Timescale domain characterization of nonWSSUS wideband channels
 Uche AK ChudeOkonkwo^{1}Email author,
 Razali Ngah^{1} and
 Tharek Abd Rahman^{1}
https://doi.org/10.1186/168761802011123
© ChudeOkonkwo et al; licensee Springer. 2011
Received: 18 May 2011
Accepted: 7 December 2011
Published: 7 December 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.
Keywords
timescale domain nonWSSUS wideband diversity techniques ergodic capacity1 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
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
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 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.
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 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.
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.
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 ${X}^{\left(\tau ,s\right)}=a\left(t\right)x\left(\frac{t\tau}{s}\right)$.
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
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.
where ${\mathrm{\Xi}}_{\tau ,wssus}^{2}={\left(\right)close="">\underset{{\mathit{P}}_{wssus}}{min}\iint {\left({\mathit{P}}_{LSSUS}\left(\tau ,s\right){\mathit{P}}_{wssus}\left(\tau ,s\right)\right)}^{2}\phantom{\rule{2.77695pt}{0ex}}\frac{ds}{{s}^{2}}}_{}{\mathit{P}}_{LSSUS}\left(\tau ,s\right)=0$ and ${\mathrm{\Xi}}_{\tau ,LSSUS}^{2}={\left(\right)close="">\iint {\left({\mathit{P}}_{LSSUS}\left(\tau ,s\right){\mathit{P}}_{wssus}\left(\tau ,s\right)\right)}^{2}\phantom{\rule{2.77695pt}{0ex}}\frac{ds}{{s}^{2}}}_{}{WSSUS}_{}\left(\tau ,s\right)=0$
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.
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
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.
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
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.
holds.
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
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
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
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
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.
Stationarity parameters and effective diversity order for Scenario A
R _{1}  R _{2}  R _{3}  R _{4}  R _{5}  

B_{ s } (GHz)  ∞  0.845  0.544  1.9  0.878 
T_{ s } (ms)  ∞  ∞  8.33  5.95  8.33 
d_{ TFD } = ℵ  ∞  ∞  521  1302  842 
d_{ FD }  ∞  81  51  182  84 
d_{ TD }  ∞  ∞  10  7  10 
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).
Stationarity Parameters and effective diversity order for Scenario B
R _{1}  R _{2}  R _{3}  U _{4}  U _{5}  

B_{ s } (GHz)  ∞  0.845  0.544  0.124  0.185 
T_{ s } (ms)  ∞  ∞  8.33  5.918  8.265 
d_{ TFD } = ℵ  ∞  ∞  521  84  179 
d_{ FD }  ∞  81  51  12  18 
d_{ TD }  ∞  ∞  10  7  9 
8.3. Illustrative underwater channel simulation
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.
Channel stationarity parameters for the UWA channel
X _{.}  B_{ s }(kHz)  T_{ s }(ms)  d_{ TF } = ℵ  d _{ F }   d _{ T }  

A  ∞  ∞  ∞  ∞  ∞ 
B  5.254  2048  107236  769  139 
C  1.791  683  12193  262  46 
D  0.926  342  3156  135  23 
E  0.535  187  996  78  12 
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.
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.
Declarations
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.
Authors’ Affiliations
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