# Modelling and Analysis of Nonstationary Vehicle-to-Infrastructure Channels with Time-Variant Angles of Arrival

Authors: Matthias Pätzold; Carlos A. Gutierrez
Journal: Wireless Communications and Mobile Computing
 (2018)

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## Abstract

In mobile radio channel modelling, it is generally assumed that the angles of arrival (AOAs) are independent of time. This assumption does not in general agree with real-world channels in which the AOAs vary with the position of a moving receiver. In this paper, we first present a mathematical model for the time-variant AOAs. This model serves as the basis for the development of two nonstationary multipath fading channels models for vehicle-to-infrastructure communications. The statistical properties of both channel models are analysed with emphasis on the time-dependent autocorrelation function (ACF), time-dependent mean Doppler shift, time-dependent Doppler spread, and the Wigner-Ville spectrum. It is shown that these characteristic quantities are greatly influenced by time-variant AOAs. The presented analytical framework provides a new view on the channel characteristics that goes well beyond ultra-short observation intervals over which the channel can be considered as wide-sense stationary.


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## Body

## 1. Introduction

In a typical downlink scenario, where plane waves travel from a base station (BS) to a mobile station (MS) via a large number of fixed scattering objects, the angles of arrival (AOAs) of the received signals are changing along the moving route of the MS. Only for very short observation intervals in which the MS travels a few tens of the wavelengths [1], the temporal variation of the AOAs can be neglected justifying the wide-sense stationary assumption of multipath fading channels. The lengths of the stationary intervals during which the mobile radio channel can be considered as wide-sense stationary or quasi-stationary have been investigated (e.g., in [2–4] and the references therein). By pushing the observation interval beyond the stationary interval, the received signal captures nonstationary effects that call for new channel modelling approaches using time-frequency analysis techniques [5]. One of the effects that come with long observation intervals is that the AOAs and thus the Doppler frequencies are changing with time along the MS’s moving route.Attempts to include the temporal variations of the AOAs in mobile radio channel models have been made in [6–8]. In [6], a nonstationary multiple-input multiple-output (MIMO) vehicle-to-vehicle (V2V) channel model has been derived by assuming that the AOAs and AODs are piecewise constant. In [7], a proposal has been made for the extension of the IMT-Advanced channel model [9] by replacing the time-invariant model parameters, such as the propagation delays, AOAs, and the angels of departure (AODs) by time-variant parameters. In [8], a nonstationary one-ring model has been introduced in which the time-variant AOAs have been modelled by stochastic processes rather than random variables.This paper is an extended version of our conference paper [10]. It expands on the recent results by studying the impact of time-variant AOAs on the statistical properties of multipath fading channels. It is shown that the multipath fading channel becomes non-wide-sense stationary if the AOAs change with time. Two new nonstationary channel models with time-variant AOAs are derived. The first one has an instantaneous channel phase that is related to the instantaneous Doppler frequency via the phase-frequency relationship [11], while the second one is based on a sum-of-cisoids (SOC) model in which the time-independent Doppler frequencies are replaced by time-dependent Doppler frequencies. The latter approach is simple, straightforward, and intuitive but results in a less accurate nonstationary channel model. The statistical properties of both channel models are investigated with emphasis on the time-dependent autocorrelation function (ACF), time-dependent mean Doppler shift, time-dependent Doppler spread, and the Wigner-Ville spectrum. Our analysis shows that our first proposed nonstationary channel model is consistent with respect to the mean Doppler shift and the Doppler spread, while this consistency property is not fulfilled by the SOC model with time-variant Doppler frequencies. The two proposed nonstationary channel models provide a trade-off between accuracy and complexity concerning the mathematical expressions.One of the main differences between [6–8] and our paper is that the AOAs are modelled in different ways. For example, in [6], the AOAs are modelled as piecewise constant functions, that is, these parameters are considered as constant apart from a finite number of jumps, while in our paper the AOAs are modelled in our paper as continuous time-variant functions. Another difference is that the models in [6–8] have been developed for different propagation scenarios. The V2V channel model in [6] has been developed to simulate propagation scenarios which are typical for T-junctions. The BS-to-MS channel model in [7] covers basically the same scenarios as the IMT-Advanced channel model [9], while the model in [8] is restricted to scenarios that can be generated by the one-ring model under the assumption of isotropic scattering. This contrasts with our nonstationary generic model which is not restricted to any specific propagation scenario. The drawback of the models in [6–8] is that they are inconsistent with respect to the mean Doppler shift and the Doppler spread. Our preferred model avoids this drawback by using an integral relationship between the instantaneous channel phases and the corresponding instantaneous Doppler frequencies.The organization of this paper is as follows. Section2 presents the derivation of two nonstationary multipath fading channel models with time-variant AOAs. Their statistical properties will be analysed in Section 3. The numerical key results of our study are visualized in Section 4. Section 5 provides guidelines for various extensions of the model. Finally, Section 6 draws the conclusion and suggests possible future research topics in relation to the issues addressed in this paper.


## 2. Derivation of the Nonstationary Multipath Channel Models

### 2.1. Time-Variant AOAs

We consider a downlink non-line-of-sight (NLOS) propagation scenario in which a fixed BS operates as transmitter, and an MS acts as receiver. It is supposed that the BS and the MS are equipped with omnidirectional antennas. The BS antenna is elevated and unobstructed by any object, whereas the MS antenna is surrounded by a large number ofN fixed scattering objects called henceforth scatterers Sn(n=1,2,…,N). The coordinate system has been chosen such that the MS is located at the origin (0,0) of the xy-plane at t=0. Furthermore, it is assumed that the MS moves with constant velocity v→ in the direction determined by the angle of motion αv as indicated in Figure 1. For reasons of clarity, this figure highlights only the location of the scatterer Sn from which the MS receives the nth multipath component (plane wave) μn(t) in the form of μn(t)=cnexp{jθn(t)}, where cn denotes the path gain which is supposed to be constant, and θn(t) is the associated channel phase that will be studied in Section 2.3. The corresponding AOA αn(t) is defined as the angle between the propagation direction of the nth incident plane wave and the x-axis, that is,(1)αnt=atan2⁡yn-yt,xn-xtfor n=1,2,…,N, where atan2⁡(y,x) denotes the four-quadrant inverse tangent function. It should be mentioned that the four-quadrant inverse tangent function atan2⁡(y,x) returns the angle of the vector (x,y) with the positive x-axis in the range -π,π. This function contrasts with the inverse tangent function atan⁡(y/x), whose results are limited to the interval (-π/2,π/2). In (1), the symbols xn and yn denote the coordinates of the scatterer Sn; and x(t) and y(t) indicate the position of the MS at time t. According to (1), the AOA αn(t) is a nonlinear function of time t, which can be turned into a linear function by developing αn(t) in a Taylor series around t=0 and retaining only the first two terms. This results in the following model for the time-variant AOA:(2)αnt=αn+γn·t,where(3)αn=αn0=atan2⁡yn,xn,(4)γn=ddtαntt=0=vrnsin⁡αn-αv.In (4), rn denotes the distance from the scatterer Sn to the origin of the xy-plane, that is, rn=xn2+yn2, as can be deduced from the geometrical model in Figure 1. In Section 4, it is shown that the two-term Taylor series expansion of αn(t) in (2) is sufficiently accurate for small observation intervals T.Figure 1
A multipath propagation scenario with time-variant AOAsαn(t).


### 2.2. Time-Variant Doppler Frequencies

Owing to the Doppler effect combined with the new feature that the AOAsαn(t) vary with time, it follows that the nth incident plane wave highlighted in Figure 1 experiences a time-variant Doppler shift of fn(t)=fmaxcos⁡(αn(t)-αv) that can be expressed by using (2) as(5)fnt=fmaxcos⁡αn-αv+γntfor n=1,2,…,N, where fmax stands for the maximum Doppler frequency. For a given propagation scenario with constant parameters fmax, αn, αv, and γn, the time-variant Doppler shift fn(t) is a deterministic function of time. Otherwise, if one or several model parameters, for example, αn and thus γn, are random variables, then fn(t) represents a stochastic process. If the MS moves during the time interval [0,T], then fn(t) describes a curve starting from the initial Doppler frequency fn(0)=fmaxcos⁡(αn-αv) and ending with the finishing Doppler frequency fn(T)=fmaxcos⁡(αn-αv+γnT).The time-dependent mean Doppler shiftBf(1)(t) and the time-dependent Doppler spread Bf(2)(t) can be computed according to(6)Bf1t=∑n=1Ncn2fnt∑n=1Ncn2,(7)Bf2t=∑n=1Ncn2fn2t∑n=1Ncn2-Bf1t2.


### 2.3. Instantaneous Channel Phase

The instantaneous channel phaseθn(t) of the nth multipath component μn(t)=cnexp⁡{jθn(t)} is related to the instantaneous Doppler frequency fn(t) via the phase-frequency relationship [5, Eq. (1.3.40)](8)θnt=2π∫-∞tfnxdxfor n=1,2,…,N. Using (5), the instantaneous phase θn(t) can be developed as follows:(9)θnt=2π∫-∞0fnxdx︸θn+2π∫0tfnxdx=θn+2πfmaxγnsin⁡αn-αv+γnt-sin⁡αn-αv,where θn=θn(0) denotes the initial phase at t=0. The initial phases θn are generally unknown and modelled by independent identically distributed (i.i.d.) random variables, each with uniform distribution over the interval (0,2π]: that is, θn~U(0,2π]. Equation (9) tells us that the instantaneous phase θn(t) is not only a nonlinear function of time t but also periodic with period Tn=2π/γn if the AOA αn(t) varies with time according to (2). In the limit γn→0, however, it can be shown by applying L’Hôpital’s rule to (9) that(10)limγn→0⁡θnt=θn+2πfnt,where fn=fn(0)=fmaxcos⁡(αn-αv). This result reveals a linear relationship between the instantaneous phase θn(t) and time t, which holds only for constant AOAs αn(t)=αn. It should be noticed that the expression in (10) can be identified as the standard phase term of SOC channel models for Rayleigh/Rice fading channels [12, Section 4.5].A simpler but less accurate expression than (9) can be obtained for the instantaneous phase θn(t) by developing θn(t) in a first-order Taylor series around t=0 as follows:(11)θnt≈θn0+θn′0t=θn+2πfmaxcos⁡αn-αvt=θn+2πfnt,where θn′(0) denotes the time derivative of θn(t) at t=0. By comparing the last two equations, we can conclude that the linear phase term θn(t)=θn+2πfnt can be obtained from the nonlinear phase term θn(t) [see (9)] either in the limit γn→0 or by developing the nonlinear phase θn(t) in a first-order Taylor series around t=0.


### 2.4. Complex Channel Gain

A model for the complex channel gain, denoted byμ(t), of a narrowband multipath fading channel is obtained by the superposition of all N plane wave components μn(t)=cnexp⁡jθnt, that is,(12)μt=∑n=1Ncnejθnt.Substituting the instantaneous channel phase θn(t) according to (9) in (12) results in the complex channel gain of the proposed nonstationary multipath fading channel with time-variant AOAs(13)μt=∑n=1Ncnej2πfmax/γnsinαn-αv+γnt-sinαn-αv+θn.On the other hand, starting from the SOC model for Rayleigh fading channels [12, Eq. (4.97)] and replacing there intuitively the time-independent Doppler frequencies fn by the instantaneous Doppler frequencies fn(t) according to (5) provide the complex channel gain μ(t) in a much simpler form, namely,(14)μt=∑n=1Ncnej2πfnt·t+θn.This intuitive mathematical manipulation results in a nonstationary channel model that is inconsistent with respect to the mean Doppler shift Bf(1)(t) and the Doppler spread Bf(2)(t), as we will see in Section 3.2. Although the expression in (14) is mathematically simpler than the one in (13), the difference is not significant in terms of implementation costs and simulation time.From the discussions in the previous subsection, it can be summed up that the two complex channel gainsμ(t) in (13) and (14) include the original SOC model [13](15)μt=∑n=1Ncnej2πfnt+θnas a special case that arises if the AOA αn(t) is supposed to be either constant (γn=0) or if the instantaneous phase θn(t) in (12) is approximated by a first-order Taylor series [see (11)]. The main difference between the three stochastic channel models above is that the former two are non-wide-sense stationary, whereas the third one is wide-sense stationary. The statistical properties of the SOC model have been studied in [13], while those of the new non-wide-sense stationary models will be analysed in the next section.


## 2.1. Time-Variant AOAs

We consider a downlink non-line-of-sight (NLOS) propagation scenario in which a fixed BS operates as transmitter, and an MS acts as receiver. It is supposed that the BS and the MS are equipped with omnidirectional antennas. The BS antenna is elevated and unobstructed by any object, whereas the MS antenna is surrounded by a large number ofN fixed scattering objects called henceforth scatterers Sn(n=1,2,…,N). The coordinate system has been chosen such that the MS is located at the origin (0,0) of the xy-plane at t=0. Furthermore, it is assumed that the MS moves with constant velocity v→ in the direction determined by the angle of motion αv as indicated in Figure 1. For reasons of clarity, this figure highlights only the location of the scatterer Sn from which the MS receives the nth multipath component (plane wave) μn(t) in the form of μn(t)=cnexp{jθn(t)}, where cn denotes the path gain which is supposed to be constant, and θn(t) is the associated channel phase that will be studied in Section 2.3. The corresponding AOA αn(t) is defined as the angle between the propagation direction of the nth incident plane wave and the x-axis, that is,(1)αnt=atan2⁡yn-yt,xn-xtfor n=1,2,…,N, where atan2⁡(y,x) denotes the four-quadrant inverse tangent function. It should be mentioned that the four-quadrant inverse tangent function atan2⁡(y,x) returns the angle of the vector (x,y) with the positive x-axis in the range -π,π. This function contrasts with the inverse tangent function atan⁡(y/x), whose results are limited to the interval (-π/2,π/2). In (1), the symbols xn and yn denote the coordinates of the scatterer Sn; and x(t) and y(t) indicate the position of the MS at time t. According to (1), the AOA αn(t) is a nonlinear function of time t, which can be turned into a linear function by developing αn(t) in a Taylor series around t=0 and retaining only the first two terms. This results in the following model for the time-variant AOA:(2)αnt=αn+γn·t,where(3)αn=αn0=atan2⁡yn,xn,(4)γn=ddtαntt=0=vrnsin⁡αn-αv.In (4), rn denotes the distance from the scatterer Sn to the origin of the xy-plane, that is, rn=xn2+yn2, as can be deduced from the geometrical model in Figure 1. In Section 4, it is shown that the two-term Taylor series expansion of αn(t) in (2) is sufficiently accurate for small observation intervals T.Figure 1
A multipath propagation scenario with time-variant AOAsαn(t).


## 2.2. Time-Variant Doppler Frequencies

Owing to the Doppler effect combined with the new feature that the AOAsαn(t) vary with time, it follows that the nth incident plane wave highlighted in Figure 1 experiences a time-variant Doppler shift of fn(t)=fmaxcos⁡(αn(t)-αv) that can be expressed by using (2) as(5)fnt=fmaxcos⁡αn-αv+γntfor n=1,2,…,N, where fmax stands for the maximum Doppler frequency. For a given propagation scenario with constant parameters fmax, αn, αv, and γn, the time-variant Doppler shift fn(t) is a deterministic function of time. Otherwise, if one or several model parameters, for example, αn and thus γn, are random variables, then fn(t) represents a stochastic process. If the MS moves during the time interval [0,T], then fn(t) describes a curve starting from the initial Doppler frequency fn(0)=fmaxcos⁡(αn-αv) and ending with the finishing Doppler frequency fn(T)=fmaxcos⁡(αn-αv+γnT).The time-dependent mean Doppler shiftBf(1)(t) and the time-dependent Doppler spread Bf(2)(t) can be computed according to(6)Bf1t=∑n=1Ncn2fnt∑n=1Ncn2,(7)Bf2t=∑n=1Ncn2fn2t∑n=1Ncn2-Bf1t2.


## 2.3. Instantaneous Channel Phase

The instantaneous channel phaseθn(t) of the nth multipath component μn(t)=cnexp⁡{jθn(t)} is related to the instantaneous Doppler frequency fn(t) via the phase-frequency relationship [5, Eq. (1.3.40)](8)θnt=2π∫-∞tfnxdxfor n=1,2,…,N. Using (5), the instantaneous phase θn(t) can be developed as follows:(9)θnt=2π∫-∞0fnxdx︸θn+2π∫0tfnxdx=θn+2πfmaxγnsin⁡αn-αv+γnt-sin⁡αn-αv,where θn=θn(0) denotes the initial phase at t=0. The initial phases θn are generally unknown and modelled by independent identically distributed (i.i.d.) random variables, each with uniform distribution over the interval (0,2π]: that is, θn~U(0,2π]. Equation (9) tells us that the instantaneous phase θn(t) is not only a nonlinear function of time t but also periodic with period Tn=2π/γn if the AOA αn(t) varies with time according to (2). In the limit γn→0, however, it can be shown by applying L’Hôpital’s rule to (9) that(10)limγn→0⁡θnt=θn+2πfnt,where fn=fn(0)=fmaxcos⁡(αn-αv). This result reveals a linear relationship between the instantaneous phase θn(t) and time t, which holds only for constant AOAs αn(t)=αn. It should be noticed that the expression in (10) can be identified as the standard phase term of SOC channel models for Rayleigh/Rice fading channels [12, Section 4.5].A simpler but less accurate expression than (9) can be obtained for the instantaneous phase θn(t) by developing θn(t) in a first-order Taylor series around t=0 as follows:(11)θnt≈θn0+θn′0t=θn+2πfmaxcos⁡αn-αvt=θn+2πfnt,where θn′(0) denotes the time derivative of θn(t) at t=0. By comparing the last two equations, we can conclude that the linear phase term θn(t)=θn+2πfnt can be obtained from the nonlinear phase term θn(t) [see (9)] either in the limit γn→0 or by developing the nonlinear phase θn(t) in a first-order Taylor series around t=0.


## 2.4. Complex Channel Gain

A model for the complex channel gain, denoted byμ(t), of a narrowband multipath fading channel is obtained by the superposition of all N plane wave components μn(t)=cnexp⁡jθnt, that is,(12)μt=∑n=1Ncnejθnt.Substituting the instantaneous channel phase θn(t) according to (9) in (12) results in the complex channel gain of the proposed nonstationary multipath fading channel with time-variant AOAs(13)μt=∑n=1Ncnej2πfmax/γnsinαn-αv+γnt-sinαn-αv+θn.On the other hand, starting from the SOC model for Rayleigh fading channels [12, Eq. (4.97)] and replacing there intuitively the time-independent Doppler frequencies fn by the instantaneous Doppler frequencies fn(t) according to (5) provide the complex channel gain μ(t) in a much simpler form, namely,(14)μt=∑n=1Ncnej2πfnt·t+θn.This intuitive mathematical manipulation results in a nonstationary channel model that is inconsistent with respect to the mean Doppler shift Bf(1)(t) and the Doppler spread Bf(2)(t), as we will see in Section 3.2. Although the expression in (14) is mathematically simpler than the one in (13), the difference is not significant in terms of implementation costs and simulation time.From the discussions in the previous subsection, it can be summed up that the two complex channel gainsμ(t) in (13) and (14) include the original SOC model [13](15)μt=∑n=1Ncnej2πfnt+θnas a special case that arises if the AOA αn(t) is supposed to be either constant (γn=0) or if the instantaneous phase θn(t) in (12) is approximated by a first-order Taylor series [see (11)]. The main difference between the three stochastic channel models above is that the former two are non-wide-sense stationary, whereas the third one is wide-sense stationary. The statistical properties of the SOC model have been studied in [13], while those of the new non-wide-sense stationary models will be analysed in the next section.


## 3. Analysis of the Nonstationary Multipath Channel Models

### 3.1. Time-Dependent ACF

The time-dependent ACFRμ(τ,t) of a complex stochastic process μ(t) is defined as(16)Rμτ,t=Eμt+τ2μ∗t-τ2,where E{·} denotes the expectation operator and (·)∗ stands for the complex conjugation operator. In the Appendix, it is proved that the time-dependent ACF Rμ(τ,t) of the complex channel gain μ(t) in (13) can be written as(17)Rμτ,t=∑n=1Ncn2ej2πfnt·sincγnτ/2·τ,where sinc(·) denotes the sinc function, which is defined by sinc(x)=sin(x)/x.Analogously, it can be shown that the time-dependent ACFRμ(τ,t) of the complex channel gain μ(t) introduced in (14) can be expressed by(18)Rμτ,t=∑n=1Ncn2ej2πfntcosγnτ/2+fn′tsincγnτ/2tτ,where fn(t) is the time-variant Doppler shift in (5) and fn′(t) denotes its derivative with respect to time t.For the special case that the AOAαn(t) is constant, that is, γn=0, it is obvious that the two time-dependent ACFs in (17) and (18) reduce to(19)Rμτ=Rμτ,t=∑n=1Ncn2ej2πfnτwhich represents the ACF of the SOC model described by (15). In this case, the ACF depends only on the time separation τ but not on time t, which was to be expected, because the SOC process μ(t) is wide-sense stationary.Furthermore, ifγn=0 and αn~U(0,2π], then the expressions in (17)–(19) reduce to the ACF Rμ(τ)=2σ02J0(2πfmaxτ), where 2σ02=∑n=1Ncn2 denotes the mean power of the complex channel gain μ(t), and J0(·) is the zeroth-order Bessel function of the first kind [14, Eq. (8.411-1)]. In other words, the proposed nonstationary multipath fading channel models include the classical Jakes/Clarke model [1, 15] as a special case.


### 3.2. Time-Dependent Mean Doppler Shift and Time-Dependent Doppler Spread

From the time-dependent ACFRμ(τ,t), the time-dependent mean Doppler shift Bμ(1)(t) and the time-dependent Doppler spread Bμ(2)(t) can be derived by means of(20)Bμ1t=12πjR˙μ0,tRμ0,t,(21)Bμ2t=12πR˙μ0,tRμ0,t2-R¨μ0,tRμ0,t,respectively, where R˙μ(0,t)(R¨μ(0,t)) denotes the first (second) order derivative of Rμ(τ,t) with respect to τ at τ=0. Inserting (17) in (20) and (21) results after some straightforward mathematical steps in the following closed-form solutions:(22)Bμ1t=∑n=1Ncn2fnt∑n=1Ncn2,(23)Bμ2t=∑n=1Ncn2fn2t∑n=1Ncn2-Bμ1t2.A comparison of (22) with (6) and (23) with (7) reveals that the equalities Bμ(1)(t)=Bf(1)(t) and Bμ(2)(t)=Bf(2)(t) hold, from which we can conclude that the proposed nonstationary multipath fading channel model described by (13) is consistent with respect to both the mean Doppler shift and the Doppler spread.On the other hand, if we insert (18) in (20) and (21), then we obtain(24)Bμ1t=∑n=1Ncn2fnt+fn′t·t∑n=1Ncn2,(25)Bμ2t=∑n=1Ncn2fnt+fn′t·t2∑n=1Ncn2-Bμ1t2.This result demonstrates that the simple nonstationary channel model introduced in (14) is inconsistent with respect to the mean Doppler shift and the Doppler spread, because Bμ(1)(t)≠Bf(1)(t) and Bμ(2)(t)≠Bf(2)(t) hold. Concerning the SOC process μ(t) in (15), we mention for completeness that the equalities Bμ(1)=Bf(1) and Bμ(2)=Bf(2) hold, where Bμ(1) and Bμ(2) are the same quantities as in (22) and (23), respectively, if we replace fn(t) by fn. Thus, the SOC model is consistent with respect to the mean Doppler shift and the Doppler spread. More information on the consistency of nonstationary multipath fading channels can be found in [16].


### 3.3. Wigner-Ville Spectrum

The Wigner-Ville spectrum, which is also called the time-varying spectrum or the evolutive spectrum, will be denoted bySμ(f,t). This function is defined as the Fourier transform of the time-dependent ACF Rμ(τ,t) with respect to τ [11]: that is,(26)Sμf,t=∫-∞∞Rμτ,te-j2πftdτ.Inserting (17) in (26) and using the property Rμ(τ,t)=Rμ∗(-τ,t), we can express the Wigner-Ville spectrum Sμ(f,t) of the proposed nonstationary multipath fading channel model described by (13) as(27)Sμf,t=2∑n=1Ncn2∫0∞cos⁡2πf-fntsinc⁡γnτ2τdτ.For the wide-sense stationary case, for which γn=0 holds, the Wigner-Ville spectrum Sμ(f,t) in (27) reduces to the Doppler power spectral density (PSD) of the SOC process μ(t) presented in (15), that is,(28)Sμf=∑n=1Ncn2δf-fn.Furthermore, for the isotropic scattering case, in which cn and αn are i.i.d. random variables with Ecn2=2σ02/N and αn~U(0,2π], we obtain the Jakes/Clarke PSD [1, 15] after computing the expected value of Sμ(f) in (28). Hence, the Wigner-Ville spectrum Sμ(f,t) in (27) includes the classical Jakes/Clarke Doppler spectrum as a special case.


## 3.1. Time-Dependent ACF

The time-dependent ACFRμ(τ,t) of a complex stochastic process μ(t) is defined as(16)Rμτ,t=Eμt+τ2μ∗t-τ2,where E{·} denotes the expectation operator and (·)∗ stands for the complex conjugation operator. In the Appendix, it is proved that the time-dependent ACF Rμ(τ,t) of the complex channel gain μ(t) in (13) can be written as(17)Rμτ,t=∑n=1Ncn2ej2πfnt·sincγnτ/2·τ,where sinc(·) denotes the sinc function, which is defined by sinc(x)=sin(x)/x.Analogously, it can be shown that the time-dependent ACFRμ(τ,t) of the complex channel gain μ(t) introduced in (14) can be expressed by(18)Rμτ,t=∑n=1Ncn2ej2πfntcosγnτ/2+fn′tsincγnτ/2tτ,where fn(t) is the time-variant Doppler shift in (5) and fn′(t) denotes its derivative with respect to time t.For the special case that the AOAαn(t) is constant, that is, γn=0, it is obvious that the two time-dependent ACFs in (17) and (18) reduce to(19)Rμτ=Rμτ,t=∑n=1Ncn2ej2πfnτwhich represents the ACF of the SOC model described by (15). In this case, the ACF depends only on the time separation τ but not on time t, which was to be expected, because the SOC process μ(t) is wide-sense stationary.Furthermore, ifγn=0 and αn~U(0,2π], then the expressions in (17)–(19) reduce to the ACF Rμ(τ)=2σ02J0(2πfmaxτ), where 2σ02=∑n=1Ncn2 denotes the mean power of the complex channel gain μ(t), and J0(·) is the zeroth-order Bessel function of the first kind [14, Eq. (8.411-1)]. In other words, the proposed nonstationary multipath fading channel models include the classical Jakes/Clarke model [1, 15] as a special case.


## 3.2. Time-Dependent Mean Doppler Shift and Time-Dependent Doppler Spread

From the time-dependent ACFRμ(τ,t), the time-dependent mean Doppler shift Bμ(1)(t) and the time-dependent Doppler spread Bμ(2)(t) can be derived by means of(20)Bμ1t=12πjR˙μ0,tRμ0,t,(21)Bμ2t=12πR˙μ0,tRμ0,t2-R¨μ0,tRμ0,t,respectively, where R˙μ(0,t)(R¨μ(0,t)) denotes the first (second) order derivative of Rμ(τ,t) with respect to τ at τ=0. Inserting (17) in (20) and (21) results after some straightforward mathematical steps in the following closed-form solutions:(22)Bμ1t=∑n=1Ncn2fnt∑n=1Ncn2,(23)Bμ2t=∑n=1Ncn2fn2t∑n=1Ncn2-Bμ1t2.A comparison of (22) with (6) and (23) with (7) reveals that the equalities Bμ(1)(t)=Bf(1)(t) and Bμ(2)(t)=Bf(2)(t) hold, from which we can conclude that the proposed nonstationary multipath fading channel model described by (13) is consistent with respect to both the mean Doppler shift and the Doppler spread.On the other hand, if we insert (18) in (20) and (21), then we obtain(24)Bμ1t=∑n=1Ncn2fnt+fn′t·t∑n=1Ncn2,(25)Bμ2t=∑n=1Ncn2fnt+fn′t·t2∑n=1Ncn2-Bμ1t2.This result demonstrates that the simple nonstationary channel model introduced in (14) is inconsistent with respect to the mean Doppler shift and the Doppler spread, because Bμ(1)(t)≠Bf(1)(t) and Bμ(2)(t)≠Bf(2)(t) hold. Concerning the SOC process μ(t) in (15), we mention for completeness that the equalities Bμ(1)=Bf(1) and Bμ(2)=Bf(2) hold, where Bμ(1) and Bμ(2) are the same quantities as in (22) and (23), respectively, if we replace fn(t) by fn. Thus, the SOC model is consistent with respect to the mean Doppler shift and the Doppler spread. More information on the consistency of nonstationary multipath fading channels can be found in [16].


## 3.3. Wigner-Ville Spectrum

The Wigner-Ville spectrum, which is also called the time-varying spectrum or the evolutive spectrum, will be denoted bySμ(f,t). This function is defined as the Fourier transform of the time-dependent ACF Rμ(τ,t) with respect to τ [11]: that is,(26)Sμf,t=∫-∞∞Rμτ,te-j2πftdτ.Inserting (17) in (26) and using the property Rμ(τ,t)=Rμ∗(-τ,t), we can express the Wigner-Ville spectrum Sμ(f,t) of the proposed nonstationary multipath fading channel model described by (13) as(27)Sμf,t=2∑n=1Ncn2∫0∞cos⁡2πf-fntsinc⁡γnτ2τdτ.For the wide-sense stationary case, for which γn=0 holds, the Wigner-Ville spectrum Sμ(f,t) in (27) reduces to the Doppler power spectral density (PSD) of the SOC process μ(t) presented in (15), that is,(28)Sμf=∑n=1Ncn2δf-fn.Furthermore, for the isotropic scattering case, in which cn and αn are i.i.d. random variables with Ecn2=2σ02/N and αn~U(0,2π], we obtain the Jakes/Clarke PSD [1, 15] after computing the expected value of Sμ(f) in (28). Hence, the Wigner-Ville spectrum Sμ(f,t) in (27) includes the classical Jakes/Clarke Doppler spectrum as a special case.


## 4. Numerical Results

This section presents a selection of numerical results to illustrate the main findings of this paper. In all considered propagation scenarios, we have set the number of multipath componentsN to N=10. The gains cn and initial AOAs αn=αn(0) have been computed by using the extended method of exact Doppler spread (EMEDS) [17]. According to this method, the parameters cn and αn are given by(29)cn=σ02N,(30)αn=2πNn-14,respectively, and the initial phases θn=θn(0) are considered as realizations of independent random variables, each characterized by a uniform distribution over the interval (0,2π]. If not stated otherwise, the radii rn in Figure 1 have been set to 50 m for all n=1,2,…,N. For the mean power (variance) σ02 of the in-phase and quadrature components of μ(t), we have chosen the value σ02=1. The carrier frequency was set to 5.9 GHz, and the maximum Doppler frequency fmax was supposed to be fmax=91Hz. This corresponds to a mobile speed of v=v→=16.65km/h, where we have assumed that the MS moves in x-direction, implying that the angle of motion αv equals zero, that is, αv=0.Figure2 depicts the trend of the time-variant Doppler frequencies fn(t) by using the exact expression for the AOAs αn(t) according to (1). For comparison, this figure also shows the behaviour of fn(t) for the approximate solution of αn(t) in (2). Figure 2 shows clearly that the first-order approximation is quite good over the interval from 0 to 2.162s during which the MS has covered a distance of 10 m.Figure 2
Trend of the time-variant Doppler frequenciesfn(t)(n=1,2,…,N) by using the exact solution (black solid line) and the approximate solution (blue dashed line), where N=10.Figure3 illustrates the signal envelope μt by using the SOC model [see (15), Case I], the proposed nonstationary multipath fading channel model [see (13), Case II], and the simple nonstationary model [see (14), Case III]. This figure demonstrates clearly that the temporal variations of the AOAs αn(t) have a great influence on the temporal behaviour of the signal envelope μt. It is interesting to note that the three signal envelopes are identical at t=0 and very similar for small values of t>0, but they differ considerably with increasing values of t. It should be mentioned that different realizations of the initial phases θn=θn(0) result in different sample functions of the signal envelopes, but the aforementioned trend is the same for all realizations.Figure 3
Illustration of the signal envelopeμt of a sample function of a wide-sense stationary SOC process [see (15)] in comparison with the signal envelopes μt of the nonstationary processes described by (13) and (14).Figures4 and 5 present the ACF Rμ(τ) of the SOC process μ(t) in (15) and the time-dependent ACF Rμ(τ,t) of the nonstationary process μ(t) in (13), respectively. It can be observed that both ACFs are identical at the origin t=0, but the temporal correlation properties of the nonstationary model differ more and more if time t proceeds. This means that the temporal variations of αn(t) influence greatly the fading behaviour of the signal envelope μt.Figure 4
ACFRμ(τ)=Rμ(τ,t) of a SOC process μ(t) with constant AOAs αn for N=10.Figure 5
Time-dependent ACFRμ(τ,t) of the proposed nonstationary process μ(t) with time-variant AOAs αn(t) for N=10.Figures6 and 7 depict the corresponding Doppler PSD Sμ(f) [see (28)] of the SOC process μ(t) in (15) and the Wigner-Ville spectrum Sμ(f,t) [see (27)] of the nonstationary process μ(t) in (13), respectively. A comparison of the two spectral representations shows clearly that the influence of the time-variant AOAs αn(t) cannot be neglected. This statement is obvious as the Doppler frequencies of the Wigner-Ville spectrum (Doppler PSD) associated with the stationary SOC process remain constant over time t (see Figure 6), while the spectral components of the nonstationary process experience a drift if time t proceeds (see Figure 7). Finally, we mention that the results in Figure 7 have been obtained numerically by setting the upper limit of τ in the integral of (27) to τ=τmax=1 s and evaluating the integral by considering Ns=1000 samples, which results in a resolution of τmax/Ns=0.001 s.Figure 6
Wigner-Ville spectrum (Doppler PSD)Sμ(f)=Sμ(f,t) of an SOC process μ(t) with constant AOAs αn for N=10.Figure 7
Wigner-Ville spectrumSμ(f,t) of the proposed nonstationary process μ(t) with time-variant AOAs αn(t) for N=10.Figures8 and 9 are devoted to a study on the influence of the (ring) radii rn on the time-dependent mean Doppler shift Bμ(1)(t) and the time-dependent Doppler spread Bμ(2)(t), respectively. The presented graphs show that the smaller the radii rn are, the faster the functions Bμ(1)(t) and Bμ(2)(t) are changing over time t. Figure 8 shows also a comparison between the time-dependent mean Doppler shift Bμ(1)(t) of the consistent model described by (14) and the inconsistent model according to (15). Both models have the same mean Doppler shift Bμ(1)(t) at the origin t=0, but the mean Doppler shifts deviate considerably from each other with increasing values of time t. The same statement holds for the time-dependent Doppler spread shown in Figure 9. These results underline the importance of consistency, as the deviations between (22) and (24) as well as between (23) and (25) cannot be neglected.Figure 8
Time-dependent mean Doppler shiftBμ(1)(t) of the nonstationary channel models for different values of the radii rn.Figure 9
Time-dependent Doppler spreadBμ(2)(t) of the nonstationary channel models for different values of the radii rn for n=1,2,…,N(N=10).


## 5. Model Extensions

To isolate the effect of time-variant AOAs on the Doppler characteristic, we have assumed that the channel is frequency-nonselective and that the transmitter and receiver are equipped with single omnidirectional antennas. To relax these assumptions, we will provide some guidelines on model extensions in the following subsections.


### 5.1. Extension to Frequency-Selectivity

Starting from the narrowband multipath fading channel model in (12) and taking into account the fact that the nth plane wave component μn(t)=cnexp⁡{jθn(t)} will be received after a time-variant propagation delay denoted by τn′(t), the impulse response h(τ′,t) of the resulting non-wide-sense stationary single-input single-output channel model can be expressed as(31)hτ′,t=∑n=1Ncnejθntδτ′-τn′t,where θn(t) is given by (9) for the consistent model or by θn(t)=2πfn(t)t+θn for the inconsistent model. The time-variant delay τn′(t) can be derived from the geometrical model in Figure 1 as(32)τn′t=1c0D+xn2+yn2+rnt,where c0 denotes the speed of light, D is the distance between the BS and MS, and rn(t)=(xn-x(t))2+(yn-y(t))2.


### 5.2. Extension to MIMO

Lethkl(τ′,t) denote the impulse response of a frequency-selective MIMO channel with MT transit and MR receive antennas; then the propagation link from the lth transmit antenna to the kth receive antenna can be modelled as(33)hklτ′,t=∑n=1Ngklntejθntδτ′-τn′tfor k=1,2,…,MR and l=1,2,…,MT, where gkln(t) is the same as in [18, Eq. (9)] apart from the fact that we have to replace there the time-invariant quantities rn and αnR by time-variant quantities rn(t) and αnR(t), respectively. The expression in (33) can be derived by applying the design steps of the generalized principle of deterministic channel modelling [12, Section 8.1].


### 5.3. Other Model Extensions

The proposed model is only applicable to omnidirectional antennas. The extension to directional antennas is possible through a proper adjustment of the constant path gainscn, which have to be replaced by time-variant path gains cn(t). The temporal characteristics of cn(t) depend on the antenna pattern and the direction in which the MS moves. It is also possible to consider birth-and-death effects of the scatterers. This extension results in a multiplication of each multipath component by a birth-and-death process or, equivalently, by replacing the time-invariant path gains cn by proper time-variant path gains cn(t). The analysis of the Wigner-Ville spectrum of nonstationary mobile radio channels with time-variant path gains cn(t) is substantially different from the analysis in Section 3.3 and beyond the scope of this paper.


## 5.1. Extension to Frequency-Selectivity

Starting from the narrowband multipath fading channel model in (12) and taking into account the fact that the nth plane wave component μn(t)=cnexp⁡{jθn(t)} will be received after a time-variant propagation delay denoted by τn′(t), the impulse response h(τ′,t) of the resulting non-wide-sense stationary single-input single-output channel model can be expressed as(31)hτ′,t=∑n=1Ncnejθntδτ′-τn′t,where θn(t) is given by (9) for the consistent model or by θn(t)=2πfn(t)t+θn for the inconsistent model. The time-variant delay τn′(t) can be derived from the geometrical model in Figure 1 as(32)τn′t=1c0D+xn2+yn2+rnt,where c0 denotes the speed of light, D is the distance between the BS and MS, and rn(t)=(xn-x(t))2+(yn-y(t))2.


## 5.2. Extension to MIMO

Lethkl(τ′,t) denote the impulse response of a frequency-selective MIMO channel with MT transit and MR receive antennas; then the propagation link from the lth transmit antenna to the kth receive antenna can be modelled as(33)hklτ′,t=∑n=1Ngklntejθntδτ′-τn′tfor k=1,2,…,MR and l=1,2,…,MT, where gkln(t) is the same as in [18, Eq. (9)] apart from the fact that we have to replace there the time-invariant quantities rn and αnR by time-variant quantities rn(t) and αnR(t), respectively. The expression in (33) can be derived by applying the design steps of the generalized principle of deterministic channel modelling [12, Section 8.1].


## 5.3. Other Model Extensions

The proposed model is only applicable to omnidirectional antennas. The extension to directional antennas is possible through a proper adjustment of the constant path gainscn, which have to be replaced by time-variant path gains cn(t). The temporal characteristics of cn(t) depend on the antenna pattern and the direction in which the MS moves. It is also possible to consider birth-and-death effects of the scatterers. This extension results in a multiplication of each multipath component by a birth-and-death process or, equivalently, by replacing the time-invariant path gains cn by proper time-variant path gains cn(t). The analysis of the Wigner-Ville spectrum of nonstationary mobile radio channels with time-variant path gains cn(t) is substantially different from the analysis in Section 3.3 and beyond the scope of this paper.


## 6. Conclusion

In this paper, we have developed and analysed multipath fading channel models with time-variant AOAs. Our study has shown that the effect of time-variant AOAs results in a non-wide-sense stationary multipath fading channel model. Expressions have been derived for the time-dependent ACF, time-dependent mean Doppler shift, time-dependent Doppler spread, and the Wigner-Ville spectrum of the proposed non-wide-sense stationary channel model. By comparing these statistical quantities with known results of studies assuming constant AOAs, we can conclude that the assumption of constant AOAs is only justified for very short observation intervals. The proposed nonstationary channel model allows extending the observation interval over a wider range without losing accuracy. The price for this added accuracy is a higher degree of complexity concerning the mathematical expressions.One of the remaining problems that might be tackled in an upcoming study is to develop quantitative methods for the investigation of the length of the observation interval over which the proposed nonstationary channel models are sufficiently accurate. Another topic could be to extend the presented framework to the modelling of MIMO channels with time-dependent AOAs.



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*Source: 6396173-2018-03-15.xml*