Wireless MIMO

In this Section the analysis of a MIMO wireless channel in free space is outlines by introducing an orthonormal expansion of the field. This allows us to parallel the observations drawn in case of wired MIMO systems based on the modal expansion in guiding structures. We suppose a constant channel. Furthermore the signal received by each of the RX antennas is affected by Additive White Gaussian Noise (AWGN) n(t) having variance sigma^2.

Let us consider N/2 double-polarized transmitting antennas fed by N signals and a number of scattering structures placed in a part of the space bounded by a sphere having radius a.

The field external to the sphere having radius a containing the electromagnetic sources (TX antennas and scatterers) can be represented in spherical coordinates by a spherical harmonic representation, e.g. by the superposition of TE_r and TM_r “radial modes” 

wherein P_n^m(x) is the associate Legendre function of the first kind, order n, degree m,

the spherical Hankel function of order n, and hat{h}_n is the second kind cylindrical Hankel function of order n.

Accordingly, the field in the space r>a is :

wherein the quantities c and d are the spherical expansion coefficients, obtained projecting the field on the spherical mode associated to the coefficient.

The number of modes in the expansion is theoretically infinite. However, from a practical point of view only a finite number of harmonics gives a practically relevant contribution to the field. This can be rigorously demonstrated by means of the analytical properties of the Hankel functions. However, in the following we adopt the approach developed by Chu.

Let us consider for example a TM mode. The modal impedance for a wave travelling outward from the spherical surface is:

wherein H'( ) is the first derivative of H( ).

By using the recurrence formulas for the spherical Bessel function the impedance can be represented by an electrical network of series capacitances and shunt inductances . In particular the network is the equivalent circuit of a high-pass filter with a cut-off frequency around

Consequently, instead of a cut-off frequency, in our case we have a cut-off distance r. All the TM modes having an index

have impedance with a large imaginary part, and give a negligible contribution to the radiated field.

Accordingly, all the terms associated to an index n > beta r can be neglected in the spherical harmonics representation of the field on a sphere of radius r. Furthermore, since

the number of TM modes on a sphere of radius r is almost equal to (beta r)^2 . The discussion about the modes is similar. In particular, it can be shown that the number of TE modes on a sphere of radius r is almost equal to (beta r)^2 . Consequently, the total number of modes on the sphere is almost equal to 2 (beta r)^2. 

Let us suppose now that r is the radius of the observation sphere, while the antennas and the scattering objects are bounded by a sphere of radius a concentric to the sphere of radius r. In this case the field outside the sphere of radius a requires spherical harmonics. Consequently, the field observed on the sphere of radius r contains not more than M=2(beta a)^2 spherical harmonics. 

Summarizing, also in the free-space propagation we can introduce the concept of modes, as in the guiding structures. The number of such modes depends on the smallest sphere including all the sources (RX antennas and scattering objects), and is 

wherein a is the radius of the sphere and A is the area of the surface of the sphere. This shows that the number of spatial modes depends on the electrical area of the smallest sphere including the sources (TX antennas and scattering objects).

The above discussion draws a parallelism between the MIMO systems operating in the space and the MIMO systems operating in the guiding structures. 

As a preliminary observation, we can note that, due to the linear relationship between the field and the coefficients of the modes, we can evaluate the coefficients of the M modes by M measurements, that can be carried out by using M linearly polarized antennas or also M/2 double-polarized antennas, each of them giving two voltage outputs related to the two tangent components of the field on the observation sphere. It is also useful to explicitly note that the use of more than M antennas is useless. Due to the spherical symmetry of the problem, the information on the coefficients is equidistributed on the observation surface. Consequently, the M/2 double-polarized antennas must be equidistributed on the observation surface to guarantee a stable reconstruction of the coefficients. The positions of the antennas can be found dividing the surface of the sphere into M/2 spherical sectors, each of them having an area equal to

As discussed above, we collect “independent” information on the field by placing a (double polarized) antenna at the centre of each sector. 

Let us now consider a wireless MIMO system in which the field is radiated by a non superdirective array consisting of N/2 (N/2 < M/2) double-polarized antennas, each of them fed by a signal modulated by a different data substream. In this case each coefficient of the spherical harmonics expansion is a linear combination of the N signals at the input of the TX antennas. Consequently, in order to retrieve the N signals we need to evaluate N linear (independent) combinations of the coefficients. This means that we can use N/2 (double polarized) antennas, whose output voltage turns out to be a complicated but linear combination of the coefficients of the spherical harmonics and consequently of the N transmitted signals. The antennas must be positioned at a suitable distance from each other in order to obtain a set of linearly independent output values. According to the observations carried out on the reconstruction of the spherical expansion coefficients, if we want to collect N independent information required to reconstruct the data streams sent by the N/2 double polarized antennas, we need N/2 spherical sectors, each of which having an area equal to Delta S. Accordingly, the surface on which the N/2 double-polarized RX antennas must be distributed must have an area not inferior to almost N Delta S /2.