Effective Rate Analysis of MISO Systems over α-µ Fading Channels

Transcription

1 Effective Rate Analysis of MISO Systems over α-µ Fading Channels Jiayi Zhang 1,2, Linglong Dai 1, Zhaocheng Wang 1 Derrick Wing Kwan Ng 2,3 and Wolfgang H. Gerstacker 2 1 Tsinghua National Laboratory for Information Science and Technology (TNList) Department of Electronic Engineering, Tsinghua University, Beijing , P. R. China 2 Institute for Digital Communications, University of Erlangen-Nurnberg, D Erlangen, Germany 3 School of Electrical Engineering and Telecommunications, The University of New South Wales, Australia San Diego, CA Dec 07, 2015 Zhang et al. (Tsinghua & FAU) jiayizhang@tsinghua.edu.cn IEEE GC / 16

2 Outline 1 Introduction 2 System Model 3 Effective Rate 4 Numerical Results 5 Conclusions 6 References Zhang et al. (Tsinghua & FAU) jiayizhang@tsinghua.edu.cn IEEE GC / 16

3 Figure 1: Queuing model Zhang et al. (Tsinghua & FAU) IEEE GC / 16 Introduction I Effective rate is an appropriate metric to quantify the system performance under QoS limitations and is given by [1] α(θ) = (1/θT)ln(E{exp( θtc)}), θ 0 (1) where C is the system throughput, T denotes the block duration and θ is the QoS exponent. For θ 0, the Effective Rate reduces to the standard ergodic capacity. L(t) Data Source a(t) x r(t)

4 Introduction II The α-µ distribution provides better fit to experimental data than most existing fading models and involves as special cases: Rayleigh One-sided Gaussian Nakagami-m Weibull Exponential Gamma The power PDF of α-µ variables is given by [2] ( f γ1 (γ 1 ) = α α 1γ 1 µ 1 / β α 1µ 1 /2 1 Γ(µ 1 ) exp where β 1 E{γ 1 } ( Γ(µ 1 ) Γ µ α 1 ) with E{γ 1 } =ˆr 2 1 ( γ1 β 1 ( ) Γ µ α ( 1 2 µ α1 1 Γ(µ1 ) ) α1 /2 ), (2) ), and ˆr 1 is defined as the α 1 -root mean value of the envelope random variable R, i.e., ˆr 1 = α 1 E {R α 1}. Zhang et al. (Tsinghua & FAU) jiayizhang@tsinghua.edu.cn IEEE GC / 16

5 System Model We consider a MISO system: y = hx+n (3) where h C 1 Nt denotes the channel fading vector, x is the transmit vector with covariance E{xx } = Q, and n represents the AWGN term. The effective rate of the MISO channel can be expressed as [3] ( { ( R(ρ,θ) = 1 A log 2 E 1+ ρ ) }) A hh bits/s/hz (4) N t where A = θtb ln2, with B denoting the bandwidth of the system, while ρ is the average transmit SNR. Zhang et al. (Tsinghua & FAU) jiayizhang@tsinghua.edu.cn IEEE GC / 16

6 Exact Effective Rate I Lemma 1 [4] The sum of i.i.d. squared α-µ RVs with parameters α 1, µ 1, and ˆr 1, i.e., γ = N t k=1 γ i, can be approximated by an α-µ RV with parameters α, µ and ˆr by solving the following nonlinear equations E 2 (γ) E(γ 2 ) E 2 (γ) = E 2( γ 2) E(γ 4 ) E 2 (γ 2 ) = Γ 2 (µ+1/α) Γ(µ)Γ(µ+2/α) Γ 2 (µ+1/α), Γ 2 (µ+2/α) Γ(µ)Γ(µ+4/α) Γ 2 (µ+2/α), ˆr = µ1/α Γ(µ)E(γ) Γ(µ+1/α), (5) As such, we can easily obtain the sum PDF as ( ( ) ) f γ (γ) αγαµ/2 1 γ α/2 2β αµ/2 Γ(µ) exp. (6) β Zhang et al. (Tsinghua & FAU) jiayizhang@tsinghua.edu.cn IEEE GC / 16

7 Exact Effective Rate II Proposition 1 For MISO α-µ fading channels, the effective rate is given by ( R(ρ,θ) = 1 A 1 A log α ) kl A 1 (N t β/ρ) αµ/2 2 (2π) l+k/2 3/2 Γ(A)Γ(α) ( [ 1 A log 2 G k+l,l (N t /ρ) l l,k+l ( β α/2 k ) (l,1 αµ/2) k (k,0), (l,a αµ/2) ]), (7) where G ( ) is the Meijer s G-function, (ǫ,τ) = τ ǫ, τ+1 ǫ,, τ+ǫ 1 ǫ, with τ being an arbitrary real value and ǫ a positive integer. Moreover, l/k = α/2, where l and k are both positive integers. For large values of l and k, it is not very efficient to compute (7). Zhang et al. (Tsinghua & FAU) jiayizhang@tsinghua.edu.cn IEEE GC / 16

8 Exact Effective Rate III Proposition 2 The effective rate in (7) can be further written in the form of Fox s H-functions by using the Mellin Barnes integral as R (ρ,θ) = 1 ( ) α (1 log A 2 Γ(A)Γ(µ) ( [ (Nt ) α/2 ])) log 2 H 2,1 (1,α/2) 1,2. (8) ρβ (µ,1),(a,α/2) It is worth to mention that (8) is very compact which simplifies the mathematical algebraic manipulations encountered in the effective rate analysis. Zhang et al. (Tsinghua & FAU) jiayizhang@tsinghua.edu.cn IEEE GC / 16

9 High-SNR Effective Rate Proposition 3 For MISO α-µ fading channels, the effective rate at high SNRs is given by ( ) βρ R (ρ,θ) log 2 1 ( ) Γ(µ 2A/α) N t A log 2. (9) Γ(µ) The above result indicates that the high-snr slope is S = 1, which is independent of β. The same observations were made in previous works for the Rayleigh, Rician, and Nakagami-m cases. Zhang et al. (Tsinghua & FAU) jiayizhang@tsinghua.edu.cn IEEE GC / 16

10 Low-SNR Effective Rate Proposition 4 For MISO α-µ fading channels, the effective rate at low SNRs is given by ( ) ( Eb Eb R,θ S 0 log N 2 / E ) b, (10) 0 N 0 N 0min where E b N 0min = S 0 = Γ(µ 1 )ln2 β 1 Γ(µ 1 +2/α 1 ), (11) 2N t Γ 2 (µ+2/α) (A+1)(Γ(µ+4/α)Γ(µ) Γ 2 (µ+2/α))+n t Γ 2 (µ+2/α). The minimum E b N 0 is independent of the delay constraint A, whereas the wideband slope S 0 is independent of β, and a decreasing function in A, while it is a monotonically increasing function in N t. (12) Zhang et al. (Tsinghua & FAU) jiayizhang@tsinghua.edu.cn IEEE GC / 16

11 Numerical Results I Effective Rate [bit/s/hz] AWGN Simulation Exact Analysis High-SNR Appro. = 4, 2, 1 Effective Rate [bit/s/hz] = 4, 2, SNR [db] SNR [db] The exact analytical expression is very accurate for all SNRs, The high-snr approximation is quite tight even in moderate SNRs and its accuracy is improved for larger values of the fading parameters, An increase of the effective rate is observed as α increases. Zhang et al. (Tsinghua & FAU) jiayizhang@tsinghua.edu.cn IEEE GC / 16

12 Numerical Results II Effective Rate [bit/s/hz] AWGN Simulation Exact Analysis High-SNR Appro. = 4, 2, SNR [db] SNR [db] Effective Rate [bit/s/hz] = 4, 2, 1 An increase of the effective rate is observed as α increases, Since a large value of µ results in more multipath components, A large value of α accounts for a larger fading gain, Zhang et al. (Tsinghua & FAU) jiayizhang@tsinghua.edu.cn IEEE GC / 16

13 The effective rate is a monotonically decreasing function of A, which implies that tightening the delay constraints reduces the effective rate, The change of the delay constraint A does not affect the minimum E b /N 0, which is 1.59 db in our case, Zhang et al. (Tsinghua & FAU) jiayizhang@tsinghua.edu.cn IEEE GC / 16 Numerical Results III Simulation Low E b /N 0 Effective Rate [bit/s/hz] db A = 1, 3, E b /N 0 [db]

14 Conclusions Novel and analytical expressions of the exact effective rate of MISO systems over i.i.d. α-µ fading channels have been derived by using an α-µ approximation. From high-snr approximation, the effective rate can be improved by utilizing more transmit antennas as well as in a propagation environment with larger values of α and µ. Our analysis provides the minimum required transmit energy per information bit for reliably conveying any non-zero rate at low SNRs. Our analytical results serve as a performance benchmark for our future work on the performance analysis of the multi-user scenario. Zhang et al. (Tsinghua & FAU) jiayizhang@tsinghua.edu.cn IEEE GC / 16

15 References [1] D. Wu and R. Negi, Effective Rate: A wireless link model for support of quality of service, IEEE Trans. Wireless Commun., vol. 2, no. 4, pp , July [2] D. Da Costa, M. D. Yacoub et al., Highly accurate closed-form approximations to the sum of α-µ variates and applications, IEEE Trans. Wireless Commun., vol. 7, no. 9, pp , Sep [3] M. Matthaiou, G. C. Alexandropoulos, H. Q. Ngo, and E. G. Larsson, Analytic framework for the effective rate of MISO fading channels, IEEE Trans. Commun., vol. 60, no. 6, pp , June [4] K. P. Peppas, A simple, accurate approximation to the sum of GammaC Gamma variates and applications in MIMO free-space optical systems, IEEE Phot. Technol. Lett., vol. 23, no. 13, pp , Jul [5] J. Zhang, Z. Tan, H. Wang, Q. Huang, and L. Hanzo, The effective throughput of MISO systems over κ-µ fading channels, IEEE Trans. Veh. Technol., vol. 63, no. 2, pp , Feb [6] J. Zhang, L. Dai, W. H. Gerstacker, and Z. Wang, Effective capacity of communication systems over κ-µ shadowed fading channels, Electronics Lett., vol. 51, no. 19, pp , Sep Zhang et al. (Tsinghua & FAU) jiayizhang@tsinghua.edu.cn IEEE GC / 16

16 Thank you! Zhang et al. (Tsinghua & FAU) IEEE GC / 16