Phase errors in the cross spectrum estimate due to misalignment

PHASE ERRORS I N THE CROSS SPECTRUM ESTIMATE DUE TO MISALIGNMENT

S v e r r e Holm

.

E l e c t r i c a l E n g i n e e r i n g Department Yarmouk U n i v e r s i t y , I r b i d , J o r d a n . PBSTRACT

is t h e time delay.

When t h e c r o s s s p e c t r u m i s e s t i m a t e d where t h e r e i s a d e l a y between t h e two i n p u t s i g n a l s , i t i s shown t h a t t h e c o r r e s p o n d i n g m i s a l i g n m e n t g i v e s b o t h magnitude and p h a s e e r r o r s . The e x i s t e n c e of t h e phase e r r o r i s a new r e s u l t . The i m p o r t a n c e of t h e e r r o r s depend on t h e a l g o r i t h m s used. I t is shown t h a t t h e m i s a l i g n m e n t may e i t h e r be g i v e n by t h e t o t a l o b s e r v a t i o n t i m e o r by t h e s e c t i o n l e n g t h i n the processor. I n the l a t t e r case t h e e f f e c t of p h a s e e r r o r s must be c o n s i d e r e d when t h e s e c t i o n l e n g t h i s low, t h a t i s when t h e c r o s s s p e c t r u m i s smooth, r e q u i r i n g l i t t l e r e s o l u t i o n . .

INTRODUCTION

I n most p r a c t i c a l e s t i m a t i o n problems one h a s a l i m i t e d , T s e c o n d s , l o n g segment o f e a c h i n p u t s i g n a l . The d e l a y , D , t h e n g i v e s a r e d u c e d o v e r l a p o f t h e two i n p u t s of T-D s e c o n d s . T h i s i s r e f e r r e d t o a s m i s a l i g n m e n t , and i t g i v e s an e r r o r i n t h e c r o s s s p e c t r u m e s t i m a t e . T h i s e r r o r was i n v e s t i g a t e d by C a r t e r [ 4 ] and K r o e n e r t [5], b u t they o n l y found t h e e f f e c t on t h e magnitude o f t h e c r o s s s p e c t r u m . I n t h i s p a p e r t h e e f f e c t on t h e p h a s e i s a l s o found. EXPECTED VALUE OF THE CROSS SPECTRLM The i n p u t s i g n a l s a r e s ( t ) and r ( t ) = s ( t - D ) + n ( t ) The second s i g n a l i s d e l a y e d by D s e c o n d s and h a s an a d d i t i v e n o i s e t e r m , n ( t ) . The f i n i t e F o u r i e r t r a n s f o r m s o v e r segments of l e n g t h T (assuming r e c t a n g u l a r time windowing) a r e :

.

The c r o s s s p e c t r u m i s e s t i m a t e d a s a p a r t o f t h e e s t i m a t i o n of t h e c r o s s c o r r e l a t i o n , u s i n g t h e F o u r i e r t r a n s f o r m r e l a t i o n s h i p between t h e two, I t i s more e f f i c i e n t t o u s e f a s t F o u r i e r t r a n s f o r m (FFT) methods and f i n d t h e c r o s s c o r r e l a t i o n v i a t h e c r o s s s p e c t r u m , t h a n t o f i n d i t d i r e c t l y i n t h e time domain. Besides t h i s mechod a l s o g i v e s a s t r a i g h t f o r w a r d way of e s t i m a t i n g t h e g e n e r a l i z e d c r o s s c o r r e l a t i o n between two s i g n a l s , s ( t ) and r ( t ) [I]:

I n t h i s case the cross spectrum, C s r ( f ) , i s weighted by a f i l t e r Y ( f ) b e f o r e transformation. S e t t i n g t h i s f i l t e r equal t o unity gives the cross c o r r e l a t i o n f u n c t i o n R s r ( r ) . The c r o s s s p e c t r u m i s a l s o u s e d t o d e f i n e t h e c o h e r e n c e between two s i g n a l s :

The c o h e r e n c e i s t h e c r o s s s p e c t r u m n o r m a l i z e d by t h e power s p e c t r a of t h e two i n p u t s , and i t s magnit u d e i s always l e s s o r e q u a l t o u n i t y . From t h e c o h e r e n c e one can o b t a i n i n f o r m a t i o n a b o u t t h e amount o f n o n l i n e a r i t y t h a t r e l a t e s t h e two s i g n a l s , o r t h e amount o f u n c o r r e l a t e d n o i s e i n t h e t r a n s m i s s i o n p a t h from s ( t ) t o r ( t ) . A s p e c i a l case of g r e a t p r a c t i c a l i n t e r e s t i n b e a r i n g e s t i m a t i o n i s when t h e r e i s a time d e l a y , D , between t h e two s i g n a l s i n a d d i t i o n t o n o i s e . I n t h i s c a s e t h e p h a s e of t h e c r o s s s p e c t r u m c o n t a i n s t h e d e l a y i n f o r m a t i o n . The d e l a y c a n b e f o u n d from t h e s l o p e of t h e phase a s a f u n c t i o n o f f r e q u e n c y (assuming n o n - d i s p e r s i v e p r o p a g a t i o n ) [2]. It can a l s o b e found from t h e g e n e r a l i z e d c r o s s c o r r e l a t i o n . b e c a u s e t h e l a g t h a t c o r r e s p o n d s t o i t s peak v a l u e

T-D T -j2nfu -j2~Fu ~ ( f a) e -jZTfD I s ( u ) e du + / n ( u ) e du -D 0 (3b) The c r o s s s p e c t r u m e s t i m a t e i s found e i t h e r by l e t t i n g T be l a r g e o r by a v e r a g i n g o v e r many s e g m e n t s , and i t a p p r o a c h e s t h e e x p e c t e d v a l u e :

The n o i s e i s u n c o r r e l a t e d w i t h t h e s i g n a l and d i s a p p e a r s . The s i g n a l s a r e z e r o o u t s i d e t h e l i m i t s of i n t e g r a t i o n , and t h e e q u a t i o n can b e s p l i t i n t o four different integrals: T- D

r

0

T-D

T

r + r

0

T-D

T-D

/ 0

T-D

+

I 0

0

r

-D

+

T

0

T-D

-D

r

I

(5)

w h e r e t h e i n t e g r a t i o n k e r n e l s a r e a s i n ( 4 ) . The f i r s t i n t e g r a l gives the following contribution a f t e r a change of v a r i a b l e s (v=u-g) :

21.6.1 CH2 1 18-8/85/0000-08 12 $1.00O 1985 IEEE

=

T-D ,-jZrfD T

*

Gss ( f )

T h i s term i s an e s t i m a t e o f t h e power s p e c t r u m G s s ( f ) , w i t h p h a s e g i v e n by t h e d e l a y . The m i s a l i gnment w e i g h t i s t h e same a s i n e q u a t i o n 7 a i n [4]. T h i s t e r m i s t h e o n l y o n e when t h e r e i s n o misalignment. The two n e x t i n t e g r a l s a r e s i m i l a r , a f t e r s u b s t i t u t i n g u = v t g i n t h e s e c o n d i n t e g r a l a n d v=u+g i n t h e t h i r d i n t e g r a l o f (5) we g e t :

The w e i g h t , w ( g ) , i s t r a p e z o i d a l f o r O