1.7 Effect of System Parameters and Antenna Correlation
1.7.6 The Effect of System Parameters on the Keyhole Propagation
As the expression for the channel matrix in (1.141) indicates, the behavior of the MIMO fading channel is controlled by the three matricesKR,KS andKT. MatricesKR andKT
are directly related to the respective antenna correlation matrices and govern the receive and transmit antenna correlation properties.
The rank of the overall channel matrix depends on the ranks of all three matricesKR, KS and KT and a low rank of any of them can cause a low channel matrix rank. The scatterer matrix KS will have a low rank if the receive scatterers angle spread is low, which will happen if the ratio Dt/R is low. That is, if the distance between the trans- mitter and the receiver R is high, the elements of KS are likely to be the same, so the rank of KS and thus the rank of H will be low. In the extreme case when the rank is one, there is only one thin radio pipe between the transmitter and the receiver and this situation is equivalent to the keyhole effect. Note that if there is no scattering at the transmitter side, the parameter relevant for the low rank is the transmit antenna radius, instead ofDt.
The rank of the channel matrix can also be one when either the transmit or receive array antenna elements are fully correlated, which happens if either the corresponding antenna elements separations or angle spreads are low.
The fading statistics is determined by the distribution of the entries of the matrix obtained as the product ofGRKSGT in (1.141). To determine the fading statistics of the correlated fading MIMO channel in (1.141) we consider the two extreme cases, when the channel matrix is of full rank and of rank one. In the first case, matrix KS becomes an identity matrix and the fading statistics is determined by the product of the twonR×SandS×nT complex Gaussian matrices GR and GT. Each entry in the resulting matrix H, being a sum of S independent random variables, according to the central limit theorem, is also a complex Gaussian matrix, if S is large. Thus the signal amplitudes undergo a Rayleigh fading distribution.
In the other extreme case, when the matrix KS has a rank of one, the MIMO channel matrix entries are products of two independent complex Gaussian variables. Thus their amplitude distribution is the product of two independent Rayleigh distributions, each with the power of 2σr2, called the double Rayleigh distribution. The pdf for the double Rayleigh distribution is given by
f (z)= ∞
0
z wσr4e−
w4+z2 2w2σ2
r dw, z≥0, (1.142)
For the channel matrix ranks between one and the full rank, the fading distribution will range smoothly between Rayleigh and double Rayleigh distributions.
The probability density functions for single and double Rayleigh distributions are shown in Fig. 1.30.
The channel matrices, given by (1.141), are simulated in slow fading channels for var- ious system parameters and the capacity is estimated by using (1.30). It is assumed in all simulations that the scattering radii are the same and equal to the distances between the antenna and the scatterers on both sides in order to maintain high local angle spreads and thus low antenna element correlations. It is assumed that the number of scatterers is high (32 in simulations). The capacity increases as the number of scatterers increases, but above a certain value its influence on capacity is negligible. Now we focus on examining the effect of the scattering radii and the distance between antennas on the keyhole effect. The capacity
Effect of System Parameters and Antenna Correlation on the Capacity of MIMO Channels 43
0 1 2 3 4 5 6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p.d.f (x)
x (Random variable)
Figure 1.30 Probability density functions for normalized Rayleigh (right curve) and double Rayleigh distributions (left curve)
0 5 10 15 20 25 30
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Capacity (bits/s/Hz)
a b c d
Probability of exceeding abscissa [Prob of capacity > abscissa]
Figure 1.31 Capacity ccdf obtained for a MIMO slow fading channel with receive and transmit scatterers and SNR=20 dB (a)Dr=Dt =50 m,R=1000 km, (b)Dr =Dt =50 m,R=50 km, (c)Dr = Dt = 100 m, R = 5 km, SNR = 20 dB; (d) Capacity ccdf curve obtained from (1.30) (without correlation or keyholes considered)
curves for various combination of system parameters in a MIMO channel withnR = nT = 4 are shown in Fig. 1.31. The first left curve corresponds to a low rank matrix, obtained for a low ratio ofDt/R, while the rightmost curve corresponds to a high rank channel matrix, in a system with a highDt/R ratio.
0 10 20 30 40 50 60 70 80 90 100 8
10 12 14 16 18 20 22 24
Dt (m)
Average Capacity (bits/s/Hz)
Figure 1.32 Average capacity on a fast MIMO fading channel for a fixed range of R = 10 km between scatterers, the distance between the receive antenna elements 3λ, the distance between the antennas and the scatterersRt =Rr=50 m, SNR=20 dB and a variable scattering radiusDt =Dr
The average capacity increase in a fast fading channel, as the scattering radius Dt increases, while keeping the distance R constant, is shown in Fig. 1.32. For a distance of 10 km, 80% of the capacity is attained if the scattering radius increases to 35m.