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Performance Analysis of Wideband SOS-based Channel Simulators with Respect to the Bit Error Probability of BPSK OFDM Systems with Perfect and Imperfect CSI Yuanyuan Ma Faculty of Engineering and Science University of Agder P. O. Box 509, NO-4898 Grimstad, Norway E-mail: yuanyuan.ma@uia.no Homepage: http://ikt.hia.no/mobilecommunications/ OptiMO & M2M Joint Meeting, 14 August 2009, UiA, Norway 1/18 Contents I. Aims of WP1.1 II. Promised Output and Achieved Output III. Overview of Research Topics IV. Methods V. Future Plan OptiMO & M2M Joint Meeting, 14 August 2009, UiA, Norway 2/18 I. Aims of WP1.1 Wireless MIMO Channels Problem: Channel changes fast due to the mobility and the environment. • Available channel capacity • Choice of the transmission schemes used in heterogeneous MIMO networks Motivation: It is important to develop realistic and accurate MIMO channel models for the purpose of simulate, design, and evaluate wireless communication systems. Aims: • Develop MIMO channel models for different typical propagation environments. • Develop reference and simulation models for frequency-selective MIMO mobile fading channels. Performance Analysis of Developed Channel Simulators Aim: • Analyze OFDM and STBC-OFDM system performance using the developed channel simulators. OptiMO & M2M Joint Meeting, 14 August 2009, UiA, Norway 3/18 II. Promised Output and Achieved Output Promised output: 2 journal papers and 6 conference papers. Achieved output: 5 accepted conference papers: • Y. Ma and M. Patzold, Performance Analysis of Wideband SOS-Based Channel Simulators with Respect to the Bit ¨ Error Probability of BPSK-OFDM Systems with Perfect and Imperfect CSI, in Proc. WPMC 2009, Sendai, Japan, Sept. 2009, accepted for publication. • Y. Ma and M. Patzold, Performance Analysis of Wideband Sum-of-Cisoids-Based Channel Simulators with Respect to ¨ the Bit Error Probability of DPSK-OFDM Systems, in Proc. VTC 2009-Spring, Barcelona, Spain, April 2009. • Y. Ma and M. Patzold, A Wideband One-Ring MIMO Channel Model Under Non-Isotropic Scattering Conditions, ¨ in Proc. VTC 2008-Spring, Singapore, May 2008, pp. 424–429. • Y. Ma and M. Patzold, Wideband Two-Ring MIMO Channel Models for Mobile-to-Mobile Communications, ¨ in Proc. WPMC 2007, Jaipur, India, Dec. 2007, pp. 380–384. • Y. Ma and M. Patzold, Performance Comparison of Space-Time Coded MIMO-OFDM Systems Using Different Wide- ¨ band MIMO Channel Models, in Proc. ISWCS 2007, Trondheim, Norway, Oct. 2007, pp. 762–766. 2 submitted conference papers: • Y. Ma and M. Patzold, Performance Analysis of STBC-OFDM Systems in Temporally and Spatially Correlated Fading ¨ Channels, in Proc. WCSP 2009, Nanjing, China, Nov. 2009. • C. E. D. Sterian, Y. Ma, M. Patzold, Huaqiang He and Ion Banica, Super-Orthogonal Space-Time Trellis Codes with ¨ ˇ ˇ Differential Phase Modulation for Noncoherent Mobile Communication Systems, in Proc. WCSP 2009, Nanjing, China, Nov. 2009. OptiMO & M2M Joint Meeting, 14 August 2009, UiA, Norway 4/18 III. Overview of Research Topics Mobile-to-Mobile MIMO Channel Modeling • Develop a narrowband two-ring single-bounce scattering MIMO channel model. Wideband MIMO Channel Modeling • Extend the two-ring single-bounce scattering channel model to frequency-selectivity. • Extend the two-ring double-bounce scattering channel model to frequency-selectivity. • Derive the wideband one-ring MIMO channel model for non-isotropic scattering environments. Performance Analysis of Wideband MIMO Communication Systems • Performance of wideband sum-of-sinusoids channel simulators w.r.t the BEP of OFDM systems • Performance of wideband sum-of-cisoids channel simulators w.r.t the BEP of OFDM systems • Performance of STBC-OFDM systems in temporally and spatially correlated fading channels Differential Super-Orthogonal Space-Time Trellis Codes OptiMO & M2M Joint Meeting, 14 August 2009, UiA, Norway 5/18 IV. Methods–Introduction Background: • Rice’s sum-of-sinusoids (SOS) method has emerged as one of the key approaches for de- signing channel simulators for mobile fading channels • Accurate and efﬁcient SOS channel simulators have been commonly used in practical system simulations due to their low realization expenditure. Motivations: • Fundamental statistical properties of SOS channel simulators have been studied. • So far, the performance of the SOS channel simulator w.r.t. bit error probability (BEP) of a transmission system has been studied only for narrowband fading channels. Aim: Analyze the performance a wideband SOS channel simulator w.r.t. the BEP of a BPSK OFDM system assuming both perfect and imperfect channel state information (CSI). OptiMO & M2M Joint Meeting, 14 August 2009, UiA, Norway 6/18 IV. Methods– Frequency-Selective Channel Models The Frequency-Selective Reference Channel Model • A wide-sense stationary uncorrelated scattering model is employed for modeling frequency- selective mobile fading channels. • The time-variant transfer function: L ′ ′ H(f ′ , t) = aℓµℓ(t)e−j2πf τℓ , ℓ=1 2 where µℓ(t) denotes a complex random Gaussian process, each having the variance 2σ0 = 1. ′ • The probability density function (PDF) pζ (r) of the envelope ζ(t) = |H(f0, t)| can be described by the Rayleigh distribution. The Frequency-Selective Sum-of-Sinusoids Channel Simulator • The time-variant transfer function: L ′ ′ ˜ ′ H(f , t) = aℓµℓ(t)e−j2πf τℓ ˜ ℓ=1 where µℓ(t) = µ1,ℓ(t) + j µ2,ℓ(t) represents a deterministic Gaussian process. ˜ ˜ ˜ OptiMO & M2M Joint Meeting, 14 August 2009, UiA, Norway 7/18 IV. Methods– Frequency-Selective Channel Models • The complex processes µi,ℓ(t) can be represented by a sum of Ni,ℓ sinusoids as ˜ Ni,ℓ µi,ℓ(t) = ˜ ci,n,ℓ cos(2πfi,n,ℓ + θi,n,ℓ ), i = 1, 2, n=1 where ci,n,ℓ , fi,n,ℓ , and θi,n,ℓ represent the Doppler coefﬁcient, the Doppler frequency, and the Doppler phase of the ℓ th path, respectively. • Parameter computation methods: 1) Generalized method of exact Doppler spread (GMEDS1): 2 π 1 ci,n,ℓ = σ0 , fi,n,ℓ = fmax sin (n− )+αi,ℓ , Ni,ℓ 2Ni,ℓ 2 where αi,ℓ = (−1)i−1πℓ/(4Ni,ℓL) is the angle of rotation. 2) Monte Carlo method (MCM): 2 π ci,n,ℓ = σ0 , fi,n,ℓ = fmax sin ui,n,ℓ , where ui,n,ℓ ∼ (0, 1] Ni,ℓ 2 • For both methods, we choose N1,ℓ = N2,ℓ = N for ℓ = 1, 2, . . . , L. The Doppler phases are considered as outcomes of a random generator with a uniform distribution over (0, 2π]. OptiMO & M2M Joint Meeting, 14 August 2009, UiA, Norway 8/18 IV. Methods– Frequency-Selective Channel Models The Frequency-Selective SOS channel Simulator ˜ ˜ ′ • The PDF pζ (r) of the envelope ζ(t) =| H(f0, t) |: ˜ π ∞ L pζ (r) = 4πr ˜ [h1,ℓ(y, θ) · h′1,ℓ(y, θ) · h2,ℓ(y, θ) · h′2,ℓ(y, θ)] · J0(2πry) y dy dθ, 0 0 ℓ=1 where the terms hi,ℓ(y, θ) and h′i,ℓ(y, θ) (i = 1, 2) depends on the Doppler coefﬁcient ci,n,ℓ . 0.7 pζ (r) (Theory) 0.6 ˜ pζ (r) (Theory) 0.5 ˜ pζ (r) (Simulation) 0.4 pζ (r) 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 r • It has been proved that pζ (r) → pζ (r) for Ni,ℓ → ∞. ˜ OptiMO & M2M Joint Meeting, 14 August 2009, UiA, Norway 9/18 IV. Methods– Performance Analysis of SOS channel simulators The BEP of BPSK OFDM Systems with Perfect CSI • BEP: average the conditional BEP over the fading channel statistics ∞ ˜ Pb = pζ (r)Pb|r (r)dr, ˜ 0 1 where Pb|ζ (r) = erfc r γ b is the conditional BEP of the BPSK modulation scheme 2 and γ b is the average signal-to-noise ratio per bit. • The actual BEP: the BEP of a transmission system composed of a transmitter, a receiver and a SOS channel simulator π ∞ 2 2 L ˜ 1 1 − π2γy π2y2 Pb = ·e b · M1 1 ( ) [h1,ℓ(y, θ) · h′1,ℓ(y, θ)][h2,ℓ(y, θ) · h′2,ℓ(y, θ)] dy dθ, 2π y 2,2 γb 0 0 ℓ=1 where Mλ,µ (x) is the Whittaker function. • The reference BEP: the BEP of a transmission system composed of a transmitter, a receiver and a reference channel model Since the envelope pζ (r) is Rayleigh distributed, we have 1 γb Pb = 1− . 2 1 + γb OptiMO & M2M Joint Meeting, 14 August 2009, UiA, Norway 10/18 IV. Methods– Performance Analysis of SOS Channel Simulators Simulation Results 0 10 Pb (Theory) ˜ Pb (Theory), GMEDS1 ˜ Bit error probability −1 10 Pb (Simulation), GMEDS1 ˜ Pb (Theory), MCM ˜ Pb (Simulation), MCM −2 10 −3 10 N =6 −4 10 0 5 10 15 20 25 30 γ b (dB) • The wideband SOS channel simulators designed with the GMEDS1 and the MCM are equiv- alent w.r.t. the BEP. ˜ • The actual BEP Pb is determined by the Doppler coefﬁcients ci,n,ℓ , while it is independent of the Doppler frequencies fi,n,ℓ . ˜ • It has been proved that Pb → Pb holds as N → ∞. OptiMO & M2M Joint Meeting, 14 August 2009, UiA, Norway 11/18 IV. Methods– Performance Analysis of SOS Channel Simulators Relative Error ˜ • The deviation between Pb and Pb cannot be ignored if the number of sinusoides is small. • Relative error of the BEP: ˜ Pb − Pb εBEP = Pb • The absolute value of the relative error |εBEP | is less than 4.1% if N ≥ 3. 0 −1 Relative error εBEP (%) −2 N =6 −3 N =4 −4 N =3 −5 −6 N =2 −7 0 5 10 15 20 25 30 γ b (dB) OptiMO & M2M Joint Meeting, 14 August 2009, UiA, Norway 12/18 IV. Methods– Performance Analysis of SOS Channel Simulators The BEP of BPSK OFDM Systems with Imperfect CSI • A pilot symbol is inserted before every block of P OFDM symbols. • If the m th symbol is a pilot symbol, according to the least-square estimation algorithm, the estimated CSI is determined by ˆ Hkm = Hkm + Vkm, k = 1, 2, . . . , K. which is also used for the symbol decisions for the next P OFDM symbol interval. • The BEP of (m + p) th OFDM symbol is given by [Ref] 1 ρ1 Pp = 1 − , p = 1, 2, . . . , P, 2 1+ 1 + ρ2 ¯ γb where ρ1 and ρ2 represent the real and imaginary parts of the correlation coefﬁcient of the ˆ estimated CSI Hkm and the real CSI Hk(m+p) at the (m + p) th symbol slot. [Ref] M. Chang and Y. T. Su, Performance analysis of equalized OFDM systems in Rayleigh fading. IEEE Trans. Commun., 1(4):721–732, Oct. 2002. OptiMO & M2M Joint Meeting, 14 August 2009, UiA, Norway 13/18 IV. Methods– Performance Analysis of SOS Channel Simulators • The reference BEP: P P HH r (p·Ts ) 1 1 p=1 Pb = P = 1 − 2σ2P (1+ , P p 2 1 ) p=1 0 γb where rHH (p · Ts ) denotes the value of the temporal autocorrelation function (ACF) rHH (τ ) of the reference channel model at τ = p · Ts and Ts is the OFDM symbol duration. • The actual BEP: P rHH (p · Ts ) ˜ ˜ 1 p=1 Pb ≈ 1 − 1 . 2 2 2σ0 P (1 + γ ) b For the wideband SOS channel simulator, the temporal ACF can be expressed as L rHH (τ ) = ˜ a2rµℓµℓ (τ ), ℓ˜ ℓ=1 2 Ni,ℓ 2 where rµℓ µℓ (τ ) = ˜ i=1 n=1 (ci,n,ℓ /2) cos(2πfi,n,ℓ τ ) describes the temporal ACF of the determin- istic process µℓ(t). ˜ OptiMO & M2M Joint Meeting, 14 August 2009, UiA, Norway 14/18 IV. Methods– Performance Analysis of SOS Channel Simulators Relative Error • The temporal ACF rHH (p·Ts) can be approximated by the following relation ˜ L (p·Ts)2 rHH (p·Ts) ≈ rHH (p·Ts)− ˜ a2△βℓ, ℓ 2 ℓ=1 where △βℓ denotes the model error of the SOS channel simulator, which is determined by the values of the Doppler coefﬁcients cn,ℓ and the Doppler frequencies fn,ℓ . ˜ • Thus, the actual BEP Pb can be expressed in closed-form as ˜ Pb = Pb + △Pb where P 2 L p · Ts a2△βℓ ℓ p=1 2 ℓ=1 △Pb △Pb = 2 1 , (εBEP = ), 2σ0 P (1 + γ ) Pb b represents the deviation of the BEP. • Using the deviation of the BEP △Pb or the relative error εBEP as a criterion, we can evaluate the performance of the GMEDS1 and the MCM. OptiMO & M2M Joint Meeting, 14 August 2009, UiA, Norway 15/18 IV. Methods– Performance Analysis of SOS Channel Simulators Simulation results 0 0 10 10 Pb (Theory) Pb (Theory) ˜ Pb (Theory) ˜ Pb (Theory) Bit error probability Bit error probability −1 10 ˜ Pb (Simulation) −1 10 · ˜ Pb (Simulation) fmax = 500 Hz, P = 5 −2 −2 fmax = 500 Hz, P = 10 10 10 fmax = 500 Hz, P = 5 −3 10 −3 10 fmax = 100 Hz, P = 10 fmax = 100 Hz, P = 5 N =6 N =6 fmax = 100 Hz, P = 5 −4 −4 10 10 0 5 10 15 20 25 30 0 5 10 15 20 25 30 γ b (dB) γ b (dB) • It is shown that increasing fmax or the value of P leads to BEP performance degradations. • If the GMEDS1 is applied to design the SOS channel simulator, the model error △βℓ = 0. ˜ • If the SOS channel simulator is designed by the MCM, the single realization of the BEP Pb, deviates from the reference BEP Pb in a random manner. OptiMO & M2M Joint Meeting, 14 August 2009, UiA, Norway 16/18 IV. Methods– Conclusion • The performance of the frequency-selective SOS channel simulator has been analyzed w.r.t. the BEP of BPSK OFDM systems. • Analytical expressions for the BEP have been derived for these systems in the presence of perfect and imperfect CSI. • It has been proved that the actual BEP evaluated in the presence of a SOS channel simulator converges to the reference BEP, when the number of sinusoids tends to inﬁnity. • By employing the relative error of the BEP as an appropriate criterion, the deterministic pa- rameter design method (GMEDS1) and the stochastic one (MCM) have been compared. • Under the assumption of perfect CSI, the SOS channel simulators designed by the GMEDS1 and the MCM are equivalent w.r.t. the BEP performance. • In the presence of imperfect CSI, the GMEDS1 outperform the MCM. OptiMO & M2M Joint Meeting, 14 August 2009, UiA, Norway 17/18 V. Future Plan List of Open Problems • Develop dynamic spatially correlated MIMO channel models. • Develop MIMO reference channel models for different typical propagation environments. • Develop efﬁcient channel simulators by using the concept of deterministic channel modeling. • Develop reference and simulation models for wideband MIMO mobile fading channels. • Study the MIMO-OFDM system performance using the developed channel models. • Develop multiuser MIMO channel models. Plan for Solving the Remaining Open Problems • Develop channel models for indoor propagation environments (2009.07.15-2009.12). ⇒ Analyze the statistical properties of the indoor channel models. ⇒ Extend the indoor channel models to frequency-selectivity. ⇒ Compare the system performance using indoor and outdoor channel models. • Develop channel models for keyhole MIMO fading channels (2010.01-2010.04). • Develop multiuser MIMO channel models. OptiMO & M2M Joint Meeting, 14 August 2009, UiA, Norway 18/18