In this section, eight classes of SOC simulation models for frequency-nonselective Rayleigh fading channels will be introduced, seven of which are stochastic simulation models and one is completely deterministic. Before presenting a detailed analysis of the various classes of simulation models, some constraints are imposed on the three types of parameters. These constraints are as follows.

In cases, where the gains *c*_{
n
}, frequencies *f*_{
n
}, and phases *θ*_{
n
} are random variables, it is reasonable to assume that their values are real and they are mutually independent. Also, it is assumed that the gains {\mathit{c}}_{1},{\mathit{c}}_{2},\dots ,{\mathit{c}}_{N} are i.i.d. random variables. The same is assumed for the sequences of random Doppler frequencies {\mathit{f}}_{1},{\mathit{f}}_{2},\dots ,{\mathit{f}}_{N} and phases {\mathit{\theta}}_{1},{\mathit{\theta}}_{2},\dots ,{\mathit{\theta}}_{N}.

Whenever the gains *c*_{
n
} and frequencies *f*_{
n
} are constant quantities, it is assumed that they are different from 0, such that *c*_{
n
}≠0 and *f*_{
n
}≠0 hold for all values of n=1,2,\dots ,N. Further constraints might also be imposed on the SOC model. For example, it is required that all frequencies *f*_{
n
} are different. This latter condition is introduced to avoid correlations within the inphase (quadrature) component of \stackrel{~}{\mu}\left(t\right).

### 6.1 Class I channel simulators

The channel simulators of Class I are defined by a set of deterministic processes \stackrel{~}{\mu}\left(t\right) (see (5)) with constant gains *c*_{
n
}, constant frequencies *f*_{
n
}, and constant phases *θ*_{
n
}. Since all model parameters are constants, there is no meaning to examine the stationary and ergodic properties of this class of channel simulators. However, in the following, the mean and ACF of \stackrel{~}{\mu}\left(t\right) will be investigated.

Since \stackrel{~}{\mu}\left(t\right) is a deterministic process, its mean {m}_{\stackrel{~}{\mu}} has to be determined using time averages instead of statistical averages. Using (5) and taking into account that *f*_{
n
}≠0, we obtain {m}_{\stackrel{~}{\mu}}=0. Hence, we realize that the deterministic process \stackrel{~}{\mu}\left(t\right) of the simulation model has the same mean value as the stochastic process *μ*(*t*) of the reference model, i.e.,

{m}_{\stackrel{~}{\mu}}={m}_{\mu}=0\phantom{\rule{0.3em}{0ex}}.

(14)

Similarly, the ACF {r}_{\stackrel{~}{\mu}\stackrel{~}{\mu}}\left(\tau \right) of \stackrel{~}{\mu}\left(t\right) has to be determined using time averages. By taking into account that all frequencies *f*_{
n
} are different, we can express the ACF {r}_{\stackrel{~}{\mu}\stackrel{~}{\mu}}\left(\tau \right) of \stackrel{~}{\mu}\left(t\right) in the following form

{r}_{\stackrel{~}{\mu}\stackrel{~}{\mu}}\left(\tau \right)=\sum _{n=1}^{N}{c}_{n}^{2}e{}^{j2\pi {f}_{n}\tau}\phantom{\rule{0.3em}{0ex}}.

(15)

The mean {m}_{\stackrel{~}{\mu}} and the ACF {r}_{\stackrel{~}{\mu}\stackrel{~}{\mu}}\left(\tau \right) will be intensively used in the next subsections.

### 6.2 Class II channel simulators

The channel simulators of Class II are defined by a set of stochastic processes \widehat{\mathit{\zeta}}\left(t\right) with constant gains *c*_{
n
}, constant frequencies *f*_{
n
}, and random phases *θ*_{
n
}, which are uniformly distributed in the interval (0,2*π*]. Using our notation, the complex process \widehat{\mathit{\mu}}\left(t\right)={\widehat{\mathit{\mu}}}_{1}\left(t\right)+j{\widehat{\mathit{\mu}}}_{2}\left(t\right) can be written as

\widehat{\mathit{\mu}}\left(t\right)=\sum _{n=1}^{N}{c}_{n}e{}^{j(2\pi {f}_{n}t+{\mathit{\theta}}_{n})}\phantom{\rule{0.3em}{0ex}}.

(16)

In [22], it has been shown that the density {p}_{\widehat{\zeta}}\left(z\right) of the stochastic process \widehat{\mathit{\zeta}}\left(t\right)=\left|\widehat{\mathit{\mu}}\right(t\left)\right| can be represented as

{p}_{\widehat{\zeta}}\left(z\right)={\left(2\pi \right)}^{2}z\underset{0}{\overset{\infty}{\int}}\left[\phantom{\rule{0.3em}{0ex}}\prod _{n=1}^{N}{J}_{0}\left(2\pi \right|{c}_{n}\left|y\right)\right]{J}_{0}\left(2\mathrm{\pi zy}\right)y\phantom{\rule{0.3em}{0ex}}\mathit{\text{dy}}\phantom{\rule{0.3em}{0ex}}.

(17)

Also, in [22], it has been shown that the density {p}_{\widehat{\zeta}}\left(z\right) approaches the Rayleigh density as *N*→*∞* and {c}_{n}={\sigma}_{0}\sqrt{2/N}.

The mean {m}_{\widehat{\mu}} and the ACF {r}_{\widehat{\mu}\widehat{\mu}}\left(\tau \right) of \widehat{\mathit{\mu}}\left(t\right) can be easily expressed as

\begin{array}{ll}{m}_{\widehat{\mu}}& =0,\phantom{\rule{2em}{0ex}}\end{array}

(18)

\begin{array}{ll}{r}_{\widehat{\mu}\widehat{\mu}}\left(\tau \right)& =\sum _{n=1}^{N}{c}_{n}^{2}e{}^{j2\pi {f}_{n}\tau}\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{2em}{0ex}}\end{array}

(19)

With reference to (17), we notice that the density {p}_{\widehat{\zeta}}\left(z\right) is independent of time. Furthermore, from (18) and (19), we can realize that conditions 1 and 2 (see (8) and (9)) are fulfilled, respectively. Hence, the stochastic process \widehat{\mathit{\zeta}}\left(t\right) is FOS, and the stochastic process \widehat{\mathit{\mu}}\left(t\right) is WSS. A specific realization of the random phases *θ*_{
n
} converts the stochastic process \widehat{\mathit{\mu}}\left(t\right) into a deterministic process (sample function) \stackrel{~}{\mu}\left(t\right). This allows us to interpret Class I as a subset of Class II. The stochastic process \widehat{\mathit{\mu}}\left(t\right) is ME since the identity {m}_{\widehat{\mu}}={m}_{\stackrel{~}{\mu}} holds. Also, the stochastic process \widehat{\mathit{\mu}}\left(t\right) is AE. This statement follows from a comparison of (19) and (15), which reveals that the criterion {r}_{\widehat{\mu}\widehat{\mu}}\left(\tau \right)={r}_{\stackrel{~}{\mu}\stackrel{~}{\mu}}\left(\tau \right) is fulfilled.

### 6.3 Class III channel simulators

The channel simulators of Class III are defined by a set of stochastic processes \widehat{\mathit{\zeta}}\left(t\right) with constant gains *c*_{
n
}, random frequencies *f*_{
n
}, and constant phases *θ*_{
n
}. Hence, the complex process \widehat{\mathit{\mu}}\left(t\right)={\widehat{\mathit{\mu}}}_{1}\left(t\right)+j{\widehat{\mathit{\mu}}}_{2}\left(t\right) has the following form

\widehat{\mathit{\mu}}\left(t\right)=\sum _{n=1}^{N}{c}_{n}e{}^{j(2\pi {\mathit{f}}_{n}t+{\theta}_{n})}\phantom{\rule{0.3em}{0ex}}.

(20)

To obtain the density {p}_{\widehat{\zeta}}(z;t) of the stochastic process \widehat{\mathit{\zeta}}\left(t\right), we continue as follows. In the first step, we consider a single complex cisoid at a fixed time instant *t*=*t*_{0}. Thus,

{\widehat{\mathit{\mu}}}_{n}\left({t}_{0}\right)={c}_{n}e{}^{j(2\pi {\mathit{f}}_{n}{t}_{0}+{\theta}_{n})}

(21)

describes a complex random variable. From [29], in the limit |*t*_{0}|→*∞*, the density {p}_{{\widehat{\mu}}_{1,n}}\left({x}_{1}\right) of {\widehat{\mathit{\mu}}}_{1,n}\left({t}_{0}\right)=\text{Re}\left\{{\widehat{\mathit{\mu}}}_{n}\right({t}_{0}\left)\right\}={c}_{n}cos(2\pi {\mathit{f}}_{n}{t}_{0}+{\theta}_{n}) can be expressed as

{p}_{{\widehat{\mu}}_{1,n}}\left({x}_{1}\right)=\left\{\begin{array}{ll}\frac{1}{\pi \left|{c}_{n}\right|\sqrt{1-{({x}_{1}/{c}_{n})}^{2}}},& \left|{x}_{1}\right|<{c}_{n}\\ 0,& \left|{x}_{1}\right|\ge {c}_{n}\phantom{\rule{0.3em}{0ex}}.\end{array}\right.

(22)

Now, by following the same procedure as described in [22], we can conclude that the density {p}_{\widehat{\zeta}}(z;\pm \infty ) is given by (17). If *t* is finite, we must replace the expression in (22) by ([29], Eq. (32)). In this case, it turns out that the density {p}_{\widehat{\zeta}}(z;t) depends on time *t* so that the stochastic process \widehat{\mathit{\zeta}}\left(t\right) is not FOS. Thus, we can conclude that a Class III channel simulator is an asymptotically FOS process.

In the following, we assume that the random frequencies *f*_{
n
} are given by

{\mathit{f}}_{n}={f}_{max}cos\left({\mathit{\alpha}}_{n}\right)

(23)

where the AOAs *α*_{
n
}, n=1,\dots ,N, are i.i.d. random variables, each having a density *p*_{
α
}(*α*) identical to that characterizing the reference model’s AOA statistics.

The mean {m}_{\widehat{\mu}}\left(t\right) of the stochastic process \widehat{\mathit{\mu}}\left(t\right) is obtained by computing the statistical average of (20) with respect to the random characteristics of the frequencies *f*_{
n
}. Hence, we can write

{m}_{\widehat{\mu}}\left(t\right)=\frac{{r}_{\mathrm{\mu \mu}}\left(t\right)}{2{\sigma}_{0}^{2}}\sum _{n=1}^{N}{c}_{n}e{}^{j{\theta}_{n}},

(24)

where *r*_{
μ
μ
}(*t*) is the ACF of the reference model described in (2). From (24), we see that the mean {m}_{\widehat{\mu}}\left(t\right) changes generally with time. To avoid this, we impose the following boundary condition on the phases

\sum _{n=1}^{N}{c}_{n}e{}^{j{\theta}_{n}}=0\phantom{\rule{0.3em}{0ex}}.

(25)

The condition above can easily be fulfilled if the number of cisoids *N* is even; the gains *c*_{
n
} are constants given by {c}_{n}={\sigma}_{0}\sqrt{2/N}, and *θ*_{
n
}=−*θ*_{n+N/2}=*π*/2 for n=1,\dots ,N/2. If the condition in (25) is fulfilled, then the mean {m}_{\widehat{\mu}}\left(t\right) is not only constant but also equal to zero, i.e., {m}_{\widehat{\mu}}=0.

The ACF {r}_{\widehat{\mu}\widehat{\mu}}({t}_{1},{t}_{2}) of \widehat{\mathit{\mu}}\left(t\right) is obtained by substituting (20) in (10). In general, {r}_{\widehat{\mu}\widehat{\mu}}({t}_{1},{t}_{2}) is time-shift sensitive since it is given as

\begin{array}{c}{r}_{\widehat{\mu}\widehat{\mu}}({t}_{1},{t}_{2})=\sum _{n=1}^{N}\left[\frac{{c}_{n}^{2}}{2{\sigma}_{0}^{2}}{r}_{\mathrm{\mu \mu}}\left(\tau \right)+\frac{1}{4{\sigma}_{0}^{4}}{r}_{\mathrm{\mu \mu}}^{\ast}\left({t}_{1}\right){r}_{\mathrm{\mu \mu}}\left({t}_{2}\right)\sum _{{}_{m\ne n}^{m=1}}^{N}{c}_{n}{c}_{m}e{}^{j({\theta}_{n}-{\theta}_{m})}\right]\phantom{\rule{0.3em}{0ex}}.\end{array}

(26)

However, {r}_{\widehat{\mu}\widehat{\mu}}({t}_{1},{t}_{2}) becomes time-shift insensitive if we impose the following boundary condition on the gains and phases

\sum _{n=1}^{N}\sum _{{}_{m\ne n}^{m=1}}^{N}{c}_{n}{c}_{m}e{}^{j({\theta}_{n}-{\theta}_{m})}=0\phantom{\rule{0.3em}{0ex}}.

(27)

In case that the above boundary condition is fulfilled and the gains *c*_{
n
} are equal to {c}_{n}={\sigma}_{0}\sqrt{2/N}, we easily see that the ACF {r}_{\widehat{\mu}\widehat{\mu}}\left(\tau \right) of the stochastic simulation model is identical to the ACF *r*_{
μ
μ
}(*τ*) of the reference model.

In Appendix Appendix 1, it is shown that conditions (25) and (27) cannot be simultaneously satisfied. However, if we let *t*_{1}→±*∞* and/or *t*_{2}→±*∞*, then the ACFs {r}_{\mathrm{\mu \mu}}^{\ast}\left({t}_{1}\right) and/or *r*_{
μ
μ
}(*t*_{2}) tend to zero, and hence the ACF {r}_{\widehat{\mu}\widehat{\mu}}({t}_{1},{t}_{2}) depends only on the time difference *τ*=*t*_{2}−*t*_{1}. In this case, together with the condition given in (25), we can conclude that the stochastic process \widehat{\mathit{\mu}}\left(t\right) is an asymptotically WSS process. Furthermore, if *t*→±*∞*, then \widehat{\mathit{\mu}}\left(t\right) tends to an asymptotically FOS process. The stochastic process \widehat{\mathit{\mu}}\left(t\right) is ME since the condition {m}_{\widehat{\mu}}={m}_{\stackrel{~}{\mu}} is fulfilled. However, the stochastic processes \widehat{\mathit{\mu}}\left(t\right) of Class III channel simulators are always non-AE because the inequality {r}_{\widehat{\mu}\widehat{\mu}}\left(\tau \right)\ne {r}_{\stackrel{~}{\mu}\stackrel{~}{\mu}}\left(\tau \right) holds.

### 6.4 Class IV channel simulators

The channel simulators of Class IV are defined by a set of stochastic processes \widehat{\mathit{\zeta}}\left(t\right) with constant gains *c*_{
n
}, random frequencies *f*_{
n
}, and random phases *θ*_{
n
}, which are uniformly distributed in the interval (0,2*π*]. In this case, the complex process \widehat{\mathit{\mu}}\left(t\right)={\widehat{\mathit{\mu}}}_{1}\left(t\right)+j{\widehat{\mathit{\mu}}}_{2}\left(t\right) is given by

\widehat{\mathit{\mu}}\left(t\right)=\sum _{n=1}^{N}{c}_{n}e{}^{j(2\pi {\mathit{f}}_{n}t+{\mathit{\theta}}_{n})}\phantom{\rule{0.3em}{0ex}}.

(28)

The density {p}_{\widehat{\zeta}}\left(z\right) of \widehat{\mathit{\zeta}}\left(t\right)=\left|\widehat{\mathit{\mu}}\right(t\left)\right| is still given by (17), as the random frequencies *f*_{
n
} have no effect on the density {p}_{\widehat{\zeta}}\left(z\right) in (17). The mean {m}_{\widehat{\mu}}\left(t\right) is given by (18). If the gains *c*_{
n
} are equal to {c}_{n}={\sigma}_{0}\sqrt{2/N} and the frequencies *f*_{
n
} are given by (23), then it is straightforward to show that the ACF {r}_{\widehat{\mu}\widehat{\mu}}\left(\tau \right) of the stochastic simulation model is identical to the ACF *r*_{
μ
μ
}(*τ*) of the reference model, i.e., {r}_{\widehat{\mu}\widehat{\mu}}\left(\tau \right)={r}_{\mathrm{\mu \mu}}\left(\tau \right). The stochastic processes \widehat{\mathit{\mu}}\left(t\right) of Class IV channel simulators are non-AE since the inequality {r}_{\widehat{\mu}\widehat{\mu}}\left(\tau \right)\ne {r}_{\stackrel{~}{\mu}\stackrel{~}{\mu}}\left(\tau \right) holds.

### 6.5 Class V channel simulators

A Class V channel simulator is determined by a stochastic process \widehat{\mathit{\zeta}}\left(t\right) with random gains *c*_{
n
}, constant frequencies *f*_{
n
}, and constant phases *θ*_{
n
}. Thus, the complex process \widehat{\mathit{\mu}}\left(t\right)={\widehat{\mathit{\mu}}}_{1}\left(t\right)+j{\widehat{\mathit{\mu}}}_{2}\left(t\right) has the following form

\widehat{\mathit{\mu}}\left(t\right)=\sum _{n=1}^{N}{\mathit{c}}_{n}e{}^{j(2\pi {f}_{n}t+{\theta}_{n})}\phantom{\rule{0.3em}{0ex}}.

(29)

In Appendix Appendix 2, it is shown that the density {p}_{\widehat{\zeta}}(z;t) of the stochastic process \widehat{\mathit{\zeta}}\left(t\right)=\left|\widehat{\mathit{\mu}}\right(t\left)\right| is given by

\begin{array}{rl}{p}_{\widehat{\zeta}}(z;t)& =2\mathrm{\pi z}\underset{0}{\overset{2\pi}{\int}}\underset{0}{\overset{\infty}{\int}}\left[\phantom{\rule{0.3em}{0ex}}\prod _{n=1}^{N}\underset{-\infty}{\overset{\infty}{\int}}\right.\phantom{\rule{2em}{0ex}}\\ \left(\right)close="]">\phantom{\rule{1em}{0ex}}{p}_{c}\left(y\right)e{}^{j2\mathrm{\pi ry}cos(2\pi {f}_{n}t+{\theta}_{n}-\theta )}\phantom{\rule{0.3em}{0ex}}\mathit{\text{dy}}& {J}_{0}\left(2\mathrm{\pi zr}\right)r\phantom{\rule{0.3em}{0ex}}\mathrm{drd\theta}\phantom{\rule{2em}{0ex}}\end{array}\n

(30)

where *p*_{
c
}(·) denotes the density of the gains *c*_{
n
}. From the equation above, we realize that the density {p}_{\widehat{\zeta}}(z;t) is a function of time. Hence, the stochastic process \widehat{\mathit{\zeta}}\left(t\right) is not an FOS process.

In the following, we will impose on the stochastic channel simulator that the i.i.d. random variables *c*_{
n
} have zero mean and variance {\sigma}_{c}^{2}, i.e., *m*_{
c
}=*E*{*c*_{
n
}}=0 and {\sigma}_{c}^{2}=\text{Var}\left\{{\mathit{c}}_{n}\right\}=E\left\{{\mathit{c}}_{n}^{2}\right\}. Thus, for the mean {m}_{\widehat{\mu}}\left(t\right) of the stochastic process \widehat{\mathit{\mu}}\left(t\right), we obtain {m}_{\widehat{\mu}}\left(t\right)=0. The ACF {r}_{\widehat{\mu}\widehat{\mu}}({t}_{1},{t}_{2}) of \widehat{\mathit{\mu}}\left(t\right) can be expressed as

\begin{array}{c}{r}_{\widehat{\mu}\widehat{\mu}}({t}_{1},{t}_{2})=\sum _{n=1}^{N}\left[({\sigma}_{c}^{2}+{m}_{c}^{2})e{}^{j2\pi {f}_{n}\tau}+\sum _{{}_{m\ne n}^{m=1}}^{N}{m}_{c}^{2}e{}^{j\left(2\pi \right({f}_{n}{t}_{2}-{f}_{m}{t}_{1})+{\theta}_{n}-{\theta}_{m})}\right]\phantom{\rule{0.3em}{0ex}}.\end{array}

(31)

By taking into account that the gains *c*_{
n
} have zero mean, i.e., *m*_{c=0}, we obtain

\begin{array}{ll}{r}_{\widehat{\mu}\widehat{\mu}}({t}_{1},{t}_{2})& ={r}_{\widehat{\mu}\widehat{\mu}}\left(\tau \right)\phantom{\rule{2em}{0ex}}\\ ={\sigma}_{c}^{2}\sum _{n=1}^{N}e{}^{j2\pi {f}_{n}\tau}\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{2em}{0ex}}\end{array}

(32)

Despite the fact that the ACF of \widehat{\mathit{\mu}}\left(t\right) depends only on the time difference *τ*=*t*_{2}−*t*_{1} if *m*_{
c
}=0, the stochastic process \widehat{\mathit{\mu}}\left(t\right) turns out to be non-AE as the inequality {r}_{\widehat{\mu}\widehat{\mu}}\left(\tau \right)\ne {r}_{\stackrel{~}{\mu}\stackrel{~}{\mu}}\left(\tau \right) holds.

### 6.6 Class VI channel simulators

The channel simulators of Class VI comprise a set of stochastic processes \widehat{\mathit{\zeta}}\left(t\right) with random gains *c*_{
n
}, constant frequencies *f*_{
n
}, and random phases *θ*_{
n
}. In this case, the complex process \widehat{\mathit{\mu}}\left(t\right)={\widehat{\mathit{\mu}}}_{1}\left(t\right)+j{\widehat{\mathit{\mu}}}_{2}\left(t\right) is of type

\widehat{\mathit{\mu}}\left(t\right)=\sum _{n=1}^{N}{\mathit{c}}_{n}e{}^{j(2\pi {f}_{n}t+{\mathit{\theta}}_{n})}\phantom{\rule{0.3em}{0ex}}.

(33)

In order to find the density {p}_{\widehat{\zeta}}\left(z\right) of \widehat{\mathit{\zeta}}\left(t\right)=\left|\widehat{\mathit{\mu}}\right(t\left)\right| of Class VI channel simulators, we make use of the conditional density {p}_{\widehat{\zeta}}\left(z\right|{\mathit{c}}_{n}={c}_{n}) of Class II channel simulators. The density {p}_{\widehat{\zeta}}\left(z\right) of Class VI channel simulators can then be obtained by averaging the conditional density {p}_{\widehat{\zeta}}\left(z\right|{\mathit{c}}_{n}={c}_{n}) over the distribution *p*_{
c
}(*y*) of the *N* i.i.d. random variables {\mathit{c}}_{1},{\mathit{c}}_{2},\dots ,{\mathit{c}}_{N}, i.e.,

\begin{array}{rl}\phantom{\rule{-10.0pt}{0ex}}{p}_{\widehat{\zeta}}\left(z\right)& =\underset{-\infty}{\overset{\infty}{\int}}\cdots \underset{-\infty}{\overset{\infty}{\int}}{p}_{\widehat{\zeta}}\left(z\right|{\mathit{c}}_{1}={y}_{1},\dots ,{\mathit{c}}_{N}={y}_{N}\left)\prod _{n=1}^{N}{p}_{c}\right({y}_{n})\phantom{\rule{0.3em}{0ex}}d{y}_{1}\cdots d{y}_{N}\\ ={\left(2\pi \right)}^{2}z\underset{0}{\overset{\infty}{\int}}{\left[\phantom{\rule{0.3em}{0ex}}\underset{-\infty}{\overset{\infty}{\int}}{p}_{c}\left(y\right){J}_{0}\left(2\pi \right|y\left|\right)\nu \phantom{\rule{0.3em}{0ex}}\mathit{\text{dy}}\right]}^{N}{J}_{0}\left(2\mathrm{\pi z\nu}\right)\phantom{\rule{0.3em}{0ex}}\nu \phantom{\rule{0.3em}{0ex}}\mathrm{d\nu}\phantom{\rule{0.3em}{0ex}}.\end{array}

(34)

From (34), we can conclude that the density {p}_{\widehat{\zeta}}\left(z\right) is independent of time. Hence, the stochastic process \widehat{\mathit{\zeta}}\left(t\right) is FOS. The mean {m}_{\widehat{\mu}}\left(t\right) of the stochastic process \widehat{\mathit{\mu}}\left(t\right) is still given by (18) since the behavior of the random gains *c*_{
n
} has no effect on the mean in (18).

The ACF {r}_{\widehat{\mu}\widehat{\mu}}\left(\tau \right) of the stochastic process \widehat{\mathit{\mu}}\left(t\right) can be expressed as

{r}_{\widehat{\mu}\widehat{\mu}}\left(\tau \right)=({\sigma}_{c}^{2}+{m}_{c}^{2})\sum _{n=1}^{N}e{}^{j2\pi {f}_{n}\tau}\phantom{\rule{0.3em}{0ex}}.

(35)

Without imposing any specific distribution on the random gains *c*_{
n
}, we can conclude that the stochastic process \widehat{\mathit{\mu}}\left(t\right) is non-AE since the inequality {r}_{\widehat{\mu}\widehat{\mu}}\left(\tau \right)\ne {r}_{\stackrel{~}{\mu}\stackrel{~}{\mu}}\left(\tau \right) holds.

### 6.7 Class VII channel simulators

This class of channel simulators involves all stochastic processes \widehat{\mathit{\zeta}}\left(t\right)=\left|\widehat{\mathit{\mu}}\right(t\left)\right| with random gains *c*_{
n
}, random frequencies *f*_{
n
}, and constant phases *θ*_{
n
}, i.e.,

\widehat{\mathit{\mu}}\left(t\right)=\sum _{n=1}^{N}{\mathit{c}}_{n}e{}^{j(2\pi {\mathit{f}}_{n}t+{\theta}_{n})}\phantom{\rule{0.3em}{0ex}}.

(36)

To obtain the density {p}_{\widehat{\zeta}}\left(z\right) of the stochastic process \widehat{\mathit{\zeta}}\left(t\right), we consider first the case *t*→±*∞*. The density {p}_{\widehat{\zeta}}\left(z\right) can be considered as the conditional density {p}_{\widehat{\zeta}}\left(z\right|{\mathit{c}}_{n}={c}_{n}) obtained for Class III channel simulators. By following the same procedure described in SubSection 6.6, we can conclude that the density {p}_{\widehat{\zeta}}\left(z\right) is given by (34). If *t* is finite, we have to repeat the procedure described in SubSection 6.6 using ([29], Eq. (32)) instead of (22). In this case, it turns out that the density {p}_{\widehat{\zeta}}(z;t) is a function of *t*. Hence, the stochastic processes \widehat{\mathit{\zeta}}\left(t\right) of Class VII channel simulators are asymptotically FOS processes.

The mean {m}_{\widehat{\mu}}\left(t\right) of this class of channel simulators is given by

{m}_{\widehat{\mu}}\left(t\right)=\frac{{m}_{c}{r}_{\mathrm{\mu \mu}}\left(t\right)}{2{\sigma}_{\mu}^{2}}\sum _{n=1}^{N}e{}^{j{\theta}_{n}}\phantom{\rule{0.3em}{0ex}}.

(37)

It follows from (37) that the mean value is time independent if any of the boundary conditions *m*_{
c
}=0, *t*→±*∞*, or (25) are fulfilled.

Let us assume that the random frequencies *f*_{
n
} are given by (23). Then, the ACF {r}_{\widehat{\mu}\widehat{\mu}}({t}_{1},{t}_{2}) of the stochastic process \widehat{\mathit{\mu}}\left(t\right) is obtained by computing the statistical average of (31) with respect to the random characteristics of the frequencies *f*_{
n
}. Thus, we obtain

\begin{array}{c}{r}_{\widehat{\mu}\widehat{\mu}}({t}_{1},{t}_{2})=\frac{N({\sigma}_{c}^{2}+{m}_{c}^{2})}{2{\sigma}_{0}^{2}}\phantom{\rule{0.3em}{0ex}}{r}_{\mathrm{\mu \mu}}\left(\tau \right)+\frac{{m}_{c}^{2}}{4{\sigma}_{0}^{4}}\phantom{\rule{0.3em}{0ex}}{r}_{\mathrm{\mu \mu}}^{\ast}\left({t}_{1}\right){r}_{\mathrm{\mu \mu}}\left({t}_{2}\right)\sum _{n=1}^{N}\sum _{{}_{m\ne n}^{m=1}}^{N}e{}^{j({\theta}_{n}-{\theta}_{m})}\phantom{\rule{0.3em}{0ex}}.\end{array}

(38)

From the equation above, it follows that the ACF {r}_{\widehat{\mu}\widehat{\mu}}({t}_{1},{t}_{2}) depends only on the time difference *τ*=*t*_{2}−*t*_{1} if we impose on this class of channel simulators any of the boundary conditions *m*_{
c
}=0 or (27). In addition, similar to the explanations given in SubSection 6.3, we can conclude that the ACF {r}_{\widehat{\mu}\widehat{\mu}}({t}_{1},{t}_{2}) depends only on the time difference *τ*=*t*_{2}−*t*_{1} if *t*_{1}→±*∞* and/or *t*_{2}→±*∞*. Furthermore, remember from SubSection 6.3 that the conditions (25) and (27) cannot be fulfilled simultaneously. If any of the boundary conditions *m*_{
c
}=0 or (27) is fulfilled and if {\sigma}_{c}^{2}=2{\sigma}_{0}^{2}/N-{m}_{c}^{2}, then {r}_{\widehat{\mu}\widehat{\mu}}({t}_{1},{t}_{2}) in (38) reduces to

{r}_{\widehat{\mu}\widehat{\mu}}({t}_{1},{t}_{2})={r}_{\widehat{\mu}\widehat{\mu}}\left(\tau \right)={r}_{\mathrm{\mu \mu}}\left(\tau \right),

(39)

which states that the channel simulators of Class VII and the reference model have identical correlation properties. However, the stochastic process \widehat{\mathit{\mu}}\left(t\right) is non-AE since the inequality {r}_{{\widehat{\mu}}_{i}{\widehat{\mu}}_{i}}\left(\tau \right)\ne {r}_{{\stackrel{~}{\mu}}_{i}{\stackrel{~}{\mu}}_{i}}\left(\tau \right) holds.

### 6.8 Class VIII channel simulators

The channel simulators of Class VIII are defined by a set of stochastic processes \widehat{\mathit{\zeta}}\left(t\right)=\left|\widehat{\mathit{\mu}}\right(t\left)\right| with random gains *c*_{
n
}, random frequencies *f*_{
n
}, and random phases *θ*_{
n
}, i.e.,

\widehat{\mathit{\mu}}\left(t\right)=\sum _{n=1}^{N}{\mathit{c}}_{n}e{}^{j(2\pi {\mathit{f}}_{n}t+{\mathit{\theta}}_{n})}\phantom{\rule{0.3em}{0ex}}.

(40)

The density {p}_{\widehat{\zeta}}\left(z\right) of the stochastic process \widehat{\mathit{\zeta}}\left(t\right) is given by (34) because the random behavior of the frequencies *f*_{
n
} has no influence on {p}_{\widehat{\zeta}}\left(z\right). Hence, a Class VIII channel simulator is FOS.

For this class of channel simulators, it is straightforward to show that the mean {m}_{\widehat{\mu}}\left(t\right) of the stochastic process \widehat{\mathit{\mu}}\left(t\right) is constant and equal to zero, i.e., {m}_{\widehat{\mu}}=0. From (38), we can easily obtain the ACF {r}_{\widehat{\mu}\widehat{\mu}}\left(\tau \right) for this class of channel simulators by taking into account the random characteristics of the phases *θ*_{
n
}, i.e.,

{r}_{\widehat{\mu}\widehat{\mu}}\left(\tau \right)=\frac{N({\sigma}_{c}^{2}+{m}_{c}^{2})}{2{\sigma}_{0}^{2}}\phantom{\rule{0.3em}{0ex}}{r}_{\mathrm{\mu \mu}}\left(\tau \right)\phantom{\rule{0.3em}{0ex}}.

(41)

Hence, {r}_{\widehat{\mu}\widehat{\mu}}\left(\tau \right)={r}_{\mathrm{\mu \mu}}\left(\tau \right) holds if {\sigma}_{c}^{2}=2{\sigma}_{0}^{2}/N-{m}_{c}^{2}. Note that in contrast to Class VII channel simulators, the ACF of Class VIII channel simulators only depends on the time difference *τ*=*t*_{2}−*t*_{1} even if *m*_{
c
}≠0. However, the channel simulators of Class VIII prove to be non-AE since the inequality {r}_{{\widehat{\mu}}_{i}{\widehat{\mu}}_{i}}\left(\tau \right)\ne {r}_{{\stackrel{~}{\mu}}_{i}{\stackrel{~}{\mu}}_{i}}\left(\tau \right) holds.