new frontiers in feedback for interference...

The Wireless Networking and Communications Group

New Frontiers in Feedback for Interference Alignment

Robert W. Heath Jr. & Omar El Ayach

The University of Texas at Austinhttp://www.profheath.org

November 2011

Also associated with Kuma Signals LLC and MIMO Wireless Inc.

Performed by ONR grant N000141010337

Friday, November 18, 2011

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http://www.profheath.org

(c) Robert W. Heath Jr. 2011

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OutlineIntroduction to interference alignment

Analog feedback for interference alignment

Limited feedback for interference alignment

Conclusions

3


Wireless Systems

4

Interference limits performance

Cellular MANET LAN



K-User Interference Channel

5

transmitter receiverdirect channel

interferencereceived

interferencecreated



Conventional Approach

6

transmitter receiver

Multiple access protocol enables sharing



Miracle of Interference Alignment

7

transmitter receiver

Space occupied by interferers is reduced

interferencealigned



Interference Alignment: Subspaces

8

H!!

H""

H##

H$$

Alignment in space (MIMO) or frequency (OFDM)Friday, November 18, 2011

Hot Topics

9

Theory Practice

Random access protocols

Proof-of-concept in real channelsAlgorithms

Prototyping

Impact of estimation error

Practical feedback strategies

Degrees of freedom

Blind alignment

Capacity scaling

Relays

Feasibility

NeutralizationImpact of spatial correlation

Realistic assessment of overhead

Realistic assessment of overheadSuccessive alignment

Interferencepricing

IA tutorial http://newport.eecs.uci.edu/~syed/papers/fntfinaltutorial.pdfIA bibliography http://www.profheath.org/research/interference-alignment/

System level performance

Partial connectivity

Alignment in cellular systems


http://newport.eecs.uci.edu/~syed/papers/fntfinaltutorial.pdf

http://newport.eecs.uci.edu/~syed/papers/fntfinaltutorial.pdf

http://www.profheath.org/research/interference-alignment/



Practical Challenge I:Real Propagation Channels

10

Antenna Correlation

Line of sightCorrelation across users

Good BadSome randomness

IA works!Indoor & Outdoor Testbed

3 MIMO users

IA designed for

random channels

IA will be used in

“real” channels

but

with a small performance

gap0 5 10 15 20 25 30 35 40

0

5

10

15

20

25

30

35

40

SNR (dB)

Su

m R

ate

b/s

/Hz

Interference Alignment

Time Division

Older Strategies

Real Channels

O. El Ayach, S. W. Peters, and R. W. Heath, Jr., ``Feasibility of Interference Alignment of Measured MIMO-OFDM Channels'', IEEE Trans. on Veh. Tech.. vol. 59. No. 9. pp. 4309-4321, Nov. 2010.


https://webspace.utexas.edu/oe559

https://webspace.utexas.edu/oe559

http://www.ece.utexas.edu/~peters/


http://www.profheath.org/


http://users.ece.utexas.edu/~rheath/papers/2009/TVT1/




http://www.ieee.org/organizations/pubs/transactions/tvt.htm

http://www.ieee.org/organizations/pubs/transactions/tvt.htm


Practical Challenge II:Channel Uncertainty

11

IA designed for

ideal channels

Only have

estimated channels

butDelay

Pilot contamination

Estimation error

Training Sequences

Good Bad

0 10 20 30 400

5

10

15

20

25

30

35

40

45

50

SNR (dB)

Sum

Rat

e

6 Pairs

3 Pairs

IncreasingEstimation

Error

75%

Red

uctio

n

70%

Redu

ctio

n

! = 0.02

! = 0.2

Real Systems

estimate =channel + error

channel estimate creates self-interference

0 20 40 60 80 100 120 140 160 180 200

0

2

4

6

8

time

magnitude

Errorfloorsevere for

more users

B. Nosrat-Makouei, J. G. Andrews, and R. W. Heath, Jr., ``MIMO Interference Alignment Over Correlated Channels with Imperfect CSI," IEEE Trans. on Signal Processing, vol. 59, no. 6, pp. 2783-2794, June 2011


https://webspace.utexas.edu/bn2989

https://webspace.utexas.edu/bn2989

http://www.ece.utexas.edu/~jandrews

http://www.ece.utexas.edu/~jandrews



http://users.ece.utexas.edu/~rheath/papers/2010/TSP2/

http://users.ece.utexas.edu/~rheath/papers/2010/TSP2/

http://www.ieee.org/organizations/society/sp/tsp.html

http://www.ieee.org/organizations/society/sp/tsp.html


Practical Challenge III:Overhead

12

Approach

IA designed for

no overhead overhead cost

but

TX

TX

RX

RX

RX

TX

Partition the network

FRAME k FRAME k+1

Training + Feedback

Training + Feedback

Data (all nodes)

L(M,K) symbols

T - L(M,K) symbols

Account for overheads Channel training Channel feedback

Insights

Rates 0 with many users

1 2 3 4 5 60

5

10

15

20

25

30

35

P (# of groups)

Sum

Rat

e b/

s/H

z = 0.9

= 0.7

= 0.5

= 0.3

= 0.1

Smaller partitions usually optimum

will have to pay

# partitions

sum

rat

e

S. W. Peters and R. W. Heath, Jr., ``User Partitioning for Less Overhead in MIMO Interference Channels'', submitted to IEEE Trans. on Wireless, see arxiv.






http://users.ece.utexas.edu/~rheath/papers/2010/TWC/

http://users.ece.utexas.edu/~rheath/papers/2010/TWC/

http://www.comsoc.org/pubs/jrnal/twc.html

http://www.comsoc.org/pubs/jrnal/twc.html


Practical Challenge IV:Channel State Information Feedback

13

Channel estim

ation

Cal

cula

te P

reco

ders

CSI feedback

Send payload data

1

2

3

4

Modulation

Pilot Symbols

Channel Estimation

SimpleProcessing

Pilot Symbols

Channel Estimation

Channel Quantization Channel

Index

Modulation0110100

Limited feedback

Limited feedbackBetter compression through structure

Analog feedbackGood for fast & simple CSI transfer

Analog feedback

Performed by ONR grant N000141010337

CSI is the hidden IA “killer”

Main Challenges

Feedback overhead scales with K2

Feedback quality must increase at high SNR



Related Work on CSI FeedbackExploit reciprocity [GomCadJaf08, ShiBerHon09]

Limited feedback (CSI quantization)

Grassmannian limited feedback for SISO IA over frequency [ThuBol09]

Grassmannian limited feedback for MIMO IA over frequency [KriVar10]

Exploiting network topology

User grouping algorithms [PetHea10, MaLiLiuChe11]

Partially connected channels could affect feedback [ChoChu08, GuiGes11]

Blind IA transmission strategies [Jaf10, GouJaf11, WanPapRamCai11]

14

Contribution: New approaches for analog and limited feedback






Conclusions

15



MIMO System Model

Interference alignment with MIMO signal model

For submission to IEEE Trans. Wireless Commun. 5

Under our assumptions, the received signal at sink node i can be written as

yi =

�P

diHi,iFisi +

�

� �=i

�P

d�Hi,�F�s� + vi,

where yi is the Nr × 1 received signal vector, Hi,� is the Nr ×Nt channel matrix from source �

to sink i, Fi is the Nt× di unitary precoding matrix used at source i, si is the di× 1 transmitted

symbol vector at node i, with unit norm elements, i.e. E [�si�2] = di, and vi is a complex vector

of i.i.d circularly symmetric white Gaussian noise with covariance matrix σ2INr .

We place no assumption on the reciprocity of the forward and reverse channel. This is similar

to a frequency division duplexed (FDD) system model, for example, in which the forward and

reverse channels are uncorrelated. For the reverse or feedback channel we write the received

signal at source node i as

←−y i =

�Pf

NtGi,i

←−x i +

�

��=i

�Pf

NtGi,�

←−x � + νi, (1)

where Pf is the transmit power used to transmit pilot and feedback symbols, Gi,� is the Nt×Nr

reverse channel between sink node i and source node � with i.i.d CN (0, 1) elements1,←−x i is

the symbol vector with unit norm elements sent by sink i, and νi is a complex vector of i.i.d

circularly symmetric white Gaussian noise with covariance matrix σ2INt .

B. Interference Alignment

Interference alignment for the MIMO interference channel is a linear precoding technique

that can achieve the maximum multiplexing gain, or degrees of freedom defined as limP→∞

Rsumlog2 P

,

which in this case isKNr2 , when coding over infinitely many channel extensions [10]. While

the maximum multiplexing gain may not be achieved without time extensions, interference

alignment for the constant MIMO channel can still provide an increase in sum rate [21]. To

do so, given global channel knowledge, interference alignment computes the transmit precoders

Fi to align interference at all receivers in a strict subspace of the received signal space, thus

leaving interference free dimensions for the desired signal. While interference alignment is only

one of the many precoding strategies for the interference channel [3], [22], [23], some of which

1Note the effective reversal in the indexing of the channel, while the indexing still has the form “sink,source” the transmit

receive roles have been switched.

Desired Signal Interference

IA precoders push interference to a low-dimensional subspace

16

Resulting sum rate with zero-forcing receiver


signals and treating interference as noise, is

Rsum =K�

i=1

di�

m=1

Rmi =

K�

i=1

di�

m=1

log2

�1 +

Pdi|(wm

i )∗Hi,if

mi |2

I1i,m + I2

i,m + σ2

�, (5)

where I1i,m and I2

i,m are the inter-stream and inter-user interference, respectively. These sum

interference terms are given by

I1i,m =

�

��=m

P

di

��(wmi )

∗Hi,if

�i

��2 ,

I2i,m =

�

k �=i

dk�

�=1

P

dk

��(wmi )

∗Hi,kf

�k

��2 .

In the presence of perfect channel knowledge, and for an achievable degree of freedom vector

d = [d1, d2, . . . , dK ], equations (2), (3), and (4) can be satisfied with probability one and thus

I1i = I2

i = 0. This gives

limP→∞

Rsum

log2 P= lim

P→∞

�i,m

log2

�1 +

Pdi|(wm

i )∗Hi,ifmi |2

σ2

�

log2 P

=K�

i=1

di ≤KNr

2.

It is not immediately clear, however, if the same sum rate scaling behavior can be expected

from a network with only imperfect knowledge of the channel derived from noisy feedback.

Results on single user MIMO prove an acceptable constant loss in sum rate due to imperfect

channel state information [25]. In multi-user scenarios, however, the cost of imperfect channel

knowledge may be much higher, potentially resulting in the loss of the channel’s multiplexing

gain [26] which saturates achieved sum rate at high SNR [27]. In Section III, we show that such

performance can be expected from a realistic system via interference alignment provided that the

quality of channel knowledge scales sufficiently with transmit power, or effectively the forward

channel’s SNR. This is similar to the results presented in [13] and [14] for SISO channels where

the feedback scaling is in terms of codebook size. We discuss several advantages of using the

feedback scheme proposed in this paper over limited feedback quantization.

=Leakage interference depends on CSI quality



Problem Statement

17

Design low overhead feedback strategy for MIMO IA

Prior work on analog feedback:Analog feedback for MISO broadcast [MarHoc06, SamMan06]

Comparison with quantized feedback [CaiJinKobRav07, KobJibCai10]

Solution: Analog channel state feedback

Objectives

Fast MIMO CSI transfer

Achieves multiplexing gains

Fixed complexity (independent of SNR)

Existing problems

CSI is quality and delay sensitive

Exploding codebooks in

limited feedback

Prior work cannot handle MIMO

Receivers estimate MIMO channel

Transmitters calculate the transmit directions

Receivers feedback the quantized channels

Transmitters send payload data



Limited feedback

Analog vs. Limited Feedback

18

Pilot Symbols

Channel Estimation

Channel Quantization Channel

Index

Codebook

Modulation0110100Hk,l

ModulationPilot

SymbolsChannel

EstimationSimple

ProcessingHk,l Hk,l

Analog feedback



Assumptions

Any number of antennas

Any number of streams

FDD (no reciprocity)

Block fading model

SNR is known

Analog feedback IA

Transmit power (P)

Feedback power (Pf)

Variable training and feedback

Analog Feedback

2. Train reverse channel (Gik)

3. Feedback Hik as QAM symbols

4. Use CSI estimates to do IA


III. INTERFERENCE ALIGNMENT WITH ANALOG FEEDBACK

In this section we propose a feedback and transmission strategy based on analog feedback and

interference alignment, which uses the estimated channels as if they were the true propagation

channels.

A. Analog Feedback

To feedback the forward channel matrices Hi,� reliably across the feedback channel given in

(1), we propose dividing the feedback stage into two phases. First, each source learns its reverse

channels. Second, the forward channels are fed back and estimated. We neglect the initial training

phase in which sinks learn the forward channels and, thus, assume they have been estimated

perfectly. This is since imperfect CSI at the receiver will adversely affect performance of all

feedback schemes and is not exclusive to analog feedback. In fact, any error in estimating the

forward channels will only add an error term to the forward channels in (7), which also decays

with transmit power, and thus similar results can be shown.

1) Reverse Link Training: To learn all reverse links, the K sink nodes must transmit pilot

symbols over a period τp ≥ KNr. Similar to the analysis done in [15], we let the sinks collectively

transmit a KNr×τp unitary matrix of pilots Φ shown to be optimal in [28]. Such reverse channel

training requires no assumptions other than training sequences be known to all sink nodes, to

guarantee orthogonality. Of course, if we enforce orthogonality in time, this assumption reduces

to synchronization, and exact sequences need not be known to all sinks.

Let←−Y i =

�←−y i[1] . . . ←−

y i[τp]�

be the Nt × τp received training matrix at source node i, and

let←−Yp =

�←−Y

∗1,←−Y

∗2, . . . ,

←−Y

∗K

�∗be the composite received training matrix. We write the received

training as←−Yp =

�τpPf

NrGΦ+V,

where G is the KNt × KNr composite reverse channel matrix between all sinks and sources

and V is a KNt× τp matrix with i.i.d CN (0, σ2INr) elements. Using the received training, each

source derives an MMSE estimate of its channels all of which can be together written as

�G =

�τpPf

Nr

σ2 + τpPf

Nr

←−YpΦ

∗. (6)





channels.

A. Analog Feedback















Let←−Y i =

�←−y i[1] . . . ←−

y i[τp]�


let←−Yp =

�←−Y

∗1,←−Y

∗2, . . . ,

←−Y

∗K



�τpPf

NrGΦ+V,




�G =

�τpPf

Nr

σ2 + τpPf

Nr

←−YpΦ

∗. (6)


marginally outperform it at low SNR [3], it is analytically tractable. Its complete interference

suppression properties make it especially amenable to the study of performance with feedback

and imperfect CSI.

To express the conditions for alignment, consider the K-user interference channel with pre-

coding presented in Section II-A and any corresponding achievable degree of freedom allocation

vector d = [d1 d2 . . . dK ]. Source node i sends its di spatial streams along the columns f�i of

the precoder Fi, resulting in a transmitted symbol

xi =1√diFisi =

1√di

di�

�=1

f�i s

�i i = 1, . . . , K

where we note that �f �i �2 = 1 and��s�i

��2 = 1 to satisfy the total power constraint with equality. We

assume equal power allocation since the gain observed from water-filling is at most a constant

and thus is negligible at high SNR [24].

While in general interference alignment can be used with any receiver design, the discussion

and proofs in this paper assume a linear zero-forcing receiver in which sink node i projects its

received signal on to the columns, w�i , of the Nr × di combiner Wi. Simulations in Section V

indicate that the same performance can be expected from an optimal receiver.

Writing the per stream input-output relation after projection gives

(wmi )

∗yi = (wm

i )∗√PHi,if

mi smi +

�

� �=m

(wmi )

∗√PHi,if

�i s

�i+

�

k �=i

dk�

�=1

(wmi )

∗√PHi,kf

�ks

�k+(wm

i )∗vi,

for m ∈ {1, . . . , di} and i ∈ {1, . . . , K}, where �wmi �2 = 1. At the output of these linear

receivers w�i , the conditions for perfect interference alignment can be restated as

(wmi )

∗Hi,if

�i = 0, ∀i, � �= m (2)

(wmi )

∗Hi,kf

�k = 0, ∀k �= i, and ∀m, � (3)

|(wmi )

∗Hi,if

mi | ≥ c > 0, ∀i,m (4)

where interference alignment is guaranteed by the first two conditions, and the third ensures the

decodability of the di desired streams.

The suboptimal sum rate achieved by the linear zero-forcing receiver, assuming Gaussian input

1. Train forward channels (Hik)

TX k-1

TX k

TX k+1

RX k-1

RX k

RX k+1

19



Analog Feedback

Done anyway for coherent detection Not specific to analog feedback

Let’s assume it’s done perfectly

Hk-1,k-1

Hk,k-1

TX k-1

TX k

TX k+1

RX k-1

RX k

RX k+1

20

1. Train forward channels (Hik)

Though we really don’t need to



Analog FeedbackGk-1,k-1

Gk,k-1

Feedback Power

Training Time

Training Sequence





channels.

A. Analog Feedback















Let←−Y i =

�←−y i[1] . . . ←−

y i[τp]�


let←−Yp =

�←−Y

∗1,←−Y

∗2, . . . ,

←−Y

∗K



�τpPf

NrGΦ+V,




�G =

�τpPf

Nr

σ2 + τpPf

Nr

←−YpΦ

∗. (6)





channels.

A. Analog Feedback















Let←−Y i =

�←−y i[1] . . . ←−

y i[τp]�


let←−Yp =

�←−Y

∗1,←−Y

∗2, . . . ,

←−Y

∗K



�τpPf

NrGΦ+V,




�G =

�τpPf

Nr

σ2 + τpPf

Nr

←−YpΦ

∗. (6)





channels.

A. Analog Feedback















Let←−Y i =

�←−y i[1] . . . ←−

y i[τp]�


let←−Yp =

�←−Y

∗1,←−Y

∗2, . . . ,

←−Y

∗K



�τpPf

NrGΦ+V,




�G =

�τpPf

Nr

σ2 + τpPf

Nr

←−YpΦ

∗. (6)

Step 2: Train the reverse channel Send a pilot matrix





channels.

A. Analog Feedback















Let←−Y i =

�←−y i[1] . . . ←−

y i[τp]�


let←−Yp =

�←−Y

∗1,←−Y

∗2, . . . ,

←−Y

∗K



�τpPf

NrGΦ+V,




�G =

�τpPf

Nr

σ2 + τpPf

Nr

←−YpΦ

∗. (6)





channels.

A. Analog Feedback















Let←−Y i =

�←−y i[1] . . . ←−

y i[τp]�


let←−Yp =

�←−Y

∗1,←−Y

∗2, . . . ,

←−Y

∗K



�τpPf

NrGΦ+V,




�G =

�τpPf

Nr

σ2 + τpPf

Nr

←−YpΦ

∗. (6)





channels.

A. Analog Feedback















Let←−Y i =

�←−y i[1] . . . ←−

y i[τp]�


let←−Yp =

�←−Y

∗1,←−Y

∗2, . . . ,

←−Y

∗K



�τpPf

NrGΦ+V,




�G =

�τpPf

Nr

σ2 + τpPf

Nr

←−YpΦ

∗. (6)Estimate G

Normalization: Lets not worry about it!!

TX k-1

TX k

TX k+1

RX k-1

RX k

RX k+1

21

2. Train reverse channel (Gik)



Analog Feedback

Step 3: Feedback CSI and estimate H


Since �G is an MMSE estimate of Gaussian random variables corrupted by Gaussian noise, this

results in �G ∼ CN�0,

τpPfNr

σ2+τpPfNr

�and �G = G − �G with i.i.d CN

�0, σ2

σ2+τpPfNr

�elements. No

collaboration between sources is needed in (6); each node estimates its channels based on its

own received signals which we have concatenated in (6) and written in terms of the composite

received training matrix,←−Yp, to conserve space.

2) Analog CSI Feedback: After training the reverse channels, each sink node i directly sends

its unquantized uncoded estimates of Hi,� ∀ � over a period τc. To have the sink nodes feedback

their CSI simultaneously while maintaining orthogonality, each node post multiplies its Nr×KNt

feedback matrix [Hi,1 . . .Hi,K ] with a unitary KNt × τc matrix Ψ∗i such that Ψ∗

iΨ� = IKNtδi,�,

a general orthogonal feedback structure that can capture the case of orthogonality in time [15].

This requires τc ≥ KNtNr. The transmitted Nr × τc feedback matrix←−X i from node i can be

written as←−X i =

�τcPf

KNtNr[Hi,1 . . .Hi,K ]Ψ

∗i .

The concatenated, or composite received feedback KNt × τc matrix is then given by

←−Yc =

�τcPf

KNtNr

K�

i=1

Gi,1

...

Gi,K

[Hi,1 . . .Hi,K ]Ψ

∗i +V

where V is now a KNt × τc noise matrix.

To estimate the forward channels Hi,�, the source nodes first isolate the training from sink

node i by post multiplying their received training by Ψi which gives

←−YcΨi =

�τcPf

KNtNr

Gi,1

...

Gi,K

� �� Gi

[Hi,1 . . .Hi,K ]� �� Hi

+VΨi. (7)

To simplify the analysis in Section III-B, we assume sources share the complete matrix←−Yc, and

Feedback

Spread feedback on orthogonal sequences




τpPfNr

σ2+τpPfNr


�0, σ2

σ2+τpPfNr

�elements. No












�τcPf


∗i .


←−Yc =

�τcPf

KNtNr

K�

i=1

Gi,1

...

Gi,K

[Hi,1 . . .Hi,K ]Ψ

∗i +V




←−YcΨi =

�τcPf

KNtNr

Gi,1

...

Gi,K

� �� Gi

[Hi,1 . . .Hi,K ]� �� Hi

+VΨi. (7)


Estimate H


effectively compute a common least squares estimate �Hi of Hi given by

�Hi =

�KNtNr

τcPf

��G∗

i�Gi

�−1 �G∗i

←−YcΨi

= Hi��Real Channel

+ �Hi��Error

,

where �Gi is the estimate of Gi; a KNt ×Nr block taken from �G obtained in (6). It is realized

that such node cooperation is not practical; we return to this assumption at the end of this section

and provide alternative approaches that we show in Section V perform close to this special case.

The error in the estimates of Hi,� can then be written as

�Hi =��G∗

i�Gi

�−1 �G∗i

��KNtNr

τcPfV − �GiHi

�.

which makes it clear that the error in the estimate consists of two error terms: the first due to

noisy feedback and the second due to a noisy estimate of the feedback channel. To quantify the

effect of the error on the achieved sum rate, we derive the variance of the error term introduced

by analog feedback. Recall that the elements of Hi,� are CN (0, 1), those of V are CN (0, σ2),

and those of �Gi are CN (0, σ2

σ2+τpPfNr

). As a result, the error term �GiHi due to the reverse channel

estimation has independent elements with a variance of Nrσ2

σ2+τpPfNr

. Similarly to [15] we see that

the covariance of each columns of �Hi denoted �H(�)i , conditioned on �Gi is

Cov(�H(�)i |�Gi) =

�KNtNrσ2

τcPf+

Nrσ2

σ2 + τpPf

Nr

��G∗

i�Gi

�−1.

Since the elements of the MMSE estimate �Gi are Gaussian and uncorrelated, the diagonal

elements of��G∗

i�Gi

�−1are reciprocals of scaled chi-squared random variables with 2(KNt −

NR+1) degrees of freedom [15]. As a result, the mean square error, σ2f , in the elements of �Hi,�

is given by

σ2f =

σ2

(KNt −Nr)Pf

�N2

r

τp+

KNtNr

τc

�1 +

Nrσ2

τpPf

��. (8)

At high SNR this gives

σ2f ≈

σ2�

N2r

τp+ KNtNr

τc

�

(KNt −Nr)Pf. (9)Variance of error

Feedback Power

Feedback Time

SpreadingSequence





channels.

A. Analog Feedback















Let←−Y i =

�←−y i[1] . . . ←−

y i[τp]�


let←−Yp =

�←−Y

∗1,←−Y

∗2, . . . ,

←−Y

∗K



�τpPf

NrGΦ+V,




�G =

�τpPf

Nr

σ2 + τpPf

Nr

←−YpΦ

∗. (6)




τpPfNr

σ2+τpPfNr


�0, σ2

σ2+τpPfNr

�elements. No












�τcPf


∗i .


←−Yc =

�τcPf

KNtNr

K�

i=1

Gi,1

...

Gi,K

[Hi,1 . . .Hi,K ]Ψ

∗i +V




←−YcΨi =

�τcPf

KNtNr

Gi,1

...

Gi,K

� �� Gi

[Hi,1 . . .Hi,K ]� �� Hi

+VΨi. (7)





τpPfNr

σ2+τpPfNr


�0, σ2

σ2+τpPfNr

�elements. No












�τcPf


∗i .


←−Yc =

�τcPf

KNtNr

K�

i=1

Gi,1

...

Gi,K

[Hi,1 . . .Hi,K ]Ψ

∗i +V




←−YcΨi =

�τcPf

KNtNr

Gi,1

...

Gi,K

� �� Gi

[Hi,1 . . .Hi,K ]� �� Hi

+VΨi. (7)


TX k-1

TX k

TX k+1

RX k-1

RX k

RX k+1

22

3. Feedback Hik as QAM symbols



Performance Analysis


Having computed feedback error, we return to the assumption on node cooperation. Source

cooperation simplifies analysis by effectively making a common �H known to all users, however,

this cannot be assumed in practical systems. The first alternative to cooperation is to have one user

calculate precoders and combiners, and feed them forward to the other users as in [29]. The other

alternative is for each node to calculate its precoder and combiner based on the feedback only

it receives. We refer to this as the “distributed processing” approach. Due to independent noise

across users in the feedback stage, distributed processing implies that users will calculate vectors

based on mismatched information, the effect of which can also be bounded. We elaborate on

this after presenting our results in Sections III-B and V and show that the cost of no cooperation

is limited.

B. Multiplexing Gain with Analog Feedback

To characterize the performance of interference alignment with analog feedback, we examine

the mean loss in sum-rate [26] incurred by naive interference alignment where the channel

estimates obtained at the sources are used, as if they were the true channels, to calculate the

columns of the precoders, fmi ∀i,m and combiners w

mi ∀i,m. Transmit and receive vectors are

calculated to satisfy (2), (3), and (4) using the estimated channels.

The mean loss in sum-rate is defined as

∆Rsum � EHRsum − EHRsum, (10)

where EHRsum is the average sum rate from interference alignment with perfect CSI, with

instantaneous rate given in (5), and EHRsum is the average rate with CSI obtained via feedback.

Theorem 1: Interference alignment on the K-user Nr×Nt interference channel with imperfect

channel state information obtained via the analog feedback strategy described in Section III-A

achieves the same average sum-rate scaling observed with perfect interference alignment as long

as the feedback power Pf scales with the transmit power P . Thus, the original multiplexing gain

is preserved. Moreover, the mean loss in sum rate ∆Rsum is O(1).

Proof: Let the K-user Nr × Nt interference channel use the analog feedback scheme

presented to achieve a vector of multiplexing gains d. Let the transmit precoding and receive

Rate with perfect CSI Rate with CSI from feedback


combining vectors be calculated to satisfy

(wmi )

∗ �Hi,if�i = 0, ∀i, � �= m (11)

(wmi )

∗ �Hi,k f�k = 0, ∀k �= i, and ∀m, � (12)

��(wmi )

∗ �Hi,ifmi

�� ≥ c > 0, ∀i,m. (13)

Using these precoding and combining vectors, the input-output relationship at the output of a

linear zero-forcing receiver is

(wmi )

∗yi = (wm

i )∗√PHi,if

mi smi +

�

� �=m

(wmi )

∗√PHi,if

�i s

�i+

�

k �=i

dk�

�=1

(wmi )

∗√PHi,k f

�ks

�k+(wm

i )∗vi.

(14)

Using the received signal in (2), the instantaneous rate expression in (5) , and the sum rate loss

defined in (10), this gives the following upper bound on mean loss in sum rate:

∆Rsum = EH

�

i,m

log2

�1 +

Pdi|(wm

i )∗Hi,if

mi |2

σ2

�− E

H,H

�

i,m

log2

1 +

Pdi

��(wmi )

∗Hi,if

mi

��2

I1i,m + I2

i,m� �� Ii,m

+σ2

= EH

�

i,m

log2

�1 +

Pdi|(wm

i )∗Hi,if

mi |2

σ2

�

− EH,H

�

i,m

log2

1 +Ii,m + P

di

��(wmi )

∗Hi,if

mi

��2

σ2

+ EH,H

�

i,m

log2

�1 +

Ii,m

σ2

�

(a)

≤ EHH

�

i,m

log2

�1 +

Ii,m

σ2

�

where (a) is due to the fact that wmi , wm

i , fmi , and fmi are independent of Hi,i and therefore

Pdi

��(wmi )

∗Hi,if

mi

��2

and Pdi|(wm

i )∗Hi,if

mi |2 are identically distributed. As a result P

di

��(wmi )

∗Hi,if

mi

��2+

Ii,m stochastically dominates Pdi|(wm

i )∗Hi,if

mi |2 [26]. We now apply Jensen’s inequality to the

Derived with perfect CSIDerived with imperfect CSI

Feedback error =

Leakage Interference

Recall: No interferenceIA conditions

23

What happens to sum rate?

What is the mean sum rate loss relative to perfect CSI?




upper bound in (a) to get

∆Rsum ≤�

i,m

log2

�1 +

EH,HIi,m

σ2

�. (15)

Since (11), (12), (13) are satisfied, however, the total interference term Ii,m can be simplified to

include only residual interference due to the channel estimation errors �Hi,�. Equation (15) can

be further upper bounded by noticing that��(wm

i )∗(Hi,k + �Hi,k)f

�k

��2=

��(wmi )

∗(�Hi,k)f�k

��2

∀k, ∀� �= m

≤ ��Hi,k�2F ,(16)

which gives

∆Rsum ≤�

i,m

log2

1 +

K��=1

Pd�

(d� − δi,�)EH,H�Hi,��2F

σ2

. (17)

From (8), however, we have EH,H�Hi,��2F = NtNrσ2

f = cPf

, where c is a constant, independent

of Pf at high enough SNR, given by

c = NtNr

σ2�

N2r

τp+ KNtNr(1+�)

τc

�

(KNt −Nr). (18)

Combining (17) and (18) gives the final upper bound on throughput loss

∆Rsum ≤�

i,m

log2

1 +P

σ2di

(di − 1)c

Pf+

�

� �=i

d�c

Pf

.

Therefore, if we have Pf = α−1P we get

∆Rsum ≤�

i

di log2

�1 +

(�d�1 − 1)αc

diσ2

�. (19)

The bound has been presented at high SNR for simplicity of exposition only; (18) can be adapted

for any SNR > 0 by using (8) instead of (9).

In summary, Theorem 1 states that if feedback power is equal to any constant fraction of

transmit power, the cost of imperfect CSI is a constant number of bits, independent of SNR. Since

transmit and feedback power are likely to be comparable in practice, this result is promising.

Using analog feedback allows the system to overcome the problem of exploding complexity and


combining vectors be calculated to satisfy

(wmi )

∗ �Hi,if�i = 0, ∀i, � �= m (11)

(wmi )

∗ �Hi,k f�k = 0, ∀k �= i, and ∀m, � (12)

��(wmi )

∗ �Hi,ifmi

�� ≥ c > 0, ∀i,m. (13)

Using these precoding and combining vectors, the input-output relationship at the output of a

linear zero-forcing receiver is

(wmi )

∗yi = (wm

i )∗√PHi,if

mi smi +

�

� �=m

(wmi )

∗√PHi,if

�i s

�i+

�

k �=i

dk�

�=1

(wmi )

∗√PHi,k f

�ks

�k+(wm

i )∗vi.

(14)

Using the received signal in (2), the instantaneous rate expression in (5) , and the sum rate loss

defined in (10), this gives the following upper bound on mean loss in sum rate:

∆Rsum = EH

�

i,m

log2

�1 +

Pdi|(wm

i )∗Hi,if

mi |2

σ2

�− E

H,H

�

i,m

log2

1 +

Pdi

��(wmi )

∗Hi,if

mi

��2

I1i,m + I2

i,m� �� Ii,m

+σ2

= EH

�

i,m

log2

�1 +

Pdi|(wm

i )∗Hi,if

mi |2

σ2

�

− EH,H

�

i,m

log2

1 +Ii,m + P

di

��(wmi )

∗Hi,if

mi

��2

σ2

+ EH,H

�

i,m

log2

�1 +

Ii,m

σ2

�

(a)

≤ EHH

�

i,m

log2

�1 +

Ii,m

σ2

�

where (a) is due to the fact that wmi , wm

i , fmi , and fmi are independent of Hi,i and therefore

Pdi

��(wmi )

∗Hi,if

mi

��2

and Pdi|(wm

i )∗Hi,if

mi |2 are identically distributed. As a result P

di

��(wmi )

∗Hi,if

mi

��2+

Ii,m stochastically dominates Pdi|(wm

i )∗Hi,if

mi |2 [26]. We now apply Jensen’s inequality to the


upper bound in (a) to get

∆Rsum ≤�

i,m

log2

�1 +

EH,HIi,m

σ2

�. (15)

Since (11), (12), (13) are satisfied, however, the total interference term Ii,m can be simplified to

include only residual interference due to the channel estimation errors �Hi,�. Equation (15) can

be further upper bounded by noticing that��(wm

i )∗(Hi,k + �Hi,k)f

�k

��2=

��(wmi )

∗(�Hi,k)f�k

��2

∀k, ∀� �= m

≤ ��Hi,k�2F ,(16)

which gives

∆Rsum ≤�

i,m

log2

1 +

K��=1

Pd�

(d� − δi,�)EH,H�Hi,��2F

σ2

. (17)

From (8), however, we have EH,H�Hi,��2F = NtNrσ2

f = cPf

, where c is a constant, independent

of Pf at high enough SNR, given by

c = NtNr

σ2�

N2r

τp+ KNtNr(1+�)

τc

�

(KNt −Nr). (18)

Combining (17) and (18) gives the final upper bound on throughput loss

∆Rsum ≤�

i,m

log2

1 +P

σ2di

(di − 1)c

Pf+

�

� �=i

d�c

Pf

.

Therefore, if we have Pf = α−1P we get

∆Rsum ≤�

i

di log2

�1 +

(�d�1 − 1)αc

diσ2

�. (19)

The bound has been presented at high SNR for simplicity of exposition only; (18) can be adapted

for any SNR > 0 by using (8) instead of (9).

In summary, Theorem 1 states that if feedback power is equal to any constant fraction of

transmit power, the cost of imperfect CSI is a constant number of bits, independent of SNR. Since

transmit and feedback power are likely to be comparable in practice, this result is promising.

Using analog feedback allows the system to overcome the problem of exploding complexity and

What happens to sum rate?

What is the mean sum rate loss relative to perfect CSI?

24

Performance Analysis

Sum rate loss is a function of interference power

Feedback power counteracts interference

power

For Pfb =O(P)Mux. gain preserved!!

Constant loss in sum rate!!



Simulation Results

Perfe

ct CSI

Pfb =P/2

Pfb =constant

Multiplexing gainpreserved

Multiplexing gain = 0

SNR independent gap & Tight lower bound

• 3 User• 2x2 MIMO links• Rayleigh fading

25

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

70

SNR (dB)

Su

m R

ate

(b

its/s

/Hz)

IA with Perfect CSI

IA with Pf=P/2

IA with Dist. Processing & Pf=P/2

Approx. Lower Bound (with c2(!

p!c) )

Lower Bound (with c(!p!c) )

IA with Pf = P

0.5

IA with Fixed Feedback Quality

Partial multiplexing gains achieved



Part I: SummaryAnalog feedback enables interference alignment

Flexible approach for providing high resolution channel estimates

Gives a scalable multiplexing optimal feedback strategy

Can be improved through optimization of various parameters

Further work

Feedback in partial interference channels

Analysis of distributed performance with uncertainty

26






Conclusions

27



OFDM System ModelK-user wide band interference channel, single antenna

Transmitters use OFDM to obtain virtual MIMO channels

Time domain channel taps are temporally correlated

28

TX

k-1

Hk-1,k-1

Hk,k

Hk+1,k+1

Hk,k-1

Hk+1,k

Hk-1,k

Hk,k+1Hk-1

,k+1

Hk+1,k-1

TX

k

TX

k+1

RX

k-1

RX

k

RX

k+1

Figure 1: K-User SISO interference channel model

matrix input-output relationship is

yk[t] = Hk,k[t]xk[t] +!

! !=k

Hk,![t]x![t] + vk[t]. (1)

where xk[t] is the OFDM symbol sent by userk at time t with the average power constraintE"!xk[t]!2

#= NscP , the Nsc " Nsc matrix

Hk,![t] = diag$FNsc

%h"k,![t], 0Nsc#L

&"'represents

the channel frequency response between transmitter ! andreceiver k at time t, and vk[t] is the i.i.d. CN (0,"2

nINsc)thermal noise observed by user k. The system modelassumes perfect time and frequency synchronization, and acyclic prefix longer than all channel responses.

The channels seen by the t-th OFDM symbol are as-sumed to remain constant. The channels over consecutiveOFDM symbols, however, are temporally correlated suchthat E [|hk,![t+ 1]hk,![t]|] = # # 1.

3. INTERFERENCE ALIGNMENT IN FREQUENCY

In this section we review the concept of IA over frequencyextensions and summarize the effect of imperfect CSI on theperformance of IA.

3.1 SISO IA via Frequency ExtensionsIA for the SISO interference channel can achieve the max-imum degrees of freedom when coding over infinite chan-nel extensions [1]. Using IA over Nsc frequency extensions,each transmitter k sends dk < Nsc symbols along the pre-coding vectors fmk [t]. As a result the transmitted symbol is

xk[t] =dk!

m=1

fmk [t]xmk [t], (2)

where xmk [t] are the symbols transmitted by user k at time

t. To satisfy the power constraint, we set !fmk [t]!2 = 1,and E

"xmk [t]2

#= NscP/dk, such that the total power in

each Nsc subcarriers is NscP . The transmit directions fmk [t]are calculated such that the interference from K $ 1 users isaligned at all receivers, leaving interference free dimensionsfor the desired signal.

In this section, we restrict our attention to IA with a zero-forcing receiver. At the output of the linear receiver, the re-ceived signal is

wmk [t]"yk[t] =wm

k [t]"Hk,k[t]fmk [t]xm

k [t]

+!

(i,!) !=(k,m)

wmk [t]"Hk,i[t]f

!i [t]x

!i [t]

+wmk [t]"vk[t],

(3)

for m % {1, . . . , dk} and k % {1, . . . ,K}, where!wm

k [t]!2 = 1. With a linear receiver, the conditions forperfect IA can be restated as

wmk [t]"Hk,i[t]f

!k[t] = 0, (i, !) &= (k,m) (4)

|wmk [t]"Hk,k[t]f

mk [t]| ' c > 0, (k,m (5)

where alignment is achieved by (4), and (5) ensures the de-codability of the dk desired streams.

The achievability proof in [1] showed that if fading is in-dependent on all subcarriers, then the vectors f !k[t] can befound to satisfy (4) and (5), if dk’s are chosen as in [1].Fortunately, [5] has claimed that fading on each subcarrierneed not be independent provided that the channel impulseresponse is long enough.

3.2 The Effect of Imperfect CSI FeedbackWith imperfect or limited CSI feedback, condition (4) is notsatisfied, resulting in residual interference. As the trans-mit power increases, so does the leakage interference power,which saturates the sum rate at high SNR.

In [6], it is shown that if imperfect CSI is used to calculatethe IA precoders and combiners, (fmk [t] and (wm

k [t], the meanloss in sum rate is upper bounded by

!Rsum #!

k,m

1

Nsclog2

)

*1 +EH

%I1k,m + I2

k,m

&

"2

+

, , (6)

where I1i,m[t] =

-! !=m

NscPdk

... (wmk [t]"Hk,k[t](f !k[t]

...2, and

I2i,m[t] =

-i !=k

di-!=1

NscPdi

... (wmk [t]"Hk,i[t](f !i [t]

...2

are the inter-

stream and inter-user interference respectively. The objec-tive of the feedback algorithm then becomes minimizing thetotal leakage interference by improving effective CSI accu-racy. Using the result from [5], the individual interferenceterms can be upper bounded as

!!! "wmk [t]!Hk,i[t]"f !i [t]

!!!2!

""wmk [t] # "f !i [t]"2"hk,i[t]"2

#1$

!!!!!hk,i[t]

!"hk,i[t]

"hk,i[t]"""hk,i[t]"

!!!!!

2$.

(7)

From (7) we see that leakage interference is directly relatedto the angle between the normalized channel impulse re-sponse, hk,i[t]

$hk,i[t]$ , and its quantized version!hk,i[t]

$!hk,i[t]$. As a

result, to limit performance degradation an efficient feedbackstrategy must attempt to minimize the angle between the ac-tual and quantized channels.

Desired Signal Interference

TX

k-1

Hk-1,k-1

Hk,k

Hk+1,k+1

Hk,k-1

Hk+1,k

Hk-1,k

Hk,k+1Hk-1

,k+1

Hk+1,k-1

TX

k

TX

k+1

RX

k-1

RX

k

RX

k+1

Figure 1: K-User SISO interference channel model

matrix input-output relationship is

yk[t] = Hk,k[t]xk[t] +!

! !=k

Hk,![t]x![t] + vk[t]. (1)

where xk[t] is the OFDM symbol sent by userk at time t with the average power constraintE"!xk[t]!2

#= NscP , the Nsc " Nsc matrix

Hk,![t] = diag$FNsc

%h"k,![t], 0Nsc#L

&"'represents

the channel frequency response between transmitter ! andreceiver k at time t, and vk[t] is the i.i.d. CN (0,"2

nINsc)thermal noise observed by user k. The system modelassumes perfect time and frequency synchronization, and acyclic prefix longer than all channel responses.

The channels seen by the t-th OFDM symbol are as-sumed to remain constant. The channels over consecutiveOFDM symbols, however, are temporally correlated suchthat E [|hk,![t+ 1]hk,![t]|] = # # 1.

3. INTERFERENCE ALIGNMENT IN FREQUENCY

In this section we review the concept of IA over frequencyextensions and summarize the effect of imperfect CSI on theperformance of IA.

3.1 SISO IA via Frequency ExtensionsIA for the SISO interference channel can achieve the max-imum degrees of freedom when coding over infinite chan-nel extensions [1]. Using IA over Nsc frequency extensions,each transmitter k sends dk < Nsc symbols along the pre-coding vectors fmk [t]. As a result the transmitted symbol is

xk[t] =dk!

m=1

fmk [t]xmk [t], (2)

where xmk [t] are the symbols transmitted by user k at time

t. To satisfy the power constraint, we set !fmk [t]!2 = 1,and E

"xmk [t]2

#= NscP/dk, such that the total power in

each Nsc subcarriers is NscP . The transmit directions fmk [t]are calculated such that the interference from K $ 1 users isaligned at all receivers, leaving interference free dimensionsfor the desired signal.

In this section, we restrict our attention to IA with a zero-forcing receiver. At the output of the linear receiver, the re-ceived signal is

wmk [t]"yk[t] =wm

k [t]"Hk,k[t]fmk [t]xm

k [t]

+!

(i,!) !=(k,m)

wmk [t]"Hk,i[t]f

!i [t]x

!i [t]

+wmk [t]"vk[t],

(3)

for m % {1, . . . , dk} and k % {1, . . . ,K}, where!wm

k [t]!2 = 1. With a linear receiver, the conditions forperfect IA can be restated as

wmk [t]"Hk,i[t]f

!k[t] = 0, (i, !) &= (k,m) (4)

|wmk [t]"Hk,k[t]f

mk [t]| ' c > 0, (k,m (5)

where alignment is achieved by (4), and (5) ensures the de-codability of the dk desired streams.

The achievability proof in [1] showed that if fading is in-dependent on all subcarriers, then the vectors f !k[t] can befound to satisfy (4) and (5), if dk’s are chosen as in [1].Fortunately, [5] has claimed that fading on each subcarrierneed not be independent provided that the channel impulseresponse is long enough.

3.2 The Effect of Imperfect CSI FeedbackWith imperfect or limited CSI feedback, condition (4) is notsatisfied, resulting in residual interference. As the trans-mit power increases, so does the leakage interference power,which saturates the sum rate at high SNR.

In [6], it is shown that if imperfect CSI is used to calculatethe IA precoders and combiners, (fmk [t] and (wm

k [t], the meanloss in sum rate is upper bounded by

!Rsum #!

k,m

1

Nsclog2

)

*1 +EH

%I1k,m + I2

k,m

&

"2

+

, , (6)

where I1i,m[t] =

-! !=m

NscPdk

... (wmk [t]"Hk,k[t](f !k[t]

...2, and

I2i,m[t] =

-i !=k

di-!=1

NscPdi

... (wmk [t]"Hk,i[t](f !i [t]

...2

are the inter-

stream and inter-user interference respectively. The objec-tive of the feedback algorithm then becomes minimizing thetotal leakage interference by improving effective CSI accu-racy. Using the result from [5], the individual interferenceterms can be upper bounded as

!!! "wmk [t]!Hk,i[t]"f !i [t]

!!!2!

""wmk [t] # "f !i [t]"2"hk,i[t]"2

#1$

!!!!!hk,i[t]

!"hk,i[t]

"hk,i[t]"""hk,i[t]"

!!!!!

2$.

(7)

From (7) we see that leakage interference is directly relatedto the angle between the normalized channel impulse re-sponse, hk,i[t]

$hk,i[t]$ , and its quantized version!hk,i[t]

$!hk,i[t]$. As a

result, to limit performance degradation an efficient feedbackstrategy must attempt to minimize the angle between the ac-tual and quantized channels.

Revision 3 on August 22, 2011 6

Throughout this paper, we consider temporally correlated Gaussian channels according to the

P -order autoregressive channel model defined in [23]. For such processes, each time series is

generated as

hk,![t] =P!

m=1

!mhk,![t!m] + "zk,![t], (3)

where zk,l[t] = CN (0,Rhk,!). The coefficients !m and " are calculated by fitting an autoregressive

model to the original Doppler spectrum proposed by Clarke [24] given by E[|h!k,!(t!m)hk,![t]|] =

J0(2#fDTsm) where fDTs is the channel’s normalized Doppler spread1 and J0 is the 0-th order

Bessel function of the first kind [25]. In some cases, for simplicity of exposition, we restrict our

attention to the special case of the first order autoregressive model where channels follow the

relation hk,![t] = "fhk,![t!m] +"1! "2fzk,![t] with "f = J0(2#fDTs).

III. INTERFERENCE ALIGNMENT IN FREQUENCY

In this section we review the concept of IA over frequency extensions when perfect channel

state information is available at the transmitter (CSIT), and summarize the effect of imperfect

CSIT on the performance of IA.

A. IA with Perfect CSI at the Transmitter

IA for the SISO interference channel can achieve the maximum degrees of freedom defined aslim

P"#1

Nsc

Rsum

log2 P= K

2 when coding over infinite channel extensions [2]. Using IA over N frequency

extensions, each transmitter k at time t sends dk < N symbols, xdk[t], along the N "1 precoding

vectors fk,d[t]. As a result, the input-output relation is

xk[t] =dk!

d=1

fk,d[t]xdk[t], (4)

where #fk,d[t]#2 = 1, and E#|xd

k[t]|2$= NP/dk, such that the total power in each N subcarriers

is NP . The transmit directions fk,d[t] are calculated such that the interference from K ! 1 users

is aligned at all receivers, leaving interference free dimensions for the desired signal.

Note that users need not code over all Nsc subcarriers, due to increasing algorithm complexity

and marginal gains with increasing dimensions. Users could potentially treat each N subcarriers

1fD is the channel’s Doppler spread, and Ts is the feedback interval, Ts = 1 OFDM symbol time, for example


Throughout this paper, we consider temporally correlated Gaussian channels according to the

P -order autoregressive channel model defined in [23]. For such processes, each time series is

generated as

hk,![t] =P!

m=1

!mhk,![t!m] + "zk,![t], (3)

where zk,l[t] = CN (0,Rhk,!). The coefficients !m and " are calculated by fitting an autoregressive

model to the original Doppler spectrum proposed by Clarke [24] given by E[|h!k,!(t!m)hk,![t]|] =

J0(2#fDTsm) where fDTs is the channel’s normalized Doppler spread1 and J0 is the 0-th order

Bessel function of the first kind [25]. In some cases, for simplicity of exposition, we restrict our

attention to the special case of the first order autoregressive model where channels follow the

relation hk,![t] = "fhk,![t!m] +"1! "2fzk,![t] with "f = J0(2#fDTs).

III. INTERFERENCE ALIGNMENT IN FREQUENCY

In this section we review the concept of IA over frequency extensions when perfect channel

state information is available at the transmitter (CSIT), and summarize the effect of imperfect

CSIT on the performance of IA.

A. IA with Perfect CSI at the Transmitter

IA for the SISO interference channel can achieve the maximum degrees of freedom defined aslim

P"#1

Nsc

Rsum

log2 P= K

2 when coding over infinite channel extensions [2]. Using IA over N frequency

extensions, each transmitter k at time t sends dk < N symbols, xdk[t], along the N "1 precoding

vectors fk,d[t]. As a result, the input-output relation is

xk[t] =dk!

d=1

fk,d[t]xdk[t], (4)

where #fk,d[t]#2 = 1, and E#|xd

k[t]|2$= NP/dk, such that the total power in each N subcarriers

is NP . The transmit directions fk,d[t] are calculated such that the interference from K ! 1 users

is aligned at all receivers, leaving interference free dimensions for the desired signal.

Note that users need not code over all Nsc subcarriers, due to increasing algorithm complexity

and marginal gains with increasing dimensions. Users could potentially treat each N subcarriers

1fD is the channel’s Doppler spread, and Ts is the feedback interval, Ts = 1 OFDM symbol time, for example

OFDM symbol

Note: Feedback strategy will be independent of correlation model

Can now use closed form solutions for diagonal channels or run our favorite algorithm



Problem Statement

29

Exploit time variation to improve CSI quality

Receivers quantize time varying channel

Transmitters calculate the transmit directions

Receivers feedback the quantized channels

Transmitters send payload data

Prior work on differential feedback:Differential feedback of input covariances [SacKal09]

Predictive Grassmannian feedback for MISO broadcast [InoHea11]

Solution: Differential quantization and feedback



What CSI is actually needed?

Loss in sum rate due to imperfect CSI

Manifold Structure

30


design of efficient, low overhead feedback strategies that limit the effect of imperfect CSI and

allow good IA performance.

In [12], [13], it was shown that quantizing wideband channel taps, in a manner similar to single

user MIMO quantization [28], and scaling the number of feedback bits with SNR, preserves the

system’s multiplexing gain. Scaling feedback bits, and effectively codebook size, however, leads

to increasing quantization complexity at high SNR. In [14] the authors showed that analog

feedback can be used instead of quantization based algorithms to maintain the IA sum rate

perfomance while keeping feedback complexity and overhead constant at all SNRs. This is

made possible by the fact that, in analog feedback, CSI quality improves automatically with

SNR and keeps leakage interference power bounded in the desired signal space.

While both these feedback strategies propose valid ways to maintain system performance, both

neglect the temporal correlation of the channel. This results in the inefficient use of feedback

resources. The analysis done in [12]–[14], [29], however, extends to arbitrary feedback strategies

and we therefore summarize their findings to motivate the problem and refer the reader to [12],

[14] for a more detailed derivation.

In [12], [14], it is shown that if the transmitters use the estimates ofHk,!, !k, ! " {1, 2, . . . , K},

denoted by !Hk,!, to calculate IA precoders, !fmk [t], and combiners, !wmk [t], then the mean loss in

sum rate due to imperfect CSI can be upper bounded by [14],

!Rsum #"

k,m

1

Nsclog2

#

1 +EH

$I1k,m + I2

k,m

%

"2

&

. (13)

The objective then becomes minimizing the sum leakage interference term I1k,m + I2

k,m. Using

the result from [12], the individual interference terms can be upper bounded by

NscP

dk

'''!wmk [t]

!Hk,i[t]!f !i [t]'''2=NscP

dk

'''(!wm

k [t] $!f!i [t]

)!hfk,i[t]

'''2

#NscP

dk%!wm

k [t] $ !f !i [t]%2%hfk,i[t]%

2

*

+1&

'''''hfk,i[t]

!!hfk,i[t]

%hfk,i[t]%%!h

fk,i[t]%

'''''

2,

-

=NscP

dk%!wm

k [t] $ !f !i [t]%2%hk,i[t]%2

*

+1&

'''''hk,i[t]!!hk,i[t]

%hk,i[t]%%!hk,i[t]%

'''''

2,

- ,

(14)


design of efficient, low overhead feedback strategies that limit the effect of imperfect CSI and

allow good IA performance.

In [12], [13], it was shown that quantizing wideband channel taps, in a manner similar to single

user MIMO quantization [28], and scaling the number of feedback bits with SNR, preserves the

system’s multiplexing gain. Scaling feedback bits, and effectively codebook size, however, leads

to increasing quantization complexity at high SNR. In [14] the authors showed that analog

feedback can be used instead of quantization based algorithms to maintain the IA sum rate

perfomance while keeping feedback complexity and overhead constant at all SNRs. This is

made possible by the fact that, in analog feedback, CSI quality improves automatically with

SNR and keeps leakage interference power bounded in the desired signal space.

While both these feedback strategies propose valid ways to maintain system performance, both

neglect the temporal correlation of the channel. This results in the inefficient use of feedback

resources. The analysis done in [12]–[14], [29], however, extends to arbitrary feedback strategies

and we therefore summarize their findings to motivate the problem and refer the reader to [12],

[14] for a more detailed derivation.

In [12], [14], it is shown that if the transmitters use the estimates ofHk,!, !k, ! " {1, 2, . . . , K},

denoted by !Hk,!, to calculate IA precoders, !fmk [t], and combiners, !wmk [t], then the mean loss in

sum rate due to imperfect CSI can be upper bounded by [14],

!Rsum #"

k,m

1

Nsclog2

#

1 +EH

$I1k,m + I2

k,m

%

"2

&

. (13)

The objective then becomes minimizing the sum leakage interference term I1k,m + I2

k,m. Using

the result from [12], the individual interference terms can be upper bounded by

NscP

dk

'''!wmk [t]

!Hk,i[t]!f !i [t]'''2=NscP

dk

'''(!wm

k [t] $!f!i [t]

)!hfk,i[t]

'''2

#NscP

dk%!wm

k [t] $ !f !i [t]%2%hfk,i[t]%

2

*

+1&

'''''hfk,i[t]

!!hfk,i[t]

%hfk,i[t]%%!h

fk,i[t]%

'''''

2,

-

=NscP

dk%!wm

k [t] $ !f !i [t]%2%hk,i[t]%2

*

+1&

'''''hk,i[t]!!hk,i[t]

%hk,i[t]%%!hk,i[t]%

'''''

2,

- ,

(14)

Power of “leakage” interference

Magnitude of quantized channel can be “normalized”

Phase of channel estimate doesn’t matter

Leakage interference has terms that look like

Required CSI lives on the Grassmann manifold

Structure can be used compress and improve feedback



h!"#[t]

h!"#[t-1]

e!"#[t]

h!"#[t]

h!"$[t]

Grassmannian Differential Feedback

31

Old quantized channelNew channelTangent vector between channels

Geodesic path between channelsNew quantized channel

Tangent codebook

Channels can be related by tangent vectors and

geodesic paths

^

^

^CalculateTangent

QuantizeTangent

UpdateSystem State

Observechannel

Tangentindices

Move AlongGeodesic

Move AlongGeodesic

ReceiveQuantizedTangent

QuantizedChannel

UpdateSystem State

Differential feedback decoderDifferential feedback encoder

Quantize tangent magnitude and direction separately



Scalar quantization problem

Propose uniform quantization

Adaptive range to avoid error floors

Performs well in simulations

32

Initialization: Synchronous operation of the Grassman-nian differential feedback algorithm is ensured by the factthat at each iteration, both transmitter and receiver calculate aquantized channel vector based on the same commonly avail-able knowledge. For this to hold, however, both transmitterand receiver need a common initial vector, !g(0), as input tothe algorithm, otherwise the time series observed by trans-mitter and receiver will not be coupled. This vector can bebased on a memoryless quantization of the channel [4] or ini-tialized with a common random vector.Tangent Magnitude Quantization: The tangent vector

calculated in (9) is decomposed naturally into a tangent mag-nitude and a unit norm tangent direction. In this paper, themagnitude and direction are quantized separately as it canbe shown in simulation that there is little to gain from jointquantization.

The problem of quantizing the tangent magnitude is thatof quantizing a positive scalar and is done as follows

!emag = argminei!Cmag

|!e[t]! " ei| , (11)

where Cmag is the magnitude quantization codebook. The in-dex of the minimizer is then sent to the transmitter via a delayand error free link which requires log2(|Cmag|) bits. Find-ing the exact probability density function of the magnitudesis intractable, and thus we do not seek to find an optimalquantization codebook. One solution is to uniformly quan-tize a range of magnitudes, !e[t]! # [0, 1], which is subopti-mal. For example, [9] has observed from simulations that inhighly correlated channels, where such feedback strategiesare most useful, quantization error in the magnitude domi-nates the error in tangent direction.

Motivated by the correlation between magnitudes in con-secutive iterations, we propose to adapt the quantizationrange to the dynamics of the system. Given the magni-tude of a tangent at time t, !!e[t]!, the codebook at timet+1 becomes a uniform quantization codebook in the range[!!!e[t]!,min {"!!e[t]!,#/2}], where 0 < ! < 1 < " arefixed parameters of the codebook. This allows the feedbackalgorithm to accurately track the statistics of the magnitudeand quantize the current range of magnitudeswith higher res-olution. In static channels, this allows our approach to con-verge to perfect CSI.Tangent Direction Quantization: The problem of quan-

tizing the tangent direction vector is that of quantizing a unitnorm vector which lies in the tangent space orthogonal tothe base vector !g[t " 1]. General vector quantization code-books, such as a random vector codebook, can not be usedto quantize the tangent directly for several reasons. First,traditional codebooks quantize the full L dimensional spacewhereas the tangent vector is of lower dimension. Further,traditional codebooks do not enforce the structural constraintthat requires the tangent direction codewords to be orthogo-nal to the base vector !g[t]. With such a non-orthogonal tan-gent vector, the geodesic path is undefined and the output ofG(!g[t"1],!e[t], $) does not lie on the manifold. Finally, notethat the tangent space changes for each base vector, whichnecessitates an adaptive codebook.

To respect the varying tangent space geometry and or-thogonality constraints, we propose to use a canonical gen-erating codebook to be adapted at each iteration. The code-book design provided allows perfectly projecting a canonical

codebook onto the tangent plane at each iteration. This en-sures that the output of the Grassmannian differential feed-back algorithm remains on the manifold. We define a canon-ical tangent codebook as Cgen which has |Cgen| = N unitnorm vector entries x1, . . . ,xN . This vector codebook canbe any L dimensional vector codebook whose entries spanthe full L-dimensional space, such as the random codebook.At each iteration we form a codebook, Cdir, with entries or-thogonal to the base vector by using a projection operation.

Definition 1 The normalized projection matrix function

P(x,xb) =IL " xbx

"b"

1" (x"xb)2x,

computes the closest unit vector to x that is also orthogonalto the base vector xb.

Definition 2 The tangent direction codebook, Cdir(!g[t"1]),for the base point !g[t" 1] is

Cdir(!g[t" 1]) = {P(x1, !g[t" 1]), . . . , P(xN , !g[t" 1])} .

To construct good tangent codebooks, note that if the changein the channel is assumed to be isotropic, then it can be shownthat the tangent direction vector is also isotropically dis-tributed in the tangent space. This motivates finding canon-ical codebooks that lead to an isotropic distribution in thetangent space. Further improving the direction codebook de-sign, or constructing an optimal one, is left for future work.

To formalize the tangent direction quantization, recallthat the quantized channel in the next time instant will becalculated as G(!g[t " 1],!e[t], 1). Given that the loss in sumrate is related to the chordal distance between the actual andquantized channel, the quantized tangent direction will begiven as

!edir = argminxi!Cdir(!g[t#1])

d(G(!g[t" 1], !emagxi, 1),g[t]) (12)

where the tangent magnitude, !emag, is given by the output ofthe magnitude quantization step.

5. SIMULATION RESULTSIn this section we present simulation results to demonstratethe performance of IA when channel knowledge at the trans-mitter is obtained via the Grassmannian differential feedbackstrategy detailed in Section 4. To remove the limitation of aper-stream receiver, we calculate the sum rate of a decoderwhich considers all desired symbols jointly and treats leak-age interference as colored Gaussian noise. Since the fre-quency extended system can be viewed as a virtualNsc$NscMIMO system, the sum rate achieved is given by,

Rsum =K!

k=1

1Nsc

log2

"""I+#!2I+Rk

$!1

(Hk,kFkF"

kH"

kk)""" ,

where Rk =#

i$=k Hk,iFiF"iH

"k,i is the interference co-

variance matrix and the precoders, Fk =$f1k , f

2k , . . . , f

dk

k

%,

are calculated given ideal or quantized CSI. For the resultsin this section, we use the IA algorithm in [2]. Althougha closed form solution for the IA precoders exists for theSISO frequency extended interference channel in [1], it can





|!e[t]! " ei| , (11)







P(x,xb) =IL " xbx

"b"

1" (x"xb)2x,







d(G(!g[t" 1], !emagxi, 1),g[t]) (12)



Rsum =K!

k=1

1Nsc

log2

"""I+#!2I+Rk

$!1

(Hk,kFkF"

kH"

kk)""" ,

where Rk =#

i$=k Hk,iFiF"iH



2k , . . . , f

dk

k

%,






|!e[t]! " ei| , (11)







P(x,xb) =IL " xbx

"b"

1" (x"xb)2x,







d(G(!g[t" 1], !emagxi, 1),g[t]) (12)



Rsum =K!

k=1

1Nsc

log2

"""I+#!2I+Rk

$!1

(Hk,kFkF"

kH"

kk)""" ,

where Rk =#

i$=k Hk,iFiF"iH



2k , . . . , f

dk

k

%,






|!e[t]! " ei| , (11)







P(x,xb) =IL " xbx

"b"

1" (x"xb)2x,







d(G(!g[t" 1], !emagxi, 1),g[t]) (12)



Rsum =K!

k=1

1Nsc

log2

"""I+#!2I+Rk

$!1

(Hk,kFkF"

kH"

kk)""" ,

where Rk =#

i$=k Hk,iFiF"iH



2k , . . . , f

dk

k

%,


Magnitude codebook at time t+1

Constants chosen by user

Quantizing the Tangent Magnitude



Tangent direction codebook considerationsUnit norm vector quantization

Tangent space changes with channel

Orthogonality can not be violated

33

Result: Can not use constant

“Grassmannian codebooks”

Need to adapt

Need to be on tangent space

Quantizing the Tangent Direction



Canonical direction codebook

Projection Function

Resulting codebook:

34





|!e[t]! " ei| , (11)







P(x,xb) =IL " xbx

"b"

1" (x"xb)2x,







d(G(!g[t" 1], !emagxi, 1),g[t]) (12)



Rsum =K!

k=1

1Nsc

log2

"""I+#!2I+Rk

$!1

(Hk,kFkF"

kH"

kk)""" ,

where Rk =#

i$=k Hk,iFiF"iH



2k , . . . , f

dk

k

%,






|!e[t]! " ei| , (11)







P(x,xb) =IL " xbx

"b"

1" (x"xb)2x,







d(G(!g[t" 1], !emagxi, 1),g[t]) (12)



Rsum =K!

k=1

1Nsc

log2

"""I+#!2I+Rk

$!1

(Hk,kFkF"

kH"

kk)""" ,

where Rk =#

i$=k Hk,iFiF"iH



2k , . . . , f

dk

k

%,






|!e[t]! " ei| , (11)







P(x,xb) =IL " xbx

"b"

1" (x"xb)2x,







d(G(!g[t" 1], !emagxi, 1),g[t]) (12)



Rsum =K!

k=1

1Nsc

log2

"""I+#!2I+Rk

$!1

(Hk,kFkF"

kH"

kk)""" ,

where Rk =#

i$=k Hk,iFiF"iH



2k , . . . , f

dk

k

%,


The tangent space’s normal vector

Projection adapts the codebook to the channel

Can be a random codebook

Tangent Codebook Construction



24

0 50 100 150 200 250 3000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Channel Realization

Chor

dal D

istan

ce

GPC (6+2) bitsRVQ with 8 bits

Proposed Algorithm (6+2) bits

Fig. 2. Chordal distance, d(!g(t), g(t)), plotted over time for a slowly channel with normalized Doppler fDTs = 0.003.This shows the high quality CSI acheived by the proposed algorithm with 7 and 3 bits for channel direction and magnitude

respectively, compared to [15] and memoryless quantization with a random codebook of 10 bits.

10!4 10!3 10!2 10!110!3

10!2

10!1

100

fd

Ts (Normalized Doppler)

Chor

dal D

istan

ce (Q

uant

izat

ion

Erro

r)

Chordal Distance vs. Bits for Magnitude

Proposed algorithm

GPC

Fig. 3. This shows the average chordal distance, or quantization error, versus the number of bits allocated for the tangent

magnitude (1 to 10 bits), while keeping direction bits fixed. This shows the good performance of the proposed algorithm when

compared with [15].

35

Previous work:Performance limited by magnitude quantization

Proposed work:No use spending more than 1 bit on magnitude

Very close to perfect magnitude knowledge

[AyaHea11]

• Simulation parameters‣ 3 tap channel‣ uniform power profile

Note: previous work in [InoHea11]

Performance Evaluation



25

10!4 10!3 10!2 10!110!3

10!2

10!1

100

fD

Ts (Normalized Doppler)

Chor

dal D

istan

ce (Q

uant

izat

ion

Erro

r)

Chordal Distance vs. Direction Bits (3 to 10)

Proposed algorithm

GPC

Fig. 4. This shows the average chordal distance, or quantization error, versus the number of bits allocated for the tangent

direction, while keeping magnitude bits fixed. This shows the good performance of the proposed algorithm when compared with

[15].

0 5 10 15 20 25 30 35 400

2

4

6

8

10

12

14

16

18

SNR (dB)

Sum

Rat

e (b

its/s

/Hz)

IA w/ Perfect CSIIA with GDC f

dT

s=0.01

IA with GDC fd

Ts=0.001

IA with GDC fd

Ts=0.03

IA with GDC fd

Ts=0.05

IA with RVQRandom Beamforming

Fig. 5. This figure shows the performance of IA with imperfect CSI obtained through the proposed algorithm, as well as

random vector quantization. This shows that for slowly varying channels, the proposed algorithm allows interference alignment

networks to come very close to the perfect CSI upper bound.

36

Previous work:Error floor obvious

at 0.01 Doppler

Proposed work:No error floor

for > 3 bits

Slightly diminishing returns with random

codebooks

Benefits from more bits

• Simulation parameters‣ 3 tap channel‣ uniform power profile We can also analytically characterize performance

(see journal paper)



(c) Robert W. Heath Jr. 2011 37

Approaches perfect CSI

Outperforms competingpredictive algorithms

Outperforms memoryless quantization

at significant Doppler

• 3 tap channel• uniform power profile• (7+3) feedback bits• 16 channel extensions• IA via alternating min.

Proposed strategy works well even at 40dB

Codebook sizes are manageable



(c) Robert W. Heath Jr. 2011 38

• 3 tap channel• uniform power profile• 16 channel extensions• IA via alternating min.

IA w/ Perfect CSI

IA w/ analog feedback (example)IA with RVQ

IA w/ GDC fDTs =0.01IA w/ GDC fDTs=0.03IA w/ GDC fDTs=0.05

IA w/ GDC fDTs=0.001

0 5 10 15 20 25 30 35 402

4

6

8

10

12

14

16

18

20

SNR (dB)

Sum

Rat

e (b

/s/H

z)

Incomple

te Compar

ison

Comparison depends onFeedback powerFeedback channel capacityCoding schemeFeedback bit errors

Here we use(7+3) feedback bits

orPfb =P/2

Could do analog differential feedback...

Performance Comparison



Proposed Grassmannian differential feedback

Exploits temporal correlation

Reduces feedback overhead, making limited feedback more practical

Approaches perfect CSI in slow fading channels

Outperforms other feedback schemes that exploit correlation

Main limitation: training (overhead) is still incurred

Further work

Improved magnitude and direction codebooks

Analytical performance results and characterization

39

Part #2: Summary



Introduction to interference alignment



Conclusions

40

Outline



ConclusionsHighlighted the CSI requirements in IA networks

Presented methods to reduce overhead (not just feedback)

Proposed analog feedback for MIMO IA

Exploits optimality of analog transmission

Provides a low complexity feedback method

Proposed Grassmannian differential feedback for OFDM IA

Exploits CSI structure and correlation

Works well with very small codebooks

41


What are the remaining IA killers...





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[GolPerKat09] S. Gollakota, S. Perli, D. Katabi “Interference alignment and cancellation. In Proceedings of the ACM SIGCOMM 2009 Conference on Data Communication (Barcelona, Spain, August 16 - 21, 2009). SIGCOMM '09. ACM, New York, NY,

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