9 Practices:channel modeling for modern communication systems Based on the knowledge introduced in previous chapters,this chapter will present the development of channel modelsfor variousom demonstrated and some interest observations and conclusions will be given. 9.1 Typical scenarios for vehicle-to-vehicle and cooperative communications((10.1)) corresponding typical scenario newly emerging tems,e.g.,V2 communicat ons and cooperae neral,typical scenarios for conventional cellular communications can be classified as Macro-cell scenaric Micro- celeand Picll cenaio ey emergn commnicionypical cen r V2Vnd channel characteristics of these new y emerging communication systems.This section will introduce and classify typical scenarios for the newly emerging V2V and cooperative communication systems. 9.1.1 Vehicle-to-vehicle communication scenarios((10.1.1)) channel knowledge obtaned from systems be for VV.som differentappication.Here.we will briefly review and classify some recent typical V2V scenarios according to srm流i6ta时收d设dda 。Carrier Frequencies were carried ou 1EEE802.1 1b/g banu caoaomcetnpncpaamaphonnaenpensrcomawaeihe9ueCgcaayn 528时meg,24a业o Frequency-Selectivity and Antennas
9 Practices: channel modeling for modern communication systems Based on the knowledge introduced in previous chapters, this chapter will present the development of channel models for various communication scenarios, e.g., conventional cellular scenarios, newly emerging V2V scenarios and cooperative scenarios, via different modeling approaches. From this chapter, some new channel models will be demonstrated and some interest observations and conclusions will be given. 9.1 Typical scenarios for vehicle-to-vehicle and cooperative communications ((10.1)) In general, different communication scenarios result in different channel characteristics. Therefore, to investigate underlying channel characteristics of any communication systems, it is desirable to completely classify the corresponding typical scenarios, on which the communication systems are developed. Unlike newly emerging communication systems, e.g., V2V communications and cooperative communications systems, typical scenarios for conventional cellular communication systems have been completely classified and widely accepted in public. In general, typical scenarios for conventional cellular communications can be classified as Macro-cell scenarios, Microcell scenarios, and Pico-cell Scenarios. As newly emerging communication systems, typical scenarios for V2V and cooperative communications are different compared with those for conventional cellular systems, resulting in unique channel characteristics of these newly emerging communication systems. This section will introduce and classify typical scenarios for the newly emerging V2V and cooperative communication systems. 9.1.1 Vehicle-to-vehicle communication scenarios ((10.1.1)) Knowledge of the V2V propagation channel for different scenarios is of great importance for the design and analysis of V2V systems. However, due to the large difference between the conventional cellular and V2V channels, the channel knowledge obtained from conventional cellular systems cannot be directly used for V2V systems. So far, some measurement campaigns have been conducted and others are ongoing to investigate the V2V propagation channels for different application scenarios. Here, we will briefly review and classify some recent typical V2V scenarios according to carrier frequencies, frequency-selectivity, antennas, environments, Tx/Rx direction of motion, and channel statistics, as shown in Table 10.1. • Carrier Frequencies Before the IEEE 802.11p standard ? was proposed, some measurement campaigns were conducted at carrier frequencies outside the 5.9 GHz dedicated short range communication (DSRC) band. In Acosta et al. (2004); ?, V2V measurements were carried out at 2.4 GHz, i.e., the IEEE 802.11b/g band. Some measurements were done around the IEEE 802.11a frequency band, e.g., at 5 GHz in ? and at 5.2 GHz in ?. Measurements at 5.9 GHz were presented in ?? for narrowband and wideband V2V channels, respectively. Based on the aforementioned measurement campaigns, we can observe that propagation phenomenon in similar environment with different frequencies can vary significantly. Therefore, more measurement campaigns are expected to be conducted at 5.9 GHz for the better design of safety applications for V2V systems following the IEEE 802.11p standard. On the other hand, for improved design of nonsafety applications for V2V systems, measurement campaigns performed at other frequency bands, e.g., 2.4 GHz or 5.2 GHz, are still required. • Frequency-Selectivity and Antennas This is a Book Title Name of the Author/Editor c XXXX John Wiley & Sons, Ltd
256 Practices:channel modeling for modern communication systems Table 10.1.Important V2V channel measurements 26 一0hE2oo more attra tive for V2V systems since multiple anter ements can be eas sily plac o MMO VZcrMIMO Vwdeband measuremen ampareneded for re V developments. Environments and Tx/Rx Direction of Motion the Tx-Rxdistance is normally larger than 1km or ranges from300 metersto1km,V2V systems are mainly used or broa ting most V2V applications fall into MSS or SSS scenarios,these two scenarios are currently rec ing more and more atentionewihns rent measurement campaigns taking eta (200 However,there than Ikm.For such LSS V2V applications.one ple is V2V decentralized environmental notification.which means ion with each other channels for LSSs charchom 2V scenarios can sobe categorized as urban canyon,suburban street,and expressway in terms of roadside unique vehicular sity (VTD impact of the VID.Note that i scauterins except in cases of high ID.To the cmpign cred the impact of the Dor remoiooftheand Rx also affect channel statistics,e Doppler effects.Many measurement campaigns,e.g,in Acosta et al.(2004)-7 and ?have focused on studying channel ch racteristics when the Tx and ?and,have vei y,it is desirable to conduct mor t campaigns for MSS and SSS scenarios with various 9.1.2 Cooperative communication scenarios((10.1.2)) Cooperative mimo groups multiple radio devices to form virtual antenna arrays so that they can c erate with each
256 Practices: channel modeling for modern communication systems Table 10.1. Important V2V channel measurements. SS: suburban street; EW: expressway; UC: urban canyon; Micro: Micro-cell; Pico: Pico-cell; H(L)VTD: high (low) vehicular traffic density; PDP: power delay profile; DD: Doppler-delay; PSD: power spectrum density; STF: space-time-frequency; CF: correlation function; LCR: level crossing rate; SDF: space-Doppler-frequency; PDF: probability density function; PL: path loss; CDF: cumulative distribution function; CT: coherence time. In 1999, the Federal Communications Commission (FCC) allocated 75 MHz of licensed spectrum, including seven channels, each with approximately 10 MHz instantaneous bandwidth, for DSRC in the USA. Such V2V channels are often frequency-selective (wideband) channels. A narrowband fading channel characterisation based on measurement results, e.g., in ?, is not sufficient for such V2V DSRC applications. Wideband measurement campaigns, e.g., in Acosta et al. (2004); ?); ?); ?); ?, are therefore essential for understanding the frequency-selectivity features of V2V channels and further designing high-performance V2V systems. Most V2V measurement campaigns so far have focused on single-antenna applications, resulting in SISO systems, e.g., in Acosta et al. (2004); ?); ?); ?. MIMO systems, with multiple antennas at both ends, are very promising candidates for future communication systems and are gaining more importance in IEEE 802.11 standards. Moreover, MIMO technology becomes more attractive for V2V systems since multiple antenna elements can be easily placed on large vehicle surfaces. However, until now only a few measurement campaigns, e.g., in ??, were conducted for MIMO V2V channels. Hence, more MIMO V2V wideband measurement campaigns are needed for future V2V system developments. • Environments and Tx/Rx Direction of Motion Similar to conventional cellular systems, V2V scenarios can be classified as large spatial scale (LSS), moderate spatial scale (MSS), and small spatial scale (SSS) according to the Tx–Rx distance. For LSS scenarios or MSS scenarios, where the Tx–Rx distance is normally larger than 1 km or ranges from 300 meters to 1 km, V2V systems are mainly used for broadcasting or geocasting, i.e., geographic broadcasting ?. While for SSS scenarios, where the Tx–Rx distance is usually smaller than 300 meters, V2V systems can be applied to broadcasting, geocasting, or unicasting. Since most V2V applications fall into MSS or SSS scenarios, these two scenarios are currently receiving more and more attention with several current measurement campaigns taking place, e.g., in Acosta et al. (2004)–?. However, there are still few applications that need communications between two vehicles separated by large distances, e.g., larger than 1 km. For such LSS V2V applications, one example is V2V decentralized environmental notification, which means that vehicles or drivers in a certain area share information with each other about observed events or roadway features. These applications have not gained much attention and thus no measurement results are available that explore V2V channels for LSS scenarios. V2V scenarios can also be categorized as urban canyon, suburban street, and expressway in terms of roadside environments, i.e., buildings, bridges, trees, parked cars, etc., located on the roadside. Many measurement campaigns, e.g., in Acosta et al. (2004)–? and ?, were conducted to study the channel statistics for various types of roadside environments. Due to the unique feature of V2V environments, the vehicular traffic density (VTD) also significantly affects the channel statistics, especially for MSS and SSS scenarios. In general, the smaller Tx–Rx distance, the larger impact of the VTD. Note that V2V channels usually exhibit non-isotropic scattering except in cases of high VTD. To the best of the authors’ knowledge, only one measurement campaign ? was carried out to study the impact of the VTD for expressway MSS and SSS scenarios. Directions of motion of the Tx and Rx also affect channel statistics, e.g., Doppler effects. Many measurement campaigns, e.g., in Acosta et al. (2004)–? and ?, have focused on studying channel characteristics when the Tx and Rx are moving in the same direction. Few V2V measurement campaigns, e.g., in ? and ?, have investigated channel characteristics when the Tx and Rx are moving in opposite directions. In summary, it is desirable to conduct more measurement campaigns for MSS and SSS scenarios with various VTDs when the Tx and Rx move in opposite directions. In addition, measurement campaigns for LSS scenarios are indispensable for some V2V applications that need communications between two vehicles with a large distance. 9.1.2 Cooperative communication scenarios ((10.1.2)) Cooperative MIMO groups multiple radio devices to form virtual antenna arrays so that they can cooperate with each other by exploiting the spatial domain of mobile fading channels. Note that the cooperation of the grouped devices does not mean that the base station (BS) and mobile station (MS) have to be in reach of each other. Some devices
Practices:channel modeling for modern communication systems 257 can be ted as The physical er ppuciomhererore 1 urther in 9.2 Channel characteristics((10.2)) Knowledge of channel statistics is essential for the analysis and design of a communication system.Unlike a rich and fascinating history of the important channel characteristics of v2y and cooperative communication systems 9.2.1 Channel characteristics of V2V communication systems((10.2.1)) As shown in Table 10.1,many different V2V channel statistics have been studied in recent measurement campaigns Acosta et al.( ignal as either Rayleigh or Ricean.In 5.9 GHZ tha received amplitude dis ution in a dicated (LoS)component is intermittently lost at large distances,the channel fading can become more severe than Rayleigh. a专sap通 com V2vchan the outhoreanye e bopple aprea angnincantly with dinerent timedhan e-time correlation function in terms oftime.was tigated in ?and?It is worth er PSD for V channels can be significantly different from the traditional (shaped Doppler 9.2.2 Channel characteristics of cooperative communication systems((10.2.2)) Several papers hav erties of MiMo channels for odes are all star ive MIMo ch measurements wer 22 es spectively.ll these measurement campaigns concentrated on the investigatio individual scale fadng MO systems. investigation. 9.3 Scattering theoretical channel models for conventional cellular MIMO systems((10.3))
Practices: channel modeling for modern communication systems 257 can be treated as relays to help the communication between the BS and MS. Unlike conventional point-to-point MIMO systems, cooperative MIMO systems consist of multiple radio links, e.g., BS-BS, BS-relay station (RS), RS-RS, RS-MS, BS-MS, and MS-MS links. Due to different local scattering environments around BSs, RSs, and MSs, a high degree of link heterogeneity or variations is expected in cooperative MIMO systems. In this chapter, we are interested in various cooperative MIMO environments which can be classified based on the physical scenarios and application scenarios. The physical scenarios include outdoor macro-cell, micro-cell, pico-cell, and indoor scenarios. Each physical scenario further includes 3 application scenarios, i.e., BS cooperation, MS cooperation, and relay cooperation. Therefore, 12 cooperative MIMO scenarios are considered in this chapter. 9.2 Channel characteristics ((10.2)) Knowledge of channel statistics is essential for the analysis and design of a communication system. Unlike a rich and fascinating history of the research of conventional cellular channel characteristics, the investigation of channel characteristics for emerging V2V and cooperative MIMO systems is still in its infancy. This section will introduce the important channel characteristics of V2V and cooperative communication systems. 9.2.1 Channel characteristics of V2V communication systems ((10.2.1)) As shown in Table 10.1, many different V2V channel statistics have been studied in recent measurement campaigns Acosta et al. (2004)–?. Here, we only concentrate on two important statistics, amplitude distribution and Doppler power spectral density (PSD). Analysis of amplitude distributions has been reported in ?, ?, and ?. In ?, the authors modeled the amplitude probability density function (PDF) of the received signal as either Rayleigh or Ricean. In ?, it was observed that the received amplitude distribution in a dedicated V2V system with a carrier frequency of 5.9 GHz gradually transits from near-Ricean to Rayleigh as the vehicle separation increases. When the line-of-sight (LoS) component is intermittently lost at large distances, the channel fading can become more severe than Rayleigh. A similar conclusion has been drawn in ?, where the amplitude PDF is modeled as Weibull distribution and this “worse than Rayleigh” fading is called severe fading. The reason behind the severe fading is the rapid transitions of multipath components induced by high speed and low height of the Tx/Rx and fast moving scatterers ?. The Doppler PSD has been investigated in Acosta et al. (2004), ?, and ?–?. Joint Doppler-delay PSD measurements for wideband V2V channels at 2.4 GHz, 5.2 GHz, and 5.9 GHz were reported in Acosta et al. (2004), ?, and ?, respectively. It was demonstrated that Doppler PSDs can vary significantly with different time delays in a wideband V2V channel. In ?, the authors analyzed the Doppler spread and coherence time of narrowband V2V channels and presented their dependence on both velocity and vehicle separation. Recently, the space-Doppler PSD, which is the Fourier transform of the space-time correlation function in terms of time, was investigated in ? and ?. It is worth noting that the Doppler PSD for V2V channels can be significantly different from the traditional U-shaped Doppler PSD for F2M channels. 9.2.2 Channel characteristics of cooperative communication systems ((10.2.2)) Several papers have reported measurements of various statistical properties of cooperative MIMO channels for different scenarios. A few indoor cooperative channel measurements were reported in ??, where the cooperative nodes are all static. Mobile multi-link measurements were presented in ? for indoor cooperative MIMO channels. Outdoor cooperative MIMO channel measurements were addressed in ?? and ?–? for static nodes and mobile nodes, respectively. All these measurement campaigns concentrated on the investigation of the channel characteristics of individual links for different scenarios, such as path loss, shadow fading, and small scale fading. As mentioned before, cooperative MIMO systems include multiple radio links that may exhibit strong correlations. The correlation of multiple links exists due to the environment similarity arising from common shadowing objects and scatterers contributing to different links and can significantly affect the performance of cooperative MIMO systems. The investigation of the correlations between different links is rare in the current literature and thus deserves more investigation. 9.3 Scattering theoretical channel models for conventional cellular MIMO systems ((10.3)) The well-known one-ring MIMO F2M RS-GBSM ? has been widely used for the analysis and design of narrowband MIMO cellular systems for macro-cell scenarios. To reach the high demand for high-speed communications, wideband
258 Practices:channel modeling for modern communication systems effe reaeeaoaeioncoaetsoneteaaptcadts ring.However,it is not trivil to use this mode match any given or measure PDE model can only be ed-form model w wideband application bydividing (or propagation eraction of the AoA,AoD,and ToA,the importance of which was described in ?More importantly,frequency is larger than the frequency separation of different sub-channels.This requency-correlated MIMO channel is ystems and MIMO-or in such requency-nopping MIM gating these syst hav e typically assumed that diffe ent frequency-separated channels are unco elated onhe are thre del has the virtual on struct aTD 油rame经 ch time nhi。 the spa d to be independen n e haed M haTheretore,the deivdT pread of the for the designof sity MIMO systems ??in addition,the wideband model can be reduced to the traditional one-ring m by remov the frequency-eectivityIn Sir,the e derived STF C is a reference model that assumesan number of effective scatterers.it cannot be implemented directly in ion model ba esponding narrow and simulation model can be obtained byr ing the frea cv-selectivin esp 9.3.1 A Wideband Multiple-ring Based MIMO Channel Reference Model ((10.3.1)) in macro-cell scenarios,where systems,since the propagation delays of all NC)incoming waves are much smaller than the transmitted -max T,the delay differ nces cau sed by different lcal scatterers randomly ne-rin ate than
258 Practices: channel modeling for modern communication systems MIMO cellular systems have been suggested in many communication standards, leading to the increasing requirement for wideband MIMO F2M channel models. However, the one-ring structure that assumes effective scatterers located on a one-ring is overly simplistic and thus unrealistic for modeling wideband channels ?. Therefore, in ? the authors for the first time extended the narrowband one-ring model to a wideband model by extending the location of effective scatterers on a one-ring to a circularring. However, it is not trivial to use this model to match any given or measured PDP since many parameters need adjustments via a complicated approach. In addition, the integral expressions of the derived STF CFs based on this model can only be numerically evaluated as no closed-form expressions were found. In contrast to ?, in ? the one-ring model was extended to a wideband application by dividing the one-ring into several segments in terms of different time of arrivals (ToAs). This makes the model easier to match any specific PDPs due to the TDL structure. However, the onering structure was still applied in this model, which causes this model to exhibit an unrealistic structure in that certain ToAs (or propagation delays) are always related to a certain proportion of AoAs. To obtain closed-form expressions of STF CFs, the model in ? further applied an unrealistic assumption that the mean AoA and the corresponding angle spread are exactly the same for all the scatterers in different segments. Moreover, neither of the two models considered the interaction of the AoA, AoD, and ToA, the importance of which was described in ?. More importantly, frequency correlation of sub-channels with different carrier frequencies, studied in ? for SISO channels, has not been investigated yet for MIMO channels. Chapter 3 showed that the frequency correlation appears when the coherence bandwidth Bc is larger than the frequency separation of different sub-channels. This frequency-correlated MIMO channel is commonly encountered in frequency-diversity MIMO communication systems, such as frequency-hopping MIMO systems and MIMO-orthogonal frequency division multiplexing (MIMO-OFDM) systems. For convenience, researchers investigating these systems have typically assumed that different frequency-separated channels are uncorrelated ?, which leads to positively biased performance results for the investigated systems. Therefore, accurate theoretical analysis and simulations of frequency-correlated MIMO channels are of great importance in investigating the impact of the frequency correlation on the performance of real frequency-diversity MIMO systems. The goals of this section are three-fold. First, we propose a new wideband MIMO F2M RS-GBSM that represents a reasonable compromise between physical reality and analytical tractability. The proposed model uses a concentric multiple-ring instead of one-ring around the MS to avoid the one-ring structure. Also, to easily match any specified or measured PDP, the model utilizes a virtual confocal multiple-ellipse to construct a TDL structure. Moreover, this model has the ability to consider the interaction of the AoA, AoD, and ToA. Secondly, from the proposed model, a closed-form expression of the STF CF between any two sub-channels with different carrier frequencies for each timebin signal is derived. For simplicity, the spatial and frequency correlations were assumed to be independent in ??. On the contrary, in this subsection we derive the STF CF by taking into account the dependency between them, which is a unique characteristic of frequency-correlated MIMO channels ?. Therefore, the derived STF CF can explicitly relate the frequency correlation to various environment parameters (e.g., mean AoA and angle spread of the AoA). Moreover, based on the derived CF, we reveal the inherent frequency correlation within the spatial correlation, which is important for the design of frequency-diversity MIMO systems ??. In addition, the proposed wideband model can be reduced to the traditional one-ring model by removing the frequency-selectivity. In this case, the derived STF CF shows a compact closed-form expression that is a generalisation of many existing CFs ??????. Since the proposed wideband RS-GBSM is a reference model that assumes an infinite number of effective scatterers, it cannot be implemented directly in practice. Therefore, the third goal of this section is to derive a wideband deterministic SoS simulation model based on the proposed reference model. Closed-form expressions are provided for the STF CF of the simulation model. Similarly, the corresponding narrowband simulation model can be obtained by removing the frequency-selectivity from the wideband simulation model. The statistical properties of our simulation models are verified by comparing with the corresponding statistical properties of the reference models. 9.3.1 A Wideband Multiple-ring Based MIMO Channel Reference Model ((10.3.1)) A narrowband one-ring MIMO model is suitable for describing a narrowband channel in macro-cell scenarios, where the BS is elevated and unobstructed, while the MS is surrounded by a large number of local scatterers. For narrowband systems, since the propagation delays τ ′ n of all N (N→∞) incoming waves are much smaller than the transmitted symbol duration Ts, i.e., τ ′ max=max {τ ′ n} N n=1≪Ts, the delay differences caused by different local scatterers randomly around the MS can be neglected in comparison to Ts. Therefore, it is reasonable to use the effective scatterers located on a one-ring instead of the real local scatterers to construct the low complexity one-ring model at the minor expense of accuracy ?. However, in high data rate wideband systems, Ts is much smaller than that in narrowband systems. In this case, the propagation delay differences cannot be neglected and thus the channel becomes a wideband channel
Practices:channel modeling for modern communication systems 259 1-1 R D Fig.9.1 A new wideband multiple-ring based MIMO F2M channel model. Therefore.the one-ring structure violates the basic characteristics of wideband channels as mentioned in ? To exte end the one-ring model to wideb ind applicat ions n macro-cell scenarios,the primary task is to modify the nd the we repla model the baone with the BS ated at the )Note the virtu and radii of the concentric multiple rings for different taps (i.e.,A and R)can be different in terms of different propagation envi onm ents for different delays.T erefore,the new eband m odel with the appropriare numb multiple rings(i.e be suitable for a d with antennas The e Bs des element spacing at th rings with radii(i=0,1,. A-1).It is usually assumed that the assumption Dmax .6n is multi-elemen Dby≈rctan(R /DB,/D The received complex impulse response at the carrier frequencyf for the be expressed as hog (t,)=>hLog (t)x8(-) (9.1) witho=1.2.....Mr and 1.2. M,where()anddenote the complex time-variant tap coefficient and the
Practices: channel modeling for modern communication systems 259 l τ ′ o T ′ Θ ,il R δ δ T R ,,, nril φ T ,, nil φ β T R ,il q R ′ l i on , , ε l i nq , , ε Y v l i r , , µ l i, ∆µ −1 ′ l τ +1 ′ l τ D X ,il +1 R β R R il −1, l i n , , ξ γ o T il S , nil S ,, q R Fig. 9.1 A new wideband multiple-ring based MIMO F2M channel model. Therefore, the one-ring structure violates the basic characteristics of wideband channels as mentioned in ?. To extend the one-ring model to wideband applications in macro-cell scenarios, the primary task is to modify the oversimple one-ring structure. To this end, we replace the one ring of effective scatterers by concentric multiple rings of effective scatterers around the MS to capture the basic characteristics of wideband channels. To make our model easier to match any specified or measured PDP, we utilise the confocal multiple virtual ellipses with the BS and MS located at the foci to represent the TDL structure, where different delays correspond to different virtual confocal ellipses (i.e., taps). Note that the total number of virtual confocal ellipses L and the values of major axes a of different ellipses are determined according to the specified or measured PDP. The newly developed structure is shown in FIGURE 10.2. For clarity, FIGURE 10.2 only presents the effective scatterers from three concentric rings (the total number of concentric rings in the lth tap is Λl) belong to the lth tap (l=1, 2, ..., L). Notice that the total number and radii of the concentric multiple rings for different taps (i.e., Λl and Rl,i) can be different in terms of different propagation environments for different delays. Therefore, the new wideband model with the appropriate number and major axes of virtual confocal multiple ellipses (i.e., L and a), and the appropriate number and radii of concentric multiple rings (i.e., Λl and Rl,i) should be suitable for any macro-cell scenario. We assume that uniform linear antenna arrays are used with MT=MR=2 antennas. The symbols δT and δR designate the antenna element spacing at the BS P and MS, respectively, and D denotes the distance between the BS and MS. The effective scatterers are located on L l=1 Λl rings with radii Rl,i (i=0, 1, ...,Λl − 1). It is usually assumed that the assumption D≫Rl,i≫max {δT , δR} is fulfilled. The multi-element antenna tilt angles are denoted by βT and βR. The MS moves with speed in the direction determined by the angle of motion γ. The angle spread seen at the BS is denoted by Θl.i, which is related to Rl,i and D by Θl,i ≈ arctan(Rl,i/D) ≈ Rl,i/D. The received complex impulse response at the carrier frequency fc for the link To − Rq can be expressed as hoq (t, τ′ ) = X L l=1 hl,oq (t) ×δ (τ ′ − τ ′ l ) (9.1) with o = 1, 2, ..., MT and q = 1, 2, ..., MR, where hl,oq(t) and τ ′ l denote the complex time-variant tap coefficient and the
260 Practices:channel modeling for modern communication systems cluste isieiieeihedhrhentesecuonofhevi each tap is r=1.2)and the corres sponding angle spread isas illustrated in FIGURE 10.2.Therefore,the effective cluster can be completel ly determined by of these t ation delay subintervals val G=0is partitioned intoLmutually disjoint sub-intervals G A /=1 G{-A2/网.23- (9.2) i-△i/2,Tma, 商险a。 ding A 2)as Tmnx(I (9.2 andeepondngameepea2AriCdheheieh肥 cluster in the /th tap are given as 4,r=(8-1,r+2o,+6+1./4=t[arccos(2r-1/rau-1) +2 arccos(2/Tmx-1)+arccos (2/x-1)]/4 (9.3 △4.i=|(-1r-8+1/个 =[arccos (271/Tx-1)-arccos (2T/-1)]/4. (9.4) Cexp (j [vn-2fen..+2 fpt cos (f)]) (9.5) ster S The AoA of the wave trav e MS is denoted b a uniform in?, -6r(cos(Br)+sin(Br)sin(/2 (9.6a) ELi.mg~RL-6Rcos(ofrn-BR)/2 (9.6b) respectively,whereD cos( ssume 9.5),the 01 literature,many different scatt erer distributions have been proposed to characterise the AoA such as the uniform approximate a f()exp[cos(-)]/2rlo(k) (9.7)
260 Practices: channel modeling for modern communication systems discrete propagation delay of the lth tap, respectively. Similar to the concept of effective scatterers in the narrowband one-ring model, the concept of an effective cluster is introduced in the new wideband multiple-ring model. From FIGURE 10.2, it is obvious that the position of the effective cluster Sl,i is identified by the intersection of the virtual ellipses and multiple rings, and the lth tap includes 2Λl effective clusters. The mean angle of the effective cluster in each tap is µl,i,r (r=1, 2) and the corresponding angle spread is ∆µl,i as illustrated in FIGURE 10.2. Therefore, the effective cluster can be completely determined by µl,i,r and ∆µl,i. Note that the angular range of µl,i,r in each tap is over [0, 2π), which means the effective cluster can be located around the MS over [0, 2π) for each tap. The setting of these two parameters follows a fixed rule. To establish this rule, firstly, we need to define the propagation delay subintervals Gl . The propagation delay interval G = [0, τ′ max] is partitioned into L mutually disjoint sub-intervals Gl . Here, we utilise the definition of subintervals as ? Gl= 0, ∆τ ′ l+1 2 , τ ′ l − ∆τ ′ l /2, τ′ l + ∆τ ′ l+1 2 , [τ ′ l − ∆τ ′ l /2, τ′ max] , l=1 l=2, 3, ..., L − 1 l=L (9.2) where ∆τ ′ l=τ ′ l−τ ′ l−1 and τ ′ max=2RL−1,ΛL−1−1 c (c is the speed of light). The propagation delay τ ′ l of the lth tap can be expressed according to the corresponding AoA φ R l,i,r (r=1, 2) as τ ′ l≈τ ′ max(1+cosφ R l,i,r)/2 ?. Solving this equation for φ R l,i,r gives φ R l,i,r=± arccos (2τ ′ l /τ ′ max − 1). According to (9.2), the expression of φ R l,i,r, and the geometrical relationship in FIGURE 10.2, the expression of the mean angle µl,i,r and the corresponding angle spread ∆µl,i of the effective cluster in the lth tap are given as µl,i,r = φ R l,i−1,r+2φ R l,i,r+φ R l,i+1,r4=± arccos 2τ ′ l−1 τ ′ max−1 +2 arccos (2τ ′ l /τ ′ max−1)+ arccos 2τ ′ l+1 τ ′ max−1 4 (9.3) ∆µl,i = φ R l,i−1,r − φ R l,i+1,r4 = arccos 2τ ′ l−1 τ ′ max−1 −arccos 2τ ′ l+1 τ ′ max−1 4. (9.4) Following the definition of the subintervals Gl and some geometrical relationship shown in FIGURE 10.2, we can determine the effective cluster in each tap according to the propagation delay τ ′ l . The time-variant tap coefficient at the carrier frequency fc of lth tap can be expressed as hl,oq (t)= lim N→∞ 1 √ N Λ Xl−1 i=0 X Rc r=1 X N n=1 exp j ψl,i,n−2πfcτl,i,oq,n + 2πfDt cos φ R l,i,r,n−γ (9.5) with τl,i,oq,n=(εl,i,on+εl,i,nq) /c. Here, τl,i.oq.n is the travel time of the wave through the link To−Sl,i,n−Rq scattered by the nth scatterer, Sl,i,n, Rc is the number of effective cluster for one ring in each tap (here Rc=2), and N is the number of effective scatterers Sl,i,n in the effective cluster Sl,i. The AoA of the wave traveling from the nth scatterer in the effective cluster Sl,i towards the MS is denoted by φ R l,i,r,n. The phases ψl,i,n are i.i.d. random variables with uniform distributions over [0, 2π) and fD is the maximum Doppler frequency. As shown in ?, the distance εl,i,on and εl,i,nq can be expressed as the function of φ R l,i,r,n as εl,i,on≈ξl,i,n−δT [cos(βT )+Θl,i sin(βT ) sin(φ R l,i,r,n)]/2 (9.6a) εl,i,nq≈Rl,i−δR cos(φ R l,i,r,n−βR)/2 (9.6b) respectively, where ξl,i,n≈D+Rl,i cos(φ R l,i,r,n). Since we assume that the number of effective scatterers in one effective cluster in this reference model tends to infinite (as shown in (9.5)), the discrete AoA φ R l,i,r,n can be replaced by the continuous expressions φ R l,i,r. In the literature, many different scatterer distributions have been proposed to characterise the AoA φ R l,i,r, such as the uniform ?, Gaussian ?, wrapped Gaussian ?, and cardioid PDFs ?. In this chapter, the von Mises PDF ? is used, which can approximate all the above mentioned PDFs. The von Mises PDF is defined as f (φ) ∆=exp [k cos (φ−µ)] /2πI0 (k) (9.7) where φ ∈ [−π, π), I0 (·) is the zeroth-order modified Bessel function of the first kind, µ ∈ [−π, π) accounts for the mean value of the angle φ, and k (k ≥ 0) is a real-valued parameter that controls the angle spread of the angle φ.
Practices:channel modeling for modern communication systems 261 the AoA in one effective cluster we further modify the general expression of von Mises PDF as fe ()=QLr expfkLr cos ()y2o() (9.8) ter).Therefore,the xpreson of theapped D of the tap of the proposed -A +Aur)and O,is the normalisation coefficient.Here we name the PDF in (9.8)the tapped'PDF o 26. where w(o )= (9.9)） manner.The interaction between the AoA and Ac the correlation properties in each tap.Therefore,inspired by?,the interac ion between the and ToA can be ered viantheoe parameterorthe fcthef 9.3.2 Generic Space-Time-Frequency Correlation Function((10.3.2)) 11， narrowband MIMO channels by removing the frequency-selectivity,which includes many existing CFs as special cases. Space-Time-Frequency Correlation Function for Wideband MIMO Channels ach tap.The correlation properties of two arbitrary of at d and undern processes infferent taps.Therefore,we can restrict ourinvestigations to the followingSTC Lo(Tx):-E[hiog(t)hiog(t-T)] (9.10) 步 is a functi own in Append 1-,the closed paration T,space d-form expression F pl.:.'(T X)can be presen (Aur)sin (eAm.)cos (+(A.)(-1)Ie (B)sin (m) xcos ((-1)I(A)!(B)os (/2)+sin (2tA) ×eos(24r+rf2/2+言-le(B+al
Practices: channel modeling for modern communication systems 261 For k=0 (isotropic scattering), the von Mises PDF reduces to the uniform distribution, while for k>0 (non-isotropic scattering), the von Mises PDF approximates different distributions based on the values of k ?. To better characterise the AoA in one effective cluster Sl,i, we further modify the general expression of von Mises PDF as fc φ R l,i,r =Ql,i,r exp kl,i,r cos φ R l,i,r−µl,i,r/2πI0 (kl,i,r) (9.8) where φ R l,i,r∈[µl,i,r−∆µl,i, µl,i,r+∆µl,i) and Ql,i,r is the normalisation coefficient. Here we name the PDF in (9.8) the truncated von Mises PDF. Here, truncated means that the range of AoA in this PDF is only defined within a limited interval [µl,i,r−∆µl,i, µl,i,r+∆µl,i). Therefore, the expression of the ‘tapped’ PDF of AoA in the lth tap of the proposed wideband multiple-ring channel model is given by fg φ R l,i,r = Λ Pl−1 i=0 P Rc r=1 Ql,i,r exp[kl,i,r cos(φ R l,i,r−µl,i,r)] 2πI0(kl,i,r) w φ R l,i,r, µl,i,r − ∆µl,i, µl,i,r + ∆µl,i where w (φ, φl , φu) = 1, 0, if φl < φ < φu otherwise. (9.9) Here, Ql,i,r are computed in such a way that the ’tapped’ PDF fg(φ R l,i,r) is equal to 1, i.e., R π −π fg(φ R l,i,r)dφR l,i,r=1. Note that the proposed wideband model allows one to consider the interaction of AoA, AoD, and ToA in a sensible manner. The interaction between the AoA and AoD is obtained in terms of the exact geometrical relationship, while the interaction between the AoA/AoD and ToA is calculated according to the TDL structure that allows one to investigate the correlation properties in each tap. Therefore, inspired by ?, the interaction between the AoA/AoD and ToA can be considered via setting the appropriate parameter kl,i,r for the PDF of AoA/AoD in each tap according to the PDF of ToA. 9.3.2 Generic Space-Time-Frequency Correlation Function ((10.3.2)) In this subsection, from the proposed model we first derive a new generic STF CF for wideband MIMO channels. As shown at the end of this subsection, the derived STF CF can be reduced to a compact closed-form STF CF for narrowband MIMO channels by removing the frequency-selectivity, which includes many existing CFs as special cases. • Space-Time-Frequency Correlation Function for Wideband MIMO Channels From the proposed wideband model, we derive the STF CF for each tap. The correlation properties of two arbitrary links hoq(t, τ′ ) and h ′ o′q ′ (t, τ′ ) at different frequency fc and f ′ c of a MIMO channel are completely determined by the correlation properties of hoq(t) and h ′ o ′q ′ (t) in each tap since we assume that no correlations exist between the underlying processes in different taps. Therefore, we can restrict our investigations to the following STF CF ρl,oq;l,o′q ′ (τ, χ):=E[hl,oq(t)h ′∗ l,o′q ′ (t − τ )] (9.10) where (·) ∗ denotes the complex conjugate operation and E[·] designates the statistical expectation operator. Note that the above defined CF is a function of the time separation τ, space separation δT and δR, and frequency separation χ=f ′ c−fc. As shown in Appendix 10-A, the closed-form expression of STF CF ρl,oq;l,o′q ′ (τ, χ) can be presented as ρl,oq;l,o′q ′ (τ, χ) = 2 πI0(k) Λ Pl−1 i=0 P Rc r=1 Ql,i,re jCl,i {∆µl,iI0 (Al,i,r) I0 (Bl,i,r)/2 + I0 (Bl,i,r) P∞ ℓ=1 ×Iℓ (Al,i,r) sin (ℓ∆µl,i) cos (ℓµl,i,r)/ℓ + I0 (Al,i,r) P∞ ℓ ′=1 (−1)ℓ ′ Iℓ ′ (Bl,i,r) sin (ℓ ′∆µl,i) ×cos (ℓ ′µl,i,r + ℓ ′π/2)/ℓ ′ + P∞ ℓ=1 (−1)ℓ Iℓ (Al,i,r) Iℓ (Bl,i,r) [∆µl,i cos (ℓπ/2) + sin (2ℓ∆µl,i) ×cos (2ℓµl,i,r + ℓπ/2)/2ℓ] + P∞ ℓ=1 P∞ q=1 (ℓ6=q) (−1)ℓ ′ Iℓ (Al,i,r) Iℓ ′ (Bl,i,r) [sin [(ℓ + ℓ ′ ) ∆µl,i] × cos [(ℓ+ℓ ′ )µl,i,r+ℓ ′π/2] ℓ+ℓ ′ + sin[(ℓ−ℓ ′ )∆µl,i] cos [(ℓ−ℓ ′ )µl,i,r−ℓ ′π/2] ℓ−ℓ ′
262 Practices:channel modeling for modern communication systems (9.11) where AL.r=aL..r+j(XU+ycosBR+) (9.12a) BLir=bLir+j(XVi+ysin BR+20Li sin Br+rsin) (9.12b) (9.12) with r=2fDT,y=2fe6R/c,2=2fe6r/c,X=2nx/c,a..=k tween hag(t,T' (9.13) Note that (911)and (913)are the generic expressions which apply to the STCF and the subseqen degenerate n be e Space-Time-Frequency Correlation Function for Narrowband MIMO Channels A special case of the proposed model described by (9.1)is given when L=1 and A=1.Consequently,we have ((which is the complex fac lo mak (ent we expr e complex y removing th bscripts （s（,an 1 h0=不e即66-2sfgn+2 tc(eg-训》 (9.14) with(/c,where and can be expressed as the function of as EonEn-8r[cos(BT)+0sin(Br)sin()/2 (9.15a) eng≈R-Rc0(R-3R)/2 (9.15b) where +Rcs).In order to consistent with the proposed wideband model,the von Mises PDF (9.7)is xc(( (9.16) where A=a+j(XU+ycosr+rcos） (9.17a) B=b+j(XV+ysin Br+zθsin3 r+zsin） (9.17b) C=:cosBr+XT (9.17 with a-=kcos，=ksi which applies to the FCF and the subs y degenerate CFs with thediff ence only in yalues of B.and C The corresponding expressions of.B,and Cfor the degenerate CFs can be easily obtained from (917)by settin The derived eriSTCP (916)includes many existing CFs as special cases.For a SI case,the time CF given =and =0in (9.16)with≠0.If fur ner s n(isotropic)in -0 .and
262 Practices: channel modeling for modern communication systems (9.11) where Al,i,r=al,i,r+j(XUl,i+y cos βR+x cos γ) (9.12a) Bl,i,r=bl,i,r+j(XVl,i+y sin βR+zΘl,i sin βT+x sin γ) (9.12b) Cl.i=z cos βT+XTl.i (9.12c) with x=2πfDτ, y=2πfcδR/c, z=2πfcδT /c, X=2πχ/c, al,i,r=kl,i,r cos µl,i,r, bl,i,r=kl,i,r sin µl,i,r, Tl,i=(δT /2) cos βT+D+ Rl,i, Ul,i=Rl,i+(δR/2) cos βR, and Vl,i=(δT /2)Θl,i sin βT +(δR/2) sin βR. Consequently, the STF CF between hoq(t, τ′ ) and h ′ o ′q ′ (t, τ′ ) can be shown as ρoq,o′q ′ (τ, χ) = 1 L X L l=1 ρl,oq;l,o′q ′ (τ, χ). (9.13) Note that (9.11) and (9.13) are the generic expressions which apply to the STF CF and the subsequently degenerate CFs (e.g., ST CF, frequency CF, etc.) differ only in values of Al,i,r, Bl,i,r, and Cl,i. The corresponding expressions of these three parameters for the degenerate CFs can be easily obtained by setting relevant terms (τ, δT and δT , and χ) to zero. • Space-Time-Frequency Correlation Function for Narrowband MIMO Channels A special case of the proposed model described by (9.1) is given when L=1 and Λl=1. Consequently, we have hoq (t, τ′ )=hoq (t) δ (τ ′ ), which is the complex fading envelope of the narrowband one-ring channel model. To make this evident, we express the complex fading envelop hoq (t) similarly to (9.5) by removing the subscripts (·) l , (·) i , and (·) r , i.e., hoq (t)= lim N→∞ 1 √ N X N n=1 exp j ψn−2πfcτoq,n + 2πfDt cos φ R n − γ (9.14) with τoq,n=(εon+εnq)/c, where εon and εnq can be expressed as the function of φ R n as εon ≈ξn−δT [cos(βT )+Θ sin(βT ) sin(φ R n )]/2 (9.15a) εnq≈R−δR cos(φ R n−βR)/2 (9.15b) where ξn≈D+R cos(φ R n ). In order to consistent with the proposed wideband model, the von Mises PDF (9.7) is employed to characterise the AoA φ R n of the narrowband model. In such a case, the angle spread ∆µl,i=π, which means the AoA range is over [0, 2π). As shown in Appendix 10-B, the STF CF of the narrowband one-ring model can be obtained from (9.11) after some manipulation ρoq,o′q ′ (τ, χ)=e jC I0 h A 2+B 2 1/2i.I0 (k) (9.16) where A=a+j(XU+y cos βR+x cos γ) (9.17a) B=b+j(XV+y sin βR+zΘ sin βT+x sin γ) (9.17b) C=z cos βT+XT (9.17c) with a=k cos µ, b=k sin µ, T=(δT /2) cos βT+D+R, U=R+(δR/2) cos βR, and V =(δT /2)Θ sin βT+(δR/2) sin βR. The parameters x, y, z, and X are the same as defined in (9.11). It is worth stressing that (9.16) is the generic expressions which applies to the STF CF and the subsequently degenerate CFs with the difference only in values of A, B, and C. The corresponding expressions of A, B, and C for the degenerate CFs can be easily obtained from (9.17) by setting relevant terms to zero. The derived generic STF CF (9.16) includes many existing CFs as special cases. For a SISO case, the time CF given in ? is obtained by setting δT = δR = 0 and χ = 0 in (9.16) with k 6= 0. If further setting k = 0 (isotropic scattering) in (9.16), the Clarke’s time CF in ? is obtained. For a SIMO case, the Lee’s ST CF in ? is obtained by substituting δT = 0, χ = 0, βR = 0, and k = 0 into (9.16). For a MISO case, the ST CF in ? is obtained by substituting δR = 0, χ = 0, and k = 0 into (9.16). If further substituting fD = 0 into (9.16), the space CF given in ? is obtained. For a MIMO case, the ST CF shown in ? is obtained by setting χ = 0 in (9.16) with k 6= 0.
Practices:channel modeling for modern communication systems 263 9.3.3 MIMO Simulation Models(10.3.3) d wideband MIMo to a narrowband one by removing the frequency-selectivity. .A Deterministic Simulation Model for Wideband MIMO Channels The wideband der e taps according to 方化，=∑)xi(-) (9.18 In(9.18),the complex fading envelope(t)is modeled by utilizing only a finite number of scatterers N and keeping all the model parameters fixed as ,件六写2名四bi-au+olm(依-奶 (9.19) 克 ator uniformly distributed over 1o.2)the discre averageseeSertes of the deterministic channel simulator by time averages instead of statistical Dl.ogl.oq(红，x):=(iw)iogt-t)》 (9.20) wheredenotes the time average operator.Substituting(9.19)into(920),we can get the closed-form STF CF as (9.2) with .：=XU.+ycosR+zcos (9.22a) J=XVi+ysin BR+zAsin Br+rsiny (9.22b) C单彩名2收e e(1.时nwoo1 )ge T Bov (r.x)=∑e(,0. (9.23) Similar to (9.11)and (9.13),(9.21)and (9.23)are the generic expressions which apply to all the CFs of the As addre ncanedcaiaaaaeaamo…c 1 of the
Practices: channel modeling for modern communication systems 263 9.3.3 MIMO Simulation Models (10.3.3) In this section, based on the proposed wideband MIMO channel reference model, we propose an efficient deterministic SoS simulation model for wideband MIMO channels. The proposed wideband simulation model can be further reduced to a narrowband one by removing the frequency-selectivity. • A Deterministic Simulation Model for Wideband MIMO Channels The wideband deterministic simulation model is proposed also based on the TDL structure. The impulse response of the simulation model at the carrier frequency fc for the To−Rq link are again composed of L discrete taps according to h˜ oq (t, τ′ ) = X L l=1 h˜ l,oq (t) ×δ (τ ′ − τ ′ l ). (9.18) In (9.18), the complex fading envelope h˜ l,oq (t) is modeled by utilizing only a finite number of scatterers N and keeping all the model parameters fixed as h˜ l,oq (t)= √ 1 N Λ Pl−1 i=0 P Rc r=1 P N n=1 exp n j h ψ˜ l,i,n−2πfcτl,i,oq,n +2πfDt cos φ˜R l,i,r,n − γ io (9.19) where the phases ψ˜ l,i,n are simply the outcomes of a random generator uniformly distributed over [0, 2π), the discrete AoAs φ˜R l,i,r,n will be kept constant during simulation, and the other symbol definitions are the same as those in (9.5). Therefore, we can analyze the properties of the deterministic channel simulator by time averages instead of statistical averages. The STF CF is defined as ρ˜l,oq;l,o′q ′ (τ, χ):=D h˜ l,oq (t) ˜h ′∗ l,o′q ′ (t − τ ) E (9.20) where h·i denotes the time average operator. Substituting (9.19) into (9.20), we can get the closed-form STF CF as ρ˜l,oq;l,o′q ′ (τ, χ) = 1 N Λ Xl−1 i=0 X Rc r=1 X N n=1 e j(Cl,i+Pl,i cos φ˜R l,i,r,n+Jl,i sin φ˜R l,i,r,n) (9.21) with Pl,i=XUl,i+y cos βR+x cos γ (9.22a) Jl,i=XVl,i+y sin βR+z∆ sin βT+x sin γ (9.22b) where Cl,i, x, y, z, X, Ul,i, and Vl,i are the same as defined in (9.11). By analogy to (9.13), we can further get the STF CF between h˜ oq (t, τ′ ) and h˜′ o ′q ′ (t, τ′ ) as ρ˜oq,o′q ′ (τ, χ) = 1 L X L l=1 ρ˜l,oq;l,o′q ′ (τ, χ). (9.23) Similar to (9.11) and (9.13), (9.21) and (9.23) are the generic expressions which apply to all the CFs of the deterministic simulation model with different Cl,i, Pl,i, and Jl,i. Comparing the expressions of Al,i,r and Bl,i,r with Pl,i and Jl,i, respectively, we have Al,i,r=al,i,r+jPl,i and Bl,i,r=bl,i,r+jJl,i. From (9.21) and (9.23), it is obvious that only {φ˜R l,i,r,n} N n=1 needs to be determined for this deterministic simulation model. As addressed in Chapter 5, the MEA and MEDS have been widely used to compute the important parameters of deterministic simulation models for isotropic scattering environments. However, these two methods fail to reproduce the desired statistical properties of the reference model under the condition of non-isotropic scattering ??. Therefore, the optimization method (i.e., Lp-norm method) ? is utilized here to calculate the model parameters {φ˜R l,i,r,n} N n=1 of the deterministic simulation model based on corresponding properties of the reference model. The time CF ρl,oq;l,oq(τ)
264 Practices:channel modeling for modern communication systems (9.24 ={nm-lxpdk} (9.25) (9.26) .A Deterministic Simulation Model for Narrowband MIMO Channels (t)of the deterministic simulation model is then given by 用=六立e即{6[氏.-2aen+2 fo(设-l} (9.27) where the phasesare simply the outcomes of a random generator uniformly distributed over 02),the discrete A0As the subscripts()()and in all the affected symbols.Thus (9.28) with P-XU+ycosBR+rcosy (9.29a) J=XV+ysin Br+z△si血r+csiny (9.29b) ere other parameters are the same as defined in (9.16).Similar to the wideband simulation model,we have +.From(9.28),iti5 ar that only{ needs to be determined for this deterministi obtained as follows ”=-{-i（ra} (9.30) 1/ -iaoPdx/xm (9.31 (9.32) if we replace by and respectively,the three error norms) and E can be minimized independently
264 Practices: channel modeling for modern communication systems frequency CF ρl,oq;l,oq(χ), and space CF ρl,oq;l,o′q ′ are identified as key properties. Then the optimization method requires the numerical minimization of the following three Lp-norms E (p) 1,l :=Z τmax 0 |ρl,oq;l,oq (τ)−ρ˜l,oq;l,oq (τ)| p dτ /τmax1/p (9.24) E (p) 2,l :=Z χmax 0 |ρl,oq;l,oq(χ)−ρ˜l,oq;l,oq(χ)| p dχ/χmax1/p (9.25) E (p) 3,l:=(Z δ max T 0 Z δ max R 0 |ρl,oq;l,o′q ′−ρ˜l,oq;l,o′q ′ | p dδT dδR/(δ max T δ max R ) ) 1/p (9.26) where p = 1, 2, ... Note that τmax, χmax, δ max T , and δ max R define the upper limits of the ranges over which the approximations ρ˜l,oq;l,oq (τ)≈ρl,oq;l,oq (τ), ρ˜l,oq;l,oq (χ)≈ρl,oq;l,oq (χ), and ρ˜l,oq;l,o′q ′≈ρl,oq;l,o′q ′ are of interest. For ρ˜l,oq;l,oq (χ) and ρ˜l,oq;l,o′q ′ , if we replace φ˜R l,i,r,n by φ˜′R l,i,r,n and φ˜′′R l,i,r,n, respectively, the three error norms E (p) 1,l , E (p) 2,l , and E (p) 3,l can be minimized independently. • A Deterministic Simulation Model for Narrowband MIMO Channels Analogous to Chapter 10.3.2.2, if we impose L=1 and Λl=1 on the wideband simulation model in (9.18), it reduces to a narrowband MIMO channel simulator. It follows that ˜hoq (t, τ′ )=h˜ oq (t) δ (τ ′ ) holds. The complex fading envelope h˜ oq (t) of the deterministic simulation model is then given by ˜hoq (t) = √ 1 N P N n=1 exp n j h ψ˜ n − 2πfcτn+2πfDt cos φ˜R n − γ io (9.27) where the phases ψ˜ n are simply the outcomes of a random generator uniformly distributed over [0, 2π), the discrete AoAs φ˜R n will be kept constant during simulation, and the other symbol definitions are the same as those in (9.14). The correlation properties of this narrowband simulation model can be obtained from (9.21) by simply neglecting the subscripts (·)l , (·)i , and (·)r in all the affected symbols. Thus ρ˜oq;o′q ′ (τ, χ) = 1 N X N n=1 e j(C+P cos φ˜R n +J sin φ˜R n ) (9.28) with P=XU+y cos βR+x cos γ (9.29a) J=XV+y sin βR+z∆ sin βT+x sin γ (9.29b) where other parameters are the same as defined in (9.16). Similar to the wideband simulation model, we have A=a+jP and B=b+jJ. From (9.28), it is clear that only {φ˜R n } N n=1 needs to be determined for this deterministic simulation model. The model parameters {φ˜R n } N n=1 can be calculated by using the same optimization method as the wideband simulation model. Therefore, by removing the subscript (·)l in (9.24)–(9.26), the model parameters can be obtained as follows E (p) 1 :=Z τmax 0 |ρoq;oq (τ)−ρ˜oq;oq (τ)| p dτ /τmax1/p (9.30) E (p) 2 :=Z χmax 0 |ρoq;oq(χ)−ρ˜oq;oq(χ)| p dχ/χmax1/p (9.31) E (p) 3 :=(Z δ max T 0 Z δ max R 0 |ρoq;o′q ′−ρ˜oq;o′q ′ | p dδT dδR/(δ max T δ max R ) ) 1/p . (9.32) Similarly, for ρ˜oq;oq (χ) and ρ˜oq;o′q ′ , if we replace φ˜R n by φ˜′R n and φ˜′′R n , respectively, the three error norms E (p) 1 , E (p) 2 , and E (p) 3 can be minimized independently
