Geometric Representation of Modulation Signals 2.13 2001/06/01 2005/09/19 13:36:15.537 GMT-5 Behnaam Aazhang aaz ece rice edu Dinesh Rajan dinesh ece rice edu Mohammad Jaber Borran mohammad ece rice edu Roy Ha rha rice edu Shawn Stewart mrshawn alumni rice edu Behnaam Aazhang aaz ece rice edu bases basis geometric representation orthogonal signal Geometric representation of signals provides a compact, alternative characterization of signals. Geometric representation of signals can provide a compact characterization of signals and can simplify analysis of their performance as modulation signals. Orthonormal bases are essential in geometry. Let s 1 t s 2 t s M t be a set of signals. Define ψ 1 t s 1 t E 1 where E 1 t 0 T s 1 t 2 . Define s 21 s 2 ψ 1 t 0 T s 2 t ψ 1 t and ψ 2 t 1 E 2 ^ s 2 t s 21 ψ 1 where E 2 ^ t 0 T s 2 t s 21 ψ 1 t 2 In general ψ k t 1 E k ^ s k t j 1 k 1 s kj ψ j t where E k ^ t 0 T s k t j 1 k 1 s kj ψ j t 2 . The process continues until all of the M signals are exhausted. The results are N orthogonal signals with unit energy, ψ 1 t ψ 2 t ψ N t where N M . If the signals s 1 t s M t are linearly independent, then N M . The M signals can be represented as s m t n 1 N s mn ψ n t with m 1 2 M where s mn s m ψ n and E m n 1 N s mn 2 . The signals can be represented by s m s m1 s m2 s mN ψ 1 t s 1 t A 2 T s 11 A T s 21 A T ψ 2 t s 2 t s 21 ψ 1 t 1 E 2 ^ A A T T 1 E 2 ^ 0 Dimension of the signal set is 1 with E 1 s 11 2 and E 2 s 21 2 . ψ m t s m t E s where E s t 0 T s m t 2 A 2 T 4 s 1 E s 0 0 0 , s 2 0 E s 0 0 , s 3 0 0 E s 0 , and s 4 0 0 0 E s m n d mn s m s n j 1 N s mj s nj 2 2 E s is the Euclidean distance between signals. Set of 4 equal energy biorthogonal signals. s 1 t s t , s 2 t s t , s 3 t s t , s 4 t s t . The orthonormal basis ψ 1 t s t E s , ψ 2 t s t E s where E s t 0 T s m t 2 s 1 E s 0 , s 2 0 E s , s 3 E s 0 , s 4 0 E s . The four signals can be geometrically represented using the 4-vector of projection coefficients s 1 , s 2 , s 3 , and s 4 as a set of constellation points. d 21 s 2 s 1 2 E s d 12 d 23 d 34 d 14 d 13 s 1 s 3 2 E s d 13 d 24 Minimum distance d min 2 E s