increase the quality of services (QoS) of cellular networks : Considering both short and long term effect (multipath and average power variation) In the literature Signal processing community → multipath cancellation (equalization, channel coding, smart antennas...) ”Network” community → fight again the QoS variations due to the variation of the average power Minimal research effort has been allocated to the study of both effects Our work Study of a QoS criterium considering both short and long term effects Philippe Mary 2/12/2010 3 / 32
the work Analytical performance of radio-mobile systems in fading channels and shadowing environment ++ Allows to point out quickly the behavior of a system ++ Allows to predict behavior x x Difficult for high complexe environment Coverage prediction over QoS constraint, best effort resource allocation. Philippe Mary 2/12/2010 4 / 32
Channel models and assumption 2 Quick overview of SEP approximation 3 Outage considering fading and shadowing 4 Outage considering interference 5 Conclusion and further work Philippe Mary 2/12/2010 5 / 32
shadowing Work with ”outage” [Conti03] : Symbol error outage (SEO) (non-ergodic channel) Ps (O) = Pr (SEP ≥ P∗ s ) The important network design criterium is the packet error outage (PEO) Philippe Mary 2/12/2010 7 / 32
model Flat fading channels The received power is : Pr ∝ αshad · α2 fading · Pt αfading : Short term effect (fast fading) Nakagami-m or Rice distributed Instantaneous SNR variation γs = α2 fading Es/N0 αshad : Long term effect (change after one or several packets) log-normally distributed Average SNR variation γs = αshad Es/N0, αshad = E α2 fading Average SNR −→ log-normally distributed Mean : µdB = E (10 log10 γs ) Standard deviation : σdB Philippe Mary 2/12/2010 8 / 32
in fading channels Average error probability (w.r.t. short term effects) : Classic form of the average SEP : Ps (E |γs ) = ∞ 0 Ps (E |γs, γs ) f (Q(√ γs )) pγs |γs (γs ) dγs Craig91 → alternative form for Gaussian function Q Allowed to derive closed-form SEP expressions in an unified way thanks to the moment generating function (MGF) of the SNR [Alouini04] : Ps (E |γs ) = 1 π (M−1)π/M 0 Mγs − gpsk sin2 θ dθ Philippe Mary 2/12/2010 9 / 32
shadowing Symbol error outage (SEO) : Ps (O |µ) = P (Ps (E |γs ) ≥ P∗ s (E |γs ) |µ) ⇐⇒ Pγs (O |µ) = P (γs ≤ γth |µ) Need of the SNR threshold : γth = f (P∗ s (E)) o` u f (P∗ s (E)) = P∗ s (E|γs )−1 The SEO is hence : Ps (O|µ) = Z γth=(P∗ s (E))−1 0 pγs (γs |µ) dγs = Q „µdB − 10 log10 γs (P∗ s (E)) σdB « Steps Find an accurate and simple expression for SEP Inverting this expression w.r.t. the SNR Estimate the SEO Philippe Mary 2/12/2010 10 / 32
of the SEP Asymptotic analysis [Giannakis03] : Behavior of the SEP for high SNR Ps (E) ≈ (Gc γs )−Gd Gc Coding gain → horizontal shift compared to a reference Gd Diversity gain → error probability slop in high SNR regime x x Not accurate at low SNR Bounds of Conti et al. [Conti03] ++ Tight bounds of the average SEP thanks to bounds on the MGF of the SNR x x One kind of channel (Nakagami-m) x x No channel coding x x No interferences Philippe Mary 2/12/2010 11 / 32
Can be used for Nakagami-m and Rice channels Integral approximation : I = y∈D h(y)e−λg(y)dy λ ∈ R, D ⊆ R The Laplace approximation of I is : ˜ I = 2π λ |g (y0 )| h(y0 )e−λg(y0) and I = ˜ I 1 + O λ−1 , λ → ∞ y0 = min |{z} y g(y) Philippe Mary 2/12/2010 12 / 32
in Nakagami-m channels (1/3) The exact average SEP (M-PSK, M-QAM) can be written as [Shin04] : Ppsk s (E |γs ) = x (γs )m „ k1 · 2F1 „ m, 1 2 ; m + 1; x (γs ) « + k2 · F1 „ 1 2 , m, 1 2 − m; 3 2 ; y (γs ) , 1 − gpsk «« Pqam s (E |γs ) = x1 (γs )m k3·2F1 „ m, 1 2 ; m + 1; x1 (γs ) « −x2 (γs )m k4·F1 „ 1, m, 1; m + 3 2 ; x2 (γs ) x1 (γs ) , 1 2 « Philippe Mary 2/12/2010 14 / 32
in Nakagami-m channels (2/3) The Gauss hypergeometric function can be expressed as 2F1 m, 1 2 ; m + 1; x = B(m, 1)−1 1 0 tm−1(1 − tx)−1/2dt, Laplace approximation by choosing [Wood03] : h(t) = B (m, 1)−1 t−1, g(t) = − m ln t − 1 2 ln(1 − xt) Philippe Mary 2/12/2010 15 / 32
in Nakagami-m channels (3/3) Theorem (SEP approximation) In a flat Nakagami-m fading channel, the average SEP of M-PSK/M-QAM signals is well approached by : Ps(E|γs ) ≈ kmod xm √ 1 − x˜ t ; ∀γs with ˜ t = m/(m + 1), x = 1/ (1 + gmod γs /m), gmod and kmod are modulation dependent constants. Philippe Mary 2/12/2010 16 / 32
Theorem (SEP inversion) Under the same conditions as previously, the average SNR is : γs (P∗ s (E)) = c0 P∗ s (E) − 1 m 1 − c1P∗ s (E) 1 m − 1 2m − k− 1 m mod Sketch of proof. Solving in [0, 1] : (kmod )2 x2m + (P∗ s (E))2 ˜ tx − (P∗ s (E))2 = 0 Constructing the series : 8 > < > : x0 = 0, xn+1 = „ P∗ s (E) kmod « 1 m `1 − ˜ txn ´ 1 2m We can show that {xn}n∈N is converging towards xs Philippe Mary 2/12/2010 18 / 32
systems The output SNR is : γSTBC = ||H||2 F ntR γs γs = E0/N0 The MGF of SNR can be shown to be (without correlation) : MγSTBC = (Mγs )nt nr Philippe Mary 2/12/2010 25 / 32
systems ZF linear receiver. The substream SNR is [Gore02] : γk = γs [HH H]−1 kk In Rayleigh channel : Z = HHH → CWnt (nr , 0, Σnt ) We can shown that : Mγk = (Mγs )nr −nt +1 Philippe Mary 2/12/2010 26 / 32
with one co-channel interference The exact SEP is obtained by averaging the conditional SEP w.r.t. the INR Ps (E) = ∞ 0 Ps (E|γi ) pγi (γi ) dγi The average SEP (M-PSK, M-QAM) with one co-channel interference in Rayleigh channel is bounded by : Ps (E|γd , γi ) ≤ 2kmod 1 + gmod γd gmod γd 1 + γi γi + 2 (1 + gmod γd ) Philippe Mary 2/12/2010 27 / 32
with shadowing and co-channel interference Shadowing → both SNR γd and INR γi are random variables We can show the following result : Theorem γd et γi are two random variables i.i.d. and log-normally distributed. The SEO of the desired signal with one co-channel interference is : P (Ps (E) > P∗ s ) = Z ∞ 0 10/ log (10) σi √ 2πγi e − (10 log10 γi −µi )2 2σ2 i Q „ µd − 10 log10 γth (P∗ s , γi ) σd « dγi where γth (P∗ s , γi ) is the needed average SNR to reach the QoS target P∗ s knowing the average INR γi . Philippe Mary 2/12/2010 29 / 32
to study the performance of wireless communications in a realistic environment (Fading + Shadowing) When shadowing is considered the channel is non-ergodic The PEO is a measure of the reliability that a wireless network can offer under constraint of QoS (average PEP) Applications Coverage prediction (4G Cellular networks, · · · ) Resource allocation considering an average target PEP with a certain outage probability Philippe Mary 2/12/2010 30 / 32
With coding How do we deal with capacity approaching codes (LDPC, Turbo, · · · ) ? Quantify the tradeoff Energy consumption/PEO reduction Connectivity study in access network Active links : average PEP < PEP target Defined the connectivity in function of the PEP target Extension to multi-hop and cooperative networks Philippe Mary 2/12/2010 31 / 32
P. Mary, M. Dohler, J.-M. Gorce, G. Villemaud, M. Arndt, ”BPSK Bit Error Outage over Nakagami-m Fading Channels in Lognormal Shadowing Environments”, IEEE Commun. Letters, 2007 [2] P. Mary, M. Dohler, J.-M. Gorce, G. Villemaud, M. Arndt, ”M-ary Symbol Error Outage over Nakagami-m Fading Channels in Shadowing Environments”, IEEE Trans. on Commun., 2009 [3] P. Mary, M. Dohler, J.-M. Gorce, G. Villemaud, ”Symbol Error Outage Analysis of MIMO OSTBC Systems over Rice Fading Channels in Shadowing Environments”, Minor Revisions for IEEE Trans. on Wireless Commun. [4] P. Mary, M. Dohler, J.-M. Gorce, G. Villemaud, ”Symbol Error Outage for Spatial Multiplexing Systems in Rayleigh Fading Channel and Lognormal Shadowing”, In Proc. of IEEE Spawc 2009, Italy, 2009 [5] P. Mary, M. Dohler, J.-M. Gorce, G. Villemaud, ”Packet Error Outage for Coded Systems Experiencing Fading Channels and Interference in Shadowing Environment”, In preparation Philippe Mary 2/12/2010 32 / 32
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