Properties and Applications of a Symmetric Toeplitz Matrix Generated by <italic>C</italic> &#x002B; 1/<italic>C</italic> Elements

Abstract

Utilizing derivations for the properties of a symmetric Toeplitz matrix, we obtain analytical expressions for the performance evaluation of wireless communication systems using multiple antennas at the transmitter and/or the receiver, including those for keyhole channels, beamforming, and noncoherent detection. Our derivations of the analytical expressions are based upon closed form expressions we have obtained for the eigenvalues and eigenvectors of the $L \times L$ symmetric Toeplitz matrix whose element in the $i$ th row and the $j$ th column is given by $C^{i-j}+C^{j-i}$ , where $C \in \mathbb {C} \setminus \{-1,0,1\}$ , with $\mathbb {C}$ denoting the set of complex numbers. Each element of this matrix can be expressed as a polynomial in $C + 1/C$ . Furthermore, the special cases of real nonzero $C$ and of complex $C$ with magnitude one are discussed. Using these new results, analytical expressions for the performance of wireless communication systems using multiple antennas at the transmitter and/or the receiver can be obtained.

Keywords

• Applications in wireless communications
• eigenvalues
• eigenvectors
• symmetric Toeplitz matrix