Antennas and MIMO

What is the role of antennas in MIMO ?

As first step, let us consider a simple antenna, consisting in a monocromatic electromagnetic source placed on the plane y=0 operating at wavelength lambda. All the geometrical quantities will be normalized to the wavelength. The tangential electric field on the plane y=0 is null except in a thin slit having length equal to 2a > 1 along z, and width b<<1 along x, on which the tangent electric field, let E(z) be, has only z-component. We will limit our attention to the far-field in the z-y plane. 

Let us consider the Fourier transform 

The far-field pattern of the antenna is proportional to

The range |s| <= 1 is associated to the visible domain (i.e. real observation angles theta= arccos(s) ) while the range s>1 is associated to the invisible domain. Since the invisible domain is related to reactive power, a high value of F in the invisible domain give superdirective sources.

According to the far-field expression, the radiating system can be modelled as a filter operating in the spatial domain (or more precisely in the angular domain) s,while the variable z represents a spatial frequency. Since E(z) is null for |z|>a, the bandwidth of the filter is finite. In particular, the one-side bandiwidth is equal to W=a. The field pattern can be represented by the Whittaker, Kotel’nikov-Shannon (WKS) expansion for bandlimited signals, i.e.

wherein Delta s is the Nyquist interval.

In order to obtain a specific patter we can apply the Woodward-Lawson synthesis method: the desired pattern (in amplitude and phase), let F_d (s) be, is sampled in the s_n sampling positions falling inside the visible domain, while the samples in the invisible domain are kept at a negligible value to avoid superdirectivity.

The synthesized pattern is consequently equal to:

wherein 2N+1 is the number of samples falling inside the real domain. 

In order to transmit information, the electromagnetic source and consequently the radiated field is varied according to a time-domain signal x(t) encoding information, which is usually modelled as a (time-domain) bandlimited signal. If the signal has bandwidth B and is observed on a temporal interval T, it can be represented as:

Consequently its representation requires practically 2BT number of samples. This number is the number of degrees of freedom of the temporal signal and will be denoted in the following as TNDF. Generally speaking, the TNDF coefficients d_m allow to uniquely identify a particular function x(t) , and consequently information encoded in x(t). For further discussion it is interesting to note the similarity between the representation of the time-domain signal, and the representation of the space-domain signal in. In particular the number of samples required to represent the field on an observation interval is the number of degrees of freedom of the space-domain signal (SNDF, Spatial Number of Degrees of Freedom) on the observation range. If the length of the observation domain is L, we have that the SNDF is almost equal to 2 W L. In particular, in the case of observation interval covering the whole visible range the SNDF is almost equal to 4W, i.e. 4a and will be called SNDF_A.  As first approximation the SNDF coefficients allow to uniquely identify the spatial distribution of the pattern. 

Coming back to the time domain signal x(t), a fundamental result of information theory is that in the case of (time domain) received signal affected by Additive White Gaussian Noise the supremum of the number of bits that we can send with a vanishing number of erroneous received bits under an average received power constraint P and noise power N is 0.5 log_2 (1+P/N) bits per degree of freedom, giving a total number of (TNDF/2) log_2 (1+P/N) bits. Note that the number of bits increases only logarithmically with the signal/noise ratio P/N, but linearly with the number of degrees of freedom.

In order to send information, the coefficients c_n in the temporal series are varied in the time-domain according to x(t). In this way, information encoded in x(t) is transmitted in all the angular directions. A receiving antenna placed in far-field condition along a direction, for example in the broadside direction of the transmitting antenna, will receive a signal having an average power, let P be, proportional to the power density radiated toward this direction. In order to maximize the amount of information received, we need to maximize the power received by the receiving antenna, e.g. (for a fixed receiving antenna) to maximize the directivity of the transmitting antenna toward the receiving one. By choosing a constant amplitude and phase distribution of the field on the radiating slit placed on the plane y=0, we obtain the sinc-shaped far field. This is the highest directivity that is possible to obtain keeping the samples in the invisible domain equal to zero. If we try to increase the directivity using also the samples in the invisible domain, we obtain a superdirective source. This situation must be avoided from the practical point of view. Consequently, even if the available number of samples to synthesize the antenna pattern is theoretically infinite, only SNDF_A can be used.

It is interesting to note the different conceptual role of the temporal and spatial series. The coefficients of temporal are chosen according to the information to be sent, and are not known by the receiver before the reception of the signal. The “uncertainty” associated to these coefficients allows to transmit information. Instead, after the synthesis process the values of the coefficients in the spatial series are fixed, and consequently information cannot be associated to them. Their role is simply to assure a suitable angular distribution of the radiated power density in order to maximize the signal/noise ratio at the receiver. However, this allows only a logarithmical increasing of the number of bits that we can send. This suggests to use the spatial series in the same way of series temporal series, associating information also to the spatial samples. 

Let us suppose that the observation domain L, wherein the receive antenna is placed, covers more than one spatial Nyquist intervals, e.g. more than one sample fall inside the observation domain. As an example, let us supose that three samples (e.g. three spatial degrees of freedom) fall inside the observation interval indicated by the two vertical dashed-dotted lines in the figure. We can associate information to each of these samples. For example, one pattern could encode the bits 111, while a different the pattern could encode the bits 011. 

It is interesting to note that, even if the samples falling inside the observation domain are only three, also the samples outside the observation domain contribute to the field inside the domain by means of their tales. This suggests to use also the samples falling outside the observation domain to send information. However, this possibility is strictly related to the noise level. Generally, noise makes impossible to identify the small variations in the pattern due to variations of samples placed outside the observation domain. In parctice, the presence of noise covers the variations of the observable part of the sinc function conveying information on the first bit, causing the lost of information associated to it. The example shows the strict similarity between this problem, and the problem of superdirectivity in classic antenna synthesis. Indeed, in both the cases we use samples placed outside the interval of interest (e.g. the visible domain in classic antennas, the observation domain in MIMO antennas) to modify the pattern inside the interval.

Paralleling the discussion in the time domain, we can say that the observation of the spatial distribution of a field having a spatial bandwidth W on a spatial interval L gives 2WL (spatial) degrees of freedom, that can be used to send independent information. The use of the spatial degrees of freedom increases the overall number of degrees of freedom that carry independent information, and the supremum of number of bits increases from (TNDF/2) log_2(1+P/N) to (STNDF/2) log_2(1+P/N) wherein STNDF is the Space-Time Number of Degrees of Freedom, while P'=P/SNDF is the average power associated to each spatial degrees of freedom (e.g. to each spatial sample falling inside the observation domain). The spatial bandwidth W depends on the wavelength, and consequently on the frequency. Under the hypothesis that the bandwidth B is sufficiently narrow we can consider the spatial bandwidth W constant in the bandwidth B, and the STNDF is equal to the product between the spatial number of degrees of freedom 2WL and the temporal number of degrees of freedom 2BT .

Summarizing, even if the use of more than one spatial degree of freedom to send information gives a lower signal/noise ratio for each spatial degree of freedom, the overall number of bits that is possible to send increases thanks to the use of STNDF degrees of freedom instead of only TNDF. This is basically the way of working of MIMO antennas. It is also interesting to note that when we are able to use more than one spatial degree of freedom to send information, maximization of the received energy in the observation domain does not assure the maximization of the bit rate..