Limited Feedback v3

Limited Feedback for Single- and Multi-User MIMO389.168 Advanced Wireless Communications 1

[email protected]

Contents

1 Motivation

2 Codebook based Single-User MIMO Feedback for LTE

3 Multi-User MIMO Feedback for LTE

4 Multi-User MIMO with Channel Subspace Feedback

5 Multi-User MIMO with Channel Gramian Feedback

6 Feedback Overhead Reduction through Excess Antennas

Slide 2 / 89 Contents

Contents

1 Motivation






Slide 3 / 89 Motivation

Limited Feedback: What and Why?

Value of channel state information at the transmitter (CSIT):

Channel quality information (e.g., SINR, achievable rate):

Improve robustness and efficiency through rate adaptation

Exploit channel and multi-user diversity through scheduling

Channel “matrix” information:

Single-user communication: mostly a power/beamforming gain(water-filling versus equal power)

Multi-user (point) communication: capacity/multiplexing gain(exploitation of degrees of freedom)


Performance of LTE’s Modulation and Coding Schemes

302520151050-5

5

4

3

2

1

0

SNR [dB]

Thr

ough

put [

Mbi

t/s]

SISO_1.4MHz_flatRayleigh

CQI 1

CQI 15

Frequency flat Rayleigh fading channel with 1.4 MHz bandwidth

Throughput over average SNR for transmission with fixed rate

15 MCSs with rates ranging form 0.15 bit/cu to 5.55 bit/cu


Scheduling Gains through CQI Feedback [Schwarz et al., 2010]

0 5 10 15 20 250

2

4

6

8

10

UE index (= UE SNR [dB])

Thr

ough

put [

Mbi

t/s]

RRMaxMinPFBCQI

25 users with average SNRs ranging from 1 to 25 dB

Round robin: assign resources in consecutive order

MaxMin: maximize minimum user throughput – throughput equalization

Proportional fair : maxjTj

T j

Best CQI: schedule user with highest CQI


Scheduling Gains through CQI Feedback (2)

RR MaxMin PF BCQI0

5

10

15

20

25

30

35

40

45

Scheduler

Thr

ough

put [

Mbi

t/s]

RR MaxMin PF BCQI0

0.2

0.4

0.6

0.8

1

Scheduler

Jain

’s F

airn

ess

Inde

x

Comparison of sum throughput and fairness of resource allocation

Jain’s fairness index :

J =

(∑Jj=1 T j

)2

J∑J

j=1 T2j

(1)

Ranges from 1 (highest fairness = equal throughput) to 1/J


Scheduling with Fairness Constraint [Schwarz et al., 2011]

Throughput user 1 [bits per channel use]

Thr

ough

put u

ser 2

[bits

per

cha

nnel

use

]

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

Maxmin. solution

Prop. fair solution

Max. throughput solution

Rate region J = 1

J = 0.9

J = 0.73

Two-user achievable rate region with operating points of some schedulers

A fairness constraint J ≥ J0 cuts out a convex cone

Operating points of α-fair sum-utility maximization

maximize:J∑

j=1

Uα(

T (j)), Uα(x) =

{ x1−α

1− α, α ≥ 0, α 6= 1

log(x), α = 1(2)




Thr

ough

put u

ser 2

[bits

per

cha

nnel

use

]

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

J ≥ 0.85

J ≥ 0.70

Rate region




maximize:J∑

j=1

Uα(

T (j)), Uα(x) =

{ x1−α

1− α, α ≥ 0, α 6= 1

log(x), α = 1(2)




Thr

ough

put u

ser 2

[bits

per

cha

nnel

use

]

α

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

= 1

αα = 1000

α = 0

J ≥ 0.85

J ≥ 0.70

Rate region




maximize:J∑

j=1

Uα(

T (j)), Uα(x) =

{ x1−α

1− α, α ≥ 0, α 6= 1

log(x), α = 1(2)


Limited Feedback: What and Why? (2)

Acquisition of CSIT:

TDD or full-duplex: channel estimation at the transmitter possible

Careful calibration of uplink/downlink chains to ensure reciprocity

Timing synchronization difficult

FDD: CSIT cannot be estimated by the transmitter

Currently dominates the field

Explicit CSI feedback from the receiver required

Limited uplink overhead ⇒ quantization


Contents

1 Motivation






Slide 10 / 89 Codebook based Single-User MIMO Feedback for LTE

Direct Channel Quantization

Remember the capacity-optimal SVD transceiver

H = UΣVH (3)

⇒ F = VP1/2, G = UH (4)

For precoder calculation the transmitter requires V and diag (Σ)

Direct quantization of H in Euclidean space [Zheng and Rao, 2008]

H = argminHq∈H

‖H− Hq‖ , H ⊂ CNr×Nt , (5)

H = UΣVH (6)

Inefficient , as the structure of the quantization problem is neglected


Direct Channel Quantization (2)

Separate quantization exploiting the structure of the problem[Schwarz and Rupp, 2014b]

σ = argminsi∈S

‖σ − si‖22 , σ =

[[Σ](1,1), . . . , [Σ](Nr ,Nr )

]T, (7)

V = argminQi∈Q

d2c

([Qi ](:,j) , [V](:,j)

)= argmin

Qi∈QNr −

∣∣∣[Qi ]H(:,j) [V](:,j)

∣∣∣2 , (8)

S ⊂ RNr×1+ , Q =

{Qi ∈ CNt×Nr

∣∣QHi Qi = INr

}⊂ St(Nt ,Nr ) (9)

Notice the order of the singular-vectors is important for power allocation⇒compact Stiefel manifold St(Nt ,Nr )


Direct Channel Quantization (3)

Water-filling power allocation provides mostly only a minor gain over on-offswitching of modes⇒ employ equal power allocation over L active modes

Grassmannian quantization with rank selection

V = argminQi∈Q

d2c (Qi ,VL) = argmin

Qi∈QL− tr

(QH

i VLVHL Qi), VL = [V]:,1:L , (10)

L = number of non-zero elements (diag (P)) , (11)

Q ={

Qi ∈ CNt×L∣∣QHi Qi = IL

}⊂ G(Nt , L) (12)

Only subspace information span(VL)

required⇒ Grassmannian quantization

Unitary rotations of Qi are irrelevant

Qi ≡ Qj ⇔ Qi = Qj U, UHU = UUH = IL, (13)

log2 det(INr + HQi QH

i HH) = log2 det(

INr + HQj UUHQHj HH

)=

log2 det(

INr + HQj QHj HH

)(14)

⇒ feedback overhead reduction compared to Stiefel manifold quantization


Codebook Based Precoding

Instead of quantizing the channel, let the user directly select the precoderfrom a given codebook F

Outperforms direct channel quantization [Love and Heath, Jr., 2005]

Optimal quantization codebook constructions [Love and Heath, Jr., 2005]

Maximally spaced subspace packings on the Grassmannian

Difficult to obtain in general(some pre-calculated codebooks can be found at [Love, 2006])

Algorithm for finding good codebooks [Dhillon et al., 2007]


Codebook Based Precoding (2)

More practical codebook constructions (low complexity implementation)

Constant modulus (phase-shifts only), nested codebooks

Codebooks based on DFT [Yang et al., 2010]

Codebooks based on QAM constellations [Ryan et al., 2007]

This approach is applied in LTE


LTE’s Signal Processing Chain

Coding

Interleaving

Segmentation

AMC scheme selection

Symbol

constellation

mapping

Spatial

layer

mapping

MIMO preprocessing

Spatial

precoding (Ws)

User data Code words c Symbols Layers (L)

Other user data

Receiver

Equalization

Demapping

DecodingTransmit

signal

Receive

signal

H

Nr

Feedback

calculation

Channel quality

indicator (CQI)

Precoding matrix

indicator (PMI)

Rank indicator (RI)Resource

element

mapping

Transmit signal generation

IFFT

CP insertion

IFFT

CP insertion

System

overhead

insertion

System

overhead

insertion

Nt

Resource

element

mapping

(Nt)

Adaptive modulation and coding (AMC)

Multiple MCSsM to adapt the rate to the channel conditions

Preferred MCS signalled by means of CQI feedback

Multiple code words to account for different channel qualities of different layers

In LTE one code word c is mapped onto multiple layers Lc

The CQI feedback can be subband or wideband specific (scheduling)


LTE’s Signal Processing Chain (2)

Coding

Interleaving

Segmentation


Symbol

constellation

mapping

Spatial

layer

mapping

MIMO preprocessing

Spatial

precoding (Ws)


Other user data

Receiver

Equalization

Demapping

DecodingTransmit

signal

Receive

signal

H

Nr

Feedback

calculation

Channel quality

indicator (CQI)

Precoding matrix

indicator (PMI)


element

mapping


IFFT

CP insertion

IFFT

CP insertion

System

overhead

insertion

System

overhead

insertion

Nt

Resource

element

mapping

(Nt)

MIMO preprocessing

Exploit the potential MIMO gains (beamforming, diversity, spatial multiplexing)

Feedback reduction: precoders are confined to a code book F (L) ⊂ CNt×L

The receiver selects and feeds back:

The preferred number of layers L (rank) - RI (OLSM, CLSM)

The best precoder from the code book - PMI (CLSM)

The same rank is used on all REs (L ≤ Lmax = rank(H))

Precoder feedback can be subband or wideband specific


LTE’s Signal Processing Chain (3)

Coding

Interleaving

Segmentation


Symbol

constellation

mapping

Spatial

layer

mapping

MIMO preprocessing

Spatial

precoding (Ws)


Other user data

Receiver

Equalization

Demapping

DecodingTransmit

signal

Receive

signal

H

Nr

Feedback

calculation

Channel quality

indicator (CQI)

Precoding matrix

indicator (PMI)


element

mapping


IFFT

CP insertion

IFFT

CP insertion

System

overhead

insertion

System

overhead

insertion

Nt

Resource

element

mapping

(Nt)


OFDM: the bandwidth is divided into K orthogonal subcarriers

The CP orthogonalizes OFDM symbols (in time)

Temporal frame structure: per frame R REs r are available

Consecutive REs are grouped into S ≤ R subbands

The mapping ρ maps an RE r to the corresponding subband s: ρ(r)→ s

The subband size determines the feedback granularity


Resource Elements and Subbands

Resourceelement r

Resources R

Subband s

Layers L

OFDM symbols

OFD

M s

ubca

rrie

rs

Set of REsRs

Rs . . . set of REs corresponding to subband s


Closed-Loop Precoding in LTE/LTE-A

Codebook

index

Number of layers `

1 2

0

1

2

3

1√2

11

1√2

1-1

1√2

1j

1√2

1-j

12

1 11 -1

12

1 1j -j

CLSM supports CQI, RI and PMI feedback

Codebook construction for Nt = 2 [3GPP, 2009]:

DFT matrices

Nested + constant modulus (QPSK)

Codebook construction for Nt = 4 [3GPP, 2009]:

Obtained from Householder matrices: Fi = I− 2 ui uHi , i ∈ {1, . . . , 16}

Nested + constant modulus (QPSK, some elements from 8-PSK)


Closed-Loop Precoding in LTE/LTE-A (2)

0 45 90 135 1800

2

4

6

8

Steering Angle [°]

An

ten

na

ga

in

[i1

,i2

] = [0,0]

[i1

,i2

] = [4,0]

[i1

,i2

] = [7,0]

[i1

,i2

] = [11,0]

0 45 90 135 1800

2

4

6

8

Steering Angle [°]

An

ten

na

ga

in

[i1

,i2

] = [0,0]

[i1

,i2

] = [0,1]

[i1

,i2

] = [0,2]

[i1

,i2

] = [0,3]

Wideband and Subband Beamformers of LTE-A assuming a ULA with Nt = 8

Nt = 8: product of subband and wideband precoder [3GPP, 2010]

Fs = F(1)F(2)s , (15)

F(1) ∈ F (1)L ⊂ CNt×L, F(2) ∈ F (2)

L ⊂ CNt×L (16)

Same notation for Nt ≤ 4: simply set F (1)L = I


Closed-Loop Precoding in LTE/LTE-A (2)

Rank Number of Number of Total number ofwideband precoders subband precoders precoder combinations

1 16 16 2562 16 16 2563 4 16 644 4 8 325 4 1 46 4 1 47 4 1 48 1 1 1

Nt = 8: product of subband and wideband precoder [3GPP, 2010]

Fs = F(1)F(2)s , (15)

F(1) ∈ F (1)L ⊂ CNt×L, F(2) ∈ F (2)

L ⊂ CNt×L (16)

Same notation for Nt ≤ 4: simply set F (1)L = I


Input-Output Relationship and Post-equalization SINR

Received signal vector yr ∈ CNr×1 on RE r

yr = Hr Fssr + zr , r ∈ {1, . . . ,R}, s = ρ(r)

Estimated symbol-vector sr ∈ CL×1 with linear equalizer

sr = Gr yr = Gr Hr Fs︸︷︷︸Kr

sr + Gr zr

E.g., MMSE receiver

Gr =(

(Hr Fs)HHr Fs + σ2z I)−1

(Hr Fs)H (17)

Post-equalization SINR of layer `

SINRr,` (Fs) =P`|Kr [`, `]|2∑

i 6=` Pi |Kr [`, i]|2 + σ2z |Gr [`, i]|2

(18)


Selection of Feedback Indicators [Schwarz and Rupp, 2011]

ESM SINR

averaging fmSINRr,l (Ws)

SNRs (Ws,m)c

AWGN SNR

to BLER

mapping gm Ps (Ws,m)c

Effective SNR averaging for BLER estimation

RI and PMI : maximize estimated user throughput

CQI : largest rate that ensures operation below target BLER P(t)b (common

practice)⇒ estimation of BLER for each MCS required

Static SISO-AWGN channel: BLER versus SNR is known (simulations)⇒ SNR estimation is sufficient

Fading environment: SINR changes over REs r and layers `

Linear averaging of SNRs: diversity order of the channel not represented

More accurate: effective SNR mapping (ESM) [Tsai and Soong, 2003]⇒ Mapping of SINRs onto equivalent AWGN-SNR


Effective SNR Mapping

Consider the SINRs of the REs corresponding to subband s and of the layersbelonging to code word c

SINRr,` (Fs) , r ∈ Rs, ` ∈ Lc (19)

To estimate the BLER of code word c using MCS m, we calculate an effectiveAWGN equivalent SNR

SNRcs (Fs,m) = f−1

m

1|Rs| |Lc |

∑r∈Rs,l∈Lc

fm(SINRr,l (Fs)

) (20)

fm(SNR). . . SINR averaging function of MCS m

Mutual information effective SNR mapping (MIESM): fm(SINR) is the(calibrated) BICM capacity of the corresponding modulation order(4/16/64 QAM) [Wan et al., 2006]

Exponential effective SNR mapping (EESM): fm(SINR) is an exponentialfunction [Sandanalakshmi et al., 2007]


BICM Capacity – MIESM Averaging Functions

channelcoding

modulationmapping

coded bits

bitinterleaving

modulated symbolsdata bits interleaved bits

BICM architecture

BICM architecture as applied by LTE

BICM capacity mapping functions fm(SINR) = Bm

(SINRβm

)


BICM Capacity – MIESM Averaging Functions

242220181614121086420-2-4-6

8

7

6

5

4

3

2

1

0

SNR [dB]

Spec

tral

eff

icie

ncy

[bit/

s/H

z]

SISO_AWGN

Shannon capacity

BICM 64 capacity

BICM 16 capacity

BICM 4 capacity

BICM capacity curves versus AWGN Shannon capacity

BICM architecture as applied by LTE

BICM capacity mapping functions fm(SINR) = Bm

(SINRβm

)


ESM – Calibration Sensitivity

2.52.2521.751.51.2510.750.5

100

10-1

10-2

10-3

Calibration parameter

Wei

ghte

d M

SE

SISO_1.4MHz_CQI695% confidence interval95% confidence interval

MIESM

EESM

MSE of the estimated SNR for MIESM and EESM in dependence of the calibration parameter

fm(SINR) = Bm

(SINRβm

), βm . . . calibration parameter (21)

Calibration according to [He et al., 2007, Cipriano et al., 2008]


MIESM – SNR Averaging Performance

2520151050-5-10SNR [dB]

Blo

ck e

rror

ratio

SISO_1.4MHz_AWGN_TU100

10-1

10-2

10-3

SISO AWGN BLERMIESM estimation

Comparison of the MIESM abstraction for a 1.4 MHz typical urban channel to the average BLERs of LTE’s 15 MCSsachieved over a 1.4 MHz AWGN channel.

With the AWGN equivalent SNR, the BLER of MCS m is estimated as

Pcs (Fs,m) = gm

(SNRc

s (Fs,m))

(22)

gm(SNR). . . AWGN SNR-BLER lookup table


Spectral Efficiency Estimation

ESM SINR

averaging fmSINRr,l (Ws)

SNRs (Ws,m)c

AWGN SNR

to BLER

mapping gm

Spectral

e!ciency

estimationPs (Ws,m)c

Es (Ws,m)c

Calculation steps required for spectral efficiency estimation

Define the following function

hm(P) =

{em, P ≤ P(t)

b

0, P > P(t)b

em. . . spectral efficiency of MCS m

Estimated spectral efficiency achieved with MCS m

Ecs (Fs,m) = hm(Pc

s (Fs,m)) · (1− Pcs (Fs,m)) (23)


Feedback Indicator Selection Optimization Problem

[L,{

Fs

}S, {ms}S

]= argmax

L,F(1),F(2)s ,ms

S∑s=1

CL∑c=1

Ecs (Fs,ms[c])

subject to: L ≤ Lmax

F(1) ∈ F (L)1

F(2)s ∈ F (L)

2

ms ∈MCL×1

CL. . . number of code words employed when L layers are transmittedS . . . number of feedback subsets

Non-linear combinatorial optimization problem⇒ exhaustive search

Practically infeasible within 1 ms subframe duration


Approximate Sequential Solution

1 Preselect rank and precoders from theoretical spectral efficiency (mutualinformation, BICM capacity) without considering the BLER

2 Given precoders and rank, select the highest MCS achieving the BLER target

Estimated spectral efficiency of RE r for L layers and precoder Fs

Ir (Fs, L) =

CL∑c=1

∑`∈Lc

f(SINRr,` (Fs)

), s = ρ(r)

f (SINR). . . spectral efficiency function (mutual information, BICM capacity)

Optimal subband precoder for fixed rank L and wideband precoder F(1)

F(2)s

(F(1), L

)= argmax

F(2)s ∈F

(L)2

Is (Fs, L) , (24)

Is (Fs, L) =∑

r∈Rs

Ir (Fs, L) , Fs = F(1)F(2)s (25)


Approximate Sequential Solution (2)

Optimal wideband precoder for fixed rank L

F(1)(L) = argmax

F(1)∈F(L)1

I (Fs, L) , (26)

I (Fs, L) =S∑

s=1

Is (Fs, L) , Fs = F(1)F(2)s

(F(1), L

)

Optimal rank

L = argmaxL≤Lmax

I(

Fs(L), L), Fs(L) = F(1)

(L) F(2)s

(F(1)

(L), L)

Further simplification: replace f (SINR) with the pre-equalization efficiency

I(Fs) = log2 det(

I +1σ2

zHr FsFH

s HHr

)(27)

Performs only well with maximum likelihood (ML) detection


Performance Investigation – Rate Adaptation

302520151050-5

5

4

3

2

1

0

SNR [dB]

Thr

ough

put [

Mbi

t/s]


CQI 1

CQI 15

Evaluation of CQI feedback methods for SISO transmission over 1.4 MHz bandwidth

Performance of LTE’s 15 MCSs without rate adaptation

Rate adaptation based on instantaneous SINR without feedback delay

Rate adaptation based on instantaneous SINR with outdated feedback

Rate adaptation based on long term average SINR



302520151050-5

5

4

3

2

1

0

SNR [dB]

Thr

ough

put [

Mbi

t/s]


instantaneous CQI, delay 0

CQI 1

CQI 15








302520151050-5

5

4

3

2

1

0

SNR [dB]

Thr

ough

put [

Mbi

t/s]




CQI 1

CQI 15








302520151050-5

5

4

3

2

1

0

SNR [dB]

Thr

ough

put [

Mbi

t/s]



long-term CQI, delay 1


CQI 1

CQI 15







Performance Investigation – Rate Adaptation (2)

10.10.01

4

3.75

3.5

3.25

3

2.75

2.5

2.25

2

51551.55.15

Normalized Doppler frequency

Thr

ough

put [

Mbi

t/s]

SISO_1.4MHz_flatRayleigh_20dB

User velocity [km/h] @ 2.1 GHz carrier frequency

instantaneous CQI

Impact of feedback delay on CQI adaptation methods

Rate adaptation based on instantaneous SINR

νd = fd Ts = fcvc0

Ts (28)

Rate adaptation based on instantaneous SINR with linear extrapolation

Rate adaptation based on instantaneous SINR with RLS prediction




10.10.01

4

3.75

3.5

3.25

3

2.75

2.5

2.25

2

51551.55.15


Thr

ough

put [

Mbi

t/s]



instantaneous CQI

instantaneous CQIlinear extrapolation



νd = fd Ts = fcvc0

Ts (28)






10.10.01

4

3.75

3.5

3.25

3

2.75

2.5

2.25

2

51551.55.15


Thr

ough

put [

Mbi

t/s]



instantaneous CQI


instantaneous CQIRLS prediction



νd = fd Ts = fcvc0

Ts (28)






10.10.01

4

3.75

3.5

3.25

3

2.75

2.5

2.25

2

51551.55.15


Thr

ough

put [

Mbi

t/s]



instantaneous CQI


instantaneous CQIRLS prediction

long-term CQIadaptive averaging window size



νd = fd Ts = fcvc0

Ts (28)





Performance Investigation – Rank Adaptation

4035302520151050-5-10

18

16

14

12

10

8

6

4

2

0

SNR [dB]

Thr

ough

put [

Mbi

t/s]

4x4_1.4MHz_VehA_corr0

rank 1

rank 2

rank 3rank 4

rank adaptive

4035302520151050-5-10

16

14

12

10

8

6

4

2

0

SNR [dB]

Thr

ough

put [

Mbi

t/s]

4x4_1.4MHz_VehA_corr0.9

rank adaptive

rank 4

rank 3

rank 2

rank 1

Comparison of fixed rank and rank adaptive transmission over 1.4MHz system bandwidth

Rank adaptation versus fixed rank transmission for spatially uncorrelated channel

Rank adaptation versus fixed rank transmission for spatially correlated channel

E(

vec (H) vec (H)H)

=

1 α

1/9corr α

4/9corr αcorr

α1/9corr 1 α

1/9corr α

4/9corr

α4/9corr α

1/9corr 1 α

1/9corr

αcorr α4/9corr α

1/9corr 1

⊗ INt , αcorr = 0.9

(29)


Performance Investigation – Precoder Subband Size

20151050-5-10

45

40

35

30

25

20

15

10

5

0

SNR [dB]

Thr

ough

put [

Mbi

t/s]

8x1_10MHz_TU_8bit

MRTMRT

subband size = 600 subband size = 600

Impact of the subband size with Nt × Nr = 8× 1, 10 MHz bandwidth, 400 kHz coherence bandwidth and precodercodebook size of 8 bit

Performance with a single feedback subband (600 subcarriers) compared tomaximum ratio transmission (MRT) on each subcarrier

Performance with a subband size of 300 subcarriers



Performance with a subband size of 12 subcarriers (one RB = 180 kHz)



20151050-5-10

45

40

35

30

25

20

15

10

5

0

SNR [dB]

Thr

ough

put [

Mbi

t/s]

8x1_10MHz_TU_8bit

MRTMRT











20151050-5-10

45

40

35

30

25

20

15

10

5

0

SNR [dB]

Thr

ough

put [

Mbi

t/s]

8x1_10MHz_TU_8bit

MRTMRT



subband size = 120 subband size = 120 subband size = 60 subband size = 60








Performance Investigation – Precoder Codebook Size

20151050-5-10

45

40

35

30

25

20

15

10

5

0

SNR [dB]

Thr

ough

put [

Mbi

t/s]

8x1_10MHz_VehA_50clusters

MRTMRT

1 bit1 bit

~9 dB~9 dB

Impact of the codebook size with Nt × Nr = 8× 1, 10 MHz bandwidth, 550 kHz coherence bandwidth and subbandsize of 180 kHz

Performance with a codebook size of 1 bit compared to maximum ratiotransmission (MRT) on each subcarrier

Performance with a codebook size of 2 bit


Performance with a codebook size of 8 bit (LTE codebook)




20151050-5-10

45

40

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2 bit2 bit

1 bit1 bit

~9 dB~9 dB









20151050-5-10

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~9 dB~9 dB









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8 bit8 bit4 bit4 bit

2 bit2 bit

1 bit1 bit

~9 dB~9 dB









20151050-5-10

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MRTMRT

16 bit16 bit

8 bit8 bit4 bit4 bit

2 bit2 bit

1 bit1 bit

~9 dB~9 dB








Contents

1 Motivation






Slide 37 / 89 Multi-User MIMO Feedback for LTE

Downlink Multi-User MIMO in LTE

yu = HHu fuxu + HH

u

U∑j=1j 6=u

fj xj + zu (30)

Basic multi-user MIMO already supported in Rel. 8 (transmission mode 5)

Restricted to two transmit antennas and single stream per user

Codebook based precoding using the CLSM codebook

In general, large residual multi-user interference

⇒ per user unitary rate control (PU2RC)

Extended multi-user MIMO support > Rel. 9 (modes 8, 9)

Non-codebook based precoding

Enables more sophisticated transceivers

Performance restricted by low accuracy of CSIT⇒ interference

⇒ zero forcing (ZF) beamforming


Per User Unitary Rate Control

Precoders are restricted to be selected from a predefined codebook

Codebook for transmission of L streams from Nt antennas

Q(Nt )L =

{Fi ∈ CNt×L∣∣FH

i Fi = IL, i ∈ {1, . . . , np}}⊂ G (Nt , L) , (31)

yu = HHu

√PL

Fx + zu = HHu [f1, . . . , fL] x + zu (32)

Each column is assigned to serve a different user

E.g., column ν is assigned to user u

yu = HHu

√PL

fνxu + HHu

√PL

L∑µ=1µ6=ν

fµxµ + zu , (33)


Per User Unitary Rate Control (2)

Optimal linear receiver: interference-aware MMSE

gu =

σ2z IMu +

PL

HHu

L∑µ=1µ6=ν

fµfHµHu

−1

PL

HHu fν (34)

Can be calculated because the precoders are restricted to the codebook

Post-equalization SINR

SINRu =PL

∣∣gHu HH

u fν∣∣2

PL∑Lµ=1, µ 6=ν

∣∣gHu HH

u fµ∣∣2 + σ2

z ‖gu‖2(35)


Per User Unitary Rate Control – CSI Feedback

CSI feedback calculation at user u:

For each Fi ∈ Q(Nt )L , find the best column ν in terms of SINR

⇒ np =∣∣∣Q(Ni )

L

∣∣∣ potential beamformers

Out of the np potential beamformers, feedback the n` best performers

Feedback overhead:

n∑`=1

dlog2 (np − (`− 1))e︸︷︷︸selected precoder

+ log2 (L)︸︷︷︸selected column

(36)

Feedback the corresponding SINRs as CQI


Per User Unitary Rate Control – Scheduling

Multi-user scheduling and precoder selection:

Find compatible user sets Sj :

Same Fi ∈ Q(Ni )L selected

Different column fν from Fi chosen

Determine the best compatible user set , e.g., to maximize sum rate

Rj =∑u∈Sj

log2 (1 + SINRu) (37)

Difficulty of PU2RC: selection of feedback parameter n`, np , L

Large np : + find a good precoder; − compatible user sets shrink

Large n`: + find a good compatible user set; − linear increase in overhead

Large L: + large multiplexing gain; − large multi-user interference


ZF Beamforming

Consider for simplicity Mu = Nr = 1

yu =√

puhHu fuxu + hH

u

∑i∈S,i 6=u

√pi fi xi + zu , (38)

S . . . set of scheduled userspj = P

|S| ‖fj‖2 for equal power allocation

SINR of user u

SINRu =pu |hH

u fu |2

σ2z +

∑i∈S\{u} pi |hH

u fi |2(39)

Problem: cannot be estimated accurately by the users during feedbackcalculation as precoders are not yet known


ZF Beamforming with Limited Feedback

To calculate the ZF beamformer the transmitter requires the normalizedchannel vectors (see multi-user MIMO lecture)

hu =hu

‖hu‖(40)

Channel vector quantization

hu = q(hu) =⇒ hu = hu hHu hu + (I− hu hH

u )hu = cos θuejϕu hu + eu , (41)

eu = sin θuejψu eu

θu . . . principle angle between span(

hu

)and span

(hu

)SINR in terms of hu

SINRu =pu ‖hu‖2 ‖fu‖2

∣∣∣cos θue−jϕu hHu fu + eH

u fu

∣∣∣2σ2

z + ‖hu‖2 sin θu2∑

i∈S\{u} pi∣∣eH

u fi∣∣2 , (42)

hHu fi = 0 and

∣∣∣hHu fu

∣∣∣2 =1

‖fu‖2 due to ZF onto hu


ZF Beamforming with Limited Feedback (2)

SINR lower bound assuming eHu fu ≈ 0

SINRu ≥pu ‖hu‖2 cos θu

2

σ2z + ‖hu‖2 sin θu

2∑i∈S\{u} pi

∣∣eHu fi∣∣2 (43)

Tight when U →∞ because orthogonal users are scheduled: fu ⊥ eu

During feedback calculation fi unknown

⇒ consider the expected value of the SINR and apply Jensen’s inequality

E (SINRu) ≥pu ‖hu‖2 cos θu

2

σ2z + ‖hu‖2 sin θu

2E(∑

i∈S\{u}P

|S|‖fi‖2 ‖fi‖2∣∣∣eH

u fi

∣∣∣2) (44)


ZF Beamforming with Limited Feedback (3)

Assuming fi isotropically distributed in the Nt − 1 dimensional null-space of hu

E(∣∣∣eH

u fi

∣∣∣2) =1

Nt − 1, (45)

E (SINRu) ≥pu ‖hu‖2 cos θu

2

σ2z + P

|S||S|−1Nt−1 ‖hu‖2 sin θu

2(46)

⇒ Feedback ‖hu‖2 and cos θu2 as two separate CQIs [Trivellato et al., 2007]

Further simplification: assume |S| = Nt (worst-case interference)

E (SINRu) ≥PNt‖hu‖2 cos θu

2

σ2z + P

Nt‖hu‖2 sin θu

2= CQIu , (47)

ˆSINRu = CQIuNt

|S| ‖fu‖2 (48)


ZF Beamforming – Scheduling

Exhaustive search: estimate the achievable rate of all possible usercombinations

Consider set Sk = {s1, . . . , sK }

1 Calculate the ZF beamformers for Sk (see multi-user MIMO lecture)

2 Estimate the achievable rate

ˆSINRsi = CQIsi

Nt

|Sk |∥∥fsi

∥∥2 , (49)

RSk =∑

si∈Sk

log2

(1 + ˆSINRsi

)(50)

3 Select the set with maximal estimated rate

Suboptimal greedy scheduling: see, e.g., [Trivellato et al., 2007]


Multi-User MIMO in LTE

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flatRayleigh_1.4MHz_multi-user_spatial_multiplexing

ZF beamformingperfect CSIT

8 × 1

8 × 4PU2RC

PU2RC

ZF beamforming

CLSM

CLSM

ZF beamforming

Comparison of ZF beamforming, PU2RC and LTE’s CLSM single-user mode.

Nt × Nr ∈ {8× 4, 8× 1} with B = 8 bit/TTI of feedback and U = 20 users

Interference-aware MMSE receiver

Subspace selection for ZF feedback [Schwarz and Rupp, 2014a]

Similar performance of ZF beamforming and CLSM based single-user MIMO


Contents

1 Motivation






Slide 49 / 89 Multi-User MIMO with Channel Subspace Feedback

Recap: Downlink Multi-User MIMO

Remember the downlink multi-user MIMO input-output relationship

yu = GHu HH

u Fu xu︸︷︷︸intended signal

+ GHu HH

u

∑s∈Ss 6=u

Fs xs

︸︷︷︸interference

+ GHu zu︸︷︷︸

noise

(51)

Channel matrix Hu ∈ CNt×Nr ,

Linear transceivers Gu ∈ CNr×L, Fu ∈ CNt×L

Effective channel Heffu = HuGu ∈ CNt×L

We consider the case Mu = Nr , ∀u and Nr ≤ Nt


Recap: Transceiver Design

Assume the schedule S to be given

Optimal transceiver: dirty paper coding [Costa, 1983]

Vector-perturbation precoding [Hochwald et al., 2005]

Tomlinson-Harashima precoding [Mezghani et al., 2006]

Disadvantage: complexity

Practically more relevant: linear transceivers

Block-diagonalization precoding [Spencer et al., 2004]

Iterative joint optimization, e.g., based on MMSEcriteria [Shi et al., 2008]


Considered Transceiver Architecture

Problems of iterative approaches:

Large signalling overhead

Slow convergence

We consider non-iterative linear transceiver designs:

Selfish selection of Gu [Schwarz and Rupp, 2013]

Block-diagonalization precoding at base station

Selection of S based on achievable rate estimate[Schwarz and Rupp, 2014a]

Advantages of this approach:

Reduced computational complexity (closed-form solutions)

Decreased signalling overhead when L < Nr

Heffu = HuGu ∈ CNt×L versus Hu ∈ CNt×Nr (52)


Recap: Block-Diagonalization (BD) Precoding

Assume for now Gu as given and S = {1, . . . ,S}

yu =(Heff

u)H Fu xu +

(Heff

u)H S∑

s=1s 6=u

Fs xs + GHu zu

Goal of BD precoding: eliminate multi-user interference(Heff

s)HFu = 0, ∀s, u ∈ S and s 6= u, (53)

rank((

Heffu)HFu

)= L, ∀u ∈ S (54)

This can be achieved by selecting the precoders as follows ∀u ∈ S

Hu =[Heff

1 , . . . ,Heffu−1,H

effu+1, . . . ,H

effS

]H∈ C(S−1)L×Nt ,

Fu ∈ null(Hu), rank (Fu) = L (55)


Channel Subspace Quantization – Grassmannian Feedback

Notice, Heffj can be replaced with any matrix spanning the same subspace

Heffj ≡ Hj ∈ CNt×L ⇐⇒ span

(Heff

j

)= span

(Hj

), (56)(

Heffj)HFu = 0⇐⇒ HH

j Fu = 0 (57)

⇒ the users have to convey span(

Heffj

)∈ G (Nt , L) to the base station

Precoding is based on channel subspace information

Grassmannian quantization for limited feedback operation

Hj = argminQi∈Q

d2c

(Heff

j ,Qi

)= argmin

Qi∈QL− tr

(HH

j Qi QHi Hj

), (58)

Q ={

Qi ∈ CNt×L ∣∣ QHi Qi = IL, i ∈ {1, . . . , 2B}

}


Why Chordal Distance Quantization?

Achievable rate with perfect CSIT

RBD = E log2

∣∣∣IL + ρ(

Heffu

)HFuFH

u Heffu

∣∣∣ , ρ =P

σ2z S L

. (59)

assuming Gu to be semi-unitary GHu Gu = IL

and equal power allocation FHu Fu = IL

Achievable rate with limited feedback

RBD-Quant = E log2

∣∣∣∣∣∣IL + ρS∑`=1

(Heff

u

)HF`FH

`Heffu

∣∣∣∣∣∣−E log2

∣∣∣∣∣∣IL + ρS∑

`=1, 6=u

(Heff

u

)HF`FH

`Heffu

∣∣∣∣∣∣ (60)

Interference only over the null-space component due to BD

Heffu = HuHH

u Heffu +

(INt − HuHH

u

)Heff

u = HuHHu Heff

u + H⊥u(

H⊥u)

H Heffu , (61)


Why Chordal Distance Quantization? (2)

The rate loss RBD − RBD-Quant is determined by d2c

(Heff

u , Hu

)Upper bounds on the expected rate loss for iid Rayleigh fading

Rate loss of ZF beamforming with L = Nr = 1 [Jindal, 2006]

RZF − RZF-Quant ≤ log2

(1 +

Pσ2

nD), D = 2

− BNt−1 , (62)

Rate loss of BD precoding with L = Nr [Ravindran and Jindal, 2008]

RBD − RBD-Quant ≤ Nr log2

(1 +

Pσ2

n NrD), (63)

D = CBD 2− B

Nr (Nt−Nr ) .

D average chordal distance distortion with random isotropic codebooks


Memoryless Grassmannian Quantization Codebooks

Codebooks for quantization of the full Nr dimensional subspace (Gu = I)

Random isotropic codebook for isotropic channels (e.g., iid Rayleigh fading)

[Hu ]m,n ∼ NC (0, 1) ∈ CNt×Nr ,

Q(iso)u =

{Q(iso)

j

∣∣∣∣Q(iso)j ΣVH = Qj , Qj ∈ CNt×Nr , [Qj ]m,n ∼ NC (0, 1)

}, (64)

Optimal codebook: maximally spaced subspace packing

Random codebook: asymptotically optimal in the codebook size⇒ random vector quantization (RVQ)

Random correlated codebook for spatially correlated channels

Hu = Γ1/2u Hu , [Hu ]m,n ∼ NC (0, 1) ∈ CNt×Nr ,

Q(corr)u =

{Q(corr)

j

∣∣∣∣Q(corr)j ΣVH = Γ

1/2u Qj , Qj ∈ CNt×Mu , [Qj ]m,n ∼ NC (0, 1)

},

The codebook has the same distribution as the channel

Receive-side correlation does not impact the subspace distribution!


Memoryless Grassmannian Quantization – Performance

10.80.60.40.20

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Correlation coefficient

Cho

rdal

dis

tanc

e M

SE

Memoryless_Grassmannian_quantization

4 × 2 (6 bit)

correlated codebook

isotropic codebook

8 × 2 (10 bit)

isotropic codebook

correlated codebook

Comparison of isotropic and correlated codebooks in dependence of the channel correlation.

Consider Nt × Nr = 8× 2 with B = 10 bit and Nt × Nr = 4× 2 with B = 6 bit

Kronecker correlation model

Γu =

1 αcorr . . . αcorr

αcorr 1 . . . αcorr...

. . ....

αcorr . . . αcorr 1

(65)


Predictive Grassmannian Quantization [Schwarz et al., 2013]

hk-1hk-2

hk

1D Grassmannian

Illustration of Grassmannian predictive quantization.

Exploit temporal correlation of the channel

Predict current channel subspace from previously quantized subspaces⇒ Grassmannian prediction exploiting geodesics

Generate a local codebook around the prediction

Volume covered by local codebook depends on prediction accuracy



hk-1hk-2

hk-2

hk-1

hk

^

^

1D Grassmannian








hk-1hk-2

hk-2

hk-1

hp,k

hk

^

^

1D Grassmannian








minimumchordal distance

quantization

Quantization

H[n,k] H[n,k]^ codebookindex extraction

codebook index

scale index

codebookgeneration

Adaptive codebook generation

H(p)[n,k]

subspaceprediction

~

codebooks

Encoder Decoder

feedbackchannel

subspacereconstruction

adaptivecodebookgeneration

H[n,k]^

codebooks

Structure of predictive quantization.






Predictive Grassmannian Quantization – MSE Performance

100

10-1

10-2

10-3

10-4

10-2

10-5

10-6

5·10-3


Cho

rdal

dis

tanc

e M

SE

95% confidence interval95% confidence interval 4x1_1stream_flat

10-1 0.5

4 bit

8 bit

robust prediction (Zhang)adaptive prediction (Schwarz)

differential (El Ayach)

10-110-210-3

100

10-1

10-2

10-3

10-4

10-6

10-5


Cho

rdal

dis

tanc

e M

SE

95% confidence interval95% confidence interval 8x2_2stream_flat

7 bit

11 bit

predictive quant. (vector)predictive quant. (matrix)

differential quant.

Performance of differential and predictive Grassmannian quantization (4× 1 and 8× 2).

Nt × Nr ∈ {4× 2, 8× 2} with varying speed (Doppler frequency)

Differential quantization [Ayach and Heath, Jr., 2011]

Robust predictive quantization [Zhang and Lei, 2012]

Adaptive predictive quantization [Schwarz et al., 2013]


Predictive Grassmannian Quantization – Throughput Performance

302724211815129630

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10

8

6

4

2

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SNR [dB]

Sum

thro

ughp

ut [M

bit/s

]

2UE_10Hz_AR_4x2_2streams

RSQ 8 bit

SU-MIMO 4 bit

Perfect CSI

ACSQ 8 bit

ACSQ 3 bit

ACSQ 2 bit

302724211815129630

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SNR [dB]

Sum

thro

ughp

ut [M

bit/s

]

Perfect CSI

4UE_10Hz_8x2_2streams

ACSQ 11 bit

ACSQ 9 bit

ACSQ 7 bit

SU-MIMO 8 bit

RSQ 11 bitACSQ 5 bit

Throughput with predictive Grassmannian quantization (4× 2 and 8× 2).

Nt × Nr ∈ {4× 2, 8× 2} at low mobility νd = 0.01 (walking speed at 1 GHz)

Memoryless quantization (RSQ) versus predictive quantization (ACSQ)

Notice: no scheduling applied→ always 2 (resp. 4) users served in parallel

Single-user MIMO using LTE’s CLSM mode with best CQI scheduler


Other Subspace Transceivers

Single-user MIMO with unitary precoding (equal power allocation)

Interference alignment [Cadambe and Jafar, 2008, Maddah-Ali et al., 2008]

Rate-loss of interference alignment with quantized CSIT is also determined bythe chordal distance quantization error [Rezaee and Guillaud, 2012]


Contents

1 Motivation






Slide 63 / 89 Multi-User MIMO with Channel Gramian Feedback

Recap: Regularized Block Diagonalization (RBD) Precoding

Precoder calculation consist of two parts (see multi-user MIMO lecture):

Trade-off residual interference and noise through MMSE precoding

Optimize transmission over effective single-user channel treating residualinterference as noise through SVD precoding

Necessary CSI at the transmitter [Schwarz and Rupp, 2014b]

Hu = UuΣuVHu ∈ CNt×Nr , (66)

Individual columns of Uu ,

Singular values in Σu

Both can be obtained from an eigen-decomposition of the channel Gramian

Ru = HuHHu = UuΣ2

uUHu (67)


Channel Gramian Quantization – Stiefel Manifold Feedback

Two possibilities for CSI feedback

Direct quantization of the Gramian Ru [Sacristan-Murga et al., 2012]

Separate quantization of Uu and diag (Σu) [Schwarz and Rupp, 2014b]

Separate quantization:

Quantization of Uu ∈ St (Nt ,Nr ) on the Stiefel manifold

d2s (U,Qi ) =

Nr∑j=1

d2c

(uj ,q

(j)i

)∈ [0,Nr ], q(j)

i = [Qi ]:,j (68)

d2c (A,B) = Nr − tr

(AHBBHA

), AHA = BHB = INr ,

Q ={

Qi ∈ St (Nt ,Nr )∣∣ i ∈ {1, . . . , 2B}

}(69)

Quantization of singular values

σu = argminsi∈Su

‖σu − si‖22 , (70)

σu =[[Σu ](1,1), . . . , [Σu ](Nr ,Nr )

]T (71)


Singular Value Codebook Su

7654321

5

4

3

2

1

0

First singular value

Seco

nd s

ingu

lar v

alue

8x2_uncorrelated_SV_CB

7654321

5

4

3

2

1

0

First singular value

Seco

nd s

ingu

lar v

alue

8x2_correlated_SV_CB

Training set

Codebook

Training sets and Lloyd codebooks for singular value quantization with uncorrelated and correlated receive antennas.

Codebook optimized using Lloyd’s algorithm (k-means clustering) [Lloyd, 1982]

Nt × Nr = 8× 2 with uncorrelated and correlated receive antennas (αcorr = 0.9)

More accurate quantization with increasing correlation

⇒ Exploit correlation!


Quantization on the Stiefel Manifold

Memoryless quantization:

Optimal: maximally spaced subspace packings on the Stiefel manifold

Random: same codebooks as with Grassmannian quantization

Different quantization metrics!

⇒ seamless transition between subspace and Gramian quantization

Predictive quantization:

Same principle applicable as with subspace quantization

Just replace manifold calculations appropriately

[Schwarz and Rupp, 2015b]


Predictive Quantization on the Stiefel Manifold


Cho

rdal

dis

tanc

e M

SE8x2_2stream_flat_corr0

11 bit

7 bit

predictive quant. differential quant.memoryless quant.

100

10-1

10-2

10-3

10-4

10-6

10-5

10-110-210-3


Cho

rdal

dis

tanc

e M

SE

8x2_2stream_flat_corr0.9

100

10-1

10-2

10-3

10-4

10-6

10-5

10-110-210-3

11 bit

7 bit

predictive quant. differential quant.memoryless quant.

Performance of Stiefel manifold quantization with uncorrelated and correlated receive antennas.

Nt × Nr = 8× 2 with varying speed (Doppler frequency)

Substantial gain of differential/predictive quantization over memoryless scheme

Larger slope of MSE curve of predictive quantization

MSE improvement with correlated receive antennas (αcorr = 0.9)

Notice: the prediction order needs to be adapted to the speed


BD versus RBD Precoding with Limited Feedback

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SNR [dB]

6x2_10Hz_MU-MIMO_RXcorr0.995% confidence interval95% confidence interval

BD quantized CSITBD perfect CSIT

RBD quantized CSITRBD perfect CSIT

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10

5

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SNR [dB]

6x2_10Hz_MU-MIMO_RXcorr0

6 bit

6 bit

10 bit

10 bit A

chie

vabl

e su

m ra

te [b

its/c

hann

el u

se]

6 bit 6 bit

10 bit 10 bit


RBD quantized CSITRBD perfect CSIT

95% confidence interval95% confidence interval

Comparison of the achievable rate of block-diagonalization and regularized block-diagonalization precoding.

Nt × Nr = 6× 2 at low mobility νd = 0.01 (walking speed at 1 GHz)

Negligible overhead for singular value quantization: 4 bit/10 TTI

Uncorrelated and strongly correlated receive antennas: αcorr ∈ {0, 0.9}

⇒ Switch between the two schemes depending on SNR


BD versus RBD Precoding with Limited Feedback

302520151050

100

90

80

70

60

50

40

30

SNR [dB]

Rel

ativ

e su

m ra

te [%

]6x2_10Hz_MU-MIMO_RXcorr0

302520151050

100

90

80

70

60

50

40

30

SNR [dB]

6x2_10Hz_MU-MIMO_RXcorr0.9

6 bit

6 bit

10 bit

10 bit


RBD quantized CSIT

6 bit

6 bit

10 bit

10 bit

Comparison of the rate relative to regularized block-diagonalization precoding with perfect CSIT.

Nt × Nr = 6× 2 at low mobility νd = 0.01 (walking speed at 1 GHz)

Negligible overhead for singular value quantization: 4 bit/10 TTI

Uncorrelated and strongly correlated receive antennas: αcorr ∈ {0, 0.9}

⇒ Switch between the two schemes depending on SNR


Other Gramian-Based Transceivers

Single-user MIMO with water-filling power allocation

MMSE based precoding schemes (iterative)

Interference leakage based schemes, e.g., signal to leakage and noise ratio(SLNR) beamforming

SLNRu =

∥∥HHu fu∥∥2

F

Nr σ2z +

∑j 6=u

∥∥∥HHj fu

∥∥∥2

F

(72)


Contents

1 Motivation






Slide 71 / 89 Feedback Overhead Reduction through Excess Antennas

Receive Antenna Combining – Maximum Eigenmode Transmission

Excess receive antennas: Nr > L

Heffu = HuGu ∈ CNt×L (73)

Minimize feedback overhead: selfishly select Gu and quantize Heffu

⇒ How to select Gu?

Maximum eigenmode transmission (MET)

G(MET)u = [Vu ]:,1:L , (74)

Heffu = [Uu ]:,1:L [Σu ]1:L,1:L

Achieves maximum rate in the absence of interference


MET with Quantized CSIT [Schwarz and Rupp, 2013]

Performance of MET combining with BD precoding, L streams per user, S usersand B bit Grassmannian quantization (iid Rayleigh fading and RVQ))

RMET − RMET-Quant ≤L∑`=1

log2

(1 + ρ σ2

`,uS − 1Nt − L

D), ρ =

Pσ2

z S L

σ2`,u expected value of `th eigenvalue ofWC

Nr

(Nt , INr

)D = CMET 2

− BL (Nt−L) average quantization distortion with RVQ

With fixed D the rate loss grows with SNR⇒ interference limitation

Avoid interference limitation:

B ∝ log (ρ) L(Nt − L) =⇒ 2− B

L (Nt−L) ∝1ρ

(75)


Subspace Quantization Based Combining [Schwarz and Rupp, 2013]

Select the effective channel to minimize the quantization error{G(SQBC)

u , H(SQBC)u

}= argmin

G,Qj

d2c

(Heff

u ,Qj

)= argmin

G,Qj

d2c(HuG,Qj

)H(SQBC)

u = argminQj∈Q

(Nt )

L

d2c(Hu ,Qj

), (76)

G(SQBC)u =

(HH

u Hu)−1 HH

u H(SQBC)u (77)

Rate loss with respect to perfect CSIT (same assumptions as before)

R(L)BD − R(L,Nr )

SQBC ≤ L log2

(1 + ρ

Nt − Nr + L

Nt − L(S − 1) D

)+ log2 (e)

L−1∑k=0

Nt−1∑`=Nt−Nr +L

1

`− k

D = CSQBC 2− B

L (Nt−Nr ) average quantization distortion with RVQ

Bit-scaling law: B ∝ log (ρ) L(Nt − Nr )


MET versus SQBC

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SQBC MET

Nr = 2


Nr = 5

Nr = 5

Achievable rate with perfect CSIT

dSQBC(L,Nr)

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SQBC MET Nr = 2

Nr = 5

Nr = 5

L (Nt-L)

L (Nt-Nr)

Feedback bit-scaling to achieve a loss of 1 bit/s/Hz

Nt = 6 transmit antennas, Nr ∈ {2, . . . , 5} receive antennas, L = 2 data streams

Throughput loss of SQBC with perfect CSI at the base station

Significant reduction of feedback overhead with SQBC for moderate SNR loss


SQBC Performance

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SQBC quant. CSIT SQBC perfect CSIT


Nr = 2

Nr = 4

Nr = 2

Nr = 3

Nr = 4

Nr = 5

Achievable rate with feedback bits scaled to achieve a loss of 1 bit/s/Hz with Nr = 5.


Feedback overhead growing from 0 to 17 bits per TTI

Significant throughput gain with growing Nr at same feedback overhead

Substantial gain over MET at same overhead


SQBC Performance

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MET quant. CSIT Nr = 5


Nr = 2

Nr = 4

Nr = 2

Nr = 3

Nr = 4

Nr = 5

Achievable rate with feedback bits scaled to achieve a loss of 1 bit/s/Hz with Nr = 5.


Feedback overhead growing from 0 to 17 bits per TTI

Significant throughput gain with growing Nr at same feedback overhead

Substantial gain over MET at same overhead


SQBC Performance (2)

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SQBC quant. CSIT (scaled bits)SQBC perfect CSIT

3 streams per user

2 streams per user

1 stream per user


Achievable rate with different number of streams per user.

Nt = 6 transmit antennas, Nr = 5 receive antennas, L ∈ {1, 2, 3} data streams

Feedback overhead per user

L = 1, U = 6: B ∈ [0, 8] bits per TTI

L = 2, U = 3: B ∈ [0, 16.1] bits per TTI

L = 3, U = 2: B ∈ [0, 18.3] bits per TTI


Maximum Expected Achievable Rate Combining (MERC)

Exploit the BD construction to further improve performance⇒ Precoding-specific combiner [Schwarz and Rupp, 2015a]:

Estimate the expected achievable rate with BD precoding, antennacombiner G and quantized channel subspace Qi : Ru(Qi ,G)

Expectation over input covariance matrices

Interference lies in the null-space (INt − Qi QHi )

MERC optimization and quantization problem:{Hu ,G

(MERC)u

}= argmax

Qi∈Q(Nt )

L ,G∈CNr ×L

Ru(Qi ,G), (78)

Hu = argminQi∈Q

(Nt )

L

log2 det(σ2

z INr +1Nt

HHu (INt − Qi QH

i )Hu

), (79)

G(MERC,1)u =

(σ2

z INr +1Nt

HHu (INt−HuHH

u )Hu

)−1HH

u Hu (80)

MMSE solution versus ZF as with SQBC


MERC Performance (1)

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MERCMET perfect CSITSQBC

Achievable rate with interference unaware antenna combining.

Nt × Nr = 4× 2, L = 1 and B = 10 bits per TTI with αcorr = 0.9

Comparison of MET, SQBC and MERC feedback and antenna combining⇒ MERC performs equal to MET and SQBC at low and high SNR, resp.

MET, SQBC and MERC feedback with interference-aware MMSE combining⇒ Overall better performance; difference reduces



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MET

MERCSQBC

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Comparison of MET, SQBC and MERC feedback and antenna combining⇒ MERC performs equal to MET and SQBC at low and high SNR, resp.




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MET perfect CSIT

Achievable rate with interference unaware antenna combining.


Comparison of MET, SQBC and MERC feedback and antenna combining⇒ MERC strictly outperforms MET and SQBC




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Comparison of MET, SQBC and MERC feedback and antenna combining⇒ MERC strictly outperforms MET and SQBC



Limited Feedback for Single- and Multi-User MIMO389.168 Advanced Wireless Communications 1

[email protected]

Abbreviations I

AMC adaptive modulation and coding

AWGN additive white Gaussian noise

BD block diagonalization

BICM bit-interleaved coded-modulation

BLER block error ratio

CLSM closed loop spatial multiplexing

CoMP coordinated multipoint transmission/reception

CP cyclic prefix

CQI channel quality indicator

CSI channel state information

CSIT channel state information at the transmitter

EESM exponential effective SNR mapping

ESM effective SNR mapping

FDD frequency division duplex

LTE long term evolution

MCS modulation and coding scheme

MERC maximum expected achievable rate combining

MET maximum eigenmode transmission

Slide 82 / 89 Abbreviations

Abbreviations IIMIESM mutual information effective SNR mappingMIMO multiple-input multiple-output

ML maximum likelihoodMMSE minimum mean squared error

MRT maximum ratio transmissionOFDM orthogonal frequency division multiplexingOLSM open loop spatial multiplexing

PMI precoding matrix indicatorPU2RC per user unitary rate control

RB resource blockRBD regularized block diagonalization

RE resource elementRI rank indicator

RVQ random vector quantizationSINR signal to interference and noise ratioSISO single-input single-outputSLNR signal to leakage and noise ratio

SNR signal to noise ratioSQBC subspace quantization based combining

SVD singular value decompositionTDD time division duplex

ZF zero forcing

Slide 83 / 89 Abbreviations

References I

3GPP (2009).Technical Specification Group Radio Access Network; Evolved Universal Terrestrial Radio Access(E-UTRA); Physical Channels and Modulation (Release 8).[Online]. Available: http://www.3gpp.org/ftp/Specs/html-info/36211.htm.

3GPP (2010).Technical Specification Group Radio Access Network; Evolved Universal Terrestrial Radio Access(E-UTRA); Physical Channels and Modulation (Release 10).[Online]. Available: http://www.3gpp.org/ftp/Specs/html-info/36211.htm.

Ayach, O. E. and Heath, Jr., R. (2011).Grassmannian differential limited feedback for interference alignment.CoRR, abs/1111.4596.

Cadambe, V. and Jafar, S. (2008).Interference alignment and degrees of freedom of the K-user interference channel.IEEE Transactions on Information Theory, 54(8):3425 –3441.

Cipriano, A., Visoz, R., and Salzer, T. (2008).Calibration issues of PHY layer abstractions for wireless broadband systems.In IEEE 68th Vehicular Technology Conference, pages 1–5, Calgary, Alberta.

Costa, M. (1983).Writing on dirty paper (corresp.).IEEE Transactions on Information Theory, 29(3):439 – 441.

Dhillon, I. S., Heath, Jr., R., Strohmer, T., and Tropp, J. A. (2007).Constructing packings in Grassmannian manifolds via alternating projection.ArXiv e-prints.

Slide 84 / 89 References

References II

He, X., Niu, K., He, Z., and Lin, J. (2007).Link layer abstraction in MIMO-OFDM system.In International Workshop on Cross Layer Design, pages 41–44.

Hochwald, B., Peel, C., and Swindlehurst, A. (2005).A vector-perturbation technique for near-capacity multiantenna multiuser communication-part II:perturbation.IEEE Transactions on Communications, 53(3):537–544.

Jindal, N. (2006).MIMO broadcast channels with finite-rate feedback.IEEE Transactions on Information Theory, 52(11):5.

Lloyd, S. (1982).Least squares quantization in PCM.IEEE Transactions on Information Theory, 28(2):129–137.

Love, D. (2006).Grassmannian subspace packing.https://engineering.purdue.edu/˜djlove/grass.html.

Love, D. and Heath, Jr., R. (2005).Limited feedback unitary precoding for spatial multiplexing systems.IEEE Transactions on Information Theory, 51(8):2967–2976.

Maddah-Ali, M., Motahari, A., and Khandani, A. (2008).Communication over MIMO X channels: Interference alignment, decomposition, and performance analysis.IEEE Transactions on Information Theory, 54(8):3457 –3470.


https://engineering.purdue.edu/~djlove/grass.html

References III

Mezghani, A., Hunger, R., Joham, M., and Utschick, W. (2006).Iterative THP transceiver optimization for multi-user MIMOsystems based on weighted sum-MSEminimization.In IEEE 7th Workshop on Signal Processing Advances in Wireless Communications, pages 1–5.

Ravindran, N. and Jindal, N. (2008).Limited feedback-based block diagonalization for the MIMO broadcast channel.IEEE Journal on Selected Areas in Communications, 26(8):1473 –1482.

Rezaee, M. and Guillaud, M. (2012).Interference alignment with quantized Grassmannian feedback in the K-user MIMO interference channel.CoRR, abs/1207.6902.

Ryan, D., Vaughan, I., Clarkson, L., Collings, I., Guo, D., and Honig, M. (2007).QAM codebooks for low-complexity limited feedback MIMO beamforming.In IEEE International Conference on Communications, pages 4162–4167, Glasgow, Scotland.

Sacristan-Murga, D., Payaro, M., and Pascual-Iserte, A. (2012).Transceiver design framework for multiuser MIMO-OFDM broadcast systems with channel Gram matrixfeedback.IEEE Transactions on Wireless Communications, 11(5):1774–1787.

Sandanalakshmi, R., Palanivelu, T. G., and Manivannan, K. (2007).Effective SNR mapping for link error prediction in OFDM based systems.In International Conference on Information and Communication Technology in Electrical Sciences, pages684–687.

Schwarz, S., Heath, Jr., R., and Rupp, M. (2013).Adaptive quantization on a Grassmann-manifold for limited feedback beamforming systems.IEEE Transactions on Signal Processing, 61(18):4450–4462.


References IV

Schwarz, S., Mehlfuhrer, C., and Rupp, M. (2010).Low complexity approximate maximum throughput scheduling for LTE.In Conference Record of the Forty Fourth Asilomar Conference on Signals, Systems, and Computers, pages1563–1569, Pacific Grove, California.

Schwarz, S., Mehlfuhrer, C., and Rupp, M. (2011).Throughput maximizing multiuser scheduling with adjustable fairness.In International Conference on Communications ICC 2011, Kyoto, Japan.

Schwarz, S. and Rupp, M. (2011).Throughput maximizing feedback for MIMO OFDM based wireless communication systems.In Signal Processing Advances in Wireless Communications SPAWC 2011, pages 316–320, San Francisco,CA.

Schwarz, S. and Rupp, M. (2013).Subspace quantization based combining for limited feedback block-diagonalization.IEEE Transactions on Wireless Communications, 12(11):5868–5879.

Schwarz, S. and Rupp, M. (2014a).Evaluation of distributed multi-user MIMO-OFDM with limited feedback.IEEE Transactions on Wireless Communications, 13(11):6081–6094.

Schwarz, S. and Rupp, M. (2014b).Subspace versus eigenmode quantization for limited feedback block-diagonalization.In 6th International Symposium on Communications, control and signal processing, pages 1–4.

Schwarz, S. and Rupp, M. (2015a).Maximum expected achievable rate combining for limited feedback block-diagonalization.Submitted to ICASSP 2015.


References V

Schwarz, S. and Rupp, M. (2015b).Predictive quantization on the Stiefel manifold.IEEE Signal Processing Letter, 22(2):234–238.

Shi, S., Schubert, M., and Boche, H. (2008).Downlink MMSE transceiver optimization for multiuser MIMO systems: MMSE balancing.IEEE Transactions on Signal Processing, 56(8):3702–3712.

Spencer, Q., Swindlehurst, A., and Haardt, M. (2004).Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels.IEEE Trans. on Signal Processing, 52(2):461 – 471.

Trivellato, M., Boccardi, F., and Tosate, F. (2007).User selection schemes for MIMO broadcast channels with limited feedback.In 65th IEEE Vehicular Technology Conference, Spring 2007, Dublin.

Tsai, S. and Soong, A. (2003).Effective-SNR mapping for modeling frame error rates in multiple-state channels.Technical Report 3GPP2-C30-20030429-010, 3GPP2.

Wan, L., Tsai, S., and Almgren, M. (2006).A Fading-Insensitve Performance Metric for a Unified Link Quality Model.In Proc. IEEE Wireless Communications & Networking Conference WCNC, volume 4, pages 2110–2114.

Yang, D., Yang, L.-L., and Hanzo, L. (2010).DFT-based beamforming weight-vector codebook design for spatially correlated channels in the unitaryprecoding aided multiuser downlink.In IEEE International Conference on Communications, pages 1–5, Cape Town, South Africa.


References VI

Zhang, Y. and Lei, M. (2012).Robust Grassmannian prediction for limited feedback multiuser MIMO systems.In IEEE Wireless Communications and Networking Conference (WCNC), pages 863 –867.

Zheng, J. and Rao, B. (2008).Capacity analysis of MIMO systems using limited feedback transmit precoding schemes.IEEE Transactions on Signal Processing, 56(7):2886–2901.


Limited Feedback v3

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