Modelling line-of-sight coupled MIMO systems with generalised scattering matrices and spherical wave translations J. Co´rcoles, J. Pontes, M.A. Gonza´lez and T. Zwick A model to rigorously characterise line-of-sight MIMO systems is introduced. It is based on the generalised scattering matrix of each antenna, considered as isolated, and the rotation and translation coeffi- cients of spherical modes. The resulting channel matrix rigorously includes the exact spherical vector nature of electromagnetic propa- gation (which may exert a significant influence over short-range links), mutual coupling effects and real antenna reflection, trans- mission, reception and scattering features. Numerical results are pre- sented for MIMO systems made up of ideal dipoles. Introduction: Multiple-input multiple-output (MIMO) systems have attracted the attention of the research community in recent years. A great number of antenna and propagation prediction models for MIMO wireless communications are currently available. A very fruitful and interesting current line of research is focused on the effects of mutual coupling on signal correlation and, therefore, on the capacity of a wireless MIMO system. It has been shown that inter-element coup- ling effects at the transmitting and/or the receiving array can play a major role in MIMO communication performance [1, 2]. Apart from that, one of the usual assumptions taken for granted in most MIMO wire- less communication models is the plane-wave nature of the propagating electromagnetic (EM) signal. When the transmitting and receiving arrays are far away from each other and from scatterers, this does not pose any problems. However, the real spherical nature of the propagating field may exert a significant influence on the characteristics of short- range MIMO systems [3, 4]. In this Letter, we propose an EM-based MIMO system model based on rotation and translation coefficients of spherical modes [5], which describes the true spherical nature of the wave propagation and in addition considers mutual coupling. Furthermore, the real features from antennas characterised by means of a generalised scattering matrix (GSM) are also included to account for the matching, transmitting, receiving and scattering antenna response in an array environment [6]. Numerical validation is accomplished by showing results from a single-input single-output (SISO) system and different MIMO systems made up of ideal dipoles. SISO system: The GSM of an antenna i relates the complex amplitude of the input v i and the output w i signal at the feeding port (assuming one single mode of excitation) with coefficients of incoming a i and outgoing b i spherical vector modes used to characterise the radiating region as: w i b i   ¼ G i R i T i S i   v i a i   ð1Þ Incoming spherical vector waves at the receiving antenna (i ¼ 2) can be related by means of a general translation matrix (GTM) [6] to scattered (outgoing minus incoming) spherical modes at the transmitting antenna (i ¼ 1). A reciprocal relation can also be established. Thus, with G ij standing for the corresponding GTM between transmitter and receiver and vice versa, we have: a 2 ¼ G 21 ðb 1  a 1 Þ; a 1 ¼ G 12 ðb 2  a 2 Þ ð2Þ By inserting (2) in (1) for i ¼ 1, 2 and solving the matrix system, the 1  1 channel matrix, which relates the input signal at the transmitter with the output signal at the receiver as w 2 ¼ H SISO v 1 , can be expressed as [6]: H SISO ¼ R 2 fI  G 21 ðS 1  I ÞG 12 ðS 2  I Þg 1 G 21 T 1 ð3Þ To validate the model, let us consider a SISO system made up of two ideal dipoles facing each other, both parallel to the z-axis. The analytical expression for the ideal electric dipole GSM is given in [6]. The com- puted value of H SISO with the proposed model is shown in Fig. 1 against the link distance. It is compared with the classic transmission equation H SISO ¼ exp (2j2pr 12 /l 0 ) ffiffiffiffiffiffiffiffiffiffiffi D 1 D 2 p l 0 =4pr 12 , where r 12 is the distance between dipoles, l 0 is the free-space wavelength and D i the directive gain, in natural units, of dipole i (¼1.5). As expected, in this case, both models disagree only for very small distances. distance between dipoles, 0 real (H SISO ) image (H SISO ) 0.1 0.9 0.6 0.3 0.0 –0.3 –0.6 0.2 0.3 0.4 0.5 0.6 0.7 0.9 1.0 0.8 1.1 1.2 H SISO Fig. 1 Channel matrix value for SISO system of two ideal dipoles –– case 1: classic transmission equation —— case 2: proposed model MIMO system: In this case a MIMO system with M transmitting anten- nas and N receiving antennas is considered. To take mutual coupling effects into account, it must be taken into consideration that the incoming field at any antenna i (transmitter or receiver) results from the field scattered by all transmitting and all receiving antennas, so that: a i ¼ P MþN j¼1 G ij ðb j  a j Þ ð4Þ where G ii ¼ 0. By inserting (4) into (1) for i ¼ 1, ... , M þ N, a direct relationship between the input v ¼ (v 1 , ... , v MþN ) T and output w ¼ (w 1 , ... ,w MþN ) T signals of every antenna in the MIMO system can be derived as w ¼ Pv. The closed-form expression for matrix P is [6]: P ¼ G þ RGfI ðS  I ÞGg 1 T ð5Þ where G, R, S and T are diagonal block matrices made up of G i , R i , S i and T i in (1) for i ¼ 1, ... , M þ N, and G is a full matrix whose subma- trix i, j is G ij , for i ¼ 1, ... , M þ N and j ¼ 1, ... , M þ N. Matrix P is known in the GSM analysis as the reflection matrix of an array made up of all the antennas composing the MIMO system (transmitters and recei- vers). By grouping the input and output signals according to their mem- bership to either the receiving or the transmitting array, so that v ¼ (v T Rx , v TTx ) T and w ¼ (w T Rx , w T Tx ) T , and by setting the input signals at the receiv- ing antennas to be null v Rx ¼ 0, it is demonstrated that the MIMO channel matrix H MIMO can be found as the upper right N  M submatrix of this reflection matrix P, since we have: w Rx w Tx   ¼ P RxRx H MIMO P RxTx P TxTx   v Rx v Tx   ð6Þ As an example, a 4  4 MIMO system made up of ideal dipoles parallel to the z-axis is considered. The inter-element distance (between dipoles) is the same in the transmitting and in the receiving array. The open-loop capacity, for a signal-to-noise ratio (SNR) of 20 dB, obtained from our proposed method and from the spherical-wave model in [3] is shown in Fig. 2 for various-sized arrays against the link distance. Normalised channel matrices are calculated by multiplying the original ones by ffiffiffiffiffiffiffiffi MN p =kH k F , where kH k F is the Frobenius norm of the channel matrix obtained with the spherical-wave model in [3]. Three different inter-element distances, and thus array sizes, are considered. As expected, for larger inter-element distances coupling is not too relevant and both models are in good agreement. However, in the case of an inter-element distance of 0.25 l 0 , mutual coupling effects in the trans- mitting and the receiving arrays result in a severe reduction of the capacity. It should be noted that for very short link distances, both models are always in disagreement as seen in the SISO example since in this case coupling between transmitting and receiving antennas is rel- evant. Fig. 3 shows the capacity, obtained with the waterfilling pro- cedure with a 20 dB SNR, for a 2  2 MIMO dipole system with a fixed link distance of 100 l 0 and various inter-element distances at the transmitting and receiving arrays (same for both). Again, it is seen how as the inter-element distance increases both models are in good agreement, since mutual coupling becomes less relevant. For very short inter-element distances, the proposed model rigorously takes coup- ling effects into account to yield lower capacity values than those pre- dicted by the model in [3]. Finally, we consider equal arrays at the transmitter and the receiver with a fixed length of 5 l 0 . Fig. 4 shows the open-loop capacity, for a 20 dB SNR, obtained from the proposed model and the model in [3] for a link distance of 100 l 0 when the number of antennas is increased and therefore inter-element distance becomes smaller. As stated in [2], it can be seen how increasing the number of antennas above a given threshold (in this case, about 12) is futile since the capacity actually drops with respect to the case where mutual coupling is not taken into account. ELECTRONICS LETTERS 4th June 2009 Vol. 45 No. 12