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Robust Optimal Cross Layer Designs for TDD-OFDMA Systems with Imperfect CSIT and

Unknown Interference— State-Space Approach based on 1-bit ACK/NAK Feedbacks

Wang Rui, Vincent K. N. LauDepartment of EEE, The Hong Kong University of Science &

Technology

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Outline

Slow Fading OFDMA System ModelAverage System Goodput &Problem

FormulationCross-layer Scheduler Design with ARQ

FeedbackSimulation Results and Conclusions

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System Model• We Consider the downlink transmission of

a TDD-OFDMA system with slow fading channel and Gaussian interference.

• The channel is quasi-static within a scheduling slot which consists of N packet bursts.

• The channel model is given by the following equation, where the k, m, n are the indices of user, subcarrier and packet bursts respectively.

nmknmknmmknmk IZXhY ,,,,,,,,

Received Symbol

NoiseInterference

Transmitted Symbol

BS

...User 1

User 2

User 3

1 N...432 1 N...432

Scheduling Slot n Scheduling Slot n+11 N...432

Scheduling Slot n+2

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CSIT Error Model

• The CSIT estimation is done (at the base station) at the very beginning of each scheduling slot only based on dedicated uplink pilots from the mobiles.

• The channel state information at the transmitter (CSIT) is imperfect.– Due to CSIT estimation noise from uplink pilots.– Due to Outdatedness of CSIT estimation (duplexing delay).– Although the CSIT is imperfect, it still can be used to improve the system

performance.

• A general model of CSIT error is given below:

mkmkb

mk hh ,,,

CSIT ErrorActual CSI

Estimated CSIT

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Packet Error Model (1)• In slow fading channels, there are two reasons of packet error

– Finite block length of channel coding [channel noise effect]– Transmitted data rate exceeding the instantaneous mutual information of the channel [channel outage]

• By applying strong channel coding (e.g. LDPC) with reasonable block length (e.g. 2k byte), it can be shown that Shannon’s limit can be achieved to within 0.05dB for a target FER of 10^{-2}. the effect of channel noise can be ignored with strong coding.

• Yet, the second factor (channel outage) is systematic and will be the major contributor of packet error (esp when strong coding is used). Hence, we assume Packet Error Rate = Pr [r > mutual information].

• In most existing cross layer design [Lau:06,Wong:99], perfect knowledge of CSIT is assumed. System ergodic capacity is used as optimization objective and the potential penalty of packet errors is completely ignored in cross layer design. This is reasonable because packet error can be ignored through strong channel coding and rate adaptation.

• However, it is shown [Lau:06b] that “naïve cross layer designs” (designed for perfect CSIT) all have very poor performance when we have imperfect CSIT. The key contributions to the performance penalty is packet errors.

• Hence, when we have imperfect CSIT, the issue of potential packet errors cannot be ignored and must be taken into the cross layer design.

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Packet Error Model (2)• The instantaneous mutual information of user k at the n-th subcarrier is given by:

– With imperfect CSIT, the BS does not know the exact mutual information.– It’s possible that the scheduled data rate is larger than the mutual information, which leads to channel outage.

• Hence, when strong coding (LDPC) is used, the packet error probability (PER) is given by:

• To account for penalty of packet errors, we shall consider system goodput (b/s/Hz successfully delivered to the mobiles) as our optimization objective.

• In [Palomar:06, Lau:06c], the authors proposed a cross layer design that optimize the system goodput (rather than ergodic capacity) and the cross layer design achieves very good performance even at high CSIT errors.

– Yet, the design requires knowledge of the CSIT error statistics (which is sometimes difficult to obtain and time varying as well).

– In this paper, we shall extend the work to consider a robust cross layer design which account for potential packet errors in the cross layer but without requiring the knowledge of CSIT error variance

– Utilizing the ACK/NAK feedbacks, the cross layer design is modeled as a Closed-Loop State Space Control Problem

22

2,,

,,2,,

||1log

Iz

nmknmknmk

hpC

b

Iz

nmknmknmk

b hh

prhPER |||

1logPr)( 22

2,,

,,2,,

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Cross Layer System Model (1)• There are ACK/NAK feedbacks after each packet transmission.• MAC layer is responsible to select the appropriate user on each subcarrier in

each packet burst and schedule the corresponding data rate and transmit power according to the channel estimation and the ACK/NAK feedback.

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Outline

Slow Fading OFDMA System ModelAverage System Goodput &Problem

FormulationCross-layer Scheduler Design with ARQ

FeedbackSimulation Results and Conclusions

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Problem Formulation --- Goodput

• Due to the potential packet outage, we use the goodput, which measures bit/sec/Hz successfully delivered to the receiver, to measure the system performance.

• The instantaneous goodput of user k in the m-th subcarrier and n-th packet burst is

• The average system goodput (Optimization objective) is

• Given the CSIT, the conditional average goodput is

,

b bh , ,

1 1

h | hm n

N M

A m nn m

G

][ ,,,,,,,, nmknmknmknmk rCIr I[] is 1 when the event is true and 0 otherwise.

b b, ,

b b, , h , ,h h

1 1 1 1

, , | h hm n m n

N M N M

A m n A m nn m n m

U A P R G

Am,n denotes the selected user

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Problem Formulation --- Policies• The average system goodput U is a function of the user selection poli

cy A, power allocation policy P and rate allocation policy R.

• User selection policy A: determine the active user for each subcarrier and each packet burst according to the CSIT and ACK/NAK feedbacks

• Power allocation policy P: determine the transmit power for active users according to the CSIT and ACK/NAK feedbacks.

• Rate allocation policy R: determine the transmit data rate for active users according to the CSIT and ACK/NAK feedbacks.

packets previous theof feedbacks ,,b

nm hA

, , , ,feedbacks of the previous packets

m n

bA m np h

, , , R ,feedbacks of the previous packets

m n

bA m nr h

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}},...,{,{ 1,,1,,,,, nmkmkb

mknmk ffhs

Problem Formulation --- States• Define the system states to be the combination of C

SIT and the feedbacks.• The state of user k in the m-th subcarrier before the

n-th packet transmission is give by:

• The states are updated after each packet burst.• The selected users, transmit power and data rate are

completely determined by the system states Sn:},|{ ,, mksS nmkn

The ACK/NAK feedback of the (n-1)-th packet

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Problem Formulation --- Objective

• The optimal user selection policy A*, the optimal power allocation policy P* as well as the optimal rate allocation policy R* are given by:

– subject to the following constraint:• Total transmit power constraint:

• The conditional packet error probability of all users is less than a target ε

* * *

, ,, , arg max , ,

A P RA P R U A P R

0,,

,, Ppnmk

nmk

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Outline

Slow Fading OFDMA System ModelAverage System Goodput &Problem

FormulationCross-layer Scheduler Design with ARQ

FeedbackSimulation Results and Conclusions

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Solutions --- Recursive Relationship• Let Fn be the conditional average goodput from the n-th pa

cket burst to the N-th packet burst. It can be expressed recursively.

• F1 equals to the conditional average goodput G. • The optimization on F1 can be solved by Markov Decision

Process in two steps.– Backward recursive: derive the close form of F and the user select

ion, power and rate allocation in terms of the arbitrary system state.

– Online strategy: according to the current system state and the result of backward recursion, determine the current user selection, power and rate allocation.

1

),h,()h,|Pr(),h,(),h,( b1

b1

bb

nSnnnnnnnnnn SpPFSSSpgSPF

Conditional average goodput of the n-th packet burst

Total Goodput (nN) = goodput (n) + f(Total Goodput (n+1 N)

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Solutions• The optimal user selection policy, rate allocation policy and p

ower allocation policy is given below:

• θ is the scaling factor to guarantee the target outage probability.

• λn is the Lagrange multiplier to guarantee the total power of the n-th packet burst to be pn:

nmnmkn

nmk

nmknmknmk

N

niimk

knm

Akp

pr

A

,,,

,,

,,,,2,,

,,,

for 11

)1(log

maxarg

M

m nmk

N

ni

M

mimAn imnNnN

Pp1 ,,1

,,1

11

1 ,

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Solutions

• For sufficiently large number of feedbacks N, we can prove that

• Hence, the system will perform as if the CSIT were perfect in steady state Zero Steady State Error.

• The ACK/NAK feedbacks can compensate the inaccuracy of the CSIT.

22

2,,

,,,

||maxlim

Iz

nmknmkknmn

hpA

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Outline

Slow Fading OFDMA System ModelAverage System Goodput &Problem

FormulationCross-layer Scheduler Design with ARQ

FeedbackSimulation Results and Conclusions

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Simulation Results

}2,1,1.0{ 4 1.02 IMH

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Simulation Results

}2,1{ 4 1.02 IMH

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Conclusions

• The performance of the proposed closed-loop cross layer design is very robust with respect to imperfect CSIT, unknown interference as well as channel variation due to Doppler.

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Reference• [Lau:06] V. K. N. Lau, “Coverage-optimized downlink scheduling design for

wireless systems with multiple antennas”, IEEE Trans. On Wireless Communication , Oct. 2006.

• [Wong:99] C. Y. Wong, R. S. Cheng, K. B. Lataief, R. D. Murch, “Multiuser OFDM with adaptive subcarrier, bit, and power allocation”, IEEE J. Sel. Areas Commun. , Oct. 1999.

• [Lau:06b] V. K. N. Lau, M. L. Jiang, Y. J. Liu, “Cross layer design of uplink multi-antenna wireless systems with outdated CSI”, IEEE Trans. On Wireless Communication , June 2006.

• [Lau:06C] V. K. N. Lau, M. L. Jiang, “Performance analysis of multiuser downlink space-time scheduling for TDD systems with imperfect CSIT”, IEEE Trans. On Vehicular Technology , Jan. 2006.

• [Palomar:06] A. Pascual-Iserte, D. P. Palomar, A. I. Perez-Neira, M. A. Lagunas, “A robust maximin approach for MIMO communications with imperfect channel state information based on convex optimization”, IEEE Trans. On Signal Processing, Jan. 2006.

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Thank You !