Abstract

Expressions are deriυed for real filters that haυe a maximum correlation signal to noise ratio. Both continuous and discrete cases are treated and shown to haυe similar forms. The signal can be complex, and the case of a real signal is considered and related to preυious results.

© 1991 Optical Society of America

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References

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  1. B. V. K. Vijaya Kumar, “Signal to Noise Ratio Loss in Correlators Using Real Filters,” Appl. Opt. 28, 3287–3288 (1989).
    [CrossRef]
  2. R. D. Juday, “Optical Correlation with a Cross-Coupled Spatial Light Modulator,” Spatial Light Modulators and Applications, 1988 Technical Digest Series, Vol. 8 (Optical Society of America, Washington, DC, 1988), pp. 238–241.
  3. M. W. Farn, J. W. Goodman, “Optimal Maximum Correlation Filter for Arbitrarily Constrained Devices,” Appl. Opt. 28, 3362–3366 (1989).
    [CrossRef]
  4. R. D. Juday, “Correlation with a Spatial Light Modulator Having Phase and Amplitude Cross Coupling,” Appl. Opt. 28, 4865–4869 (1989).
    [CrossRef] [PubMed]

1989 (3)

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Equations (39)

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SNR = | E { C ( 0 ) } | 2 var { C ( 0 ) } .
SNR = | S ( f ) H ( f ) d f | 2 P n ( f ) | H ( f ) | 2 d f .
S ( k Δ f ) = A k exp ( j ϕ k ) ; H ( k Δ f ) = H k ; P n k = P n ( k Δ f ) .
SNR = [ l A l H l exp ( + j ϕ l ) ] [ k A k H k exp ( j ϕ k ) ] k H k 2 P n k ,
H m ( SNR ) = 0 .
u ν H m = ν u H m .
u = k l A k A l H k H l exp [ j ( ϕ l ϕ k ) ] .
u H m = A m exp ( + j ϕ m ) k A k H k exp ( j ϕ k ) + A m exp ( j ϕ m ) k A k H k exp ( + j ϕ k ) .
B exp ( j β ) = k A k H k exp ( j ϕ k )
exp ( j x ) exp ( j y ) + exp ( j x ) exp ( j y ) = 2 cos ( x y ) ,
u H m = 2 A m B cos ( ϕ m β ) .
ν = k H k 2 P n k ν H m = 2 P n m H m .
k l x k x l exp [ j ( ϕ k ϕ l ) ] k l x k x l cos ( ϕ k ϕ l ) ,
H m P n m k l A k A l H k H l cos ( ϕ k ϕ l ) = A m ( k H k 2 P n k ) B cos ( ϕ m β ) .
H m A m P n m cos ( ϕ m β ) ,
H m = A m P n m cos ( ϕ m β 1 ) , β 2 = arg { k H k A k exp ( j ϕ k ) } .
β = arg { k A k P n k cos ( ϕ k β ) A k exp ( j ϕ k ) } = arg { k A k P n k [ exp ( j ϕ k ) exp ( j β ) + exp ( j ϕ k ) exp ( j β ) ] A k exp ( j ϕ k ) } = arg ( exp ( j β ) { [ k A k 2 P n k exp ( j 2 ϕ k ) ] exp ( j 2 β ) + k A k 2 P n k } ) .
β = ( 1 2 ) arg { k [ A k exp ( j ϕ k ) ] 2 P n k } + n π 2 , where n is an integer .
SNR ( β ) = k l [ A k 2 P n k cos ( ϕ k β ) ] [ A l 2 P n l cos ( ϕ l β ) ] exp [ j ( ϕ k ϕ l ) ] k A k 2 cos 2 ( ϕ k β ) .
Γ exp ( j γ ) k A k 2 exp ( j 2 ϕ k ) P n k , R k A k 2 P n k ; R r 0 .
SNR ( β ) = R { 1 1 2 ( R Γ Γ R ) [ R Γ + cos ( 2 β γ ) ] } .
β = 1 2 γ
S ˆ R k = A k cos ( ϕ k ) P n k , S ˆ I k = A k sin ( ϕ k ) P n k , S ˆ k = A k exp ( j ϕ k ) P n k ,
β = 1 2 arg { k [ ( S ˆ R k ) 2 ( S ˆ I k ) 2 ] + j 2 k S ˆ R k S ˆ I k }
u = { H ( f ) A ( f ) exp [ + j ϕ ( f ) ] d f } { H ( f ) A ( f ) exp [ j ϕ ( f ) ] d f } ; ν = H 2 ( f ) P n ( f ) d f ; SNR = u / ν .
δ ( SNR ) = α [ α ( u α ν α ) ] α = 0 .
δ ( SNR ) = 0 , implying ν δ u = u δ ν .
H ( f ) = A ( f ) P n ( f ) cos [ ϕ ( f ) β 1 ] ; β 2 = arg H ( f ) A ( f ) exp [ j ϕ ( f ) ] d f .
β = ( 1 2 ) arg { [ S ˆ ( f ) ] 2 d f } = ( 1 2 ) arg { [ S ˆ R ( f ) ] 2 d f [ S ˆ I ( f ) ] 2 d f + j 2 S ˆ R ( f ) S ˆ I ( f ) d f }
S ˆ ( f ) = A ( f ) P n ( f ) exp [ j ϕ ( f ) ] = S ˆ R ( f ) + j S ˆ I ( f ) .
S ˆ R ( f ) S ˆ I ( f ) d f = 0 ; β = 0 .
[ S ˆ R ( f ) ] 2 d f = [ S ˆ I ( f ) ] 2 d f , then β is arbitrary .
[ S ˆ R ( f ) ] 2 d f = [ S ˆ I ( f ) ] 2 d f , then β = π 4 .
SNR RMF = 1 2 | S ˆ ( f ) | 2 d f + 1 2 | [ S ˆ ( f ) ] 2 d f | = R + Γ 2 ,
SNR RMF SNR CMF = 1 2 + 1 2 | [ S ˆ ( f ) ] 2 d f | | S ˆ ( f ) | 2 d f = 1 2 + 1 2 Γ R ,
SNR ( β ) = | [ S ˆ R ( f ) + j S ˆ I ( f ) ] [ cos β S ˆ R ( f ) + sin β S ˆ I ( f ) ] d f | 2 [ cos β S ˆ R ( f ) + sin β S ˆ I ( f ) ] 2 d f .
SNR ( β ) = cos 2 β [ S ˆ R 2 ( f ) d f ] 2 + sin 2 β [ S ˆ I 2 ( f ) d f ] 2 cos 2 β [ S ˆ R 2 ( f ) d f ] + sin 2 β [ S ˆ I 2 ( f ) d f ] .
η ˆ R = S ˆ R 2 ( f ) d f , η ˆ I = S ˆ I 2 ( f ) d f .
SNR ( β ) = cos 2 β η ˆ R 2 + sin 2 β η ˆ I 2 cos 2 β η ˆ R + sin 2 β η ˆ I .

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