**5.2 Deterministic SoS model**

The model proposed above may require several simulation trials to converge to the desired properties. A low-complexity alternative is described in this section. It was recently used for IVC channels with no LOS component (Patel et al., 2005) and called the MEDS model. The complex faded envelope generated by the MEDS model is given by

$$Y(t) = Y\_c(t) + jY\_s(t) \tag{24}$$

$$Y\_{\mathbf{c}}(t) = \sqrt{\frac{2}{N\_{\mathbf{c}}M\_{\mathbf{c}}}} \sum\_{n,m=1}^{N\_{\mathbf{c}}M\_{\mathbf{c}}} \cos\left(2\pi f\_{1,n}^{\mathbf{c}}t + 2\pi f\_{2,m}^{\mathbf{c}}t + \phi\_{nm}^{\mathbf{c}}\right) \tag{25}$$

$$Y\_{\mathbf{s}}(t) = \sqrt{\frac{2}{N\_{\mathbf{s}}M\_{\mathbf{s}}}} \sum\_{n,m=1}^{N\_{\mathbf{s}}M\_{\mathbf{s}}} \cos\left(2\pi f\_{1,n}^{\mathbf{s}}t + 2\pi f\_{2,m}^{\mathbf{s}}t + \phi\_{nm}^{\mathbf{s}}\right) \tag{26}$$

$$f\_{1,n}^{c/s} = f\_1 \cos\left(\frac{\pi(n - 0.5)}{2N\_{c/s}}\right) \qquad n = 1, 2, \dots \\ N\_{c/s} \tag{27}$$

$$f\_{2,m}^{c/s} = f\_2 \cos\left(\frac{\pi(m - 0.5)}{M\_{c/s}}\right) \qquad m = 1, 2, \dots \\ M\_{c/s} \tag{28}$$

where the phase *φnm* ∼ *U*[−*π*,*π*) is independent for all *n*, *m* and the in-phase and quadrature components.

Similarly, with reference to (2),(25),(26),(27),(28), the complex signal of the IVC channel model are expressed as

$$Z(t) = \frac{Y(t) + \sqrt{K} \exp(j2\pi f\_0 t + \phi\_0)}{\sqrt{1+K}}\tag{29}$$

As described in (Patel et al., 2005), all the frequencies, *f <sup>c</sup>* 1,*n*, *<sup>f</sup> <sup>c</sup>* 2,*<sup>m</sup>* and *<sup>f</sup> <sup>s</sup>* 1,*k*, *<sup>f</sup> <sup>s</sup>* 2,*<sup>l</sup>* must be distinct. In addition, *f <sup>c</sup>* 1,*n*, *<sup>f</sup> <sup>c</sup>* 2,*<sup>m</sup>* and *<sup>f</sup> <sup>c</sup>* 1,*k*, *<sup>f</sup> <sup>c</sup>* 2,*<sup>l</sup>* have also to be distinct. From simulations, we found that with *Nc* = *Mc* = *NC* and *Ns* = *Ms* = *NC* + 1, the Doppler frequencies are indeed distinct for practical ranges varying from 5 to 60. Under these assumptions, it can be shown that the time-average correlations are equal to the statistical correlations.

10 Will-be-set-by-IN-TECH

*<sup>N</sup>*0*<sup>M</sup>* <sup>−</sup> <sup>4</sup> *fc*(2*<sup>π</sup> <sup>f</sup>*1*τ*, 2*<sup>π</sup> <sup>f</sup>*2*τ*)]/(<sup>1</sup> <sup>+</sup> *<sup>K</sup>*)<sup>2</sup>

The model proposed above may require several simulation trials to converge to the desired properties. A low-complexity alternative is described in this section. It was recently used for IVC channels with no LOS component (Patel et al., 2005) and called the MEDS model. The

cos (2*π f <sup>c</sup>*

cos (2*π f <sup>s</sup>*

1 + *K* |

2]

(<sup>1</sup> <sup>+</sup> *<sup>K</sup>*) <sup>|</sup>]

2

*nm*) (25)

*nm*) (26)

2,*<sup>l</sup>* must be distinct.

(<sup>1</sup> <sup>+</sup> *<sup>K</sup>*) <sup>−</sup> *<sup>K</sup>* exp *<sup>j</sup>*2*<sup>π</sup> <sup>f</sup>*0*<sup>τ</sup>*

*Y*(*t*) = *Yc*(*t*) + *jYs*(*t*) (24)

2,*m<sup>t</sup>* <sup>+</sup> *<sup>φ</sup><sup>c</sup>*

2,*m<sup>t</sup>* <sup>+</sup> *<sup>φ</sup><sup>s</sup>*

) *n* = 1, 2, ...*Nc*/*<sup>s</sup>* (27)

) *m* = 1, 2, ...*Mc*/*<sup>s</sup>* (28)

<sup>√</sup><sup>1</sup> <sup>+</sup> *<sup>K</sup>* (29)

1,*k*, *<sup>f</sup> <sup>s</sup>*

2,*<sup>m</sup>* and *<sup>f</sup> <sup>s</sup>*

2,*<sup>l</sup>* have also to be distinct. From simulations, we found that

1,*n<sup>t</sup>* <sup>+</sup> <sup>2</sup>*<sup>π</sup> <sup>f</sup> <sup>c</sup>*

1,*n<sup>t</sup>* <sup>+</sup> <sup>2</sup>*<sup>π</sup> <sup>f</sup> <sup>s</sup>*

1,*n*, *<sup>f</sup> <sup>c</sup>*

Similarly, we can validate the second part of (20) and equation (21). Thus, we have

<sup>=</sup> *<sup>E</sup>*[|*R*<sup>ˆ</sup> *ZkZk* (*τ*) <sup>−</sup> <sup>2</sup>*J*0(2*<sup>π</sup> <sup>f</sup>*1*τ*)*J*0(2*<sup>π</sup> <sup>f</sup>*2*τ*) + *<sup>K</sup>* exp (*j*2*<sup>π</sup> <sup>f</sup>*0*τ*)

<sup>=</sup> *<sup>E</sup>*[|2*R*<sup>ˆ</sup> *ZckZck* (*τ*) + *<sup>j</sup>*2*R*<sup>ˆ</sup> *ZckZsk* (*τ*) <sup>−</sup> <sup>2</sup>*J*0(2*<sup>π</sup> <sup>f</sup>*1*τ*)*J*0(2*<sup>π</sup> <sup>f</sup>*2*τ*)

complex faded envelope generated by the MEDS model is given by

 2 *NcMc*

 2 *NsMs*

1,*<sup>n</sup>* = *f*<sup>1</sup> cos (

2,*<sup>m</sup>* = *f*<sup>2</sup> cos (

As described in (Patel et al., 2005), all the frequencies, *f <sup>c</sup>*

1,*k*, *<sup>f</sup> <sup>c</sup>*

time-average correlations are equal to the statistical correlations.

2,*<sup>m</sup>* and *<sup>f</sup> <sup>c</sup>*

*Nc*,*Mc* ∑ *n*,*m*=1

*Ns*,*Ms* ∑ *n*,*m*=1

> *π*(*n* − 0.5) 2*Nc*/*<sup>s</sup>*

*π*(*m* − 0.5) *Mc*/*<sup>s</sup>*

where the phase *φnm* ∼ *U*[−*π*,*π*) is independent for all *n*, *m* and the in-phase and quadrature

Similarly, with reference to (2),(25),(26),(27),(28), the complex signal of the IVC channel

*<sup>Z</sup>*(*t*) = *<sup>Y</sup>*(*t*) + <sup>√</sup>*<sup>K</sup>* exp(*j*2*<sup>π</sup> <sup>f</sup>*0*<sup>t</sup>* <sup>+</sup> *<sup>φ</sup>*0)

with *Nc* = *Mc* = *NC* and *Ns* = *Ms* = *NC* + 1, the Doppler frequencies are indeed distinct for practical ranges varying from 5 to 60. Under these assumptions, it can be shown that the

*Var*{*R*<sup>ˆ</sup> *ZkZk* (*τ*)}

This completes the proof.

components.

In addition, *f <sup>c</sup>*

model are expressed as

1,*n*, *<sup>f</sup> <sup>c</sup>*

**5.2 Deterministic SoS model**

= [ <sup>1</sup> <sup>+</sup> *<sup>J</sup>*0(4*<sup>π</sup> <sup>f</sup>*1*τ*)*J*0(4*<sup>π</sup> <sup>f</sup>*2*τ*)

*Yc*(*t*) =

*Ys*(*t*) =

*f <sup>c</sup>*/*<sup>s</sup>*

*f <sup>c</sup>*/*<sup>s</sup>*

$$\hat{R}\_{\mathbf{Z}\_{\mathbf{c}}\mathbf{Z}\_{\mathbf{c}}}(\tau) = \frac{1}{1+K} [\frac{1}{N\_{\mathbf{C}}^2} \sum\_{n,m=1}^{N\_{\mathbf{c}}N\_{\mathbf{c}}} \cos\left\{2\pi f\_{1,n}^{\mathbf{c}}\tau + 2\pi f\_{2,m}^{\mathbf{c}}\tau\right\} + \frac{K\cos\left(2\pi f\_0\tau\right)}{2}] \tag{30}$$

$$\hat{R}\_{Z\_i Z\_i}(\tau) = \frac{1}{1+K} [\frac{1}{(N\_\mathbb{C}+1)^2} \sum\_{n,m=1}^{N\_\mathbb{C}+1, N\_\mathbb{C}+1} \cos\left\{2\pi f\_{1,n}^\circ \tau + 2\pi f\_{2,m}^\circ \tau\right\} + \frac{K \cos\left(2\pi f\_0 \tau\right)}{2}] \tag{31}$$

$$
\hat{R}\_{Z\_i Z\_s}(\tau) = -\hat{R}\_{Z\_i Z\_t}(\tau) = \frac{K \sin(2\pi f\_0 \tau)}{2(1+K)} \tag{32}
$$

$$\hat{R}\_{ZZ}(\tau) = \frac{1}{1+K} [\frac{1}{N\_{\mathbb{C}}^2} \sum\_{n,m=1}^{N\_{\mathbb{C}}N\_{\mathbb{C}}} \cos\left\{2\pi f\_{1,n}^{\mathbb{C}}\tau + 2\pi f\_{2,m}^{\mathbb{C}}\tau\right\} + $$

$$\frac{1}{(N\_{\mathbb{C}}+1)^2} \sum\_{n,m=1}^{N\_{\mathbb{C}}+1, N\_{\mathbb{C}}+1} \cos\left\{2\pi f\_{1,n}^{\mathbb{C}}\tau + 2\pi f\_{2,m}^{\mathbb{C}}\tau\right\} + K \exp\left(2\pi f\_0 \tau\right)] \tag{33}$$

*Remark*: The expressions for variances of the correlation functions for the proposed MEDS model cannot be obtained in a simplified form, here are not provided.
