Properties and Applications of a Symmetric Toeplitz Matrix Generated by <italic>C</italic> + 1/<italic>C</italic> Elements
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 noncoheren...
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| Format: | Article |
| Language: | English |
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IEEE
2023-01-01
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| Series: | IEEE Access |
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| Online Access: | https://ieeexplore.ieee.org/document/10219045/ |
| _version_ | 1797736934243565568 |
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| author | Ranjan K. Mallik Ross Murch |
| author_facet | Ranjan K. Mallik Ross Murch |
| author_sort | Ranjan K. Mallik |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-12T13:21:06Z |
| format | Article |
| id | doaj.art-1d97d4ec816e44c2a78552e95dca0974 |
| institution | Directory Open Access Journal |
| issn | 2169-3536 |
| language | English |
| last_indexed | 2024-03-12T13:21:06Z |
| publishDate | 2023-01-01 |
| publisher | IEEE |
| record_format | Article |
| series | IEEE Access |
| spelling | doaj.art-1d97d4ec816e44c2a78552e95dca09742023-08-25T23:01:01ZengIEEEIEEE Access2169-35362023-01-0111884768848810.1109/ACCESS.2023.330543010219045Properties and Applications of a Symmetric Toeplitz Matrix Generated by <italic>C</italic> + 1/<italic>C</italic> ElementsRanjan K. Mallik0https://orcid.org/0000-0002-0210-5282Ross Murch1https://orcid.org/0000-0002-2527-7693Department of Electrical Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, IndiaDepartment of Electronic and Computer Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong KongUtilizing 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.https://ieeexplore.ieee.org/document/10219045/Applications in wireless communicationseigenvalueseigenvectorssymmetric Toeplitz matrix |
| spellingShingle | Ranjan K. Mallik Ross Murch Properties and Applications of a Symmetric Toeplitz Matrix Generated by <italic>C</italic> + 1/<italic>C</italic> Elements IEEE Access Applications in wireless communications eigenvalues eigenvectors symmetric Toeplitz matrix |
| title | Properties and Applications of a Symmetric Toeplitz Matrix Generated by <italic>C</italic> + 1/<italic>C</italic> Elements |
| title_full | Properties and Applications of a Symmetric Toeplitz Matrix Generated by <italic>C</italic> + 1/<italic>C</italic> Elements |
| title_fullStr | Properties and Applications of a Symmetric Toeplitz Matrix Generated by <italic>C</italic> + 1/<italic>C</italic> Elements |
| title_full_unstemmed | Properties and Applications of a Symmetric Toeplitz Matrix Generated by <italic>C</italic> + 1/<italic>C</italic> Elements |
| title_short | Properties and Applications of a Symmetric Toeplitz Matrix Generated by <italic>C</italic> + 1/<italic>C</italic> Elements |
| title_sort | properties and applications of a symmetric toeplitz matrix generated by italic c italic x002b 1 italic c italic elements |
| topic | Applications in wireless communications eigenvalues eigenvectors symmetric Toeplitz matrix |
| url | https://ieeexplore.ieee.org/document/10219045/ |
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