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Timescale domain characterization of nonWSSUS wideband channels
EURASIP Journal on Advances in Signal Processing volume 2011, Article number: 123 (2011)
Abstract
To account for nonstationarity, channel characterization and system design methods that employ the nonwidesense stationary uncorrelated scattering (nonWSSUS) assumption are desirable. Furthermore, the inadequacy of the Doppler shift operator to properly account for the frequency shift in wideband channel implies that the timefrequency characterization methods that employ the Doppler shift operator are not appropriate for most wideband channels. In this article, the statistical timescale domain characterization of the nonWSSUS wideband channel is presented. This approach employs the time scaling operator in order to account for frequency spreading, and also emphasizes on the nonstationarity of the wideband channel. The nonWSSUS statistical assumption termed localsense stationary uncorrelated scattering (LSSUS) is presented and employed in characterizing the nonstationary property of the timevarying wideband channel. The LSSUS channel model is then parameterized to provide useful coherence and stationarity/nonstationarity parameters for optimal system design. Some application relevance of the developed model in terms of channel capacity and diversity techniques are discussed. Measurement and simulation results show that the assumption of ergodic capacity and the performance of various diversity techniques depend on the degree of channel stationarity/nonstationarity. It is shown that the quantification of this degree of stationarity through the channel parameters can provide a way of tracking channel variation and allowing for adaptive application of diversity techniques and the channel capacity.
1 Introduction
Timevarying channels are often modeled as stationary random processes using the concept of the widesense stationary uncorrelated scattering (WSSUS) assumption [1]. This stationarity assumption leads to simplification of transceiver design and may in some circumstances be reasonable on physical grounds. In this case, the multiple channel states presented by the mobility of the communication terminal(s)/scatterer(s) have varying channel statistics over the transmission duration, and it is assumed that the variances among the statistics of the multiple channels are insignificant, so they can be averaged out over a wide range of interval. The assumption of the statistical stationarity of the timevarying channel allows for the definition of some channel parameters that are employed in system designs.
Unfortunately, in practice the WSSUS assumption is not often met. The nature of the timevarying channel is such that the spatial structures of the multipath components, i.e., their number, timeofarrivals, angleofarrivals (AOA), and magnitudes, change with time and location, leading to nonstationary statistics [2, 3]. More also scattering by the same object as well as variation in the AOA caused by mobility with respect to a statistical stationary duration of reference may result in correlation among scatterers. This condition violates the uncorrelated scattering assumption. And like stationarity, nonstationarity also carries informative features of the channel. Hence, developing approaches that would tap into the information that can be obtained from nonstationary analysis of the channel will be of great merit to optimal system design.
Typically, timevarying channels are characterized by time and frequency dispersions. When using Doppler shift as a measure of the channel's frequency dispersion, it is presumed that composite (multitone) signals or subcarriers (in the case of multicarrier systems) passing through the channel, experience the same amount of frequency shift obtained with respect to the carrier frequency f_{c}. For narrow bandwidth composite signals, this approximation may be practically true and the presentations in [2], sufficient. However, for widebandwidth signals, the Doppler approximation wholly fails since the composite signals experience different Doppler shifts. This is the case of ultrawideband (UWB) and underwater acoustic (UWA) channels. Therefore, channel characterization methods that do not depend on the carrier frequency to obtain the measure of frequency dispersion in wideband channels are much desired. One of such method is the timescale domain channel characterization [4–10].
The delayDoppler effects of timevarying channel are often modeled as taps of a timevarying filter [11, 12]. In the case of the narrowband channel, one tap is the sum of many paths. Hence, it is acceptable to characterize the time evolution of the individual taps using an autocorrelation function (ACF) model which is assumed to be independent across delay. Therefore, in nonWSSUS characterization, the time evolution model of the narrowband channel can be decomposed into the time evolution of individual taps. This is not the case for the UWB channel where taps are often composed of few or no paths due to the fine time resolution. The fine time resolution implies narrow delay bins which enable paths to move fast from one tap to another [13]. Hence, the time evolution of taps becomes correlated across delays. In this case, the narrowband assumption of modeling the stochastic process of taps independent across delay no longer holds. So, the time evolution of the UWB channel cannot be decomposed into the time evolution of the individual taps, but of the individual paths. Therefore, the nonWSSUS characterization of the narrowband channel will differ significantly from that of the wideband channel.
In this article, we attempt to answer the following questions. (1) How can the timevarying nonWSSUS wideband channel be characterized in the timescale domain? (2) What are the necessary parameters and information that can be obtained from the nonWSSUS model? (3) How can such information be used in system design in order to optimize performance? To address these questions, we present the nonWSSUS timescale domain characterization method for timevarying wideband channel which employs the assumption that the channel statistics are locally stationary. This method is considered appropriate for wideband nonstationary channels. The main contributions of this article which sort to answer the three questions posed above are outlined as follows.

A method of characterizing the nonWSSUS wideband channels using a statistical concept termed LSSUS assumption is presented in the timescale domain. The LSSUS concept results primarily from the designation of statistical intervals of quasistationarity and quasinonstationarity over which the WSSUS and LSSUS assumptions can be jointly defined. The closedform expression for the LSSUS channel is derived bearing in mind that the stationarity/nonstationarity interval is dependent on the properties of the transmit signal and the wireless channel. We note that this LSSUS closedform expression can be applied to most channels, but in terms of merit, it is more appropriate for the characterization of the wideband channels like acoustic, sonic, UWB channels, and other emerging systems operating at high fractional bandwidths; in which case the concept of time scaling is more suitable than Doppler shift. In the narrowband channel, the LSSUS expression when presented in the timefrequency domain can be deem to be equivalent to the channel correlation function (CCF) in [2] as will be explained in Section 4.

To parameterize the nonWSSUS channel in timescale domain using the LSSUS assumption, a stationarity degree estimation method is presented. This estimator uses the concept in [14] to quantify the extent to which an 'instantaneous' scattering functions deviates from a given one assumed to be statistically stationary (WSSUS). The stationarity degree is used to obtain condensed coherence and stationarity parameters that are equivalent to those presented in [2]. However, unlike in [2] the approach here emphasizes on the frequency dependence of these parameters, particularly, the coherence/stationarity time.

The illustration and application relevance of the LSSUS concept to the timevarying wideband communication systems is presented. Illustrative measurement and simulation examples for the timevarying UWB channel typical of the infostation [15–17] environment are provided. And simulation with numerical results in the case of a simplified underwater communication scenario is also presented. These simulations and measurements are used to show the merit and application of the LSSUS concept to improving system performance by the optimal application of ergodic capacity assumption and diversity techniques.
The rest of this article is organized as follows. In Section 2, the relevant studies that are related to the idea developed in the article are presented. The basic timescale channel model is specified in Section 3. In Section 4, the WSSUS characterization of the wideband channel in the timescale domain is presented. Section 5 is devoted to the nonWSSUS characterization of the timevarying wideband channel using the LSSUS assumption. The estimation of the nonWSSUS channel stationarity degree and the derivation of LSSUS channels parameters are presented in Section 6. In Section 7, the application relevance of the LSSUS model in relation to channel ergodic capacity assumption and diversity techniques is presented. In Section 8, measurement and simulation examples are used to illustrate the concept and application of the LSSUS model. Conclusions are given in Section 9.
2. Related studies
Some of the existing studies related to the ideas presented in this article are as follows. In his seminal paper [1], Bello highlighted on the discrepancy between the WSSUS and nonWSSUS channels using the term QuasiWSSUS (QWSSUS). The QWSSUS assumption implies that the channels statistics do not change within a specific time and frequency interval. A more rigorous and classical theoretical framework for the description of the nonWSSUS channels was introduced by Matz [2], where instead of the WSSUS scattering function, the local scattering function (LSF), in consonance with the concept of timedependent spectrum for nonstationary analysis, was defined. And the timefrequency CCF appropriate for the nonWSSUS case was also delineated. The LSF and CCF are given respectively as [2]
and
where R_{ L }(t, f; Δt, Δf) is the autocorrelation of the timevarying transfer function which in this case is viewed as a nonstationary random process.
Equation 2 implies that A_{ H }(·) is the 4D Fourier transform of (1). Hence, the function A_{ H }(·) is a complete characterization of the narrowband nonWSSUS channel. While the coherence parameters were defined using the conventional approach in [11, 12], the stationarity parameters in time and frequency were defined as the inverse of some normalized maximum delay and Doppler spread weighted integrals, respectively. The direct computation and measurement of A_{ H }(·) and subsequently, the stationarity parameters, are somewhat obdurate. Therefore, to obtain stationarity time, the collinearity measure was used in [18] in order to characterize the nonWSSUS typical of highway and urban scenarios. But, the collinearity measure falls short of taking the strict positivity of the LSF into account. Hence, in [19], the time and frequency divergences of the nonWSSUS vehicular channel at 5.2 GHz were characterized using spectral divergence measure. However, for wide bandwidth applications the LSF and CCF do not suffice since they are Doppler operator based and narrowband oriented; and the obtained coherence and stationarity parameters will vary with frequency for every single realization. And of course as stated earlier, the fast movement of paths from one delay bin to another suggests that the nonWSSUS model of narrowband channels may be different from that of the wideband channels. In [20], the concept of local regions of stationarity (LRS) of the mobile radio channel was presented and used to obtain nonstationary information on the powerdelay profile of the channel. We note that this concept of LRS inspired the definition of the stationarity region adopted in this study.
Some of the existing studies on timescale domain channel characterization are available in [4–10]. In [4], Weiss presented the use of wavelet theory in wideband correlation processing, and the importance of wideband processing in the wavelet (timescale) domain. In [5, 6], the use of the concepts from wavelet transforms and group theory to derive a linear timevarying system characterization for wideband input signal was presented. The canonical timescale representation of the timevarying channel was proposed in [7]. In [8], the Mellin transformbased timescale was applied to address the issue of jointmultipath scale diversity gain over dyadic timescale framework. A similar work directed toward achieving joint scalelag (delay) diversity in wideband mobile direct spread spectrum systems was presented by Margetts et al. [9]. In [10], the timescale channel characterization was presented in the wavelet domain. However, while all these literatures projected the notion of timevarying channel characterization using the time scaling operator, the application of this method to actual channel parameterization was not discussed. More also, the authors of [4–10] did not address the issue of statistical channel characterization especially the case of nonWSSUS which is the main focus of this study.
3. System model
In general, for transmit signal x(t) and received signal y(t), the continuous timescale and timefrequency representation of the linear timevarying (LTV) channel H are, respectively, given by
where a(t) is the amplitude. The terms {\mathit{W}}_{\mathbf{H}}\left(\tau ,s\right)=\int y\left(t\right)\phantom{\rule{2.77695pt}{0ex}}a\left(t\right)\phantom{\rule{2.77695pt}{0ex}}x\left(\left(t\tau \right)\u2215s\right)\phantom{\rule{2.77695pt}{0ex}}dt and {\mathit{S}}_{\mathbf{H}}\left(\tau ,\upsilon \right)=\int h\left(\tau ,t\right)\phantom{\rule{2.77695pt}{0ex}}{e}^{j2\pi \upsilon t}\phantom{\rule{2.77695pt}{0ex}}dt denote the delayscale (wideband) spreading function [8–10, 21] and the delayDoppler spreading function [21], respectively. While the latter is interpreted as the reflectivity of the scatterers associated to propagation delay τ and Doppler shift υ, the former is interpreted as the reflectivity of the scatterers associated to delay τ and scale shift (or time scaling) s.
The scale s = (c ± v)/(c ∓ v) is related to frequency by
The term {f_{ p } } in (5) is the frequency vector of length P comprising stepwise of all the frequency components of the transmitted signal. Indeed, the pairs of Equations 3 and 4 are equivalent in some applications, but are not in some others. The distinction lies with the interaction between the transmit signal x(t) and the channel H. This interaction determines whether a system is narrowband or wideband.
The narrowbandwideband assumption is often made in two aspects: (1) the relationship between signal bandwidth B_{sig} and coherence bandwidth B_{c}[12]. (2) the relationship between B_{sig} and f_{c}. In the first aspect, a general constraint B_{ sig } < < f_{ c } is imposed, and the narrowband assumption is then upheld when the inequality B_{sig} < B_{c} is satisfied, otherwise, the system under consideration is wideband. With the imposition of the constraint B_{ sig } < < f_{ c } , the pair of Equations 3 and 4 are equivalent and W_{ H }(τ, s) ≡ S_{ H }(τ, υ) irrespective of whether the system is wideband or narrowband in this context. In the second aspect which is the focus of this article, no constraint is imposed and narrowband assumption is then made when (B_{ sig } /f_{ c } ) < B_{ f } , where B_{ f } (typically 0.2), is called the fractional bandwidth [22, 23], otherwise, the system is wideband. For the rest of this article, narrowband and wideband are referred to in the context of this second aspect.
For the narrowband assumption, {f_{ p } } can be approximated to f_{c} in (5) so that time scaling is equivalent to Doppler shift and W_{ H }(τ, s) ≡ S_{ H }(τ, υ) is valid. In the case of wideband, this equivalence is invalid and W_{ H }(τ, s) is the appropriate valid channel response. Wireless communication systems like UWB [22, 23] and UWA channels [24, 25] are wideband in this context.
It can be seen from (3) that the timescale representation establishes a onetoone correspondence between the received signal and the delayscale spreading function. This onetoone correspondence allows one to read up condense parameters and useful features directly from the channel response. However in practice, such correspondence is not feasible due to the stochastic nature of the channel. In order to obtain condensed and useful parameters from this random process, some assumptions can be made in order to define an interval over which the randomness of the channel can be deemed to be statistically stationary. In view of that, let ∪ be the universal set of all stochastic processes, there exist the subsets of ∪ whose statistical properties vary with certain degrees in respect to the variations within some intervals J_{ k } , k = 1, 2,.., K. If we define the partition J_{ k } as the countable collection of subintervals, then we can denote the interval for which some statistical properties of a process under observation are assumed to be stationary, as J_{(v)}. One of the popular statistical properties used in channel characterization is the mean and the ACF [1].
4. WSSUS characterization
Definition 1: A process is called widesense stationary (WSS) if it's first two moments are independent of absolute time t on a defined interval J_{(v)}. Such process is delineated if there is some partition for which the expression J_{(v)}= J_{ k } , ∀k is valid and provides time independence with respect to the mean and ACF.
If we denote the observation interval in general by J, then from Definition 1, we can identify two different time instants, t and t' over which the WSS assumption is defined. Let J_{(v)}= t't, if we incorporated the US assumption into the model, then for two given scale instants s ≠ 0 and s' ≠ 0, the ACF depends only on Δτ = τ' τ and Δs = s's. The ACF of the timescale channel is then given by
In (6), '*' denotes conjugation and E[.] is the expectation operator. However, for simplicity we assume that x(t) is real so that for the rest of this article, the conjugation '*' can be neglected. Hence,
It can easily be shown that
where
{X}_{\Delta \tau ,\Delta s}^{\left(\tau ,s\right)}=a\left(\Delta t\right)\phantom{\rule{2.77695pt}{0ex}}x\left(\frac{\Delta t\Delta \tau \tau}{\Delta s+s}\right), and {X}^{\left(\tau ,s\right)}=a\left(t\right)x\left(\frac{t\tau}{s}\right).
The first inner product on the righthandside (RHS) of (8) is called the delayscale scattering function (DSSF)
The inner product {\u27e8\delta ,{X}^{\left(\tau ,s\right)}\u27e9}_{t} is an approximation of δ(Δ τ)δ(Δs) which implies that at each different delay the distribution is simply a scaled version of the transmitted signal. The function P _{ wssus } (τ, s) has compact support defined on the set η = {s_{min} ≤ s ≤ s_{max}, τ_{min} ≤ τ ≤ τ_{max}} where s_{min} and s_{max} are the minimum and maximum scale spreads, respectively, and, τ_{min} and τ_{max} are the minimum and maximum delay spreads, respectively.
Equation 9 is comparable to the delayDoppler scattering function {\mathit{S}}_{\mathbf{H}}\left(\tau ,\upsilon \right)=\int {R}_{\mathbf{H}}\left(\Delta t,\tau \right)\phantom{\rule{2.77695pt}{0ex}}{e}^{j2\pi \upsilon \Delta t}d\Delta t, where R_{ H }(Δt, τ) ≡ ∏(R_{ y } (Δt)) is the delay crosspower density. The term ∏ is a filter or window operator whose output is dependent on the particular filter, τ and x(t). The conventional WSSUS condensed parameters [12] like the coherence bandwidth B_{c} and coherence time T_{c} can be used to quantify the channel dispersion bearing in mind the relation between frequency and scale as stated in (5). While the delay spread/B_{c} in both S_{ H }(τ, υ) and P _{ wssus } (τ, s) are equivalent, their similarity in terms of frequency spread/T_{c} is derivable from the inverse relation between frequency and scale [4]. Thus, it suffices that S_{ H }(τ, υ) ≡ Λ^{}( P _{ LSSUS } (τ, s)) where Λ^{} is the scaletoDoppler conversion operator. This implies that while the computation of S_{ H }(τ, υ) is dependent on the carrier frequency, the realization of P _{ wssus } (τ, s) is independent of the carrier frequency or any reference frequency. Therefore, different values of T_{c} can be obtained for different values of frequencies, from a single P _{ wssus } (τ, s) realization. The variation of coherence time with frequency is depicted in Figure 1 for a typical wideband channel at a reference mobile speed of 5 m/s. This figure shows that T_{c} computed at f_{c} vary significantly with T_{c} at other frequencies within a defined wide bandwidth.
5. NonWSSUS characterization
Definition 2: A process is called localsense stationary (LSS) if there exist some partitions for which at least one interval say J_{ i } is considered to be WSS, J_{(v)}= J_{ i } . Within this 'locally' stationary interval J_{ i } the secondorder statistics are approximately independent of time, but vary slowly in time across all other intervals for which J_{(v)}≠ J_{k ≠ i}. Thus, the autocorrelation is WSS at J_{ i } but nonWSS at all other intervals J_{ k } ≠ J_{ i } .
For all other processes with gross time varying statistical properties over all J for which no J_{(v)}can be ascertained for practical purposes, the nonstationary process is defined.
Definition 2 is related to that of the locally stationary random processes introduced by Silverman [26], and the uniformly bounded linearly stationary (u.b.l.s) processes introduced by Tjøstheim and Thomas [27]. As it was pointed out in [27], the above definition is the same as saying that the u.b.l.s processes can be obtained by filtering WSS processes. Hence, the LSS process has the desirable property of including WSS process as a special case.
The relation among the various subsets of the statistical processes is shown in Figure 2. Thus, a little above the upper bound of strictsense stationarity lies the quasistationary region and a little below the lower bound of nonstationarity lies the quasinonstationary region.
From Definition 2, we can identify three different time instants, t, t', and t" over which the LSS assumption is defined. Within the quasistationary intervals for two time instants t and t', the secondorder channel statistics are constant over Δt = t't. However, the statistics vary across the quasinonstationary interval J = t"t over which LSS is defined. If we extend the above statements to the concept of uncorrelated scattering (US), then it can easily be shown that, t"t = Δ(t + Δt)+ Δt, Δs = s's, s"s = Δ(s + Δs)+ Δs, Δ τ =  τ' τ and τ" τ = Δ (τ + Δτ) + Δτ. Therefore, the ACF for the LSSUS is given for some spatial displacement \Delta \stackrel{\u20d7}{r} by
With the assumption of a real x(t) it can easily be shown that
where {X}_{\Delta \left(\tau +\Delta \tau \right)+\Delta \tau ,\Delta \left(s+\Delta s\right)+\Delta s}^{\left(\tau ,s\right)}=a\left(\Delta \left(t+\Delta t\right)+\Delta t+t\right)
.\phantom{\rule{2.77695pt}{0ex}}x\left(\frac{\left(\Delta \left(t+\Delta t\right)+\Delta t+t\right)\left(\Delta \left(\tau +\Delta \tau \right)+\Delta \tau +\tau \right)}{\Delta \left(s+\Delta s\right)+\Delta s+s}\right) and {X}^{\left(\tau ,s\right)}=a\left(t\right)x\left(\frac{t\tau}{s}\right).
The first inner product term in (9) is called the localsense scattering function (LSSF):
The scattering function P _{ LSSUS } (τ, s, Δτ, Δs) completely characterizes the LSSUS channel. In relation to (8), there exist some Γ > 0 such that
Implicitly for small values of spatial displacement \Delta \stackrel{\u20d7}{r} the terms Δ(t + Δt), Δ(s + Δs), and Δ(τ + Δτ) in (8) become insignificant, Γ = 1, and (13) simplifies to (9)
Therefore, the measure of the channel stationarity or nonstationarity is with regard to the extent to which P _{ LSSUS } (τ, s) deviates from P _{ wssus } (τ, s).
6. Estimation of stationarity degree
6.1. Stationarity test
The LSSUS scattering function(s) completely characterizes the wireless wideband channel. Let \mathrm{\Psi}=\left\{{\mathit{P}}_{LSSUS}{\left(\tau ,s\right)}_{\theta}:\theta \in \Theta \subset \Re \right\} denote the set of all LSSUS scattering functions for a particular case where P _{ wssus } (τ, s) is a subset. The main issue is how to estimate the deviation of the set Ψ from its subset P _{ wssus } (τ, s). Though there are several proposed methods in the literature for modeling locally stationary processes, the problem of testing the stationarity of such processes has attracted less attention in the literature. Some existing work in stationarity test can be found in [28, 29]. The weaknesses of some of the test module proposed in literature include the dependency of the test on the choice of the regularization parameter [28], and the complexity of the solutions [29]. In [14], a simplified alternative method for measuring deviation from stationarity in locally stationary processes was presented. This method measures stationarity degree by the best L^{2} approximation of the spectral density of the underlying process by the spectral density of a stationary process. Hence, using the minimal distance deviation measure proposed in [14] we define the stationarity measure {\mathrm{\Xi}}_{\tau ,s}^{2} as the measure of the deviation of P _{ LSSUS } (τ, s) from P _{ wssus } (τ, s):
with the limit of integration practically taken over [s_{min}s_{max}] and [τ_{min} τ_{max}]. The term {\mathrm{\Xi}}_{\tau ,s}^{2} is a measure of the deviation of P _{ LSSUS } (τ, s) from P _{ wssus } (τ, s).
6.2. NonWSSUS condensed channel parameters
As stated earlier, wireless channels are often characterized using the condensed parameters, B_{c} and T_{c}. However, these parameters do not completely characterize all classes of timevarying processes. It should be noted that coherency is a concept developed for WSSUS channels to describe their nonselectivity [12]. In this sense, the stationarity attribute of the WSSUS channel is infinite. Thus, the channel is nonselective (coherence) in time and frequency over certain bounded values, and is assumed to remain statistically invariant (stationary) in time and frequency for infinite extent determined by the choice of the interval of observation. Therefore, apart from the coherence parameters, the stationarity parameters [2] are also required in order to fully characterize the LSSUS channel.
In [2], the stationarity parameters were introduced and obtained as the inverse of some normalized weighted integrals which defines the spread of the CCF. The weight function is a function of the spread in time, frequency, Doppler, and delay. In this study, we opt for a different approach to obtain the stationarity parameters. To do so, we consider the stationarity parameters as quantifying the deviation of the LSSUS from WSSUS. In consonance with (13), we can define the minimal delay profile deviation (MDPD) {\mathrm{\Xi}}_{\tau}^{2} and the minimal scale profile deviation {\mathrm{\Xi}}_{s}^{2} by
The minimal r.m.s delay spread deviation Δτ _{ rms } and minimal scale spread deviation Δs_{ rms } are given by
and
where {\mathrm{\Xi}}_{\tau ,wssus}^{2}={\left(\right)close="">\underset{{\mathit{P}}_{wssus}}{min}\iint {\left({\mathit{P}}_{LSSUS}\left(\tau ,s\right){\mathit{P}}_{wssus}\left(\tau ,s\right)\right)}^{2}\phantom{\rule{2.77695pt}{0ex}}\frac{ds}{{s}^{2}}}_{}{\mathit{P}}_{LSSUS}\left(\tau ,s\right)=0\n and {\mathrm{\Xi}}_{\tau ,LSSUS}^{2}={\left(\right)close="">\iint {\left({\mathit{P}}_{LSSUS}\left(\tau ,s\right){\mathit{P}}_{wssus}\left(\tau ,s\right)\right)}^{2}\phantom{\rule{2.77695pt}{0ex}}\frac{ds}{{s}^{2}}}_{}{WSSUS}_{}\left(\tau ,s\right)=0\n
Hence, the stationarity bandwidth B_{ s } ≈ 1/Δτ_{ rms } , stationarity time T_{ s } ≈ 1/Δs_{ rms } {f_{ p } }, coherence bandwidth B_{c} and coherence time T_{c} provide complete parameterization of the LSSUS channel. The stationarity parameters tend to infinity in the case of WSSUS for which {Δτ_{ rms } , Δs_{ rms } }→ 0. It can also be seen that using this approach, different values of T_{s} can be obtained from a single wideband channel realization for different values of frequencies.
It is important to note the use of Δτ_{max} in [2] in defining the stationarity bandwidth. The parameter τ_{max} is a singular value whose statistics across different channel realizations are independent of other delay values in a particular channel response. On the other hand, the parameter τ_{rms} is obtained taking into consideration the statistics of all the delay values associated with a particular channel response. Hence, while Δτ_{max} may be appropriate value for determining the channel coherence parameters, it is not quite suitable for the statistical measure of stationarity. For instance, let us consider two channel responses h_{1}(τ, t) and h_{2}(τ, t) at two different time instants as shown in Figure 3.
From Figure 3, it can be seen that τ_{max} is the same for both h_{1}(τ, t) and h_{2}(τ, t), hence Δτ_{max} is zeros. But it is obvious that both responses are not equivalent, in this case we do not know for how much they deviate from each other using the information provided by Δτ_{max} = 0. On the other hand Δτ_{ rms } ≠ 0 hence, some value of the statistical deviation of h_{1}(τ, t) from h_{2}(τ, t) is readily available as it is truly the case in this example. For instance, the employment of multipath diversity means that h_{1}(τ, t) provides better channel diversity than h_{2}(τ, t) since it offers more fingers.
6.3. Stationarity distance
An important issue in the characterization of the LSSUS is how to determine the interval T_{(v)}over which WSSUS is defined. This interval depends basically on the spatial displacement \Delta \stackrel{\u20d7}{r} and terminal velocity v. Let the displacement at a time instant t be given by \stackrel{\u20d7}{r}\left({t}^{\prime}\right)\stackrel{\u20d7}{r}\left(t\right)=\Delta \stackrel{\u20d7}{r}, where t' is some instantaneous time. For a terminal velocity v, \Delta \stackrel{\u20d7}{r}=J.v. The WSSUS assumption requires that \Delta \stackrel{\u20d7}{r} must be small enough to ensure statistical stationarity. Succinctly, J = t't should not exceed some time interval J_{(v)}= t't, where t_{ v } ' is a time instant within which statistical stationarity is guaranteed. In order to determine J_{(v)}, consider the fact that the channel response to any excitation can be seen as a 'snapshot' of the channel. To adequately account for changes in the channel, these snapshots need to be taken sufficiently often at some intervals called the repetition duration T_{rep}. The repetition duration encompasses the interval J, and some time t_{ps} for data processing and storage. Intuitively, J_{(v)}should be smaller than the time duration T_{Δ} over which the channel changes, J_{(v)}< T_{Δ}. There exists a minimum sampling rate required to be able to identify a timevariant process with a bandlimited spectrum. The temporal sampling frequency must be twice the maximum frequency shift υ_{max}[12]. Thus, we can express the interval J_{(v)}as
where v_{max} is the maximum constant velocity of the mobile unit, f_{up} is the upper band frequency, and χ is arbitrarily chosen to ensure that a reasonable measurement/sampling distance is obtained. The value of χ must be carefully chosen to ensure that the J_{(v)}is not too small (to ensure reasonable acquisition time) nor too big (as to violate stationarity condition). The corresponding stationarity distance \Delta \stackrel{\u20d7}{r}={X}_{s} is given by
7. Application relevance of LSSUS assumption
The practical illustrations of the LSSUS concept that stem directly from physical considerations in typical communication scenarios are presented in this section. In essence, the LSSUS scattering functions can be viewed as a set of evolutionary functions that are more or less the instantaneous responses of the channel to an input. Although the coherence parameters of these instantaneous channel realizations vary from one to another, for practical rationality we consider channel coherency only with respect to the reference channels response taken to be WSSSU. Hence, all other sets of \mathrm{\Psi}=\left\{{\mathit{P}}_{LSSUS}{\left(\tau ,s\right)}_{\theta}:\theta \in \Theta \subset \Re \right\} are defined only by the stationarity parameters.
While statistical stationarity are desired in order to achieve simple transceiver designs, the knowledge of statistical nonstationarity can also be employed to improve the performance of the transceiver. For instance, nonstationarity information can give insight into the longterm behavior of the system in terms of channel estimation/prediction, ergodic capacity, and diversity. Hence, the joint knowledge of stationarity and coherence parameters can be used to access the longterm behavior of the channel and adjust the transceiver parameters. In [2], brief discussion on the relevance of nonWSSUS characterization on ergodic capacity assumption was presented, and further application potential to delayDoppler diversity was mentioned. In this study, we provide broadbased formulation of the application relevance of LSSUS to ergodic channel capacity assumption, time diversity, frequency diversity, joint timefrequency diversity, and delayDoppler/delayscale diversity.
Let us consider the case of the doubly spread flatfading channel with timefrequency coherence/stationarity subspace as shown in Figure 4. If we assumed that independent and identically distributed (i.i.d) channel realizations occur every T_{ c } second for the corresponding B_{ c } , then averaging over T_{ s } = NT_{ c } , B_{ s } = KB_{ c } , {n, k} = 1, 2, 3.., {N, K}, {N, K} → ∞ gives a convergent value. The number of i.i.d channel realizations ℵ can be given by [2]
Thus, the stationarity dimension T_{ s } × B_{ s } determines the number of effective i.i.ds available for a given channel with respect to the coherence dimension T_{ c } × B_{ c } . We will show that the value of ℵ determines the validity of ergodic assumptions. And it also affects the effective timefrequency diversity order of a given timevarying channel.
7.1. Ergodic capacity
The ergodic capacity C _{ erg } of the channel is often desired in order to reveal the 'longterm' properties of an arbitrary fading process say χ(t, τ) which is assumed to be an ergodic process in t. It is well known that in order to achieve ergodic capacity, averaging by coding over numerous i.i.d. fades is required [30]. Thus, whether sufficient averaging can be achieved to guarantee ergodic capacity depends on the number of i.i.d fading coefficients offered by the channel. The channel capacity C in the case where the channel state information is available to the receiver can be expressed as [30]
where P_{ av } is the average power, N_{0} is the noise variance, and χ(t, f) is the response of the channel, and by analogy χ(t, f) ≡ W(τ, s) in the timescale domain.
Under the ergodic assumption, the statistics of χ are independent of either t or f in χ(t, f) and subsequently τ or s in W(τ, s). If P_{av} is nonnegative, then the relation
holds.
As defined by the LSSUS statistics, the time and scale independency of the statistics of W(τ, s) are accessed over the stationarity dimension. Therefore, for flatfading, the ℵ dependent ergodic capacity can then be given by
where {\mathit{q}}_{\aleph}=\frac{{P}_{av}.\phantom{\rule{2.77695pt}{0ex}}\left[{\left{\chi}_{\aleph}\left(t,f\right)\right}^{2}\right]}{{N}_{0}}, with probability distribution {p}_{\wp}\left(\mathit{q}\right).
The expression (26) implies that when B_{ s }T_{ s } → ∞ (WSSUS case), the number of independent realizations is large enough and ergodic capacity is achieved. However, as T_{ s } decreases the number of i.i.ds reduces and the ergodic capacity can only be defined for sufficiently large realizations. In the case where there is insufficient number of i.i.ds, the use of C_{erg} as a measure of channel capacity becomes unreliable.
7.2. Effective diversity
Diversity techniques are often employed to combat the fading effect of the channel. The basic idea is to transmit the signal over multiple i.i.d channels, while keeping the total power constant by transmitting at a lower power in each channel [31]. It is evident that diversity performance improves monotonically with increasing number of i.i.d [32]. In fact as the number of i.i.d ℵ approaches infinity, the performance of coherent diversity reception converges to the performance over a nonfading AWGN channel [33–37]. In practice, diversity is physically implemented in a variety of ways such as time diversity, frequency diversity, joint timefrequency diversity, delay diversity, Doppler diversity, scale diversity, joint delayDoppler diversity, and joint delayscale diversity. In the case of time, frequency, and joint timefrequency diversities, decoupling the stationarity dimension and (23) gives the number of effective diversity order d_{.} as
Thus, as the stationarity dimension changes (by virtue of the variation in the degree of the correlation among channel realizations at different time instants), the diversity order varies too. Hence, the stationarity dimension sets an upper limit for the above diversity schemes.
The above relationships can be extended to the case of delayDoppler diversity [32] and that of the delayscale diversity [8, 9]. In order to do this, it should be noted that timefrequency/scale coherence dimension is inversely related to the delayDoppler/scale diversity in the doubly spread channel. Let us consider the case of delayDoppler diversity [32]. The number of delay diversity and Doppler diversity branches depends on the maximum delay spread and maximum Doppler spread, respectively. Since these parameters vary by virtue of LSSUS assumption, it means that the achievable effective delay/Doppler diversity should also vary with time. We restrict our analysis in this article to delayDoppler diversity.
8. Illustrative measurements and simulation
In this section, illustrative measurement and simulation examples for the timevarying UWB channel typical of the infostation [15] environments are provided. This environment depicts a typical scenario where the timescale domain LSSUS channel suffices. The concept of infostation [15–17] illustrated in Figure 5 presents a new way to look at the problem of providing high data rate wireless access. It is an isolated pocket area with small coverage (about 100 m) of high bandwidth connectivity that collects information requests from mobile users and delivers data while users are going through the coverage area. One of the technologies that have the potential to deliver the envisaged highdata rate infostation services is the UWB signalling [17].
Most existing UWB channel characterization and measurement have been limited to the case where the channel is assumed to be fixed over the transmission duration [38–41]. However, for many infostation channels, time variation is expected due to the mobility of one of the communication terminals. In this case, the WSSUS assumption and the assumption of uniform Doppler shift across the operating bandwidth all composite frequencies are no more valid. In the illustrative measurement and simulation below, we consider the LSSUS analysis with respect to both time and frequency dispersive effects.
8.1. Illustrative measurement
The complex channel response is measured with a vector analyzer (VNA) R&S^{®} ZVL13. Measurements were carried out at various locations along a road within the vicinity of Wireless Communication Centre (WCC) complex, Universiti Teknologi Malaysia, as shown in Figure 6 for the frequency range 3.13.6 GHz. The speed of the mobile is about 2 m/s, and measurements were taken at each location marked A_{1}A_{ 6 } . At each location, the measurement is repeated 50 times. The VNA records the variation of 601 complex tones within the band. This recording is done by sweeping the spectrum in about J_{ v } time interval. The time J_{ v } is obtained from (21) where χ is taken to be 20. Apart from the mobile antenna, all the objects (potential scatterers) are kept stationary throughout the duration of the measurement. The antennas (monopoles) are of the same height, 1.5 m and the transmit power is 10 dB for all measurements. The signal from the receiving antenna is passed through a low noise amplifier with a gain of 20 dB. The distances between the locations are A_{1}A_{2} = 1 m, A_{2}A_{3} = 1 m, A_{3}A_{4} = 4 m, A_{4}A_{5} = 3 m, and A_{5}A_{6} = 3 m. The distances between the infostation antenna and the mobile antenna at locations A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, and A_{6} are 6, 5.7, 5.4, 5, 5.5, and 6 m, respectively.
In order to obtain the statistical model of the timevariant response from the measured complex channel responses, we apply the autoregressive (AR) model proposed in [42]. Let H(f_{ p } , t; A) be the timevarying complex transfer function measured at a location A and time t. Then the first and secondorder statistics of the measured channel are captured by the model
where V(f_{ p } ) is a complex white noise process and a_{ n } is the function representing the n th timevarying AR coefficient. The MDPD for the positions A_{2}, A_{3}, A_{4}, A_{5}, and A_{6} (with reference to A_{1}) are shown in Figure 7.
The computed B_{ s } for the positions A_{2}, A_{3}, A_{4}, A_{5}, and A_{6} are approximately 12, 3.32, 0.56, 0.77, and 0.29 GHz, respectively. These B_{ s } values are computed at 30 dB threshold. For this same threshold B_{ c } value of 23.2 MHz is obtained using the WSSUS scattering function (9) at reference position A_{1}. The implication of the ratio B_{ s } /B_{ c } can be observed in the case of multiband orthogonal frequency division multiplex (MBOFDM) UWB in this channel. If we consider the MBOFDM system designed with N_{ s } number of subcarriers and subcarrier spacing of F_{ s } MHz. In order to combat fading in MBOFDM, the bandwidths of the subcarriers should be equal or less than the B_{ c } to ensure flatfading. However, the choice of small value for F_{ s } implies that the system will be more susceptible to intercarrier interference (ICI) [43]. Hence, the choice of the value of F_{ s } should be optimal between combating frequency selective fading and ICI. If we consider a total bandwidth of 528 MHz, the value of F_{ s } for 128, 64, 32, and 16 subcarriers are 4.125, 8.25, 16.5, and 33 MHz, respectively. Hence, the choice of 128 subcarrier ensures good ISI performance but with increased error due to ICI, and the choice of 16 subcarriers ensures good ICI performance but with increased susceptibility to ICI. The optimal choice will be to choose the number of subcarriers such that kF_{ s } = B_{ c } , where the value of k should be chosen to take care of the timevarying nature of B_{ c } . Conventionally, k is chosen to be fixed and of low value, hence a fixed number of subcarriers. The signaltonoise ratio (SNR) degradation caused by ICI is given by [44]
where f_{ e } is the frequency offset. If we consider the above channel measurement, then the SNR degradations for 128, 64, 32, and 22 subcarriers are shown in Figure 8.
Figure 8 shows that the SNR degradation for 22 subcarriers (k = 1) is better than the performance at k < 1 (128, 64, and 32 subcarriers). However, when k is greater than unity, ISI degradation sets in. This implies that instead of a fixed value for k, some form of adaptive subcarrier bandwidth can be employed. The value of stationarity bandwidth can provide information that can be used to adjust k for optimal performance. The values of B_{ s } /B_{ c } at A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, and A_{6} are approximately, ∞, 538, 143, 23, 33, and 12, respectively. We can define the time/bandwidth utilization parameter {U}^{\left(\Im \right)} by
where {\Im}_{TF}=\left({B}_{s}{T}_{s}\right)\u2215\left({B}_{c}{T}_{c}\right), {\Im}_{F}={B}_{s}\u2215{B}_{c}, {\Im}_{T}={T}_{s}\u2215{T}_{c} and B_{ co } is the reference coherence bandwidth. Hence, the values of {U}^{\left({\Im}_{F}\right)} at A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, and A_{6} are approximately, 0, 0.19, 0.7, 4.6, 3.13, and 9.1%, respectively. The negative sign in {U}^{\left({\Im}_{F}\right)} indicates that k is greater than 1 by the given percentage and the positive sign in {U}^{\left({\Im}_{F}\right)} indicates that k is less than 1 by the given percentage. Therefore, this nonstationarity information can be employed to adjust the values of k in order to optimize the system performance at any time instant.
Since the velocity of the mobile is constant throughout the measurement run, and LOS propagation exists for all measurements, the coherence time at all measurement with reference to s_{max} is approximately constant. Hence, the values of T_{ s } /T_{ c } at A_{2}, A_{3}, A_{4}, A_{5}, and A_{6} are infinity as long as constant velocity is maintained and LOS propagation exists. In this particular case, the achievable i.i.d given by ℵ is large enough to validate the assumption of ergodic capacity. Also, large time diversity and joint timefrequency diversity gains are obtainable.
8.2. Illustrative UWB channel simulation
Let us consider a typical UWB highway infostation propagation channel as shown in Figure 9. The UWB operating frequency band is 3.13.6 GHz and the signaling waveform is the Mexican hat wavelet mathematically expressed by {e}_{I}\left(t\right)=\left(1{t}^{2}\right){e}^{{t}^{2}}. We assume that the power P_{ T } and duration T_{pluse} of this function is about 100 mW and 10 ns, respectively.
The geometricalbased single bounce elliptical model [45] is employed in the simulation of this channel. The values of the major axis halflength a_{max} and minor axis halflength b_{max} are 33 and 12 m, respectively. The number of potential scatterers is about 5000 and their positions are assumed fixed over the duration of the simulations. The initial distance D_{0} between the infostation I_{0} and the car C_{0} is 25 m, and the initial speed of the car is 10 m/s. The gains of the infostation and car antennas are taken to be unity. Let us assume that the car is moving from R_{1} through points R_{2}, R_{3}, R_{4}, and R_{5} (scenario A) and from R_{1} through points R_{2}, R_{3}, U_{4}, and U_{5} (scenario B) as shown in Figure 9.
Scenarios A and B are typical cases that involve LOS and nonLOS propagation. The LSSUS concept can directly be related to the physical propagation scenario in Figure 6 and to the various phases of the mobile's movement.
Scenario A: The distances of I_{0} from R_{1}, R_{2}, R_{3}, R_{4}, and R_{5} are approximately 25, 23, 20, 16, and 12 m, respectively. And the instantaneous velocities at R_{1}, R_{2}, R_{3}, R_{4}, and R_{5} are 10, 10, 5, 3, and 5 m/s, respectively. The LSSUS scattering functions at points R_{1} to R_{1} are shown in Figure 10. The values of B_{ c } and T_{ c } at the reference point R_{1} (WSSUS) are 11 MHz and 1.67 ms, respectively. The stationarity parameters for points are shown in Table 1. In Figure 10, the change in the scattering function as the mobile move through the points can be clearly observed, and the LOS propagation path is obvious at all points.
Table 1 shows the computed stationarity parameters and effective diversity order for the positions R_{1}, R_{2}, R_{3}, R_{4}, and R_{5}. In this particular case, the validity of the assumption of ergodic capacity is appropriate mostly at R_{1}, R_{2}, and R_{4}. The effective time, frequency, and joint timefrequency diversities obtainable at R_{1}, R_{2}, R_{3}, R_{4}, and R_{5} are shown in Figure 11.
The diversity gains are obtained for the case of quadrature phaseshift keying (QPSK). The variation of diversity order with respect to the points R_{1}, R_{2}, R_{3}, R_{4}, and R_{5} suggests that the use of some adaptive method can improve the performance of the wideband system. The values of {U}^{\left({\Im}_{F}\right)} at R_{1}, R_{2}, R_{3}, R_{4}, and A_{5} are approximately, 0, 1.25, 2, 0.55, and 1.21%, respectively. And using the same argument discussed in Section 8.1, this nonstationarity information can be employed to adjust the values of k in order to optimize the system performance at any time instant. The values of {U}^{\left({\Im}_{T}\right)} with respect to time can also be computed using (26).
Scenario B: The distances of I_{0} from R_{1}, R_{2}, R_{3}, R_{4}, and R_{5} are approximately 25, 23, 20, 18, and 19 m, respectively. And the instantaneous velocities at R_{1}, R_{2}, R_{3}, R_{4}, and R_{5} are 10, 10, 5, 3, and 5 m/s. The LSSUS scattering functions at points R_{1} to R_{1} are shown in Figure 12. The values of B_{ c } and T_{ c } at the reference point R_{1} (WSSUS) are 11 MHz and 1.96 ms, respectively. The computed stationarity parameters and effective diversity order for the positions R_{1}, R_{2}, R_{3}, U_{4}, and U_{5} are shown in Table 2.
As can be inferred from Table 2, the validity of the assumption of ergodic capacity may not be appropriate since the values of ℵ at the different points may not be enough to ensure longterm averaging. The diversity gains obtained for the case of QPSK at the points R_{1}, R_{2}, R_{3}, R_{4}, and R_{5} are shown in Figure 13. These figures also provide information about the nonstationarity of the channel and can be employed in providing some form of adaptation to diversity processes in order to improve the performance of the system. The values of {U}^{\left({\Im}_{F}\right)} at R_{1}, R_{2}, R_{3}, U_{4}, and U_{5} are approximately, 0, 1.25, 2, 9.1, and 5.9%, respectively. Hence, bandwidth utilization can be very low in the nonLOS scenario compared to the LOS case.
8.3. Illustrative underwater channel simulation
Let us consider the following example for a single carrier underwater communication between a transmitter Tx submerged at a fixed depth of h = 40 m, and a mobile receiver Rx at the same depth as shown in Figure 14. We consider signaling using the Mexican wavelet. The operational bandwidth B is 10 kHz (100 Hz10.1 kHz). Let us assume that Rx is moving at a constant speed of 5 m/s from a point A through B, C, D, to E, where all points are at the same depth h. The temperature of the volume is taken to be constant over the simulation period. For a negligible wind speed, we also assume that the water volume is isotropic, and the floor is smooth and nonabsorptive. Hence, the angle of incidence can be assumed to be equal to the angle of reflection.
Let the time of travel from A to E be sampled at the rate of the valid stationary interval J_{(v)}≈ c/2f_{ ref } , where the distance vector X is [X_{ A }X_{ B } X_{ C } X_{ D } X_{ E } ] = [1 5 10 15 25] × J_{(v)}. To be on a safe side, we assume f_{ref} = 11.1 kHz. For the water depth of H = 100 m and the initial TxRx distance of 50 m, the time correlation ρ = J_{0}(2πυ_{max, X}Δt_{ sym } ) at the different points A to E with respect to the T_{ c } obtained at J_{(v)}is shown in Figure 15. The term J_{0}(·)is the zeroorder Bessel function of the first kind.
The term Δt_{sym} is a vector of length {T}_{c,{J}_{\left(v\right)}}\u2215{T}_{sym} with a step size of T_{sym} (symbol duration). The symbols υ_{max}, _{ X }and {T}_{c,{J}_{\left(v\right)}} represent the maximum frequency spread at a particular point A to E and the coherence time at the interval J_{(v)}, respectively.
In Figure 15, we can see that the effect of channel variation within the transmission duration denoted by {T}_{c,{J}_{v}} can be assumed to be negligible up to the point marked 'Ω'. This implies that the transmission performance of this system using fixed frame size or assuming channel invariance may degrade with time.
To access the available number of effective diversity branches and the validity of assuming an ergodic capacity for this channel, we consider Table 3.
In Table 3, we can see the trend of the variation in the stationarity parameter as Tx moves. The number of available independent fade channels is given by ℵ. Thus ergodic capacity can be assumed up to the point for which ℵ is large enough to average out both the AWGN and the channel fluctuations. When ℵ is not large enough, the outage capacity will be a preferred measure of capacity.
The parameter ℵ also provides the number of diversity branches for timefrequency diversity (different from delayDoppler diversity) technique. The values ℵ _{ T } and ℵ _{ F } provide the available number of time and frequency diversity branches, respectively. Let us consider the influence of the assumption of nonWSSUS on time, frequency, Doppler, and delay diversities.
In order to achieve time diversity, a codeword is ideally separated by {T}_{c,{J}_{\left(v\right)}}. For a frame length L={T}_{c,{J}_{\left(v\right)}}\times {\aleph}_{T}, it is required that {T}_{c,{J}_{\left(v\right)}} should be approximately constant over the segments of L. This supposition stems from the WSSUS assumption. In this case we can say that with respect to the {T}_{c,{J}_{\left(v\right)}} (without update), the number of diversity branches is fixed. On the other hand, if {T}_{c,{J}_{\left(v\right)}} varies across L, then correlation among the initially independent fade segments sets in, thus affecting diversity gain. For low SNR which is typical of the covert UWA communications, small variation in diversity order can be meaningful. Hence, when the variation in {T}_{c,{J}_{\left(v\right)}} becomes quite significant, update on the {T}_{c,{J}_{\left(v\right)}} is required at the transmitter.
We can consider an effective stationarity time T_{ s, eff } ≤ T_{ s } within which {T}_{c,{J}_{v}} is constant as being one in which the ratio ℵ _{ T } at A to ℵ _{ T } at AE is for instance, 10:\wp where \wp \in {\Re}^{+} is chosen nontrivially. Hence, when the variation in {T}_{c,{J}_{v}} becomes quite significant, an update on the {T}_{c,{J}_{\left(v\right)}} is required at the transmitter. This may of course increase or decrease the number of the available diversity branches depending on whether the difference between the initial {T}_{c,{J}_{\left(v\right)}} and the updated one has a positive or negative value. Of course, when ℵ _{ T } = 0, the application of time diversity is of no obvious advantage. This situation occurs when {T}_{c,{J}_{\left(v\right)}}=\infty \left({\upsilon}_{max}=0,s=1\right) or when the channel variation is so rapid that gross nonstationarity of the process has to be considered. In such situation a different diversity technique may be applied. Hence, the LSSUS parameters can be used to track channel variation and ensure that the benefit of time diversity is optimally obtained as shown in Figure 16.
In the case of frequency diversity, a codeword is sent over different frequencies separated by {B}_{c,{J}_{\left(v\right)}}. The same argument made for the time diversity can also be applied to frequency diversity in which case the effective stationarity bandwidth B_{ s, eff } ≤ B_{ s } defines the frequency segment over which {B}_{c,{J}_{v}} is constant. Of course the value of \wp in time diversity may not necessarily be the same in the case of frequency diversity.
The LSSUS argument can be extended to the delay (multipath) diversity, Doppler diversity, delayDoppler diversity, and the delayscale diversity. To do this, the duality in terms of time/Doppler spread and frequency/delay spread has to be taken into account. The number of delay diversity N and Doppler diversity Q branches are given by [τ_{max}B] and [2υ_{max}T_{ sym } ], respectively [32]. We note that these expressions of the number of diversity branches are made bearing in mind the assumption of WSSUS. However, it should be noted that both τ_{max} and υ_{max} varies with time. Hence, the variation can appropriately be taken into account using the LSSUS concept where N(t) = [τ_{max}(t)B] and Q\left(t\right)=2\u2308{\upsilon}_{max}\left(t\right)\phantom{\rule{2.77695pt}{0ex}}{T}_{sym}\u2309. We can also use the same argument above to determine how long τ_{max} and υ_{max} can be considered approximately invariant (WSSUS).
9. Conclusion
We presented the timescale domain characterization of the timevarying wideband propagation channel using the concept of LSSUS which emphasized on the nonstationary properties of the channel. The channel characterization in timescale domain provides the leverage of carrier frequencyindependent computation of channel responses. The statistical assumption termed the LSSUS was also presented and employed in order to evaluate and quantify the degree of nonstationarity of the wideband channel. The LSSUS channel parameters were obtained. By the way of measurement and simulation, these channel parameters were employed in order to analyze the performance of the real and synthesized wideband channels in terms of diversity and channel capacity. Results show that as the assumption of WSSUS becomes violated, the assumption of ergodic capacity and its application becomes unreliable. More also, the gain of the effective diversity varies with the degree of channel stationarity/nonstationarity for different techniques like time, frequency, delay, Doppler and joint delayDoppler diversities. Hence, it is obvious that the optimal performance of a communication system can be obtained where the instantaneous channel condition is considered. Since the effective diversity gain and channel capacity assumptions depend on the degree of stationarity/nonstationarity, it is therefore necessary to consider some form of adaptive methods for choosing a particular diversity technique or/and channel capacity type. Hence, wideband communication systems that incorporate algorithms based on the LSSUS concept will greatly improve the performance of such systems.
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Acknowledgements
The authors thank the Ministry of Higher Education (MOHE), Malaysia, for providing financial support for this study through the Grants (4D040 and Q.J130000.7123.02H31) managed by the Research Management Center (RMC), Universiti Teknologi Malaysia (UTM). We also thank the reviewers of this manuscript for their constructive remarks.
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ChudeOkonkwo, U.A., Ngah, R. & Abd Rahman, T. Timescale domain characterization of nonWSSUS wideband channels. EURASIP J. Adv. Signal Process. 2011, 123 (2011). https://doi.org/10.1186/168761802011123
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DOI: https://doi.org/10.1186/168761802011123
Keywords
 timescale domain
 nonWSSUS
 wideband
 diversity techniques
 ergodic capacity