Morning Session Capacity-based Power Control. Department of Electrical and Computer Engineering University of Maryland

Transcription

1 Morning Session Capacity-based Power Control Şennur Ulukuş Department of Electrical and Computer Engineering University of Maryland

2 So Far, We Learned... Power control with SIR-based QoS guarantees Suitable for delay-intolerant services, e.g., voice Satisfy all SIR constraints with minimum transmit power Leading to energy-efficient communications Power control for capacity Suitable for delay-tolerant services, e.g., data Maximize rate with a given average transmit power Equivalently, support a given rate with minimum power Leading to energy-efficient communications

3 Capacity-Based Power Control Fading: random fluctuations in channel gains. Perfect CSI at both the transmitter and the receiver Maximize ergodic capacity subject to average power constraints Main operational difference: QoS based power control: compensate for channel fading Capacity-based power control: exploit the channel fading

4 Channel Fading r = hx + n 2 Change in Channel State as a Function of Time 6 Distribution of the States Assumed by the Channel Fading Level Density of Fading Levels (States) Time Fading Levels (States)

5 Channel Fading r = hx + n 4 Change in Channel State as a Function of Time 2 Distribution of the States Assumed By the Channel 2 Fading Level Density of Fading Levels (States) Time Fading Levels (States)

6 Single-User Fading Channel (Goldsmith-Varaiya 94) Channel capacity for single user C = log ( + SNR) 2 = ( 2 log + p ) σ 2 In the presence of fading, the capacity for a fixed channel state h, C(h) = ( 2 log + p(h)h ) σ 2 Ergodic (expected) capacity under an average power constraint [ ( max E h p(h) 2 log + p(h)h )] σ 2 s.t. E h [p(h)] P p(h), h

7 Optimal Power Allocation using Waterfilling The Lagrangian function ( E h [log + p(h)h )] λ(e σ 2 h [p(h)] P) + µ(h)f (h)dh Optimality conditions where f (h) is the PDF of h. h p (h)h + σ + µ(h) 2 f (h) = λ The complementary slackness conditions µ(h)p (h) = for all h. If p (h) >, we get Otherwise, p (h) =. p (h) = λ σ2 h

8 Optimal Power Allocation using Waterfilling Optimal power allocation: waterfilling of power over time p(h) = ( ) + λ σ2 h Illustration of Waterfilling Over Inverse of the Channel States 3 Single User Optimum Power Allocation vs Fading States σ 2 /h 5 p(h) h h

9 Optimal Power Allocation using Waterfilling Optimal power allocation: waterfilling of power over time p(h) = ( ) + λ σ2 h 3 Single User Optimum Power Allocation vs Fading States p(h) h

10 Optimal Power Allocation using Waterfilling Optimal power allocation: waterfilling of power over time p(h) = ( ) + λ σ2 h 3 Single User Optimum Power Allocation vs Fading States p(h) h

11 Optimal Power Allocation using Waterfilling Optimal power allocation: waterfilling of power over time p(h) = ( ) + λ σ2 h 3 Single User Optimum Power Allocation vs Fading States p(h) h

12 Optimal Power Allocation using Waterfilling Optimal power allocation: waterfilling of power over time p(h) = ( ) + λ σ2 h 3 Single User Optimum Power Allocation vs Fading States p(h) h

13 Differences Between QoS-Based and Capacity-Based Power Control Single-user system SIR-based y = hx + n p(h)h γ p(h) = γσ2 σ 2 h Channel inversion; more power if bad channel, less if good channel Compensate for channel fading via power control Capacity-based [ ( max E h 2 log + p(h)h )] σ 2 p(h) = ( ) + λ σ2 h Waterfilling; more power if good channel, less if bad channel Exploit variations, opportunistic transmission

14 Fading Gaussian Multiple Access Channel user x h y user 2 x 2 h 2 n Channel model y = h x + h 2x 2 + n Simultaneously achievable ergodic rates for both users (R, R 2) Channel state vector h = (h, h 2) Adapt powers as functions of h: p (h) and p 2(h)

15 R Fading Gaussian Multiple Access Channel (Tse-Hanly 98) Union of pentagons [ R < E 2 log [ R 2 < E 2 log [ R + R 2 < E 2 log ( + p(h)h σ 2 ( + p2(h)h2 ( + over all feasible power distributions σ 2 )] )] ( C ) ( C 2) )] p(h)h + p2(h)h2 σ 2 ( C s) E [p (h)] P, E [p 2(h)] P 2, p (h), p 2(h) R 2

16 Determining the Boundary of the Capacity Region R 2 (R *, R* 2 ) µ + µ 2 R 2 = C * R R Capacity region is a convex region. To determine the boundary, maximize µ R + µ 2R 2 for all µ, µ 2. Any (R, R2 ) on the boundary is a corner of one of the pentagons. If µ 2 > µ then the upper corner; if µ > µ 2 then the lower corner.

17 Achieving Arbitrary Rate Tuples on the Boundary For given µ i, maximize C µ µ R + µ 2R 2 s.t. E h [p i (h)] P i, R C. Wlog, µ 2 > µ. Given power policy, the optimum R is the upper corner. The coordinates of the upper corner are: and R 2 = C 2, R = C s C 2 C µ = µ (C s C 2) + µ 2C 2 = (µ 2 µ )C 2 + µ C s R 2 (R *, R* 2 ) µ + µ 2 R 2 = C * R R

18 Achieving Arbitrary Rate Tuples on the Boundary Therefore, the optimum power allocation policy is the solution of: ( ) ( )] max E h [(µ 2 µ ) log + p2(h)h2 p(h)h + p2(h)h2 + µ p(h) σ 2 log + σ 2 s.t. E h [p i (h)] P i, i =, 2 p i (h), h, i =, 2 Objective function is concave, and constraint set is convex in powers. R 2 (R *, R* 2 ) µ + µ 2 R 2 = C * R R

19 Optimality Conditions p (h) achieves the global maximum of C µ iff it satisfies the KKTs, µ h h p (h) + h 2p 2(h) + σ 2 λ, µ h 2 (µ2 µ)h2 + h p (h) + h 2p 2(h) + σ2 h 2p 2(h) + σ λ2, 2 h h with equality at h, if p (h) > and p 2(h) >, respectively. Solution based on utilities in Tse-Hanly 98. Solve KKTs iteratively: generalized iterative waterfilling (Kaya-Ulukus)

20 Generalized Iterative Waterfilling Given p 2(h), find p (h) and λ such that µ h h p (h) + h 2p 2(h) + σ 2 λ, h with equality at h, if p (h)>. Waterfilling for user given the power of user 2 contributing to noise p (h) = Given p (h), find p 2(h) and λ 2. ( ) + σ2 + h 2p 2(h) λ h µ h 2 (µ2 µ)h2 + h p (h) + h 2p 2(h) + σ2 h 2p 2(h) + σ λ2, 2 h with equality at h, if p 2(h) >. A second order equation to solve.

21 Optimal Power Allocation via Generalized Iterative Waterfilling KKT conditions for the K-user case k (µ i µ i ) h k i j= p j(h)h j + p k (h)h k + σ λ k, h, k =,, K 2 i= with equality at h, if p k (h) >. Generalized iterative waterfilling Given p j (h), j < k, find p k (h) and λ k such that k (µ i µ i )h k i i= j= p j(h)h j + p k (h)h k + σ λ k, h, k =,, K 2 with equality at h, if p k (h) >. One-user-at-a-time algorithm, converges to the optimum. A Gauss-Seidel type of iteration.

22 Special Case: Sum Capacity (Knopp-Humblet 95) Ergodic sum capacity (µ = µ 2 = ) ( E h [log + max p(h) KKT conditions )] p(h)h + p2(h)h2 σ 2 s.t. E h [p i (h)] P i, p i (h), i =, 2 h p (h)h + p 2(h)h 2 + σ 2 λ, h h 2 λ2, h, p (h)h + p 2(h)h 2 + σ2 with equality at h, if p (h) > and p 2(h) >, respectively. For both users to transmit simultaneously at channel state h = (h, h 2), h h 2 = λ λ 2 For continuous channel gains, this is a zero-probability event. Only the strongest (after some scaling) user transmits at any given time.

23 Closed-Form Solution Single-user waterfilling over favorable channel states { ( ) + p k (h) = λ k σ2 h k, if hk /λ k > h j /λ j, j k, otherwise Power Distribution of User Power Distribution of User p.5 p h h h h 2.8

24 Issue of Simultaneous Transmissions Sum capacity achieving pentagon: a rectangle (orthogonal transmissions) R 2 Transmit Regions for Identical Sequences (R *, R* 2 ) User.6 R + R 2 = C * h User 2. None This result is specific to sum capacity and scalar channel. R h 2

25 Differences Between QoS-Based and Capacity-Based Power Control in MAC SIR-based p ( h γ γ p p p 2h 2 + σ2 2h 2 + σ 2) I (p) h p ( 2h 2 γ2 γ2 p2 p p h + σ2 h + σ 2) I 2(p) h 2 p > I (p) p 2 > I 2 (p) p γ 2σ 2 h 2 p γ 2σ 2 h 2 γ σ 2 h γ σ 2 h Both users transmit simultaneously More power if bad channels, less if good channels.

26 Differences Between QoS-Based and Capacity-Based Power Control in MAC Sum-capacity-based Power Distribution of User Power Distribution of User 2 p p h h h h 2.8 User with the stronger channel transmits Stronger user transmits with more power at better channels Multi-user diversity, multi-user opportunistic transmission

27 Single-User MIMO Channel (Telatar 99) x y x 2 y 2 x t Tx H y r Rx Channel gain matrix H: r t matrix Transmitted signal t-dim. vector x and received vector r-dim. vector y y = Hx + n n is i.i.d. zero-mean Gaussian noise vector with equal variance H is deterministic or random and known perfectly at the receiver Average power constraint E[x T x] P

28 Single-User MIMO Channel (Telatar 99) Use singular value decomposition to express H as H = UDV T U and V are unitary matrices, D is a diagonal matrix of singular values Diagonal entries of D are square roots of the eigenvalues of HH T Columns of U are the normalized eigenvectors of HH T Columns of V are the normalized eigenvectors of H T H

29 Single-User MIMO Channel (Telatar 99) Obtain an equivalent channel y = UDV T x + n Let x = V T x, ỹ = U T y and ñ = U T n ỹ = D x + ñ ñ is also i.i.d. zero-mean Gaussian Equivalent channel: min{r, t} parallel channels with (squared) gains d i ỹ i = d i x i + ñ i N(, ) x d ỹ N(, ) x 2 d2 ỹ 2 N(, ) x i di ỹ i

30 Single-User MIMO Channel (Telatar 99) Independent signalling is optimal over parallel channels Let P i E[ x 2 i ], power over the ith parallel channel. MIMO capacity min{r,t} max 2 log ( + d ip i ) i= min{r,t} s.t. i= P i P Optimal power allocation: waterfilling ( P i = λ ) + d i

31 MIMO Multiple Access Channel H Tx Tx 2 H 2 Rx Tx k H k The received vector at the receiver, y = H x + H 2x 2 + n H and H 2 are r t matrices Additive noise n is i.i.d. Gaussian with covariance I

32 Capacity Region of MIMO MAC Channel (Yu-Rhee-Boyd-Cioffi ) Q = E[x x T ] and Q 2 = E[x 2x T 2 ] are covariances Transmit power constraints tr(q ) P, tr(q 2) P 2 For fixed Q, Q 2, define B(Q,Q 2) R HQ 2 log H T + I R 2 H2Q 2 log 2H T 2 + I R + R 2 HQ 2 log H T + H 2Q 2H T 2 + I The capacity region is C = tr(q i ) P i B(Q,Q 2)

33 Boundary of the Capacity Region of MIMO MAC (Yu-Rhee-Boyd-Cioffi ) Solve the following wlog for µ µ 2 max µ log H Q H T + H 2Q 2H T 2 + I + (µ 2 µ ) log H 2Q 2H T 2 + I Q,Q 2 s.t. tr (Q ) P, tr (Q 2) P 2 R 2 A B C slope = µ µ D R

34 Sum Capacity of the MIMO MAC Channel (Yu-Rhee-Boyd-Cioffi ) Maximize sum rate max Q,Q 2 log H Q H T + H 2Q 2H T 2 + I s.t. tr (Q ) P, tr (Q 2) P 2 Sum rate optimal allocation Q and Q 2 Necessary condition: Q is the single user waterfilling over the colored noise H 2Q 2 H T 2 + I. Q 2 is the single user waterfilling over the colored noise H Q H T + I. Multi-dimensional water-filling:

35 Iterative Waterfilling (Yu-Rhee-Boyd-Cioffi ) Perform single-user optimizations: Iterative waterfilling log H Q H T + H 2Q 2H T 2 + I }{{} effective colored noise Given Q 2, user waterfills over the effective colored noise and updates Q Q = arg max log Q HQHT + H 2Q 2H T 2 + I Given Q, user 2 waterfills over the effective colored noise and updates Q 2 Q 2 = arg max log H 2Q 2H T 2 + H Q H T + I Q 2

36 Iterative Waterfilling (Yu-Rhee-Boyd-Cioffi ) An illustration of the trajectory followed during the iterative waterfilling R 2 A B C D R

37 Conclusions So Far Power adaptation for maximizing the capacity in single-user and MAC fading channels, and MIMO and MIMO MAC channels Common tool: waterfilling Fading channel is equivalent to parallel channels over channel states. MIMO channel is equivalent to min{t, r} parallel channels. Fading scalar and MIMO multiple access channels Sum-rate optimal operating points are reached by iterative waterfilling. Any arbitrary point is reached by generalized iterative waterfilling.

38 Power Adaptation for Energy Minimal Transmission (Uysal-Biyikoglu, Prabhakar, El Gamal 2) Rate-power relation in the AWGN channel R = 2 log ( + P σ 2 ) Energy-per-bit (E pb ) in an AWGN channel E pb = σ2 (2 2R ) R E pb monotonically decreases as R E pb σ 2 2ln(2) q = R

39 Power Adaptation for Energy Minimal Transmission Serving bits with slower rates is more energy-efficient. q: Number of transmissions to send a bit Codebook of fixed but sufficiently large block length. Code rate is adapted by changing its average power. q R Given B bits at the transmitter, decreasing transmit power yields longer transmission durations smaller transmission energy This is true for any system with a concave rate-power relation. E pb σ 2 2ln(2) q = R

40 Energy Minimal Packet Scheduling b b 2 b 3 b 4 t t 2 T t 3 t 4 Packets arriving at different times, deadline constrained by T Scheduling packets: τ i is the time allocated for packet i b b 2 b 3 b 4 τ τ 2 τ 3 τ 4 T Deadline constraint τ i T i Energy of a schedule τ: ω(τ) Find the energy minimal schedule τ to send all packets by T Adapt the transmission times of the packets. Equivalently, adapt the rate and hence the power.

41 Offline Packet Scheduling Offline schedule: bit arrival times are known a priori Bits cannot be served before they arrive at the data buffer: causality A necessary condition for optimality in b i = b case: τi τi+ with τi = T The necessity is due to the convexity of ω(τ). Assume τ i < τ i+ for some i. This is a contradiction since σ i = σ i+ = τ i+τ i+ 2 and σ j = τ j elsewhere ω(σ) ω(σ) = ω(τ i ) + ω(τ i+ ) ω(σ i ) ω(σ i+ ) i = ω(τ i ) + ω(τ i+ ) 2ω( τ i + τ i+ ) < 2 Equate τ i subject to feasibility constraints.

42 Optimal Offline Packet Scheduling A schedule with idle intervals is suboptimal. b b 2 = Not optimal τ τ idle 2 idle T Equate τ i subject to feasibility constraints. Split the transmission times to the available deadline as much as possible. b = b 2 = B b b 2 = Less equalized τ τ 2 T b b 2 = More equalized τ τ 2 T

43 Optimal Offline Packet Scheduling The optimal policy has the structure: τ i = b i max k {,...,M k i } k i = k i + arg k j= d k i +j max k j= b j k {,...,M k i } k j= d k i +j k j= b j Optimal offline schedule is called lazy scheduling. The lazy schedule start slowly and work harder as the deadline approaches Key reason: convexity of energy per bit in transmission time.

44 A Calculus Approach for Energy Minimal Packet Scheduling (Zafer-Modiano 9) Uses a geometric framework Let h(t) be the last time a packet arrived before t h(t) i= B i is the cumulative data arrival curve A policy is feasible if its cumulative service curve lies below h(t) i= B i The optimal rate policy is the tightest string. B B B 2 B 3 T h(t) i= Bi r 4 r 3 r 2 r s s 2 s 3 T

45 Algorithm to Find the Optimal Policy Connect the points at data arrival times to form lines Select the line which is feasible and which has minimum slope Deciding r and r 2 Deciding r 3 h(t) i= Bi h(t) i= Bi selected line selected line T T s s2 s3 s s2 s3 h(t) i= Bi r 4 r 3 r 2 r T s s2 s3

46 Conclusions Energy per bit monotonically decreases as increases, i.e., R decreases. R The slower the transmission, the more energy-efficient it is. Scheduling packets that arrive at different times. Optimal offline schedule has a majorization structure. Serve bits with a rate as constant as possible subject to bit feasibility.

47 References () A. Goldsmith and P. Varaiya, Capacity of fading channels with channel side information, IEEE Trans. on Info. Theory, Nov. 997 (2) R.Knopp and K. Humblet, Information capacity and power control in single-cell multiuser communications ICC 95, IEEE International Conference on Communications, June 8-22, 995, Seattle, USA (3) D.Tse and S. Hanly, Multi-Access Fading Channels: Part I: Polymatroid Structure, Optimal Resource Allocation and Throughput Capacities, IEEE Trans. on Info. Theory, Nov (4) E. Telatar, Capacity of multi-antenna Gaussian channels. European Transactions on Telecommunications, (6): , 999 (5) W.Yu, W. Rhee, S.Boyd and J.Cioffi, Iterative Water-filling for Gaussian Vector Multiple Access Channels, IEEE Trans. on Info. Theory, Jan. 24. (6) O. Kaya and S. Ulukus, Achieving the Capacity Region Boundary of Fading CDMA Channels via Generalized Iterative Waterfilling, IEEE Trans. on Wireless Commun., Nov. 26. (7) E. Uysal-Biyikoglu, B. Prabhakar and A. El-Gamal, Energy-efficient Packet Transmission over a Wireless Link, IEEE/ACM Trans. on Networking, Aug. 22. (8) M. Zafer and E. Modiano, A calculus approach to energy-efficient data transmission with quality-of-service constraints., IEEE/ACM Trans. on Networking, July 29.