期刊
IEEE TRANSACTIONS ON COMMUNICATIONS
卷 59, 期 5, 页码 1309-1320出版社
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TCOMM.2011.040111.100137
关键词
Demmel condition number; probability distribution; Wishart matrix
资金
- Engineering and Physical Science Research Council (EPSRC), United Kingdom [EP/G026092/1]
- UKIERI [SA08-080]
- European Commission [CROWN-233843]
- [DAG_S08/09.EG04]
- Engineering and Physical Sciences Research Council [EP/G033528/1] Funding Source: researchfish
- EPSRC [EP/G033528/1] Funding Source: UKRI
This paper studies the Demmel condition number of Wishart matrices, a quantity which has numerous applications to wireless communications, such as adaptive switching between beamforming and diversity coding, link adaptation, and spectrum sensing. For complex Wishart matrices, we give an exact analytical expression for the probability density function (p.d.f.) of the Demmel condition number, and also derive simplified expressions for the high tail regime. These results indicate that the condition of complex Wishart matrices is dominantly decided by the difference between the matrix dimension and degree of freedom (DoF), i.e., the probability of drawing a highly ill conditioned matrix decreases considerably when the difference between the matrix dimension and DoF increases. We further investigate real Wishart matrices, and derive new expressions for the p.d.f. of the smallest eigenvalue, when the difference between the matrix dimension and DoF is odd. Based on these results, we succeed to obtain an exact p.d.f. expression for the Demmel condition number, and simplified expressions for the high tail regime.
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