Outline - Part III: Co-Channel Interference

Transcription

1 General Outline Part 0: Background, Motivation, and Goals. Part I: Some Basics. Part II: Diversity Systems. Part III: Co-Channel Interference. Part IV: Multi-Hop Communication Systems.

2 Outline - Part III: Co-Channel Interference 1. Co-Channel Interference (CCI) Analysis Effect of Shadowing Effect of Multipath Fading Single Interferer Multiple Interferers Minimum Desired Signal Requirement Random Number of Interferers 2. CCI Mitigation Diversity Combining Optimum Combining/Smart Antennas Optimized MIMO Systems in Presence of CCI

3 Effect of Shadowing Finding the statistics of the sum of log-normal random variables. No known exact closed-form available. Several analytical techniques have been developed over the years. As an example, we cover the Farley bounds on the sum of log-normal random variables.

4 Effect of Multipath Fading (1) Single Interferer Let the carrier-to-interference ratio (CIR) λ = s d s i, where s d = α 2 d (with average Ω d) and s i = α 2 i (with average Ω i ) are the instantaneous fading powers of desired and interfering users. Outage probability P out = Prob[λ λ th ] = 0 p sd (s d ) Prob [ s i s ] d s λ d th ds d = p sd (s d ) p si (s i ) ds i ds d 0 s d /λ th Exp: Rician (Rician factor K d )/Rayleih case P out = λ ( th λ th + b exp Kb ), λ th + b where Ω b = d Ω i (K d + 1).

5 Effect of Multipath Fading (2) N I independent identically distributed interferers Let the carrier-to-interference ratio λ = s d s I, where s I = N I i=1 s i and all the s i have the same average fading power Ω i. Outage probability P out = Prob[λ λ th ] = p sd (s d ) p si (s I ) ds I ds d 0 s d /λ th Exp: Nakagami/Nakagami scenario P out = I x (m, mn I ) where ( x = 1 + Ω ) d Ω i λ th and Γ(a + b) x I x (a, b) = t a 1 (1 t) b 1 dt Γ(a)Γ(b) 0 is the incomplete Beta function ratio.

6 Effect of Multipath Fading (3) N I independent non-identically distributed interferers Let the carrier-to-interference ratio λ = s d s I, where s I = N I i=1 s i and all the s i can have different average fading power Ω i. Outage probability P out = Prob[λ λ th ] N I = Prob s d λ th s i 0. i=1 Define α = λ th NI i=1 s i s d. α 0 corresponds to an outage. α 0 corresponds to satisfactory transmission. Find characteristic function of α and then use Gil-Palaez lemma.

7 Gil-Palaez Lemma and Application Let X be a random variable with cumulative distribution function (CDF) P X (x) = Prob[X x] and characteristic [ function (CF) φ X (t) = E e jxt] then P X (x) = Im [φ X (t)e jxt] dt. π t Application to outage probability 0 P out = Prob[α 0] = 1 Prob[α 0] = 1 P α (0) = Im [φ α (t)] π t where for the Nakagami/Nakagami fading case 0 dt, φ α (t)=φ sd ( t) N I ( = 1 + jtω d m d φ si (λ th t) i=1 ) md N I i=1 ( 1 jtω ) mi iλ th. m i

8 Random Number of Interferers Depending on traffic conditions, the number of active interferers n I is a random variable from 0 to N I which is the maximum number of active interferers. Outage probability is given by P out = N I n I =0 Prob[n I ] P out [n I ]. Assume N c available channels per cell each with activity probability p a. The blocking probability B = p N c a. PDF of n I is Binomial ) NI Prob[n I ]=( p n I a (1 p n a ) N I n I, n I = 0,, N I. I ) NI ( =( B n I/N c 1 B 1/N NI n c) I. n I

9 Outline - Part III: Co-Channel Interference 1. Co-Channel Interference (CCI) Analysis Effect of Shadowing Effect of Multipath Fading Single Interferer Multiple Interferers Minimum Desired Signal Requirement Random Number of Interferers 2. CCI Mitigation Diversity Combining Optimum Combining/Smart Antennas Optimized MIMO Systems in Presence of CCI

10 Diversity Combining Reduce the effect via selective (or switched) antenna diversity combining techniques. Consider a dual-antenna diversity system with one co-channel interferer. Let α 11 denote the fading amplitude from desired user to antenna 1, α 12 denote the fading amplitude from desired user to antenna 2, α 21 denote the fading amplitude from interfering user to antenna 1, and α 22 denote the fading amplitude from interfering user to antenna 2. Three main decision algorithms for selective diversity.

11 Decision Algorithms CIR algorithm: picks and process the information from the antenna with the highest CIR. For the scenario describe previously: [ (α11 ) 2 ( ) ] 2 α12 Max,. α 21 α 22 Desired signal algorithm: picks and process the information from the antenna with the highest desired signal, i.e., [ ] Max α11 2, α2 12. Signal plus interference algorithm: picks and process the information from the antenna with the highest desired plus interfering signal, i.e., [ ] Max α α2 21, α α2 22.

12 Interference Mitigation More advanced interference mitigation techniques Optimum combining Optimized MIMO systems

13 General Outline Part 0: Background, Motivation, and Goals. Part I: Some Basics. Part II: Diversity Systems. Part III: Co-Channel Interference. Part IV: Multi-Hop Communication Systems.

14 Multi-Hop Communication Systems Advantages of transmission with relays: Broader coverage Lower transmitted power (higher battery life and lower interference) Cooperative/Collaborative/Multi-user diversity. Pionnering work on this topic: Sendonaris, Erkip, and Aazhang, [ISIT 98]. Laneman, Wornell, and Tse [WCNC 00, Allerton 00, ISIT 01]. Emamian and Kaveh, [ISC 01]. Goal: Develop an analytical framework for the exact end-to-end performance analysis of dual-hop then multi-hop relayed transmission over fading channels.

15 Dual-Hop Systems Consider the following dual-hop communication system with a relay B α1 α2 A C Two relaying options: Non-regenerative relaying (known also as analog or amplify-and-forward relaying) Regenerative relaying (known also as digital or decode-and-forward relaying)

16 Non-Regenerative Systems Received signal at the relay input (B) is r b (t) = α 1 s(t) + n 1 (t). Received signal at the destination (C) is r c (t) = α 2 Gr b (t) + n 2 (t) = α 2 G(α 1 s(t) + n 1 (t)) + n 2 (t). Equivalent end-to-end SNR γ eq = α 2 1 α2 2 G2 α 2 2 G2 N 01 + N 02 = α 2 1 N 01 α 2 2 N 02 α2 2. N G 2 N 01

17 Choice of the Relay Gain One possible choice of the relay gain is just channel inversion, i.e., G 2 = 1/α 2 1, Resulting equivalent end-to-end SNR γ eq 2 = γ 1γ 2 γ 1 + γ 2. Lower bound on the performance of practical relays Related to the Harmonic Mean of γ 1 and γ 2

18 A Second Choice of the Relay Gain Another possible choice of the relay gain [Laneman et al. 00] G 2 1 = α1 2 + N. 0 Limits the gain of the relay when first hop is deeply faded Resulting equivalent end-to-end SNR γ eq 1 = γ 1 γ 2 γ 1 + γ 2 + 1,

19 Monte Carlo Simulation Comparison of the outage probability for the two choices of the relay gain 10 0 Comparison of the Two Choices of Relay Gain Using Monte Carlo Simulaion Relay Gain (Laneman) Relay Gain (Channel Inversion) 10 1 Outage Probability P out Normalized SNR [db]

20 Harmonic Mean Given two numbers X 1, X 2 : Arithmetic Mean µ A (X 1, X 2 ) = X 1 + X 2 2 Geometric Mean µ G (X 1, X 2 ) = X 1 X 2 Harmonic Mean µ H (X 1, X 2 ) = 2X 1X 2 X 1 + X 2 = Relation with end-to-end SNR 2 1 X X 2 γ eq 2 = 1 2 µ H(γ 1, γ 2 ) γ eq 1, where γ 1 and γ 2 are the instantaneous SNRs of hops 1 and 2, respectively.

21 Harmonic Mean of Exponential Variates Theorem : Let X 1 and X 2 be two independent exponential variates with parameters β 1 and β 2 respectively. Then, the PDF of X = µ H (X 1, X 2 ), p X (x), is given by p X (x)= 1 [( ) 2 β 1β 2 xe x 2 (β 1+β 2 ) β1 + β ( 2 K 1 x ) β 1 β 2 β1 β 2 +2K 0 ( x β 1 β 2 )] U(x), where K i ( ) is the ith order modified Bessel function of the second kind and U( ) is the unit step function. The CDF and MGF of the harmonic mean of two independent exponential variates are also available in closed-form.

22 Validation by Monte-Carlo Simulations Monte Carlo Simulation for p Γ (γ) 1.6 Analysis Simulation 1.4 Probability Density Function, p Γ (γ) γ

23 Derivation of the CDF of X = µ H (X 1, X 2 ) Let Z = 1 X = 1 ( ). 2 X 1 X 2 The CDF of X, P X (x), is given by P X (x) = Pr (X ( < x) 1 = Pr X > 1 ) ( = Pr Z > 1 ) ( x x = 1 Pr Z < 1 ) ( ) 1 = 1 P x Z x, where P Z ( ) is the CDF of Z.

24 Derivation of the CDF of X = µ H (X 1, X 2 ) (Continued) If X is an exponential random variable with parameter β then the MGF of Y = 1/X can be shown to be given by M Y (s) = E [e ] sy = 2 ( βs K 1 2 ) βs. Using the differentiation property of the Laplace transform, P Z (z) can be written as ( ) P Z (z) = L 1 MZ (s) s = 1 L 1( 2 β 1 β 2 K 1 (2 ) ( β 1 s K 1 2 )) β 2 s z= 1, x which is a tabulated inverse Laplace transform leading to ( ) 1 P X (x)=1 P Z x =1 x ( β 1 β 2 e x 2 (β 1+β 2 ) K 1 x ) β 1 β 2.

25 Derivation of the PDF of X = µ H (X 1, X 2 ) Taking the derivative of the CDF of X with respect to x results in d [ (P dx X (x)) = β1 β 2 e x 2 (β 1+β 2 ) K 1 (x ) β 1 β 2 + x ( β 1 β (β 1 + β 2 ) e x 2 (β 1+β 2 ) K 1 ( x β 1 β 2 ) Using + e x 2 (β 1+β 2 ) d dx [K 1 ( x β 1 β 2 )] )]. z d dz K v(z) + vk v (z) = zk v 1 (z) leads to the final desired result p X (x) = 1 [( ) 2 β 1β 2 xe x 2 (β 1+β 2 ) β1 + β 2 β1 β 2 K 1 ( x β 1 β 2 ) + 2K 0 ( x β 1 β 2 )].

26 Formulas for the Outage Probability For non-regenerative systems, P out is given by P out = 1 2γ ( ) ( th 2γth K γ1 γ 1 e γ 1 th γ γ1 γ 2 γ 2 ), where γ 1 and γ 2 are the average SNRs of hops 1 and 2, respectively. For regenerative systems, P out is given by P out = 1 e γ th ( 1 γ1 + γ 1 2 ). Both formulas are equivalent at high average SNR since for small x K 1 (x) 1 x.

27 Outage Probability: Numerical Example 10 0 Comparison of Outage Probability for Regenerative and Non Regenerative Systems Non Regenerative System,γ 1 =γ 2 Regenerative System,γ 1 =γ 2 Non Regenerative System,γ 1 =2γ 2 Regenerative System,γ 1 =2γ 2 Outage Probability P out Normalized SNR [db]

28 Outage Probability of Collaborative Systems Consider a wireless communication system with one direct link and L collaborating paths. Assume direct link with average SNR γ 0 and that the two hops in collaborating path l have the same average SNR γ l. Assume that the strongest path is selected at any given time. Resulting outage probability ( ) P out = 1 e γ th γ 0 ( L 1 2γ th e γ l l=1 ( ) ) γ 2γth l K 1. γ l 2γ th

29 Formulas for the Average BER The MGF of γ eq, E(e γs ), for identical and independent faded hops, i.e. γ 1 = γ 2 = γ, is given by M Γ (s) = ( γ 4 s γ4 s + 1) + arcsinh ( )3 2 γ4 s γ4 2 s + 1 ( γ 4 s ) For non-regenerative systems with DPSK the average BER is P b (E) = 1 2 M γ(1). For regenerative systems with DPSK over identical and independent faded hops P b (E) = 1 + 2γ 2 (1 + γ) 2.

30 Average BER: Numerical Example 10 0 Comparison of Bit Error Rates for Regenerative and Non Regenerative Systems Non Regenerative System Regenerative Systems 10 1 Bit Error Rate P b (E) Average SNR per Hop [db]

31 Average BER with Collaboration Consider one direct link and L i.i.d. faded collaborating paths. Using maximal-ratio combining at the receiver, the overall SNR can be written as L γ t = γ 0 + γ l. l=1 Under these conditions the MGF of the overall combined SNR γ t is given by L M γt (s) = M γ0 (s) M γl (s). l=1

32 Diversity Gain due to Collaboration 10 0 Effect of Collaborative Diversity on Average BER Performance 10 1 L=0 L=1 L=2 L=3 L= Average Bit Error Rate Average SNR per bit

33 Formulas for the Shannon Capacity For non-regenerative systems C/W = log 2 (1 + γ eq )bps/hz Capacity PDF is given by p C (c)= 2c+1 ln 2(2 c 1)e (2c 1) [( ) ( γ 1 γ 2 γ1 + γ 2 2(2 c ) 1) K γ1 γ 1 2 γ1 γ 2 For regenerative systems C eq = min(c 1, C 2 ) ( 1 γ1 + 1 γ 2 ) ( 2(2 c )] 1) +2K 0. γ1 γ 2 Capacity PDF ( is given by 1 p C (c)= ln ) 2 c e (2c 1) γ 1 γ 2 ( 1 γ1 + γ 1 2 ).

34 Capacity: Numerical Example 10 0 Comparison of Capacity Outage for Regenerative and Non Regenerative Systems Non Regenerative System Regenerative Systems 10 1 Capacity Outage Average SNR per Hop [db]

35 Extension to Systems with N Hops Analog relaying with channel inversion of the previous link 1 N γ eq 2 = 1 n=1 γ n Related to the harmonic mean of the hop s SNRs. Analog relaying with bounded relay gains N ( γ eq 1 = ) 1 1. γ n n=1 Example of a triple-hop system: 1 = γ 1 γ 2 γ 3 γ 1 γ 2 γ 1 γ 3 γ 2 γ 3 γ eq 1

36 Where to Regenerate? 10 0 Effect of the Regeneration Position Non regenerative System Regeneration after hop 1 or 3 Regeneration after hop Outage Probability Normalized Average SNR per Hop [db]

37 Increasing the Number of Hops 10 0 Effect of Number of Hops on Outage Probability 10 1 Outage Probability 10 2 N=1 N= Normalized Average SNR per Hop [db]

38 Analog versus Digital Relaying 10 0 Comparison Between Regenerative and Non regenerative Systems Non regenerative System Regenerative System 10 1 Outage Probability Number of Hops

39 Other Topics of Interest Analog relaying with fixed relay gain. Variable-power and/or variable rate relays. Latency and delay associated with multi-hop systems. Global optimization versus local optimization. Power consumption and fairness issues.