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 100084, P. R. China 2 Institute for Digital Communications, University of Erlangen-Nurnberg, D-91058 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 GC2015 1 / 16
Outline Introduction 1 System Model 2 Effective Rate 3 Numerical Results 4 Conclusions 5 References 6 Zhang et al. (Tsinghua & FAU) jiayizhang@tsinghua.edu.cn IEEE GC2015 2 / 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 /θ T ) ln ( E { exp ( − θ TC ) } ) , 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) r(t) Data a(t) Source x Figure 1: Queuing model Zhang et al. (Tsinghua & FAU) jiayizhang@tsinghua.edu.cn IEEE GC2015 3 / 16
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] � � α 1 / 2 � f γ 1 ( γ 1 ) = α 1 γ 1 α 1 µ 1 / 2 − 1 � γ 1 exp − , (2) 2 β α 1 µ 1 / 2 β 1 Γ ( µ 1 ) 1 � � µ 1 + 2 Γ Γ( µ 1 ) � with E { γ 1 } = ˆ r 2 where β 1 � E { γ 1 } α 1 � , and ˆ r 1 is 1 2 � � µ 1 + 2 Γ α 1 Γ( µ 1 ) µ 1 α 1 defined as the α 1 -root mean value of the envelope random variable R , i.e., � E { R α 1 } . ˆ r 1 = α 1 Zhang et al. (Tsinghua & FAU) jiayizhang@tsinghua.edu.cn IEEE GC2015 4 / 16
System Model We consider a MISO system: y = hx + n (3) where h ∈ C 1 × N t 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] � �� � − A �� R ( ρ, θ ) = − 1 1 + ρ hh † A log 2 bits/s/Hz (4) E N t where A = θ TB ln 2 , 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 GC2015 5 / 16
Exact Effective Rate I Lemma 1 [4] The sum of i.i.d. squared α - µ RVs with parameters α 1 , µ 1 , and ˆ r 1 , i.e., γ = � N t k =1 γ i , can be approximated by an α - µ RV with parameters α , µ and ˆ r by solving the following nonlinear equations E 2 ( γ ) Γ 2 ( µ + 1 /α ) E ( γ 2 ) − E 2 ( γ ) = Γ ( µ ) Γ ( µ + 2 /α ) − Γ 2 ( µ + 1 /α ) , Γ 2 ( µ + 2 /α ) E 2 � γ 2 � E ( γ 4 ) − E 2 ( γ 2 ) = Γ ( µ ) Γ ( µ + 4 /α ) − Γ 2 ( µ + 2 /α ) , r = µ 1 /α Γ ( µ ) E ( γ ) ˆ Γ ( µ + 1 /α ) , (5) As such, we can easily obtain the sum PDF as � � α/ 2 � αγ αµ/ 2 − 1 � γ f γ ( γ ) ≈ 2 β αµ/ 2 Γ ( µ ) exp − . (6) β Zhang et al. (Tsinghua & FAU) jiayizhang@tsinghua.edu.cn IEEE GC2015 6 / 16
Exact Effective Rate II Proposition 1 For MISO α - µ fading channels, the effective rate is given by √ � � kl A − 1 ( N t β/ρ ) αµ/ 2 R ( ρ, θ ) = 1 A − 1 α A log 2 (2 π ) l + k / 2 − 3 / 2 Γ ( A ) Γ ( α ) � � �� ( N t /ρ ) l � − 1 ∆ ( l , 1 − αµ/ 2) G k + l , l � A log 2 , (7) � l , k + l � k ∆ ( k , 0) , ∆ ( l , A − αµ/ 2) � β α/ 2 k � where G ( · ) is the Meijer’s G-function, ∆ ( ǫ, τ ) = τ ǫ , τ +1 ǫ , · · · , τ + ǫ − 1 , with ǫ 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 GC2015 7 / 16
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 2 A Γ ( A ) Γ ( µ ) � �� N t � α/ 2 � �� � (1 , α/ 2) H 2 , 1 � − log 2 . (8) 1 , 2 � ( µ, 1) , ( A , α/ 2) ρβ � 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 GC2015 8 / 16
High-SNR Effective Rate Proposition 3 For MISO α - µ fading channels, the effective rate at high SNRs is given by � βρ � − 1 � Γ ( µ − 2 A /α ) � R ∞ ( ρ, θ ) ≈ log 2 A log 2 . (9) N t 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 GC2015 9 / 16
Low-SNR Effective Rate Proposition 4 For MISO α - µ fading channels, the effective rate at low SNRs is given by � E b � � E b / E b � R , θ ≈ S 0 log 2 , (10) N 0 N 0 N 0 min where E b Γ ( µ 1 ) ln 2 = β 1 Γ ( µ 1 + 2 /α 1 ) , (11) N 0 min 2 N t Γ 2 ( µ + 2 /α ) S 0 = ( A + 1) (Γ ( µ + 4 /α ) Γ ( µ ) − Γ 2 ( µ + 2 /α )) + N t Γ 2 ( µ + 2 /α ) . (12) 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 . Zhang et al. (Tsinghua & FAU) jiayizhang@tsinghua.edu.cn IEEE GC2015 10 / 16
Numerical Results I 9 AWGN Simulation 8 Exact Analysis High-SNR Appro. 7 Effective Rate [bit/s/Hz] 6 5 Effective Rate [bit/s/Hz] 6.6 � = 4, 2, 1 4 6.4 6.2 3 6 � = 4, 2, 1 19.5 20 20.5 SNR � [dB] 2 10 15 20 25 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 GC2015 11 / 16
Numerical Results II 9 AWGN Simulation 8 Exact Analysis High-SNR Appro. 7 Effective Rate [bit/s/Hz] 6 5 Effective Rate [bit/s/Hz] 7.6 4 7.5 � = 4, 2, 1 7.4 3 7.3 � = 4, 2, 1 22.6 22.8 23 23.2 23.4 SNR � [dB] 2 10 15 20 25 SNR � [dB] 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 GC2015 12 / 16
Numerical Results III 1 Simulation 0.9 Low E b /N 0 0.8 Effective Rate [bit/s/Hz] 0.7 0.6 0.5 0.4 0.3 A = 1, 3, 5 0.2 -1.59 dB 0.1 0 -2 -1.5 -1 -0.5 0 0.5 1 E b /N 0 [dB] 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 GC2015 13 / 16 The accuracy of low- E b / N 0 approximate solution is improved for
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 GC2015 14 / 16
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