HetNets: what tools for analysis?

Transcription

1 HetNets: what tools for analysis? Daniela Tuninetti (Ph.D.)

2 Motivation Seven Ways that HetNets are a Cellular Paradigm Shift, by J. Andrews, IEEE Communications Magazine, March 2013

3 Outline Part I: MWBM MWBM is a secial LP for which efficient olynomial-time algorithms exist, for examle, the Hungarian algorithm. Useful to aroximate at high-snr MIMO terms. Ex: gdof of Gaussian broadcast networks with relays and user scheduling decisions [arxiv: , M. Cardone et al] Part II: MixIn + TIN To reduce control-lane overhead, design PHY robust to asynchronism and lack of user coordination. Ex: Mixed inuts' and treat interference as noise : otimal to within a log(log(snr)) ga [arxiv: A. Dytso et al]

4 Part I Maximum Weight Biartite Matching

5 Definitions A matching, or indeendent edge set, in a grah G=(V,E) is a set of edges without common vertices. In a weighted biartite grah, each edge has an associated value. A maximum weighted biartite matching (MWBM), or assignment roblem, is a matching where the sum of the values of the edges in the matching have a maximal value. The Hungarian algorithm solves the assignment roblem in O(V^2 E); it uses a modified shortest ath search in the augmenting ath algorithm.

6 Examle Each link (from a Tx antenna to a Rx antenna) has a `weight given by its ower exressed in db

7 MIMO-tye Setting Y = HX + Z 2 C n R 1, H 2 C n R n T, Z N(0, I nr ) indeendent of X, C 1 :X 2 C n T 1 : E[ X i 2 ] ale 1, i 2 [1 : n T ], indeendent, C 1 :X 2 C n T 1 : E[ X i 2 ] ale 1, i 2 [1 : n T ], X C 2 :X 2 C n T 1 : E[ X i 2 ] ale n T, i2[1:n T ] I(X; Y ) := log I nr + H x H H x := E[XX H ] 0 nt

8 MIMO-tye Setting Tx1 Tx Rx Tx3 53 Classical MAC C 1 = log I nr + HH H Tx1 Tx Rx ale C 2 ale C 3 ale log I nr + HH H n T Tx3 53 Cooerative MAC C 3 C 1 ale rank[h] log(n T ) Tx x2 MIMO Rx ale min(n T,n R ) log(n T ) U to a constant, Tx ower constraint does not matter

9 Examle: SIMO (or SISO BC) [H] ij := SNR ij ex(j ij ) H := [h 1,...,h nr ] T! B := [ 1,..., n R ] T Tx Rx log 1+SNR max i{ i } ale C 1 = C 2 = C 3 = log 1+ X! SNR i i ale log 1+SNR max i{ i } + log(n R ). U to a constant, Rx rocessing does not matter

10 Examle: SIMO (or SISO BC) [H] ij := SNR ij ex(j ij ) H := [h 1,...,h nr ] T! B := [ 1,..., n R ] T Tx Rx log 1+SNR max i{ i } ale C 1 = C 2 = C 3 = log 1+ X! SNR i i ale log 1+SNR max i{ i } + log(n R ). U to a constant, Rx rocessing does not matter

11 log I + HH H MWBM(B) log 1+SNR

12 log I + HH H MWBM(B) log 1+SNR

13 log I + HH H MWBM(B) log 1+SNR

14 log I + HH H MWBM(B) log 1+SNR

15 log I + HH H MWBM(B) log 1+SNR

16 Examle: FD SISO BC+relay (W 1, W 2 ) Z 1 Tx X 0 S1 + Y 1 Rx1 S2 C N 1 + T 1 I1 e j 1 RN X 1 I2 e j 2 + Y 2 Rx2 Z 2

17 Examle: FD SISO BC+relay (W 1, W 2 ) Z 1 Tx X 0 S2 S1 + Y 1 Rx1 R 1 + R 2 ale I (X 0 ; Y R,Y 1,Y 2 X 1 )! max { 10, 20, } C N 1 + T 1 I1 e j 1 RN X 1 I2 e j 2 + Y 2 Rx2 Z 2

18 Examle: FD SISO BC+relay (W 1, W 2 ) Z 1 Tx X 0 S2 S1 + Y 1 Rx1 R 1 + R 2 ale I (X 0 ; Y R,Y 1,Y 2 X 1 )! max { 10, 20, } C N 1 + T 1 I1 e j 1 RN X 1 I2 e j 2 + Y 2 Rx2 R 1 + R 2 ale I (X 0,X 1 ; Y 1,Y 2 )! max { , } Z 2

19 Examle: FD SISO BC+relay (W 1, W 2 ) Z 1 Tx X 0 S1 + Y 1 Rx1 S2 C N 1 + T 1 I1 e j 1 RN X 1 I2 e j 2 + Y 2 Rx2 n o min max { 10, 20, }, max { , } Z 2 : 0 ale ale max { 10, 20} : serve best user without relay max { 10, 20} < ale max { 11, 21} :serve best userwithrelay max { 11, 21} < ale max { , } : serve both users max { , } < : serve both users == MISO BC

20 HetNets: BCIC + relays Rx1b Rx1a Tx2 Tx1 Rx2a Tx3 Rx2b

21 Recie Take your favorite outer or lower bound, ossibly further uer or lower bound so as to only have channel gains Pre-log == gdof == MWBM The overall channel matrix must be full rank (i.e., aroximation too crude to cature small variations in channel gains, ale examle (1 + ) S S ) H = S S

22 Disclaimer SISO comound MAC + relays: NNC achieves 0.63 x 2 x N bits of cut-set bound SISO rivate msgs BC + relays: can achieve O(N log(n)) bits of cut-set bound IC (+ relays): oen, and cut-set bound known to be insufficient...

23 Part II Mixed Inuts and Treat Interference as Noise: mixed inut refers to a random variable that is a mixture of a continuous and a discrete art, i.e., a Gaussian and a uniform PAM

24 Oblivious Processing X n 1 W 1 Encoder Decoder Ŵ 1 Y1 n C 1 C 1 X n 2 P Y 1,Y2 X1,X2 (y 1,y2 x1,x2) W 2 Encoder Decoder Ŵ 2 Y2 n C 2 C 2 C 1 C 2

25 Oblivious Processing X n 1 W 1 Encoder Decoder Ŵ 1 Y1 n C 1 C 1 X n 2 P Y 1,Y2 X1,X2 (y 1,y2 x1,x2) W 2 Encoder Decoder Ŵ 2 Y2 n C 2 C 2 C 1 C 2

26 Past Work W O. Simeone, E. Erki, and S. Shamai, On codebook information for interference relay channels with out-of-band relaying, IT May Primitive relay channel: caacity with comress forward 2. IC+R+Oblivious receivers: caacity with comress forward and TIN 3. Gaussian noise: otimizing inut unknown W Encoder C Encoder C X n X n P Y 1,Y2 X P Y 1,Y2 X Y n 1 Y n 2 Y n 1 Y n 2 Relay Relay Relay Decoder C Decoder C Ŵ Ŵ W 1 A. Sanderovich, S. Shamai, Y. Steinberg, and G. Kramer, Communication via decentralized rocessing, IT July Uer and lower bounds, which coincide for deterministic channels 2. Gaussian noise: otimizing inut unknown 3. Gaussian noise: examle where BPSK outerforms Gaussian inuts Encoder C1 X n 1 X2 n W 2 Encoder C 2 P Y 1,Y2,Y3 X1,X2,X3 (y 1,y2,y3 x1,x2,x3) Y n 1 Y n 3 Y n 2 C 1 Decoder Relay Decoder C 2 Ŵ 1 Ŵ 2

27 Past Work (discrete inuts) Y. Wu and S. Verdu, The imact of constellation cardinality on Gaussian channel caacity, Allerton 2010 (oint to oint) E.Calvo et al On the totally asynchronous interference channel with SU receivers, ISIT 2009 E.Abbe and L.Zheng, A coordinate system for Gaussian networks, IT Feb Achievable for any (i-stable) IC Continuous inuts are bad interferers -- esecially if one treats them as noise R k ale I(X k ; Y k ), k 2 [1 : K]

28 Main Tool Z G N(0, 1) indeendent of X D discrete : I d N,d 2 min(x D ) ale I(X D ; X D + Z G ) ale 1 2 log min N 2, 1+E XD I d (n, x) := ale log(n) 1 e 2 log 2 log 1+(n 1)e 4x + N 2 =1+E XD, N e 4d2 min(x D ) < constant I(X D ; X D + Z G )= 1 2 log (1 + E X D ) constant Lower bound holds for any constellation but may be arbitrary lose for a secific one, i.e., for PAM.

29 Examle: AWGN 10 Achievable rate vs. SNR db 9 Caacity of PTP Achievable with finite N Achievable with N=f(SNR) R bit/sec/hz N=128 N=64 N=32 N=16 Y = SNR X + Z : Z N(0, 1), X PAM (N), d 2 min(x) = 12 N 2 1, N = j1+snr 1 k 1 2 log 1 6 ln(snr) N=8 N=4 N=2 = 1 2 log(snr) ga = 2 log(snr)+1 2 log (8e) SNR db

30 Main Result Choice of inuts X i = 1 i X id + i X ig, i 2 [0, 1], X id PAM (N i ), X ig N(0, 1), where X ij are indeendent for i 2 [1 : 2],j 2{D, G}. Discrete art = `common message j N = 1+x 1 k 1 2 log 1 a ln(x) + = R = log(snr) log(1 + x) ga ga = 2 log(snr)+1 2 log (8e)

31 How about TDMA? R = X i log i 1 2 log (1 + SNR i) Y i SNRi i! SNRi! N i N = Y i N i i No need to time-share / coordinate: the same effect (u to a ga) can be obtained by varying the number of oints of the discrete art

32 Recie Common message <--> discrete inut Private message <--> Gaussian inut TIN is otimal to within log(log(snr)) No need of joint decoding No need of synchronous communication TDMA by aroriately varying number of oints in the discrete art of the inut

33 Thank you