣
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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)
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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
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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
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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
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Le 3: MIMO receivers
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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
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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
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Linkoping University, ISY, Communication Systems, E. G. Larsson 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 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
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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Linkoping University, ISY, Communication Systems, E. G. Larsson 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)84
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Linkoping University, ISY, Communication Systems, E. G. Larsson 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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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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Linkoping University, ISY, Communication Systems, E. G. Larsson 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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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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Linkoping University, ISY, Communication Systems, E. G. Larsson 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 = −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
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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.
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
“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
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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)
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➮ 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
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
➮ 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
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
➮ 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.
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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
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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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Linkoping University, ISY, Communication Systems, E. G. Larsson MIMO Fundamentals and Signal Processing
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
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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
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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
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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 =
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
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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)
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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)
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 0 1 2 3
Fra
me-e
rror-
rate
(FER)
Normalized signal-to-noise-ratio (SNR) [dB]
Practical detector
Perfect CSI
Imperfect CSI
Brute-force
Practical detector, mismatched metricPractical detector, optimal metric
Brute-force, optimal metric
Brute-force, mism.
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Last slide
117