Introduction System Model Energy efficiency function

Energy efficient communications over MIMO channels Elena-Veronica Belmega1 , Samson Lasaulce1 and M´erouane Debbah2 1 Laboratoire des signaux et syst` emes (joint lab of CNRS, Sup´ elec, Univ. Paris-Sud 11) Gif-sur-Yvette, France 2 Alcatel-Lucent Chair on Flexible Radio, Sup´ elec Gif-sur-Yvette, France

10/05/2010

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Introduction System Model Energy efficiency function

Energy efficient communications

The performance of a system is often measured as the ratio between what the system delivers and what it consumes In communication theory: transmit power and Shannon transmission rate are two measures of the cost and benefit of a communication The ratio transmission rate (say in bit/s) to transmit power (in J/s) appears to be a natural energy efficiency measure Question: what is the maximum amount of information (in bits) that can be reliably conveyed per Joule consumed?

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Introduction System Model Energy efficiency function

Pragmatic vs. Information theoretic view Pragmatic approach: [Shah-pimrc-1998][Goodman-pcom-2000] Energy efficiency function for SISO quasi-static channel with perfect channel state information F (p, R) =

LRf (γ) , Mp

where L the information bits, M the packet size, R the transmission rate, γ the received signal-to-noise ratio (SNR), f (γ) = (1 − e −γ )M the probability of correct reception Using the S-shaped property of f (γ), F (p, R) is quasi-concave w.r.t. p and the optimal is not trivial p ∗ > 0 [Rodriguez-globecom-2003] Extension to multi-carrier CDMA [Meshkati-jsac-2006], MIMO channel with beamforming receiver [Buzzi-eusipco-2008] Information theoretic approach: [Verdu-it-1990] Capacity per unit cost for the SISO staticGaussian channel is given by   log2 1+ P2

log2 1+ P2

1 σ σ supP>0 . Similarly to minimum = limP→0 P P σ 2 ln 2 energy-per-bit approach in [ElGamal-it-2006][Verdu-it-2002].

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Introduction System Model Energy efficiency function

Multiple input multiple output (MIMO) channels Received baseband signal: y (τ ) = H(τ )x(τ ) + z(τ ) H is the nr × nt channel matrix with i.i.d. CN (0, 1) entries, z the nr -dimensional Gaussian noise CN (0, σ2 I) Three classes of channels: 1 static 2 fast fading 3 slow fading H is perfectly known at the receiver (CSIR) and only statistics are available at the transmitter (CDIT) Transmit power constraint: Tr(Q) ≤ P where Q = E[xx H ] input covariance matrix Objective: Study the power allocation policy (Q∗ ) that maximizes an information theoretic energy-efficiency function.

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Introduction System Model Energy efficiency function

Static channels Fast fading channels Slow fading channels

H is a constant matrix The energy efficiency function log2 I + ρHQHH Gstatic (Q) = Tr(Q) where ρ =

1 σ2

The capacity per unit cost [Verdu-it-1990] for static MIMO channel corresponds to the optimal energy efficiency function ∗ The optimal covariance matrix Q∗ → 0 and Gstatic →

H 1 Tr(HH ) ln 2 σ2

The optimal power strategy is to transmit with very small power. Problem: the transmission rate will also be very small.

Applicable in the case of sensor networks Not acceptable for applications where a minimum transmission rate is required (cellular networks, satellite communications,...) 5 / 18

Introduction System Model Energy efficiency function

Static channels Fast fading channels Slow fading channels

H varies with each channel use The energy efficiency function:   E log2 I + ρHQHH Gfast (Q) = Tr(Q) The capacity per unit cost for fast fading MIMO channel corresponds to the optimal energy efficiency function ∗ The optimal covariance matrix Q∗ → 0 and Gfast →

H 1 Tr(E[HH ]) ln 2 σ2

For static and fast fading cases possible solution: introduce a minimum rate constraint.

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Introduction System Model Energy efficiency function

Static channels Fast fading channels Slow fading channels

H varies with each codeword/packet Shannon achievable rate is zero The energy efficiency function: Γ(Q, R) =

R[1 − Pout (Q, R)] Tr(Q)

where R(1 − Pout (Q, R)) is the expected throughput [Katz-it-2005] and Pout (Q, R) is the outage probability [Ozarow-vt-1994]:   Pout (Q, R) = Pr log2 I + ρHQHH < R

The outage probability Pout (Q, R) minimization problem w.r.t. Q is still an open problem. Main difficulty: intractable p.d.f. and c.d.f. of the instantaneous mutual information.

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Introduction System Model Energy efficiency function

Static channels Fast fading channels Slow fading channels

Conjectures Telatar’s conjecture [Telatar-ett-1999]: the optimal transmit policy that minimizes Pout (Q, R) is to spread all the available power over a subset of k = k(R, σ2 ) antennas: Q∗ = Pk diag(e k ), where e k is any nt −dimension vector with zero/one entries s.t. the number of ones equals to k. . Our conjecture: Conjecture (Optimal precoding matrices) There exists a power threshold P 0 such that: if P ≤ P 0 then Q∗ ∈ arg minPout (Q, R) ⇒ Q∗ ∈ arg max Γ(Q, R) (maximizing Q

Q

Γ(Q, R) is equivalent to minimizing Pout (Q, R)); ∗

if P > P 0 then Γ(Q, R) has a unique maximum in Q∗ = pn I where p ∗ ≤ P t (uniform power allocation over all the antennas is optimal but not necessarily using all the available power). Remark: there is no loss of optimality by restricting the search for optimal precoding matrices to diagonal matrices, Q = UDUH with U unitary and D = Diag(p) diagonal matrix with p = (p1 , . . . , pnt ). 8 / 18

Introduction System Model Energy efficiency function

Static channels Fast fading channels Slow fading channels

Particular cases The conjectures have been solved for: Outage probability: MISO (nt ≥ 1, nr = 1) in [Jorswieck-ett-2006], TISO (nt = 2, nr = 1) in [Katz-it-2007] Energy efficiency: MISO and SIMO (nt = 1, nr ≥ 1) cases Proposition (Optimum precoding matrices for MISO channels) The optimum precoding matrices that maximize the energy efficiency function have the following form: P Diag(e ` )  ` Q∗ = 2 R σ (2 −1) P min , I νn t nt

if P ∈ if P ≥

h

c , c c`−1 c`

c cnt −1



(1)

where = σ2 (2R − 1), ifor all h` ∈ {1, ..., nt − 1}, h cP i c` is the unique solution w.r.t. x of P` 1 2 ≤ x − Pr 1 2 ≤ x = 0 where X are i.i.d. zero-mean Pr `+1 `+1 |X | |X | i i i i =1 i =1 ` Gaussian random variables with unit variance. By convention c0 = +∞, cnt = 0 and nt P t −1 y i = 0. νnt is the unique solution w.r.t. y of (n y−1)! − in=0 i! t

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Introduction System Model Energy efficiency function

Static channels Fast fading channels Slow fading channels

Exemple MISO case, nt = 3, nr = 1, uniform power allocation over a subset of ` ∈ {1, 2, 3} antennas, ρ = 10 dB, R = 3 bpcu.

2.5

0.8

0.6

Energy Efficiency Γ [bit/Joule]

Success probability 1 − Pout

1

c/c =0.5571 W 1

c/c =0.612 W 2

0.4

l=1 l=2 l=3

0.2

0 0

0.2

0.4

0.6

0.8

1

1.2

Power constraint P [W]

1.4

1.6

1.8

2

2

c/c1=0.5571 W c/c =0.612 W 2

1.5

1

l=3 l=2 l=1

0.5

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Power constraint P [W]

The outage probability and energy-efficiency function have similar optimum points.

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Introduction System Model Energy efficiency function

Static channels Fast fading channels Slow fading channels

Numerical simulation

MIMO, nt = nr = 2, R = 1 bpcu, ρ = 3 dB.

2.5

Energy Efficiency Γ [bit/Joule]

Success probability 1 − Pout

1 P0=0.16 W

0.8

0.6

0.4 Uniform PA Beam−forming PA Optimal PA

0.2

0 0

0.1

0.2

0.3

0.4

0.5

0.6

Power constraint P [W]

0.7

0.8

0.9

1

P =0.16 W 0

2

1.5

1 Uniform PA Beam−forming PA Optimal PA

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0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Power constraint P [W]

The conjecture is valid for the 2x2 MIMO channel.

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Introduction System Model Energy efficiency function

Static channels Fast fading channels Slow fading channels

Uniform Power Allocation (UPA), Q =

p nt I

If Telatar’s conjecture is proven, then Q∗ has the same structure as the covariance matrix minimizing the outage probability except that using all the ∗ available power is not necessarily optimal, p ∗ ∈ [0, P], D∗ = p`∗ Diag(e `∗ ) To find the optimal p ∗ one has to study the UPA case for a smaller size MIMO system (i.e., `∗ × nr instead of nt × nr ) Conjecture (UPA, quasi-concavity for MIMO channels) Assume the UPA, Q =

p nt

h i I then ΓUPA (p, R) is quasi-concave w.r.t. p ∈ 0, P .

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Introduction System Model Energy efficiency function

Static channels Fast fading channels Slow fading channels

Quasi-concavity

Particular cases where the result is proven: MISO case Large systems in terms of antennas (e.g., nt → +∞, nr → +∞ with nr = β < +∞ where the mutual information can be lim ni →+∞,i ∈{t,r} nt approximated with a Gaussian random variable [Debbah-it-2003]) Low and high SNR regimes (ρ → 0, ρ → +∞) Distributed multiuser channels: MIMO multiple access channels with single-user decoding at the receiver, the corresponding distributed power allocation game where the transmitters’ utility functions are their energy efficiency is guaranteed to have a Nash equilibrium following the Debreu-Fan theorem [Fundenberg-book-1991]

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Introduction System Model Energy efficiency function

Static channels Fast fading channels Slow fading channels

Numerical results Energy efficiency vs. transmit power p ∈ [0, 1] W for MIMO nr = nt = n ∈ {1, 2, 4, 8}, UPA D = np I, ρ = 10 dB, R = 1 bpcu. Energy Efficiency ΓUPA(p, R) [bit/Joule]

t

90

*

ΓUPA(p ,R) =80,3 bit/Joule

n=8 n=4 n=2 n=1

p* = 12 mW

80 70 60 50 *

ΓUPA(p ,R) =31,3 bit/Joule

40

*

p = 26 mW *

ΓUPA(p ,R) =10,6 bit/Joule

30

*

p = 64 mW

20

(p*,R) =3,7 bit/Joule

Γ

UPA

10 0 0

p* = 100 mW

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Transmit power p [W]

The energy efficiency is a quasi-concave w.r.t. p. The optimal point p ∗ is   ∗ function decreasing and Γ pn , R is increasing with n. t

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Introduction System Model Energy efficiency function

Static channels Fast fading channels Slow fading channels

Related publications:

E.V. Belmega, S. Lasaulce, M. Debbah and A. Hjorungnes, “A New Energy Efficiency Measure for Quasi-Static MIMO Channels”, invited paper, International Wireless Communications and Mobile Computing Conference (IWCMC), Leipzig, Germany, Jun. 2009. E.V. Belmega, S. Lasaulce, “An information-theoretic look at MIMO energy-efficient communications”, ACM Proc. of the International Conference on Performance Evaluation Methodologies and Tools (VALUETOOLS), Pisa, Italy, Oct. 2009. To be done: Solve the conjectures for the general MIMO slow fading channels Study energy efficiency games for multiuser MIMO multiple access channels and MIMO interference channels For the slow fading and fast fading cases introduce minimum rate constraints

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Introduction System Model Energy efficiency function

Static channels Fast fading channels Slow fading channels

References (1)

[Verdu-it-1990] S. Verd´ u, “On channel capacity per unit cost”, IEEE Trans. on Inf. Theory, vol. 36, no. 5, pp. 1019–1030, Sep. 1990. [Katz-it-2005] M. Katz and S. Shamai, “Transmitting to colocated users in wireless ad hoc and sensor networks”, IEEE Trans. on Inf. Theory, vol. 51, no. 10, pp. 3540–3562, Oct. 2005. [Goodman-pcom-2000] D. J. Goodman, and N. Mandayam, “Power Control for Wireless Data”, IEEE Personal Communications, vol. 7, pp. 48–54, Apr. 2000. [ElGamal-it-2006] A. El Gamal, M. Mohseni, S. Zahedi, “Bounds on capacity and minimum energy-per-bit for AWGN relay channels”, IEEE Trans. on Inf. Theory, vol. 52, no. 4, pp. 1545–1561, Apr. 2006. [Shah-pimrc-1998] V. Shah, N. B. Mandayam and D. J. Goodman, “Power control for wireless data based on utility and pricing”, IEEE Proc. of the 9th Intl. Symp. Personal, Indoor, Mobile Radio Communications (PIMRC), Boston, MA, pp. 1427–1432, Sep. 1998. [Saraydar-tcom-2002] C. U. Saraydar, N. B. Mandayam, and D. J. Goodman, “Efficient power control via pricing in wireless data networks”, IEEE Trans. on Communications, vol. 50, No. 2, pp. 291–303, Feb. 2002. [Meshkati-jsac-2006] F. Meshkati, M. Chiang, H. V. Poor, and S. C. Schwartz, “A Game-Theoretic Approach to Energy-Efficient Power Control in Multi-Carrier CDMA Systems”, IEEE Journal on Selected Areas in Communications, vol. 24, no. 6, pp. 1115–1129, Jun. 2006.

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Introduction System Model Energy efficiency function

Static channels Fast fading channels Slow fading channels

References (2) [Ozarow-vt-1994] L. H. Ozarow, S. Shamai, and A. D. Wyner, “Information theoretic conisderations for cellular mobile radio”, IEEE Trans. on Vehicular Technology, vol. 43, no. 2, pp. 359–378, May 1994. [Verdu-it-2002] S. Verd´ u, “Spectral efficiency in the wideband regime”, IEEE Trans. on Inf. Theory, vol. 48, no. 6, pp. 1319–1343, Jun. 2002. [Buzzi-eusipco-2008] S. Buzzi, H. V. Poor and D. Saturnino, “Energy-efficient resource allocation in multiuser MIMO systems: A game-theoretic framework”, Proc. of 16th European Signal Processing Conference (Eusipco), Lauzanne, Switzerland, Aug. 2008. [Telatar-ett-1999] E. Telatar, “Capacity of multi-antenna gaussian channels”, European Transactions on Telecommunications, vol. 10, no. 6, pp. 585–596, Nov./Dec. 1999. [Katz-wc-2007] M. Katz, and S. Shamai, “On the outage probability of a multiple-input single-output communication link”, IEEE Trans. on Wireless Comm., vol. 6, no. 11, pp. 4120–4128, Nov. 2007. [Jorswieck-ett-2007] E. A. Jorswieck and H. Boche, “Outage probability in multiple antenna systems”, European Transactions on Telecommunications, vol. 18, pp. 217–233, 2006. [Debbah-it-2005] M. Debbah, and R. R. Muler, “MIMO channel modeling and the principle of maximum entropy”, IEEE Trans. on Inform. Theory, Vol. 51, No. 5, pp. 1667–1690, May 2005. [Fundenberg-book-1991] D. Fudenberg and J. Tirole, “Game Theory”, MIT Press, 1991. [Rodriguez-globecom-2003] V. Rodriguez, “An Analytical Foundation for Ressource Management in Wireless Communication”, IEEE Proc. of Globecom, San Francisco, CA, USA, pp. 898–902, Dec. 2003.

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Introduction System Model Energy efficiency function

Static channels Fast fading channels Slow fading channels

Backup: MISO UPA case Proposition Assume the UPA, Q = npt I, then Γ(p, R) is quasi-concave w.r.t. n R o   2 tσ p ∈ 0, P and has a unique maximum point in p ∗ = min (2 −1)n ,P νnt where νnt is the solution of: nX t −1 i y nt y − = 0. (nt − 1)! i!

(2)

i =0

The sum

nt X

|hk |2 is a 2nt − Chi-square distributed random variable

k=1

The SIMO case (nt = 1, nr ≥ 2) follows directly since |I + ρphhH | = 1 + ρphH h 18 / 18