T he Scaling Law of SNR-Monitoring in Dynamic Wireless Networks

1. The Scaling Law of SNR-Monitoring in Dynamic Wireless Networks Hongyi Yao Xiaohang Li Soung Chang Liew

2. Channel Gain or Single-Noise-Ratio (SNR) • The channel gain H of a wireless channel (S,R) is defined by: Y= H X + Z, where X is the signal sent by S, Y is the signal received by R and Z ~ N(0,1) is the noise term. • For the simplicity, both noise power and transmit power are normalized to be 1. Z Channel Model H S R 1

3. Channel Gain Monitoring • In a wireless network, the knowledge of channel gains are needed to design high performance communication schemes. • Due to fading, node mobility and node power instability, channel gains vary with time. • Thus, tracking and estimating channel gains of wireless channels is fundamentally important • This work seeks the answer of the following question: • What is the minimum communication overhead such that all network channels can be tracked? 2

4. S1 S2 S3 H1 H2 H3 R Toy Example Prior Knowledge: H1=1 and H2=1 and H3=1. Update There exists i in {1,2,3} such that Hi varied. Monitoring Object: The receiver R wants to recover i and Hi. 3

5. Toy Example Hi is unknown, Hj = 1 for • Recovering i and x: Unit Probing S1 S2 S3 Time Slot 1: 1 1 1 Time Slot 2: Time Slot 3: R Three time slots are required for probing. 4

6. S1 S1 S2 S2 S3 S3 1 1 1 2 3 1 R R Toy Example (Differential Group Probing) Hi is unknown, Hj = 1 for Time Slot 1: Time Slot 2: Receive Y[1]=3+(Hi-1) Receive Y[2]=6+(Hi-1)i Using the a priori knowledge of the channel gains, R computes [Y’[1],Y’[2]]=[3,6] and then the difference: [Y[1],Y[2]] - [Y’[1],Y’[2]]=(Hi-1)[1,i]. Since [1,1], [1,2] and [1,3] are linear independent, R can decode i and then Hi. - One time slot saving ! 5

7. Motivation Raised by the Toy Example • Unit Probing VS. Differential Group Probing. • Unit Probing (Scheduling Interference): Since we do not know which channel varied, all channels must be sampled one by one. • Differential Group Probing (Embracing Interference): All channels are sampled simultaneously to explore the a prior knowledge. • Question: Does differential group probing suffice to achieve the minimum communication overheads? • Answer: YES! 6

8. Outline of the Talk • Fundamental setting: multiple transmitters and one receiver. • The scaling law of tracking all channel gains. • Achieving the scaling law by ADMOT. • General setting: multiple transmitters, relay nodes and receivers. • The scaling law of above fundamental setting still holds. • Achieving the scaling law by ADMOT-GENERAL. • Simulation results. 7

9. Fundamental Setting • Multiple transmitters and one receiver: For Si, the probe in the s’th time slot is Xi[s]. … S1 S2 Sn R receives: H1 H2 Hn The term Z[s] is the noise i.i.d. (of s) ~ N(0,1). R Definition (State): The state H is a length n vector, with the i’th component equaling Hi. The vector H’ is the a priori knowledge of H preserved by R. 8

10. State Variation • The state variation H-H’ is said to be approx-k-sparse if there are at most k “significant” nonzero components in H-H’. • Practical interpretation: Approx-k-sparse state variation means there are at most k channels suffering significant variations, while the variations of other channels are negligible. • Details about “approx” can be found in paper [1]. 9

11. Main Theorem • Theorem: When the state variation H-H’ is approx-k-sparse, we have: • Scaling Law: At least time slots are required for reliably estimating all the n channels. • Achievability: There exists a monitoring scheme using time slots, such that R can estimate all the n channels in a reliable and computational efficient manner. 10

12. Proof of the Scaling Law • Assuming T time slots are used for allowing R estimating H from the a priori knowledge H’. • For the clarity, we simplify the problem by assuming the noise term Z[s]=0 for each time slot s. • Thus, R receives for s={1,2,…T}. 11

13. Proof of the Scaling Law • Using H’, R computes and • Note that recovering H is the same as recovering H-H’ by using the linear samples D[s] for s={1,2,…,T}. • Using the results in [2], at least linear samples are required for reliably recovering a approx-k-sparse vector H-H’ [1]. Key Idea: Wireless interference only provides linear samples. 12

14. Achieve the Scaling Law by ADMOT • Systematical View of ADMOT: • Core techniques in ADMOT: Differential Group Probing+ Compressive Sensing. 13

15. The Training Data of ADMOT • The matrix of dimensions consists of the training data of ADMOT. Here, N is the maximum number of time slots allowed by ADMOT, and n is the number of transmitters. • Each component of is i.i.d. chosen from {-1,1} with equal probability. • The i’th column of is the training data of transmitter Si. To be concrete, in the s’th time slot, Si sends , as: 14

16. Construction of ADMOT • ADMOT(m, H’) • Variables Initialization: H* is the estimation of H. Vector Y is of dimension m. Matrix consists of the 1,2,…,m’th rows of . • Step A (Probing): For s = 1, 2,…m, in the s’th time slot: • For each i in {1,2,…,n}, Si sends • Receiver R sets Y[s] (i.e., the s’th component of Y) to be the received sample. Thus, • Then we have 15

17. Construction of ADMOT • ADMOT(m, H’) Continued from previous slide • Step B (Computing Differences): Receiver R computes • Step C (Norm-1 Sparse Recovering): Receiver R finds the solution E* of the following convex program: • Minimize , subject to • Step D (Estimating) : Receiver R estimates H as H*=H’+E*. • Step E: Terminate ADMOT. 16

18. Comments for ADMOT • The computational complexity of R is dominated by a norm-1 minimization convex program. • If H-H’ is approx-k-sparse, using the results of Compressive Sensing[3], E* is a reliable estimation of H-H’ provided that m=Cklog(n/k) for a constant C. • The receiver can adapt the system parameter m for future rounds of ADMOT by analyzing the square-root estimation error |H-H*|2. Details can be found in [1]. Tightly Match the Scaling Law! 17

19. Outline of the Talk • Fundamental setting: multiple transmitters and one receiver. • The scaling law of tracking all channel gains. • Achieving the scaling law by ADMOT. • General setting: multiple transmitters, relay nodes and receivers. • The scaling law of above fundamental setting still holds. • Achieving the scaling law by ADMOT-GENERAL. • Simulation results. 18

20. General Communication Networks • There are multiple transmitters in S, multiple relay nodes in V and multiple receivers in R. • For each node , all its incoming channels (from S and V) require monitoring. • In the following toy network, the directed lines denote the channel requiring monitoring. S R V1 V2 19

21. Simplified Model • The challenging of general communication network rises from the existence of relay nodes in V. • For the simplicity, we consider a network with only relay nodes V={v1,v2,…,vn}. • Thus, for each node vi in V, it wants to track the channel (vj,vi) for each j=1,2,…,n. Complete Network! 20

22. The Scaling Law of General Setting • Assume for each node vi in V, the incoming channels of vi suffer approx-k-sparse variation. • Directly using the scaling law of the single receiver scenario, at least time slots are required. • Surprisingly, this scaling law is also tight for general communication networks. 21

23. Achieving the Scaling Law • Full-Duplex model: Any node in V can transmit and receive in the same time slot. • Due to the broadcast nature of wireless medium, each node in V can probe under ADMOT, and in the mean time receive the probes of other nodes in V. • In the end, each node in V can estimate its incoming channels following ADMOT. • Thus, the overall overhead is • Half-Duplex Model: Any node in V can not transmit and receive in the same time slot. • The generalization is non-straightforward and shown in the following slides. • For both models, the achievability schemes are implemented in a distributed manner, i.e., no centralized controller is needed. 22

24. Achieving the Scaling Law for Half-Duplex Model • We construct ADMOT-GENERAL to achieve overheads for a constant C’. • The matrix of dimensions consists of the training data. • Each component of is i.i.d. chosen from {0,-1,1} with probability {1/2,1/4,1/4}. • The i’th column of is the training data of vi. 23

25. High-level Construction of ADMOT-GENERAL • ADMOT-GENERAL runs m time slots. • In the s’th time slot, if node vi receives in the time slot; Otherwise, vi sends in the time slot. • In the end, with large probability (Chernoff Bound), each node, say vi, received at least m/3 data. • Let the vector Yi consist of the received data of vi, and Hi be the vector consisting of all incoming channel gains of vi. • Each component of Yi is a linear sample (with noise) of Hi. That is, , where consists of at least m/3 rows of . 24

26. High-level Construction of ADMOT-GENERAL • Node vi computes the difference using the a priori knowledge Hi’ for its incoming channel gains. • Note each component of is i.i.d. sampled from {0,-1/2,1/2} with probability {0.5, 0.25, 0.25}, which are therefore sub-Gaussian ensembles. • Approx-k-sparse Hi-Hi’ can be recovered provided that RowNumber( ) for a constant C’ [4]. Tightly Match the Scaling Law! 25

27. Outline of the Talk • Fundamental setting: multiple transmitters and one receiver. • The scaling law of tracking all channel gains. • Achieving the scaling law by ADMOT. • General setting: multiple transmitters, relay nodes and receivers. • The scaling law of above fundamental setting still holds. • Achieving the scaling law by ADMOT-GENERAL. • Simulation results. 26

28. Simulations • Setting: • n=500 transmitters. • One receiver. • Average SNR = 20 dB. • Approx-k state variation. Define channel stability=1-k/n. • ADMOT is implemented as the consecutive manner: 27

29. Simulations 28

30. Future Works • General Setting: Network Tomography + Channel Gain Estimation? • Current ADMOT-GENERAL requires the internal nodes in V performing sophisticated protocol (ADMOT) for channel gain estimation. • Can we estimate internal channel gains as “tomography”, in which relay nodes do normal network transmission, only the transmitters and receivers perform sophisticated protocols? 29

31. Thanks! & Questions? • [1]. H. Yao and X. Li and S. C. Liew, “Achieving the Scaling Law of SNR-Monitoring for Dynamic Wireless Networks”, arxiv 1008.0053. • [2]. K. D. Ba, P. Indyk, E. Price, and D. P. Woodruff, “Lower bounds for sparse recovery,” in Proc. of SODA, 2010. • [3]. E. Cand´es, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Communications on Pure and Applied Mathematics, 2006. • [4]. S. Mendelson, A. Pajor, and N. T. Jaegermann, “Uniform uncertainty principle for bernoulli and subgaussian ensembles,” Constructive Approximation, 2008. 30