A generalised multi-receiver radio network and its decomposition - PowerPoint PPT Presentation

General models of radio network Technical development and results Comparative case study: Macro-diversity Conclusions A generalised multi-receiver radio network and its decomposition into independent transmitter-receiver pairs: Simple feasibility condition and power levels in closed form Virgilio RODRIGUEZ, R. Mathar Theoretische Informationstechnik RWTH Aachen Aachen, Germany email: vr@ieee.org IEEE ICC, Dresden, 16 June 2009 Virgilio RODRIGUEZ, R. Mathar General radio network: QoS, power, simplif. (ICC’09) 1/20

General models of radio network Technical development and results Comparative case study: Macro-diversity Conclusions Outline General models of radio network 1 Technical development and results 2 3 Comparative case study: Macro-diversity Conclusions 4 Virgilio RODRIGUEZ, R. Mathar General radio network: QoS, power, simplif. (ICC’09) 2/20

General models of radio network Technical development and results Comparative case study: Macro-diversity Conclusions Power, interference and QoS: 2 questions In many interesting situations, user’s QoS increases with the power in its signal, and decreases with the interfering power present at the concerned receiver(S) Typically each terminal “aims” for certain level of QoS Two fundamental questions: Are the QoS targets feasible (achievable)? ⇐ CRITICAL for admission control! If yes, which power vector achieves the QoS targets? Ideally, one would like to answer these questions for a generallised network that includes many past, present, and future networks as special cases. Virgilio RODRIGUEZ, R. Mathar General radio network: QoS, power, simplif. (ICC’09) 3/20

General models of radio network Technical development and results Comparative case study: Macro-diversity Conclusions Abstract model (Yates’95) N terminals whose power choices affect each other Terminal i chooses a power p i given by a function g i ( p − i ) , with p − i denoting the power levels of the others p i = g i ( p − i ) leads to terminal i its desired QoS for given p − i All details of the network (the QoS targets, number of receivers, interference functions, etc) are assumed “hidden” inside the power functions These functions are assumed to satisfy some simple mathematical properties (monotonicity, homogeneity, etc) Considering the functions properties the analyst addresses some of the fundamental questions about QoS achievability[1] Virgilio RODRIGUEZ, R. Mathar General radio network: QoS, power, simplif. (ICC’09) 4/20

General models of radio network Technical development and results Comparative case study: Macro-diversity Conclusions Generalised multi-receiver radio network N transmitters, K receivers i ’s QoS requirement given by � P i h i , 1 P i h i , K � Q i , ··· , ≥ κ i (1) Y i , 1 ( P )+ σ 1 Y i , K ( P )+ σ K h i , k is the known channel gain from TX i to RX k Q i , and Y i , k are general functions obeying certain simple properties (monotonicity, homogeneity, etc) Virgilio RODRIGUEZ, R. Mathar General radio network: QoS, power, simplif. (ICC’09) 5/20

General models of radio network Technical development and results Comparative case study: Macro-diversity Conclusions An example: macro-diversity macro-diversity: definition cellular structure is removed all transmitters are jointly decoded by all receivers equivalently, ‘one cell’ with a distributed antenna array i ’s QoS is given by [2]: P i h i , 1 / ( Y i , 1 + σ 1 )+ ··· + P i h i , K / ( Y i , K + σ K ) with Y i , k = ∑ n � = i P n h n , k Thus, Y i , k ( P ) = ∑ n � = i P n h n , k and Q i ( x ) = Q MD ( x ) = x 1 + ··· + x K (notice that same function works for all i ) Other examples: all scenarios from (Yates 1995)[1] Virgilio RODRIGUEZ, R. Mathar General radio network: QoS, power, simplif. (ICC’09) 6/20

General models of radio network Technical development and results Comparative case study: Macro-diversity Conclusions Motivation: Why a new model? Both models can be useful (think macroeconomics vs. microeconomics) Abstract model is more general (powerful?) Detailed model is closer to ’real’ world (easier to interpret) separates QoS function from Interference function (conceptually different... may have different properties) may provide insights/opportunities not otherwise available (e.g., we provide a simple closed-form solution for this model... see below) Virgilio RODRIGUEZ, R. Mathar General radio network: QoS, power, simplif. (ICC’09) 7/20

General models of radio network Technical development and results Comparative case study: Macro-diversity Conclusions Main result Let κ i denote i ’s QoS target, and � � q i = Q i h i , 1 / Y i , 1 ( 1 ) , ··· , h i , K / Y i , K ( 1 ) ⇐ QoS with each power level equal to unity. Theorem If the functions Q i and Y i , k are non-negative, non-decreasing, and homogeneous , and additionally, random noise is negligible, then κ i ≤ q i ∀ i implies that (i) each QoS target can be achieved, in particular, (ii) with the power levels P ∗ i = κ i / q i . Virgilio RODRIGUEZ, R. Mathar General radio network: QoS, power, simplif. (ICC’09) 8/20

General models of radio network Technical development and results Comparative case study: Macro-diversity Conclusions Network simplification Consider ‘network’ with N independent (orthogonal) transmitter-receiver pairs. Each transmitter has a power limit ¯ P i = σ i := 1 and wants QoS (SNR) of κ i . Let the channel gain of transmitter i be h i := q i . The maximal QoS that i can achieve is ¯ P i h i / σ i = h i = q i . Thus κ i is achievable provided κ i ≤ q i . Furthermore, if κ i / q i ≤ 1 then P i = κ i / q i is feasible ( ≤ ¯ P i = 1), and yields an SNR exactly equal to κ i . The “solution” to this simple ‘network´ works for the original one! Virgilio RODRIGUEZ, R. Mathar General radio network: QoS, power, simplif. (ICC’09) 9/20

General models of radio network Technical development and results Comparative case study: Macro-diversity Conclusions A simple and useful Lemma Let f : ℜ M → ℜ , and 1 M denote the “all ones” M -vector. Definition f if positively quasi-homogeneous (of degree one) if for all r ∈ ℜ + , f ( r 1 ) = rf ( 1 ) Definition f if quasi-non-decreasing if f ( x ) ≤ f ( � x � 1 ) , where � x � denotes the largest absolute value of the components of x . Fact If f satisfies both definitions, f ( x ) ≤ f ( � x � 1 ) = � x � f ( 1 ) Virgilio RODRIGUEZ, R. Mathar General radio network: QoS, power, simplif. (ICC’09) 10/20

General models of radio network Technical development and results Comparative case study: Macro-diversity Conclusions Solution applied to macro-diversity For macro-diversity, Y i , k ( 1 ) = ∑ n � = i h n , k . Since Q MD ( x ) = x 1 + ··· + x K , then i K h i , k q MD ∑ = i ∑ n � = i h n , k k = 1 Thus, the feasibility condition is κ i ≤ q MD and a solution is i P i = κ i / q MD i If all h i , k are of the same order of magnitude q MD ≈ ∑ K k = 1 1 / ( N − 1 ) = K / ( N − 1 ) i Then the condition becomes κ i ≤ K / ( N − 1 ) Thus, ∑ N k = 1 κ i ≤ KN / ( N − 1 ) ≈ K Virgilio RODRIGUEZ, R. Mathar General radio network: QoS, power, simplif. (ICC’09) 11/20

General models of radio network Technical development and results Comparative case study: Macro-diversity Conclusions Other macro-diversity formulae (Hanly, 1996 [2]) provides the condition N ∑ κ n < K n = 1 (Rodriguez, et al., 2008 [3]) provides a condition that — with each transmitter “equidistant to each receiver (and with κ N ≤ κ n ∀ n for convenience) — simplifies to: N − 1 ∑ κ n < K n = 1 Virgilio RODRIGUEZ, R. Mathar General radio network: QoS, power, simplif. (ICC’09) 12/20

General models of radio network Technical development and results Comparative case study: Macro-diversity Conclusions Macro-diversity formulae compared Table: Macro-diversity formulae under symmetry Herein Rodriguez08 Hanly96 ∑ N − 1 ∑ N ∑ N k = 1 κ i ≤ KN / ( N − 1 ) n = 1 κ n < K n = 1 κ n < K Virgilio RODRIGUEZ, R. Mathar General radio network: QoS, power, simplif. (ICC’09) 13/20

General models of radio network Technical development and results Comparative case study: Macro-diversity Conclusions Macro-diversity achievable regions TX 1, 2 & 3 at (0,0), (-1,0), (1,0) RX 1, 2 are at (0,-1), and (0,1) h i , k ∝ d − 2 i , k with d i , k the distance from i to k d 1 , k = 1; √ d 2 , k = d 3 , k = 2 h 1 , 1 = h 1 , 2 ∝ 1; h 2 , k = h 3 , k ∝ 1 / 2 Virgilio RODRIGUEZ, R. Mathar General radio network: QoS, power, simplif. (ICC’09) 14/20

General models of radio network Technical development and results Comparative case study: Macro-diversity Conclusions Recapitulation: strengths Model seems to be new Explicit (conservative) feasibility condition given ( κ i ≤ q i ) Matching power vector given ( P i = κ i / q i ) Interpretation: Generalised radio network can be (conservatively) associated with set of independent transmitter receiver pairs Analysis already extended to consider noise (Submitted) Solution is technology/application independent (useful for present and future networks) Analysis BOTH generalises AND simplifies (these are usually contrary aims) Provides specific/detailed information (formulae) applicable to wide variety of networks (result-specificity and result-generality tend to be contrary aims) Virgilio RODRIGUEZ, R. Mathar General radio network: QoS, power, simplif. (ICC’09) 15/20