90 Geometry based stochastic channel modeling 4.5 Sum-of-sinusoids simulation model(5.5) From Chapter 5.4,it is clear that the filter method belongs to a purely mathematical method.By considering that the received signals are the sum at the receiver side of all scatterers's,SoS me 5.8 and h(t)=en cos(2xfnt+0n) (4.7) n=1 odehquenndpha ofon mode Ioroern r for MIMO Therefore So methods are chos the mainmlaion method trouhout this ic motal. el has at least one parameter (gains,frequenc ophases)as a random for each simulation trial but conver to the desired ones when averaged over a sufficient number of trials.Itis properties 4.5.1 Sum-of-sinusoids simulation models for cellular channels (5.5.1) Many approaches have been suggested for SoS simulation of SISO F2M Rayleigh fading channels.Jakes Jakes(1994) ulation mod SISO F2M Rayleigh fading channe Pand Beaulieu(2001).Therefor various modifica s of Jakes model have been proposed in the 2002);Pop and Bea Wang et al.(2008) Pop and B eu have osed a n modelstill caorisf most of the other saristical properies of Rayleigh fading channels odand hisc ies are uiareal.This ete quasi-optimal cedure is the met very similar perfe ance to the optimization method (ie i orm method)Patzold eral.(1996b.1998).Compared tha ece the drawback of the MEDS nerating multiple uncorrelated Rayleigh fading waveforms has been resolved by Wang and his (2008 dela and xiao have pr e parameters (gains,frequencie be rane in the n hame )to cheng and Xiao's mod de but they cony propert eoaetomPcicahearxarceteroretaeraerehaoandgtestada2oOsnaroatiadmMmMo mod 90 Geometry based stochastic channel modeling 4.5 Sum-of-sinusoids simulation model (5.5) From Chapter 5.4, it is clear that the filter method belongs to a purely mathematical method. By considering that the received signals are the sum at the receiver side of all scatterers’s contributions, SoS methods generate the channel waveform by superimposing a finite number of properly designed sinusoids as shown in FIGURE 5.8 and the channel waveform can be expressed as h (t)=X N n=1 cn cos(2πfnt + θn) (4.7) where cn, fn, and θn are the gains, frequencies, and phases of a SoS simulation model. In contrast to filter simulation models, SoS simulation models have low complexity and produce channel waveforms with high accuracy and a perfectly band-limited Doppler PSD. Furthermore, it is easy to extend the SoS models to develop ST correlated simulators for MIMO systems. Therefore, SoS methods are chosen as the main simulation method throughout this book. A SoS simulation model can be either deterministic or stochastic in terms of the underlying parameters (gains, frequencies, and phases) xiang Wang et al. (2008). For a deterministic model, all the parameters are fixed for all simulation trials. In contrast, a stochastic model has at least one parameter (gains, frequencies, or phases) as a random variable which varies for each simulation trial. Therefore, the relevant statistical properties of a stochastic model vary for each simulation trial but converge to the desired ones when averaged over a sufficient number of trials. It is worth noting that a stochastic model with only phases as random variables is actually an ergodic process. Due to the ergodicity, such a stochastic simulation model needs only one simulation trial to converge to the desired statistical properties. 4.5.1 Sum-of-sinusoids simulation models for cellular channels (5.5.1) Many approaches have been suggested for SoS simulation of SISO F2M Rayleigh fading channels. Jakes Jakes (1994) was among the first to propose a deterministic simulation model for SISO F2M Rayleigh fading channels. However, Jakes’ model does not satisfy most of the statistical properties of Rayleigh fading channels Patzold et al. (1998a) and it is not WSS Pop and Beaulieu (2001). Therefore, various modifications of Jakes’ model have been proposed in the literature Pätzold (2002); Pop and Beaulieu (2001); xiang Wang et al. (2008). Pop and Beaulieu have proposed a new deterministic simulation model Pop and Beaulieu (2001) to solve the non-stationarity of Jakes’ model. However, this model still cannot satisfy most of the other statistical properties of Rayleigh fading channels. Pätzold and his co-workers have developed several deterministic simulation models with different parameter computation methods Patzold et al. (1994, 1996a,b, 1998b). The method of equal area (MEA) Patzold et al. (1994, 1996a) is characterized by the fact that the area under the Doppler PSD between two neighboring discrete frequencies are equiareal. This parameter computation method presents acceptable performance with low complexity. A quasi-optimal procedure is the method of exact Doppler spread (MEDS) Patzold et al. (1996b, 1998b). This method outperforms the MEA and even shows the very similar performance to the optimization method (i.e., Lp-norm method) Patzold et al. (1996b, 1998b). Compared to the MEA and MEDS, the arising numerical complexity of the optimization method is comparatively high, so that the simulation of such isotropic scattering Rayleigh fading channels is often not worth the effort Pätzold (2002). More recently, the drawback of the MEDS in generating multiple uncorrelated Rayleigh fading waveforms has been resolved by Wang and his co-workers in Wang et al. (2009); xiang Wang et al. (2008). In order to satisfy more statistical properties and/or match the desired properties over longer time delays, Zheng and Xiao have proposed several new stochastic simulation models Xiao et al. (2006); Zheng and Xiao (2002, 2003). By allowing all three parameters (gains, frequencies, and phases) to be random variables, Zheng and Xiao’s model obtain the statistical properties similar to the ones of Rayleigh fading channels. Since the models are no longer ergodic process, the statistical properties of these models vary for each simulation trial, but they converge to the desired properties over sufficient number of simulation trials (normally 50 to 100). A detailed comparison of the statistical properties for Zheng and Xiao’s models is presented in Patel et al. (2005a). However, all the aforementioned deterministic and stochastic simulation models are limited to isotropic scattering SISO F2M Rayleigh fading channels, while simulation models for MIMO F2M channels under a more realistic scenario of non-isotropic scattering are scarce in the current literature. In Patzold and Hogstad (2004), a narrowband MIMO F2M deterministic simulation was proposed for macro-cell scenarios. In Hogstad et al. (2005), new deterministic simulation models were developed for both narrowband and wideband MIMO F2M channels of micro-cell scenarios. Up to now, only one wideband MIMO F2M deterministic simulation model was proposed in Patzold and Hogstad