Optimum detection of weak Gaussian signals in interdependent nonGaussian noises

Abstract

W e consider the problem of d e t e c t i n g a Gaussian s i g n a l i n n o n G a u s s i a n n o i s e u s i n g two s e t s of d a t a , where one contains the signal (if present) and noise w h i l e t h e o t h e r c o n t a i n s n o i s e o n l y . The n o i s e i n t h e second data is e s s e n t i a l l y a delayed version of t h e one i n t h e f i r s t which i s Gaussian. What makes t h e noise in the second nonGaussian i s t h e randomness of t h i s d e l a y . I f t h e s i g n a l w e r e s t r o n g , it would be more economical to ignore the second data, which cont a i n s n o i s e o n l y , and do t h e s i m p l e q u a d r a t i c d e t e ct i o n u s i n g t h e f i r s t d a t a o n l y s i n c e b o t h t h e s i g n a l and t h e n o i s e a r e G u a s s i a n i n t h i s c a s e . However, i f t h e s i g n a l i s weak and t h e n o i s e s i n two d a t a a r e interdependent, i t i s worth attempting to cancel the n o i s e i n t h e f i r s t d a t a by u s i n g t h e s e c o n d d a t a , t h u s e f f e c t i v e l y i n c r e a s i n g t h e s i g n a l -t o -n o i s e r a t i o . Unlike the phase-matching of simple sinusoids, matching two broad-band Gaussian noises with random r e l a t i v e d e l a y i s g r e a t l y i n v o l v e d . A n a l y t i c d e r i v a t i o n of t h e approximately optimum d e t e c t i o n s t a t i s t i c i s made f e a s i b l y by t h e weak signal approximation and the small parameter variation approximation. The random delay T i s decomposed i n t o t h r e e p a r t s i n t h i s d e t e c t i o n s t r u c t u r e : V , 8 and p. Li t e l l s which sampling interval T f a l l s i n , and t h e e f f e c t of where i n t h a t i n t e r v a l i s expressed by t h e c o r r e l a t i o n c o e f f i c i e n t peie between the sampled noise in the f i r s t d a t a and t h e v t h l a t e r sampled n o i s e i n t h e second. The approximately optimum d e t e c t o r c o n s i s t s o f t h r e e t y p e s of quadratic forms, which are averaged over v. The f i r s t , ql, i n v o l v e s t h e f i r s t d a t a o n l y , the second, q , t h e d e l a y e d v e r s i o n of t h e s e c o n d d a t a and t h e t h i r d , q12, both. 2 I n f o r m i n g t h e q u a d r a t i c f o r m s w i t h t h e f i r s t d a t a t h e f i r s t n-v samples are more heavily weighted than t h e r e m a i n i n g s i n c e t h e n o i s e i n them are "matched" t o t h e l a s t n-v samples of the second data, thus having g r e a t e r p o s s i b i l i t y of c a n c e l l a t i o n . With the second d a t a , c o n t a i n i n g no s i g n a l a t a l l , however, t h e f i r s t v samples are simply ignored since they are independent of t h e n o i s e i n t h e f i r s t . Roughly speaking, q1 f o r m s t h e u s u a l q u a d r a t i c f o r m u s i n g t h e f i r s t d a t a , q12 a t t e m p t s t o s u b t r a c t from q1 t h e p a r t o f t h e n o i s e which match t h e o n e i n t h e s e c o n d , and q2 a c t s a s a c o r r e c t i o n term.