Citation
Schaffner, Charles Albert (1968) The global optimization of phaseincoherent signals. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:12212015144937012
Abstract
The problem of global optimization of M phaseincoherent signals in N complex dimensions is formulated. Then, by using the geometric approach of Landau and Slepian, conditions for optimality are established for N = 2 and the optimal signal sets are determined for M = 2, 3, 4, 6, and 12.
The method is the following: The signals are assumed to be equally probable and to have equal energy, and thus are represented by points ṡ_{i}, i = 1, 2, …, M, on the unit sphere S_{1} in C^{N}. If W_{ik} is the halfspace determined by ṡ_{i} and ṡ_{k} and containing ṡ_{i}, i.e. W_{ik} = {ṙϵC^{N}: ≥  ˂ṙ, ṡ_{k}˃}, then the Ʀ_{i} = ∩/k≠i W_{ik}, i = 1, 2, …, M, the maximum likelihood decision regions, partition S_{1}. For additive complex Gaussian noise ṅ and a received signal ṙ = ṡ_{i}e^{jϴ} + ṅ, where ϴ is uniformly distributed over [0, 2π], the probability of correct decoding is P_{C} = 1/π^{N} ∞/ʃ/0 r^{2N1}e^{(r2+1)}U(r)dr, where U(r) = 1/M M/Ʃ/i=1 Ʀ_{i} ʃ/∩ S_{1} I_{0}(2r  ˂ṡ, ṡ_{i}˃)dσ(ṡ), and r = ǁṙǁ.
For N = 2, it is proved that U(r) ≤ ʃ/C_{α} I_{0}(2r˂ṡ, ṡ_{i}˃)dσ(ṡ) – 2K/M. h(1/2K [Mσ(C_{α})σ(S_{1})]), where C_{α} = {ṡϵS_{1}:˂ṡ, ṡ_{i}˃ ≥ α}, K is the total number of boundaries of the net on S_{1} determined by the decision regions, and h is the strictly increasing strictly convex function of σ(C_{α}∩W), (where W is a halfspace not containing ṡ_{i}), given by h = ʃ/C_{α}∩W I_{0} (2r˂ṡ, ṡ_{i}˃)dσ(ṡ). Conditions for equality are established and these give rise to the globally optimal signal sets for M = 2, 3, 4, 6, and 12.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Electrical Engineering 
Degree Grantor:  California Institute of Technology 
Division:  Engineering and Applied Science 
Major Option:  Electrical Engineering 
Thesis Availability:  Restricted to Caltech community only 
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Thesis Committee: 

Defense Date:  23 April 1968 
Record Number:  CaltechTHESIS:12212015144937012 
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:12212015144937012 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  9342 
Collection:  CaltechTHESIS 
Deposited By:  Leslie Granillo 
Deposited On:  22 Dec 2015 17:37 
Last Modified:  22 Dec 2015 17:37 
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