| Summary: | 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 <inline-formula> <tex-math notation="LaTeX">$L \times L$ </tex-math></inline-formula> symmetric Toeplitz matrix whose element in the <inline-formula> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula>th row and the <inline-formula> <tex-math notation="LaTeX">$j$ </tex-math></inline-formula>th column is given by <inline-formula> <tex-math notation="LaTeX">$C^{i-j}+C^{j-i}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$C \in \mathbb {C} \setminus \{-1,0,1\}$ </tex-math></inline-formula>, with <inline-formula> <tex-math notation="LaTeX">$\mathbb {C}$ </tex-math></inline-formula> denoting the set of complex numbers. Each element of this matrix can be expressed as a polynomial in <inline-formula> <tex-math notation="LaTeX">$C + 1/C$ </tex-math></inline-formula>. Furthermore, the special cases of real nonzero <inline-formula> <tex-math notation="LaTeX">$C$ </tex-math></inline-formula> and of complex <inline-formula> <tex-math notation="LaTeX">$C$ </tex-math></inline-formula> 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.
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