Investigation of SAC Channel Effects on MIMO System Capacity and Optimal Coherence Distance Estimation under Different Angular Dispersions for Next-Gen Networks

Abstract

In practical 4G/5G MIMO systems, it is difficult to have independent fading among sub channels between different antenna pairs. There will be sub channel correlation between the transmit and receive antenna pairs. Spatial Antenna Correlation (SAC) is an important constraint in the performance of MIMO system capacity. It is observed that, if there is correlation then it effects the random distribution of eigenvalues and more correlation means it is probable that a few eigenvalues are large and rest are small. Therefore, correlation effects the loss in capacity and is bound by its eigenvalue distribution. In this chapter, the different SAC channel effects on Ergodic and outage MIMO capacities are analyzed and the correlation between signals received among antenna pairs are investigated to determine the optimal coherence distance between the spatial antennas under different angular dispersion conditions (rich and poor scattering phenomena).

1. Introduction

The growing demand for the high capacity wireless mobile communication systems (e.g. 5G cellular systems) under severe multipath fading conditions has drawn a great attention towards the MIMO compared to conventional SISO, SIMO or MISO communication systems. However, the Spatial Antenna Correlation (SAC) is one of the predominant factors that limit the MIMO systems performance substantially in terms of capacity which is a function of covariance of correlation coefficients obtained due to coherence distance between the antennas and angular spread of the multipath fading environment. It is observed that higher the correlation between the antennas and low angular spread in fading environment results in low capacity and vice versa. This chapter mainly focuses on investigation of various analytical MIMO correlation channel models with the optimal coherence distance parameter under different angular dispersion conditions, also for ergodic and outage capacity analysis in SAC environments. The chapter is structured as, the importance of space selectivity and angular dispersion of wireless channel with respect to spatial antenna correlation between BS and MS antenna are discussed in Section 2. Correlations among the spatial antennas with respect to coherence time, coherence distance and effects of angular spread on correlation coefficients and correlation length are presented in Section 3. Various correlation-based analytical MIMO channel models are presented in Section 4. The correlation antenna effects on its system capacity are presented in Section 5 and its results are discussed under Section 6. Finally, Section 7 concludes this chapter.

2. Space selectivity and angular dispersion of wireless channel

In this section, a brief look at different types of angular dispersion that occur in a typical multipath environment are presented. The space selectivity arises from the spatial interference patterns of the radio waves. The radio channel is said to be space selective between two antennas if their separation is larger than the coherence distance. The coherence distance is the maximum spatial separation among diversity antennas over which the channel response can be assumed constant. This can be related to the behavior of arrival directions of the reflected paths and is characterized by the angular spread of the multipath as shown in Figure 1. Angular dispersion. The angular spread and coherence distances are reciprocal to each other. Coherence distance is one of the key parameter in the characterization of wireless channel other than coherence time and coherence bandwidth [1, 2].

Figure 1.

Angular dispersion due to scattering phenomena of multipath fading channel.

3. Spatial antenna correlation (SAC) effects

In this section, the mathematical expression for channel correlation coefficient and envelope channel correlation in terms of coherence distance and coherence time are derived, which helps in understanding the SAC channel effects, correlation between antenna signals, the impact of angular spread on correlation coefficient and coherence distance.

Consider a BS and MS, which are separated by a distance x at time t, and after an instant of time (t + Δt), mobile has shifted to a new distance (x + Δx) as shown in Figure 2.

Figure 2.

Spatial separation between BS to MS and MS¹.

Δt is the time taken for MS to move to new position MS¹ covering the distance, Δx and can be related as

$$\mathrm{\Delta }x=\nu \mathrm{\Delta }t$$E1

where $\nu$ is mobile velocity and can be expressed as

$$\nu =\frac{f_{\max }}{f_c}c$$E2

on substituting Eq. (2) in Eq. (1) gives

$$\mathrm{\Delta }x=\left(\frac{f_{\max }}{f_c}c\right)\mathrm{\Delta }t$$

the coherence distance and coherence time can be related as

$$f_{\max }\mathrm{\Delta }t=\left(\mathrm{\Delta }x\frac{f_c}{c}\right)$$E3

Eq. (3) can now be expressed as

$$\boxed{f_{\max }\mathrm{\Delta }t=\frac{\mathrm{\Delta }x}{\lambda }}$$E4

first received band pass signal correlation has to be derived and then Eq. (4) has to be substituted to obtain the expression for spatial antenna correlation.

Expression for received band pass signal at time instant‘t’

the received signal under flat fading multipath channel is

$$r\left(t\right)=Re\left(\sum _{n=1}^N{c}_n{e}^{j{\varphi }_n\left(t\right)}\stackrel{\sim }{s}\left(t\right)e^{j2\pi f_{c}t}\right)$$E5

If the transmitted baseband signal is a constant amplitude tone in frequency axis i.e.

$$\stackrel{\sim }{s}\left(t\right)=1$$

the transmitted pass band signal is given as

$$s\left(t\right)=Re\left(e^{j2\pi f_{c}t}\right)=cos\left(2\pi f_{c}t\right)$$

But when it was received across the band of interest, it was observed to be flat and fluctuating amplitude with respect to time.

$$r\left(t\right)=Re\left(\sum _{n=1}^N{c}_n{e}^{j{\varphi }_n\left(t\right)}{e}^{j2\pi f_{c}t}\right)$$E6

So the received baseband signal $\stackrel{\sim }{r}\left(t\right)$is

$$\stackrel{\sim }{r}\left(t\right)=h\left(t\right)=\sum _{n=1}^N{c}_n{e}^{j{\varphi }_n\left(t\right)}$$

substituting $e^{j{\varphi }_n\left(t\right)}$ from Euler’s formula, the above equation is expressed as

$$h\left(t\right)=\sum _{n=1}^N{c}_{n}cos{\varphi }_n\left(t\right)+j\sum _{n=1}^N{c}_{n}sin{\varphi }_n\left(t\right)$$

the above channel, h(t) when it is expressed in terms of in-phase and out of phase as,

$$h\left(t\right)=h_I\left(t\right)+j{h}_Q\left(t\right)$$E7

where h(t) is a complex Gaussian random variable, ${\varphi }_n\left(t\right)$ follows uniform distribution and the envelope of h(t) is $\alpha \left(t\right)$ which follows the Rayleigh distribution with function $p_{\alpha }\left(t\right)$

$$\alpha \left(t\right)=\left|h\left(t\right)\right|$$E8

Mean of the envelope of h(t)

$$E\left[h_I\left(t\right)\right]=E\left[h_Q\left(t\right)\right]=0$$

distribution function is given by, $\begin{array}{ll}{p}_{\alpha }\left(t\right)=\frac{t}{b_o}{e}^{\raisebox{1ex}{$-t^2$}\!\left/ \!\raisebox{-1ex}{$2b_o$}\right.},& t\ge 0\\ =0,& t<0\end{array}$.

Where $b_o=\frac{{\mathrm{\Omega }}_p}{2}$, ${\mathrm{\Omega }}_p$ is the total received signal power.

by substituting Eq. (7) in Eq. (6), the received signal can be expressed as

$$r\left(t\right)=Re\left\{\left(h_I\left(t\right)+j{h}_Q\left(t\right)\right)e^{j2\pi f_{c}t}\right\}$$

Using Euler’s formula, the above equation becomes

$$r\left(t\right)=Re\left\{\left(h_I\left(t\right)+j{h}_Q\left(t\right)\right)\left(cos\left(2\pi f_{c}t\right)+jsin\left(2\pi f_{c}t\right)\right)\right\}$$

by expanding the above equation.

$$r\left(t\right)=Re\left\{\left(h_I\left(t\right)cos\left(2\pi f_{c}t\right)-h_Q\left(t\right)sin\left(2\pi f_{c}t\right)+j\left(h_I\left(t\right)sin\left(2\pi f_{c}t\right)+h_Q\left(t\right)cos\left(2\pi f_{c}t\right)\right)\right)\right\}$$

taking the real part of the above expression, the received signal becomes

$$r\left(t\right)=h_I\left(t\right)cos\left(2\pi f_{c}t\right)-h_Q\left(t\right)sin\left(2\pi f_{c}t\right)$$

on substituting ${\omega }_c=2\pi f_{c}t$, the above expression becomes

$$r\left(t\right)=h_I\left(t\right)cos\left({\omega }_{c}t\right)-h_Q\left(t\right)sin\left({\omega }_{c}t\right)$$E9

Since $h\left(t\right)$ is a random linear time-variant channel, the time-varying behavior of $h\left(t\right)$ can be characterized by its correlation properties.

3.1 Received band pass signal correlation

The correlation between signals received at time instant t and (t + Δt) can be expressed as

$${\varphi }_{\mathit{rr}}\left(\mathrm{\Delta }t\right)=E\left[r\left(t\right)r\left(t+\mathrm{\Delta }t\right)\right]$$E10

From Eqs. (9) and (10) received signal correlation is expressed as.

${\varphi }_{\mathit{rr}}\left(\mathrm{\Delta }t\right)=E\left[\left\{h_I\left(t\right)cos\left({\omega }_{c}t\right)-h_Q\left(t\right)sin\left({\omega }_{c}t\right)\right\}\left\{h_I\left(t+\mathrm{\Delta }t\right)cos\left({\omega }_c\left(t+\mathrm{\Delta }t\right)\right)-h_Q\left(t+\mathrm{\Delta }t\right)sin\left({\omega }_c\left(t+\mathrm{\Delta }t\right)\right)\right\},]\right.$which may be rewritten as,

$${\varphi }_{\mathit{rr}}\left(\mathrm{\Delta }t\right)=E\left[\begin{array}{l}{h}_I\left(t\right)h_I\left(t+\mathrm{\Delta }t\right)cos\left({\omega }_{c}t\right)cos\left({\omega }_c\left(t+\mathrm{\Delta }t\right)\right)+\dots \\ h_Q\left(t\right)h_Q\left(t+\mathrm{\Delta }t\right)sin\left({\omega }_{c}t\right)sin\left({\omega }_c\left(t+\mathrm{\Delta }t\right)\right)-\dots \\ h_I\left(t\right)h_Q\left(t+\mathrm{\Delta }t\right)cos\left({\omega }_{c}t\right)sin\left({\omega }_c\left(t+\mathrm{\Delta }t\right)\right)-\dots \\ h_Q\left(t\right)h_I\left(t+\mathrm{\Delta }t\right)sin\left({\omega }_{c}t\right)cos\left({\omega }_c\left(t+\mathrm{\Delta }t\right)\right)\end{array}\right]$$E11

Eq. (11) can be expressed as

$$\begin{array}{c}{\varphi }_{\mathit{rr}}\left(\mathrm{\Delta }t\right)=E\left[h_I\left(t\right)h_I\left(t+\mathrm{\Delta }t\right)\right]\frac{1}{2}cos\left({\omega }_c\left(2t+\mathrm{\Delta }t\right)\right)+E\left[h_I\left(t\right)h_I\left(t+\mathrm{\Delta }t\right)\right]\frac{1}{2}cos\left({\omega }_c\left(\mathrm{\Delta }t\right)\right)+\dots \\ E\left[h_Q\left(t\right)h_Q\left(t+\mathrm{\Delta }t\right)\right]\frac{1}{2}cos\left({\omega }_c\left(\mathrm{\Delta }t\right)\right)-E\left[h_Q\left(t\right)h_Q\left(t+\mathrm{\Delta }t\right)\right]\frac{1}{2}cos\left({\omega }_c\left(2t+\mathrm{\Delta }t\right)\right)-\dots \\ E\left[h_I\left(t\right)h_Q\left(t+\mathrm{\Delta }t\right)\right]\frac{1}{2}sin\left({\omega }_c\left(2t+\mathrm{\Delta }t\right)\right)-E\left[h_I\left(t\right)h_Q\left(t+\mathrm{\Delta }t\right)\right]\frac{1}{2}sin\left({\omega }_c\left(\mathrm{\Delta }t\right)\right)+\dots \\ E\left[h_Q\left(t\right)h_I\left(t+\mathrm{\Delta }t\right)\right]\frac{1}{2}sin\left({\omega }_c\left(\mathrm{\Delta }t\right)\right)-E\left[h_Q\left(t\right)h_Q\left(t+\mathrm{\Delta }t\right)\right]\frac{1}{2}sin\left({\omega }_c\left(2t+\mathrm{\Delta }t\right)\right)\end{array}$$E12

By separating the similar coefficient terms in Eq. (12) can be expressed as

$$\begin{array}{c}{\varphi }_{\mathit{rr}}\left(\mathrm{\Delta }t\right)=\left[E\left[h_I\left(t\right)h_I\left(t+\mathrm{\Delta }t\right)\right]+E\left[h_Q\left(t\right)h_Q\left(t+\mathrm{\Delta }t\right)\right]\right]\frac{1}{2}cos\left({\omega }_c\left(\mathrm{\Delta }t\right)\right)+\dots \\ \left[E\left[h_I\left(t\right)h_I\left(t+\mathrm{\Delta }t\right)\right]-E\left[h_Q\left(t\right)h_Q\left(t+\mathrm{\Delta }t\right)\right]\right]\frac{1}{2}cos\left({\omega }_c\left(2t+\mathrm{\Delta }t\right)\right)-\dots \\ \left[E\left[h_I\left(t\right)h_Q\left(t+\mathrm{\Delta }t\right)\right]-E\left[h_Q\left(t\right)h_I\left(t+\mathrm{\Delta }t\right)\right]\right]\frac{1}{2}sin\left({\omega }_c\left(\mathrm{\Delta }t\right)\right)-\dots \\ \left[E\left[h_I\left(t\right)h_Q\left(t+\mathrm{\Delta }t\right)\right]+E\left[h_Q\left(t\right)h_I\left(t+\mathrm{\Delta }t\right)\right]\right]\frac{1}{2}sin\left({\omega }_c\left(2t+\mathrm{\Delta }t\right)\right)\end{array}$$E13

Here, it is assumed that received signal is Wide Sense Stationary (WSS) [3], then the ${\varphi }_{\mathit{rr}}\left(\mathrm{\Delta }t\right)$ is independent of, ‘t’ but only depends on $\mathrm{\Delta }t$, so the coefficients of $2t+\mathrm{\Delta }t$ are equal to zero, i.e.

$$E\left[h_I\left(t\right)h_I\left(t+\mathrm{\Delta }t\right)\right]=E\left[h_Q\left(t\right)h_Q\left(t+\mathrm{\Delta }t\right)\right]$$

$$E\left[h_I\left(t\right)h_Q\left(t+\mathrm{\Delta }t\right)\right]=-E\left[h_Q\left(t\right)h_I\left(t+\mathrm{\Delta }t\right)\right]$$

So applying above solutions to Eq. (13), ${\varphi }_{\mathit{rr}}\left(\mathrm{\Delta }t\right)$ is expressed as

$$\begin{array}{c}{\varphi }_{\mathit{rr}}\left(\mathrm{\Delta }t\right)=\left[2E\left[h_I\left(t\right)h_I\left(t+\mathrm{\Delta }t\right)\right]\right]\frac{1}{2}cos\left({\omega }_c\left(\mathrm{\Delta }t\right)\right)-\left[2E\left[h_I\left(t\right)h_Q\left(t+\mathrm{\Delta }t\right)\right]\right]\frac{1}{2}sin\left({\omega }_c\left(\mathrm{\Delta }t\right)\right)\\ =E\left[h_I\left(t\right)h_I\left(t+\mathrm{\Delta }t\right)\right]cos\left({\omega }_c\left(\mathrm{\Delta }t\right)\right)-E\left[h_I\left(t\right)h_Q\left(t+\mathrm{\Delta }t\right)\right]sin\left({\omega }_c\left(\mathrm{\Delta }t\right)\right)\end{array}$$

which may be written as,

$${\varphi }_{\mathit{rr}}\left(\mathrm{\Delta }t\right)={\varphi }_{h_I{h}_I}\left(\mathrm{\Delta }t\right)cos{\omega }_c\mathrm{\Delta }t-{\varphi }_{h_Q{h}_I}\left(\mathrm{\Delta }t\right)sin{\omega }_c\mathrm{\Delta }t$$E14

Further the ${\varphi }_{\mathit{rr}}\left(\mathrm{\Delta }t\right)$ band pass correlation function is

$$\boxed{{\varphi }_{\mathit{rr}}\left(\mathrm{\Delta }t\right)=Re\left({\varphi }_{\mathit{hh}}\left(\mathrm{\Delta }t\right)e^{j2\pi f_c\mathrm{\Delta }t}\right)}$$E15

where ${\varphi }_{\mathit{hh}}\left(\mathrm{\Delta }t\right)$ is baseband equivalent of complex band pass signal

$${\varphi }_{\mathit{hh}}\left(\mathrm{\Delta }t\right)={\varphi }_{h_I{h}_I}\left(\mathrm{\Delta }t\right)+j{\varphi }_{h_Q{h}_I}\left(\mathrm{\Delta }t\right)$$E16

${\varphi }_{\mathit{rr}}\left(\mathrm{\Delta }t\right)$ is related to the I component of baseband correlation function with $cos{\omega }_c\mathrm{\Delta }t$ and cross-correlation function with $sin{\omega }_c\mathrm{\Delta }t$. The above equation describes the received signal correlation, from Eq. (15) signal correlation in band pass can be related to that of signal with baseband equivalent to the carrier frequency.

Auto correlation of the complex envelope of baseband equivalent${\varphi }_{h_I{h}_I}\left(\mathrm{\Delta }t\right)$is

$$\begin{array}{c}{\varphi }_{h_I{h}_I}\left(\mathrm{\Delta }t\right)=E\left[\sum _{n=1}^N{c}_{n}cos{\varphi }_n\left(t\right)\sum _{m=1}^N{c}_{m}cos{\varphi }_m\left(t+\mathrm{\Delta }t\right)\right]\\ =E\left[\sum _{n=1}^N\sum _{m=1}^N{c}_n{c}_{m}cos{\varphi }_n\left(t\right)cos{\varphi }_m\left(t+\mathrm{\Delta }t\right)\right]\end{array}$$

which may be expressed as,

$${\varphi }_{h_I{h}_I}\left(\mathrm{\Delta }t\right)=E\left[\sum _{n=1}^N\sum _{\begin{array}{l}m=1\\ m\ne n\end{array}}^N{c}_n{c}_{m}cos{\varphi }_n\left(t\right)cos{\varphi }_m\left(t+\mathrm{\Delta }t\right)\right]+E\left[\sum _{\begin{array}{l}n=1\\ m=n\end{array}}^N{c}_n^{2}cos{\varphi }_n\left(t\right)cos{\varphi }_n\left(t+\mathrm{\Delta }t\right)\right]$$E17

Since the received signal is assumed to be following WSS [3], which leads to Doppler shift in different directions of n^th path, which are independent to each other. i.e. ${\varphi }_m$ and ${\varphi }_n$ are independent for m ≠ n. So $E\left[cos{\varphi }_n\left(t\right)\right]$ and $E\left[cos{\varphi }_m\left(t+\mathrm{\Delta }t\right)\right]$ terms will lead to zero. So in Eq. (17) 1st coefficient term of m ≠ n will leads to zero

$$E\left[\sum _{n=1}^N\sum _{\begin{array}{l}m=1\\ m\ne n\end{array}}^N{c}_n{c}_{m}cos{\varphi }_n\left(t\right)cos{\varphi }_m\left(t+\mathrm{\Delta }t\right)\right]=0$$E18

By substituting Eq. (18) in Eq. (17), the auto correlation of the complex envelope of baseband equivalent is

$${\varphi }_{h_I{h}_I}\left(\mathrm{\Delta }t\right)=E\left[\sum _{n=1}^N{c}_n^{2}cos{\varphi }_n\left(t\right)cos{\varphi }_n\left(t+\mathrm{\Delta }t\right)\right]$$E19

Using laws of trigonometry, Eq. (19) becomes,

$${\varphi }_{h_I{h}_I}\left(\mathrm{\Delta }t\right)=E\left[\sum _{n=1}^N{c}_n^2\frac{1}{2}\left\{cos\left({\varphi }_n\left(2t+\mathrm{\Delta }t\right)\right)+cos{\varphi }_n\left(\mathrm{\Delta }t\right)\right\}\right]$$E20

Since the received signal is WSS, $E\left[cos{\varphi }_n\left(2t+\mathrm{\Delta }t\right)\right]=0$

$${\varphi }_n\left(\mathrm{\Delta }t\right)=2\pi f_{D,n}\mathrm{\Delta }t\text{and}{f}_{D,n}=f_{\max }\mathit{cps}\left(\theta \right)$$

$${\varphi }_{h_I{h}_I}\left(\mathrm{\Delta }t\right)=\sum _{n=1}^{N}E\left[c_n^2\right]\frac{1}{2}E\left[cos\left(2\pi f_{\max }cos\theta \mathrm{\Delta }t\right)\right]$$E21

Total received signal power due to N number of multipaths is

$$\sum _{n=1}^{N}E\left[c_n^2\right]={\mathrm{\Omega }}_p$$

$$E\left[c_n^2\right]=\frac{{\mathrm{\Omega }}_p}{N}$$

By substituting expected value of $\left[c_n^2\right]$ in Eq. (21)

$$\begin{array}{c}{\varphi }_{h_I{h}_I}\left(\mathrm{\Delta }t\right)=\sum _{n=1}^{N}E\left[c_n^2\right]\frac{1}{2}E\left[cos\left(2\pi f_{\max }cos\theta \mathrm{\Delta }t\right)\right]\\ =N\frac{{\mathrm{\Omega }}_p}{N}\frac{1}{2}E\left[cos\left(2\pi f_{\max }cos\theta \mathrm{\Delta }t\right)\right]\end{array}$$

Auto correlation of the complex envelope of baseband equivalent can be expressed as

$${\varphi }_{h_I{h}_I}\left(\mathrm{\Delta }t\right)=\frac{{\mathrm{\Omega }}_p}{2}{E}_{\theta }\left[cos\left(2\pi f_{\max }cos\theta \mathrm{\Delta }t\right)\right]$$E22

Similarly,

$${\varphi }_{h_Q{h}_I}\left(\mathrm{\Delta }t\right)=\frac{{\mathrm{\Omega }}_p}{2}{E}_{\theta }\left[sin\left(2\pi f_{\max }cos\theta \mathrm{\Delta }t\right)\right]$$E23

Using Clark’s model of 2D isotropic scattering, where rays coming from all directions are uniform i.e. its phase factor ${\varphi }_n\left(t\right)$ is following uniform distribution over the interval $\left[-\pi ,\pi \right]$ and its pdf is

$$p\left({\varphi }_n\right)=\frac{1}{2\pi }-\pi \le {\varphi }_n\le \pi$$E24

then.

$${\varphi }_{h_I{h}_I}\left(\mathrm{\Delta }t\right)=\frac{{\mathrm{\Omega }}_p}{2}\underset{-\pi }{\overset{\pi }{\int }}cos\left(2\pi f_{\max }cos\theta \mathrm{\Delta }t\right)p\left(\theta \right)G\left(\theta \right)\mathit{d\theta }$$E25

where $G\left(\theta \right)$ is gain of the antenna with respect to direction θ and it is assumed to be constant in all directions, therefore

$$G\left(\theta \right)=1$$

on substituting angular power distribution $p\left(\theta \right)$ and $G\left(\theta \right)$ in Eq. (25)

$${\varphi }_{h_I{h}_I}\left(\mathrm{\Delta }t\right)=\frac{{\mathrm{\Omega }}_p}{2}\frac{1}{2\pi }\underset{-\pi }{\overset{\pi }{\int }}cos\left(2\pi f_{\max }cos\theta \mathrm{\Delta }t\right)\mathit{d\theta }$$

since the integration of $cos\left(-\alpha \right)=cos\left(\alpha \right)$

$${\varphi }_{h_I{h}_I}\left(\mathrm{\Delta }t\right)=\frac{{\mathrm{\Omega }}_p}{2}\frac{1}{2\pi }2\underset{0}{\overset{\pi }{\int }}cos\left(2\pi f_{\max }cos\theta \mathrm{\Delta }t\right)\mathit{d\theta }$$

which may be expressed as,

$${\varphi }_{h_I{h}_I}\left(\mathrm{\Delta }t\right)=\frac{{\mathrm{\Omega }}_p}{2\pi }\underset{0}{\overset{\pi }{\int }}cos\left(2\pi f_{\max }cos\theta \mathrm{\Delta }t\right)\mathit{d\theta }$$

Similarly,

$${\varphi }_{h_I{h}_I}\left(\mathrm{\Delta }t\right)=\frac{{\mathrm{\Omega }}_p}{2}{J}_0\left(2\pi f_{\max }\mathrm{\Delta }t\right)$$E26

This is well known Bessel function of zero^th order of 1st kind. i.e.,

$$J_0\left(x\right)=\frac{1}{\pi }\underset{0}{\overset{\pi }{\int }}cos\left(xcos\theta \right)\mathit{d\theta }=\frac{1}{\pi }\underset{0}{\overset{\pi }{\int }}cos\left(xsin\theta \right)\mathit{d\theta }$$

on substituting $p\left(\theta \right)$ and $G\left(\theta \right)$ in Eq. (23)

$${\varphi }_{h_Q{h}_I}\left(\mathrm{\Delta }t\right)=\frac{{\mathrm{\Omega }}_p}{2}\frac{1}{2\pi }\underset{-\pi }{\overset{\pi }{\int }}sin\left(2\pi f_{\max }cos\theta \mathrm{\Delta }t\right)\mathit{d\theta }$$

since the integration of $sin\left(-\alpha \right)=-sin\left(\alpha \right)$ and it is odd function so above equation can be written as

$${\varphi }_{h_Q{h}_I}\left(\mathrm{\Delta }t\right)=0$$E27

Substituting Eqs. 26 and 27 in Eq. (15), the baseband equivalent of complex band pass signal ${\varphi }_{\mathit{hh}}\left(\mathrm{\Delta }t\right)$ is expressed as

$$\boxed{{\varphi }_{\mathit{hh}}\left(\mathrm{\Delta }t\right)=\frac{{\mathrm{\Omega }}_p}{2}{J}_0\left(2\pi f_{\max }\mathrm{\Delta }t\right)}$$E28

So, correlation among spatial antennas can be derived by substituting Eq. (4) in Eq. (28), in which instead of correlation function $\mathrm{\Delta }t$, it can be written as function of $\mathrm{\Delta }x$ and is expressed as

$$\boxed{{\varphi }_{\mathit{hh}}\left(\mathrm{\Delta }x\right)=\frac{{\mathrm{\Omega }}_p}{2}{J}_0\left(2\pi \frac{\mathrm{\Delta }x}{\lambda }\right)}$$E29

and the correlation for the envelope of $h\left(t\right)$ is ${\mu }_{\mathit{\alpha \alpha }}\left(\mathrm{\Delta }x\right)$ and is expressed as

$$\boxed{{\mu }_{\mathit{\alpha \alpha }}\left(\mathrm{\Delta }x\right)=\frac{\pi }{16}{\mathrm{\Omega }}_p{J}_0^2\left(2\pi \frac{\mathrm{\Delta }x}{\lambda }\right)}$$E30

3.2 Correlation between antenna signals

In this section the correlation between antenna signals are derived using narrowband antenna array assumption. Let a wave front arrived is z(t) and it makes an angle θ with respect to antenna array [4]. Figure 3 clearly shows that the ray arrived at 1st antenna takes certain amount of time from the time it reaches the 1st antenna to the time it reaches the 2nd antenna and let this be denoted by T_Z i.e. the propagation time of signal from 1st antenna to the 2nd antenna with reference to angle at which it arrives and the separation distance between antennas. This distance can be calculated as

Figure 3.

Spatial antenna array with separation distance d.

$$T_Z=\frac{d}{c}sin\theta$$E31

where c is the propagation velocity in the medium.

If a signal has bandwidth, B then z(t) with amplitude β(t) and carrier f_c is expressed as

$$z\left(t\right)=\beta \left(t\right)e^{j2\pi f_{c}t}$$E32

from narrowband antenna array assumption $B<<\frac{1}{T_Z}$_,$\frac{1}{T_s}<<\frac{1}{T_Z}$

$$\boxed{T_s>>T_Z}$$E33

Signal received at 1st antenna is

$$y_1\left(t\right)=z\left(t\right)=\beta \left(t\right)e^{j2\pi f_{c}t}$$

Then $y_2\left(t\right)$ is signal received at 2nd antenna and it is

$$y_2\left(t\right)=z\left(t-T_z\right)=\beta \left(t-T_z\right)e^{j2\pi f_c\left(t-T_z\right)}$$

also make assumption that there is identical antenna pattern i.e. $\beta \left(t-T_z\right)=\beta \left(t\right)$_, so $y_2\left(t\right)$ can be expressed as

$$y_2\left(t\right)=\beta \left(t\right)e^{j2\pi f_c\left(t-T_z\right)}$$

Similarly

$$y_2\left(t\right)=\beta \left(t\right)e^{j2\pi f_{c}t}{e}^{-j2\pi f_c{T}_z}$$

which may be expressed as,

$$y_2\left(t\right)=\beta \left(t\right)e^{j2\pi f_{c}t}{e}^{-j2\pi \frac{d}{\lambda }sin\theta }$$

the signal received at 2nd antenna in terms of 1st antenna can be expressed as,

$$\boxed{y_2\left(t\right)=y_1\left(t\right)e^{-j2\pi \frac{d}{\lambda }sin\theta }}$$E34

The signal received at 2nd antenna is basically the same signal which has already received at 1st antenna along with some phase delay due to effective separation between the two antennas [5]. So the signals arriving at different antennas only differ with each other in terms of relative phase delay but the rest of the signal is almost the same. The correlation matrix ‘R’ obtained due to phase delay from Eq. (34) is

$$\left[R\right]=\underset{0}{\overset{2\pi }{\int }}{e}^{-j2\pi \frac{d}{\lambda }sin\theta }{g}_1\left(\theta \right)g_2\left(\theta \right)p\left(\theta \right)\mathit{d\theta }$$E35

where p(θ) is channel power spectrum

d is the antenna spacing, g₁(θ) and g₂(θ) are antenna patterns

The correlation between two signals at m^th and n^th antenna ports x_m and x_n is given according to the definition of correlation coefficient ‘ρ_mn’ [6].

$${\rho }_{\mathit{mn}}=\frac{E\left[\left(x_m-E\left[x_m\right]\right){\left(x_n-E\left[x_n\right]\right)}^{\ast }\right]}{\sqrt{E\left[{\left|x_m-E\left[x_m\right]\right|}^2\right]E\left[{\left|x_n-E\left[x_n\right]\right|}^2\right]}}$$

Where E represents the expectation and the superscript, ‘* ‘denotes the complex conjugate. For uniform correlation, ${\rho }_{\mathit{mn}}=\rho$.

The correlation coefficient ρ is generally complex-valued with magnitude less than unity:

$$\left|\rho \right|=\left|\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}{e}^{-j2\pi \frac{d}{\lambda }sin\theta }\mathit{d\theta }\right|\le \frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}\left|e^{-j2\pi \frac{d}{\lambda }sin\theta }\right|\mathit{d\theta }\le 1$$

the correlation matrix at N × N antennas can be constructed as

$$\left[R\right]=\left[\begin{array}{cccc}1& {\rho }_{12}& \cdots & {\rho }_{1N}\\ {\rho }_{12}^{\ast }& 1& \cdots & {\rho }_{2N}\\ ⋮& ⋮& \ddots & ⋮\\ {\rho }_{1N}^{\ast }& {\rho }_{2N}^{\ast }& \cdots & 1\end{array}\right]$$

3.3 Angular spread effects on the correlation coefficient and correlation distance

Spatial correlation function determines how far the diversity antennas must be separated before the spatial fading of their signal becomes uncorrelated [7]. In multipath fading channels, the approximate correlation between two adjacent antennas averaged over all possible azimuthal orientations is given by a spatial correlation function [8] as

$${\rho }_{\mathit{\text{approx}}}\left(d\right)=e^{-23{\mathrm{\Lambda }}^2{\left(\frac{d}{\lambda }\right)}^2}$$E36

where d is the separation distance between the antennas

λ is the wavelength

Λ is the angular spread.

The angular spread Λ is expressed as,

$$\mathrm{\Lambda }=\frac{\sqrt{{\alpha }^2-2+2cos\alpha }}{2\alpha }$$E37

The limiting case of a single multipath arriving from precisely one direction corresponds to α = 0, which results in Λ = 0. The other limiting case of uniform illumination in all directions corresponds to α = 2π, which results in the maximum angular spread of Λ = 1.

Correlation length vs. angular spread:

Correlation length is defined as the change in the antenna separation distance (d) at which approximate correlation coefficient is considered sufficiently uncorrelated [9] i.e. ${\rho }_{\mathit{\text{approx}}}\left(d\right)<0.4$ and expressed as,

$$L_c=\frac{\lambda }{\sqrt{23}\mathrm{\Lambda }}$$E38

Eq. (38) explains how the correlation length increases with decreasing angular spread, in the case of omni directional channel the angular spread is equal to one.

4. Correlation based analytical MIMO channel models

In this section different narrowband analytical models are explained based on the MIMO channel coefficients, transmit and receive correlation matrices. The channel matrix can be split into a zero-mean stochastic part, ‘H_s’ and a purely deterministic part, ‘H_d ‘[10].

$$H=\sqrt{\frac{1}{1+K}}{H}_s+\sqrt{\frac{K}{1+K}}{H}_d$$

where K ≥ 0 is the rice factor, the matrix H_d is for LoS components and other non fading conditions [11]. For simplicity it is assumed that K = 0 and then H=H_s. In its most general form, the zero mean multivariate complex Gaussian distribution of h = vec {H} is given by

$$f\left(h\right)=\frac{1}{{\pi }^{\mathit{nm}}det\left\{R_H\right\}}exp\left(-h_H{R}_H^{-1}h\right)$$

The vec{.} operator returns the length nm vector for an n × m matrix $H=\left[h_1\dots h_m\right]$, $\mathrm{vec}\left\{\mathrm{H}\right\}={\left[h_1^T\dots h_m^T\right]}^T$_, The nm × nm matrix

$$R_H=E\left\{{\mathit{hh}}^H\right\}$$E39

is the full correlation matrix and describes the spatial MIMO channel statistics. It contains the correlations of all channel matrix elements. The MIMO channels with distribution can be obtained by

$$\mathrm{H}=\mathit{\text{unvec}}\left\{h\right\}\text{with}\mathrm{h}=R_H^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}g.$$E40

Where $R_H^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ denotes an arbitrary matrix square root i.e. any matrix satisfying $R_H^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{R}_H^{\raisebox{1ex}{$H$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=R_H$ and g is an nm × 1 vector with independent and identically distributed (i.i.d.) Gaussian elements with zero mean and unit variance. Here, unvec{·} is the inverse operator of vec{·}.

Note that direct use of Eq. (40) in general requires full specification of R_H which involves (nm)² real valued parameters. To reduce this large number of parameters, several different models were proposed that impose a particular structure on the MIMO correlation matrix. Some of these models will be briefly reviewed in the following sections.

For instance, consider a 2 × 2 MIMO channel model with receiver correlation matrices and transmitter correlation matrices as follows.

$$R_{\mathit{Rx}}=\left[\begin{array}{cc}1& {\rho }_R\\ {\rho }_R^{\ast }& 1\end{array}\right]R_{\mathit{Tx}}=\left[\begin{array}{cc}1& {\rho }_T^{\ast }\\ {\rho }_T& 1\end{array}\right]$$

where ${\rho }_R$ and ${\rho }_T$ are the receiver correlation and transmitter correlation coefficients respectively. These correlation matrices are sometimes given in terms of α and β, such as the notation in 3GPP [12] where

$$R_{\mathit{BS}}=R_{\mathit{Tx}}=\left[\begin{array}{cc}1& {\alpha }^{\ast }\\ \alpha & 1\end{array}\right]\text{for the}\mathrm{BS}\text{and}{R}_{\mathit{MS}}=R_{\mathit{Rx}}=\left[\begin{array}{cc}1& \beta \\ {\beta }^{\ast }& 1\end{array}\right]\text{for the}\mathrm{MS}$$

Then for Kronecker channel model, the correlation matrix is given as

$$R_H=R_{\mathit{BS}}^T\otimes R_{\mathit{MS}}$$

Similarly,

$$R_H=\left[\begin{array}{cc}{R}_{\mathit{MS}}& \alpha R_{\mathit{MS}}\\ {\alpha }^{\ast }{R}_{\mathit{MS}}& R_{\mathit{MS}}\end{array}\right]$$

which may be expressed as,

$$R_H=\left[\begin{array}{cccc}1& \beta & \alpha & \mathit{\alpha \beta }\\ {\beta }^{\ast }& 1& {\mathit{\alpha \beta }}^{\ast }& \alpha \\ {\alpha }^{\ast }& {\alpha }^{\ast }\beta & 1& \beta \\ {\alpha }^{\ast }{\beta }^{\ast }& {\alpha }^{\ast }& {\beta }^{\ast }& 1\end{array}\right]$$E41

Where the values of α and β are ranging from 0 to 1. Various values of α and β are used to represent different types of channels. One set of values is shown in Table 1.

Low Correlation		Medium Correlation		High Correlation
α	β	α	β	α	β
0	0	0.3	0.9	0.9	0.9

Table 1.

Different correlation scenarios.

The different matrices with low, medium and high correlation scenarios for 2 × 2 MIMO antenna configurations are given as

$$R_{\mathit{\text{high}}}=\left[\begin{array}{cccc}1& 0.9& 0.9& 0.81\\ 0.9& 1& 0.81& 0.9\\ 0.9& 0.81& 1& 0.9\\ 0.81& 0.9& 0.9& 1\end{array}\right]R_{\mathit{\text{medium}}}=\left[\begin{array}{cccc}1& 0.9& 0.3& 0.27\\ 0.9& 1& 0.27& 0.3\\ 0.3& 0.27& 1& 0.9\\ 0.27& 0.3& 0.9& 1\end{array}\right]R_{\mathit{low}}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

There is also a possibility that, where fading channels correlation matrices are having equi-correlated, correlated matrices. By $\alpha =\beta =\rho$ substitution in Eq. (41), the receive and transmit correlation matrices can be expressed as,

$$R_{\mathit{Rx}}=R_{\mathit{Tx}}=\left[\begin{array}{cccc}1& \rho & \rho & {\rho }^2\\ \rho & 1& {\rho }^2& \rho \\ \rho & {\rho }^2& 1& \rho \\ {\rho }^2& \rho & \rho & 1\end{array}\right]$$E42

5. SAC effects on MIMO system capacity

In this section the mathematical expression for a MIMO system capacity under SAC channel is presented and also the effect of antenna correlation on MIMO capacity is explained. The MIMO channel capacity with N_T transmitting and N_R receiving antennas can be increased by a factor N which is equal to minimum value among N_T and N_R without increasing signal power and spectral bandwidth for an i.i.d. Rayleigh fading channel.

Instantaneous capacity for MIMO can be expressed as

$$C_{\mathit{\text{MIMO}}}=W{\mathit{Log}}_{2}det\left(I_{N_R}+\frac{\overline{P}\cdot {\mathit{HH}}^{\ast }}{N_T{\sigma }_n^2}\right)$$E43

Since statistical channels like fading channels will have random channel matrix H, the predictable value of capacity must be calculated to obtain the ergodic capacity [13]. In this section capacity analysis is carried out for MIMO channels based on the ergodic and outage capacity which are obtained from the instantaneous capacity [14].

The average MIMO capacity for the ergodic fading channel is

$$\begin{array}{c}{C}_{i.i.d.\mathit{\text{MIMO}}}=E\left[W{\mathit{Log}}_{2}det\left(I_{N_R}+\frac{\overline{P}\cdot {\mathit{HH}}^{\ast }}{N_T{\sigma }_n^2}\right)\right]\mathit{\text{for}}{N}_R<N{}_T\\ C_{i.i.d.\mathit{\text{MIMO}}}=E\left[W{\mathit{Log}}_{2}det\left(I_{N_R}+\frac{\overline{P}\cdot H^{\ast }H}{N_T{\sigma }_n^2}\right)\right]\mathit{\text{for}}{N}_R\ge N{}_T\end{array}$$E44

The ergodic capacity estimated using the above equations under uncorrelated channel is the maximal rate for which reliable transmission is achieved and denoted as R_max. If the selected transmission rate R is larger than R_max for the given block, then the reliable communication is not possible and system is said to be in outage. Thus, the outage probability is defined based on targeted transmission rate R denoted as Pr_out(R) and C_out(Pr_out) is the capacity with outage is defined for a targeted outage probability, Pr_out(R). It is the maximum transmission rate for which the outage probability is smaller [14]. The outage probability is given as

$${Pr}_{\mathit{\text{out}}}\left(R\right)=Pr\left[E\left[W{\mathit{Log}}_{2}det\left(I_{N_R}+\frac{\overline{P}\cdot {\mathit{HH}}^{\ast }}{N_T{\sigma }_n^2}\right)\right]<R\right]$$E45

It is difficult to obtain the closed form expression of the capacity with outage, but can be expressed as

$$Pr\left[E\left[W{\mathit{Log}}_{2}det\left(I_{N_R}+\frac{\overline{P}\cdot {\mathit{HH}}^{\ast }}{N_T{\sigma }_n^2}\right)\right]<C_{\mathit{\text{out}}}\left({Pr}_{\mathit{\text{out}}}\right)\right]={Pr}_{\mathit{\text{out}}}$$E46

In general MIMO channel coefficients are not always independent and identically distributed. But there will be correlation among spatially separated antennas which affects the capacity of i.i.d. channel. The SAC channel matrix of MIMO systems with correlated spatial gains for Rayleigh flat fading-like channels is expressed as

$$H=R_{\mathit{Rx}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{H}_w{R}_{\mathit{Tx}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$E47

where, R_Rx is the receiver correlation and R_Tx is the transmitter correlation matrices.

The impact of spatial correlations on i.i.d. MIMO channel capacity is

$$\begin{array}{l}{C}_{\mathit{SAC}\mathit{\text{MIMO}}}=W{log}_{2}det\left(I_{N_R}+\frac{\overline{P}\cdot R_{\mathit{Rx}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{H}_w{R}_{\mathit{Tx}}{H}_w^{\ast }{R}_{\mathit{Rx}}^{\ast \left(\frac{1}{2}\right)}}{N_T{\sigma }_n^2}\right)\\ \frac{C_{\mathit{SAC}\mathit{\text{MIMO}}}}{W}={log}_{2}det\left(I_{N_R}+\frac{\overline{P}\cdot H_w{H}_w^{\ast }}{N_T{\sigma }_n^2}\right)+{log}_{2}det\left(R_{\mathit{Tx}}\right)+{log}_{2}det\left(R_{\mathit{Rx}}\right)\end{array}$$E48

The SAC capacity for unit system bandwidth (W = 1) is given as

$$C_{\mathit{SAC}\mathit{\text{MIMO}}}={log}_{2}det\left(I_{N_R}+\frac{\overline{P}\cdot H_w{H}_w^{\ast }}{N_T{\sigma }_n^2}\right)+{log}_{2}det\left(R_{\mathit{Tx}}\right)+{log}_{2}det\left(R_{\mathit{Rx}}\right)$$E49

From Eq. (48), it is observed that the MIMO channel capacity has been affected due to the correlation between the transmit and receive antennas and can be expressed as combination of

$${log}_{2}det\left(R_{\mathit{Tx}}\right)+{log}_{2}det\left(R_{\mathit{Rx}}\right)$$E50

This is always negative by the fact that ${log}_2\left(det\left(R_H\right)\right)\le 0$ for any correlation matrix R_H.

The determinant of a correlation matrix can be expressed as

$$det\left(R_H\right)=\prod _{i=1}^N{\lambda }_i$$

$$\text{Trace}\left({\mathrm{R}}_{\mathrm{H}}\right)=\mathrm{N}$$

Note that the product of all Eigen values of a matrix is equal to the determinant of the matrix. Geometric mean is bounded by the arithmetic mean, that is,

$${\left(\prod _{i=1}^N{\lambda }_i\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$N$}\right.}\le \frac{1}{N}\sum _{i=1}^N{\lambda }_i=\frac{1}{N}\left(\mathit{\text{trace}}\left(R_H\right)\right)=1$$

So it is obvious that

$${log}_2\left(det\left(R_H\right)\right)\le 0$$E51

The equality in Eq. (51) holds when the correlation matrix R_H is identity matrix. Therefore, the values of Eq. (50) are always negative. The outage probability can now be expressed as

$${Pr}_{\mathit{\text{out}}}\left(R,R_H\right)=Pr\left[E\left[W{log}_{2}det\left(I_{N_R}+\frac{\overline{P}\cdot R_{\mathit{Rx}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{H}_w{R}_{\mathit{Tx}}{H}_w^{\ast }{R}_{\mathit{Rx}}^{\ast \left(\frac{1}{2}\right)}}{N_T{\sigma }_n^2}\right)\right]<R\right]$$E52

The outage capacity with the transmission rate R

$$Pr\left[E\left[W{log}_{2}det\left(I_{N_R}+\frac{\overline{P}\cdot R_{\mathit{Rx}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{H}_w{R}_{\mathit{Tx}}{H}_w^{\ast }{R}_{\mathit{Rx}}^{\ast \left(\frac{1}{2}\right)}}{N_T{\sigma }_n^2}\right)\right]<C_{\mathit{\text{out}}}\left({Pr}_{\mathit{\text{out}}},R_H\right)\right]={Pr}_{\mathit{\text{out}}}$$E53

6. Results and discussion

This section presents the results obtained due to the SAC channel effects on MIMO system capacity for different correlation scenarios such as uncorrelated, low correlation, medium correlation, and high correlation discussed in the earlier section. The results of angular spread impact on the correlation coefficient and correlation length are also presented in this section. All simulations are done in MATLAB.

Channel correlation coefficient estimation:

The steps involved in the computation of channel correlation coefficient in terms of coherence distance and optimal coherence distance with respect to angular spread are shown in Figures 4 and 5 respectively.

Figure 4.

Block diagram for computation of channel correlation coefficient.

Figure 5.

Block diagram for computation of optimal coherence length.

The Ergodic and outage capacity computations for different correlation based analytical MIMO channel models are determined by various steps in the Block diagram as shown in Figure 6. The plot between the approximate correlation coefficients and antenna separation distance for different multipath angular distributions resulting from the simulation is shown in Figure 7. It is observed that, with increase in angular spread from 0 to 1 there is a decrease in the correlation coefficient. It can be depicted that in rich scattering environment with uniform power distribution i.e. α = 2π rad (Λ = 1, higher multipath angular spread), the separation distance in terms of λ required between antennas array is less to get the uncorrelated signals. Therefore correlation between the transmitter and receiver diversity antennas decreases with increasing the antenna separation distance depending on the different angular spread values.

Figure 6.

Block diagram for computation of ergodic and outage capacity.

Figure 7.

Correlation coefficients for different angular spreads with antenna separation lengths.

The correlation coefficient values for different angular spread values considering antenna separation distance in terms of λ are tabulated in Table 2.

S.No	Antenna separation distance in λ	Correlation coefficient ρapprox(d) for different angular spreads Λ
		Angular spread value Λ = 1 to 0
		Λ = 1	0.9	0.8	0.7	0.6	0.5	0.4	0.3	0.2	0.1	0
1	0	1	1	1	1	1	1	1	1	1	1	1
2	0.25	0.237	0.312	0.398	0.494	0.596	0.698	0.794	0.878	0.944	0.985	1
3	0.5	0.003	0.009	0.025	0.059	0.126	0.237	0.398	0.596	0.794	0.944	1
4	0.75	2.41E-06	2.81E-05	2.54E-04	0.001	0.009	0.039	0.126	0.312	0.596	0.878	1
5	1	1.03E-10	8.11E-09	4.05E-07	1.27E-05	2.54E-04	0.0031	0.025	0.126	0.398	0.794	1
6	1.25	2.47E-16	2.28E-13	1.03E-10	2.25E-08	2.41E-06	1.2E-04	0.00318	0.039	0.237	0.698	1
7	1.5	3.35E-23	6.24E-19	4.13E-15	9.71E-12	8.11E-09	2.41E-06	2.54E-04	0.0094	0.126	0.596	1
8	1.75	2.57E-31	1.67E-25	2.64E-20	1.02E-15	9.71E-12	2.25E-08	1.27E-05	0.0017	0.059	0.494	1
9	2	1.11E-40	4.33E-33	2.68E-26	2.64E-20	4.13E-15	1.03E-10	4.05E-07	2.54E-04	0.025	0.398	1

Table 2.

Correlation coefficient values for different angular spreads.

The correlation lengths as a function of different angular spread are plotted in Figure 8 and tabulated in Table 3. It shows that correlation length increases with decrease in angular spread. Therefore, as angular spread decreases, effective spatial diversity at the receiver requires a larger separation distance among diversity antennas.

Figure 8.

Correlation lengths for different angular spreads.

S.No	Angular Spreads	Correlation lengths
1	0	Inf
2	0.1	2.09
3	0.2	1.04
4	0.3	0.70
5	0.4	0.52
6	0.5	0.42
7	0.6	0.35
8	0.7	0.30
9	0.8	0.26
10	0.9	0.23
11	1	0.21

Table 3.

Correlation length values for different angular spreads.

If the spatial diversity were designed with lower angular spreads, then the correlation between the diversity antennas increases and the receiver becomes vulnerable to fading. So angle spread and coherence distances are reciprocal to each other. The larger the angle spread (rich scattering at MS), the smaller the coherence distance.

This is the major advantage of multipath channels leading to multiple antenna systems. So in wireless communication systems the smaller coherence distance among antennas is required to have uncorrelated signals. From Eq. (29) it is observed that spatial antenna correlation is also a function of λ, which means larger λ value will lead to smaller values of correlation that would affect coherence distance $\mathrm{\Delta }x$. Whereas small values of λ will lead coherence distance to be small compared to larger values of wavelengths. So frequency is also having its affect on spatial domain (i.e. mm wave and MIMO system).

The ergodic capacities computed for 4 × 4 MIMO system under different correlated environments are illustrated in the Figure 9. The results indicate that, the ergodic capacity computed for medium and high correlated fading scenarios are less than that of the uncorrelated environment. From simulated results, the percentages of change in ergodic capacity due to the other correlated scenarios compared to uncorrelated scenario are as given below.

1. At 2 dB SNR the percentage of degradation in ergodic capacities are 14.73%, and 28.12% for medium and high correlation 4 × 4 MIMO scenarios respectively.

2. At 18 dB SNR the percentage of degradation in ergodic capacities are 17.90%, and 30.42% for medium and high correlation 4 × 4 MIMO scenarios respectively.

Figure 9.

4 × 4 MIMO ergodic capacities for different SAC channel scenarios.

Ergodic capacities for different SNR in dB under low, medium and high correlation scenarios are discussed in Section 4 are computed and tabulated in Table 4.

S.no	SNR in dB	Ergodic capacity
		Low Correlation	Medium Correlation	High Correlation
1	0	3.35	2.92	2.51
2	2	4.48	3.82	3.22
3	4	5.81	4.88	4.06
4	6	7.34	6.08	5.03
5	8	9.06	7.42	6.13
6	10	10.94	8.91	7.38
7	12	12.97	10.54	8.77
8	14	15.12	12.30	10.30
9	16	17.37	14.18	11.95
10	18	19.72	16.19	13.72
11	20	22.14	18.30	15.60

Table 4.

Ergodic capacities for different correlation environments.

Cumulative distribution functions (CDFs) of the maximal achievable rate at 22 dB SNR for a 4 × 4 MIMO i.i.d. channel and a 4 × 4 MIMO correlated channel are illustrated in Figure 10. CDFs are one of the ways to illustrate the benefits of MIMO system capacity and the capacity with 10% outage can easily be determined from the CDF.

Figure 10.

10% outage capacity for different MIMO channels.

The 10% outage capacity (C_out, 10) is the information rate that is guaranteed for 90% of the channel realizations with probability Pr_out(C_targeted ≤ C_out,10) = 10%. It is observed that 14 and 20 bit/s/Hz is the 10% outage capacity value for 4 × 4 MIMO correlated and 4 × 4 MIMO i.i.d. channels. Which means 90% of time the correlated 4 × 4 MIMO system evolves a capacity values more than 14 bit/s/Hz and i.i.d. 4 × 4 MIMO system will have capacity values more than 20 bit/s/Hz. It is also observed that the outage probability decreases as the MIMO channel changes from correlated to uncorrelated scenarios. From simulated result of a selected rate 20 bit/s/Hz, the outage probability is 0.98 for correlated MIMO system and 0.1 for uncorrelated MIMO system.

The capacity with outage as a function of the SNR can also be determined as shown in the Figure 11, where a fixed outage probability of 10% is considered. The capacity with 10% outage is equal to 6.16 bit/s/Hz for the high correlated MIMO system, 7.49 bit/s/Hz for the medium correlated MIMO system and 9.3 bit/s/Hz for the low correlated MIMO system at 10 dB of SNR.

Figure 11.

4 × 4 MIMO outage capacities for different SAC channel scenarios.

Outage capacities for different SNR in dB under low, medium and high correlation scenarios are discussed under section 4 are computed and tabulated in Table 5.

S.no	SNR in dB	10% Outage capacity
		Low Correlation	Medium Correlation	High Correlation
1	0	2.62	2.15	1.76
2	2	3.58	2.92	2.40
3	4	4.71	3.80	3.09
4	6	6.04	4.87	3.99
5	8	7.63	6.13	5.03
6	10	9.30	7.49	6.16
7	12	11.13	9.02	7.42
8	14	13.07	10.62	8.82
9	16	15.18	12.41	10.33
10	18	17.36	14.29	11.99
11	20	19.60	16.28	13.75

Table 5.

Outage capacities for different correlation environments.

The ergodic capacities computed for different antenna configurations under fully correlated and uncorrelated scenarios are illustrated in the Figure 12. The result indicates that the ergodic capacity of 3 × 3 MIMO correlated system under performs than that of 2 × 2 MIMO uncorrelated system. It is observed that even there is an increase in size of MIMO antennas from 2 × 2 to 3 × 3, MIMO channel capacity is degrading due to spatial correlation among the antennas.

Figure 12.

SAC channel capacities for 3 × 3 and 2 × 2 MIMO systems.

The result tabulated in Table 6 indicates that the ergodic capacity computed for fully correlated fading scenario is less than of the uncorrelated environment. From simulated results, the percentages of change in ergodic capacity due to the other correlated scenarios compared to uncorrelated scenario are as given below.

1. At 2 dB SNR the percentage of degradation in ergodic capacities are 7.52%, and 17.94% for 2 × 2 and 3 × 3 MIMO respectively.

2. At 18 dB SNR the percentage of degradation in ergodic capacities are 29.45%, and 45.11% for 2 × 2 and 3 × 3 MIMO respectively.

S.no	SNR in dB	Ergodic Capacity in bit/s/Hz
		2 × 2		3 × 3
		Uncorrelated	Fully correlated	Uncorrelated	Fully correlated
1	2	2.672	2.471	4.007	3.288
2	4	3.292	2.974	4.999	3.93
3	6	4.037	3.414	6.092	4.57
4	8	4.864	3.991	7.296	5.147
5	10	5.704	4.518	8.617	5.753
6	12	6.694	5.45	10.04	6.456
7	14	7.746	5.942	11.66	7.065
8	16	8.897	6.533	13.28	7.565
9	18	10.13	7.146	15.04	8.255
10	20	11.22	7.808	16.77	8.937

Table 6.

Ergodic capacities for different SAC MIMO antenna configurations.

7. Conclusions

Spatial Antenna correlation is one of the practical impairments that severely limits the performance of the higher order MIMO configurations both at transmitter and receiver. The correlation based analytical methods are used to model the impact of transmitter and receiver correlations on the SAC MIMO system. The ergodic and outage capacities have been analyzed in detail and the optimal coherence distance that reduces the SAC effects at both MS/BS antennas under different angular dispersion conditions are also investigated. It is found that even though higher MIMO configuration gives greater system capacities under normal conditions (e.g., 3 × 3 gives 16.77 bit/s/Hz and 2 × 2 gives 11.22 bit/s/Hz) but due to correlation among antennas at higher MIMO configurations, the capacity degrades when compared to the uncorrelated lower MIMO configuration (e.g., 3 × 3 correlated capacity is 8.937 bit/s/Hz). It is found that higher correlation degrees of MIMO configuration with high SNR limit the potential capacity of MIMO due to the channel matrix rank deficient. The correlation affect due to the coherence distance between the antennas is also studied with respect to the angular spread of the reflected paths. From the analysis it is concluded that the correlation between the transmitter and receiver diversity antennas decreases with increasing the antenna separation distance.

Acknowledgments

The work presented in this chapter is supported by the Ministry of Social Justice and Empowerment, Govt. of India, New Delhi, under the UGC NFOBC Fellowship Vide Sanction letter No. F./2016-2017/NFO-2016-2017-OBC-AND-26194/(SAIII/Website) dated February, 2016.