Abstract

The objective is to distinguish the true target from point-target imitators and from extended-target clutter in the exoatmospheric regime. Matched filters are carefully studied from the viewpoint of SNR enhancement and pulse recognition. The matched filter structure takes into account photon noise, modulation noise, generation-recombination (GR) noise, contact noise, and various thermal noise sources. A multicolor radiant-intensity structure for target discrimination is developed by analyzing the uncertainties in such target irradiance parameters as range, temperature, projected area, and emissivity. Bias terms, variances, and other statistical descriptors are derived. Certain statistical discrimination techniques are discussed that exploit the radiant-intensity format. Helstrom’s method for processing radar signals is adapted to a four-channel pulse-recognition system for which degradation due to arrival time delays and mismatched filters is discussed.

© 1980 Optical Society of America

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References

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  1. J. Salz, IEEE Trans. Inf. Theory IT-15, 644 (1969).
    [CrossRef]
  2. B. Mandelbrot, IEEE Trans. Inf. Theory IT-13, 289 (1967).
    [CrossRef]
  3. R. C. Jones, Proc. IRE 47, 1481 (1959).
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  4. G. L. Turin, IRE Trans. Inf. Theory IT-6, 311 (1960).
    [CrossRef]
  5. R. E. Paley, N. Wiener, Fourier Transforms in the Complex Domain (American Mathematical Society, Chicago, Colloquium Pub. 10, 1934).
  6. J. L. Doob, Stochastic Processes (Wiley, New York, 1953).
  7. N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series (Wiley, New York, 1949).
  8. B. P. Lathi, Signals, Systems and Communication (Wiley, New York, 1965).
  9. F. McNolty, Math. Comp. 27, 495 (1973).
    [CrossRef]
  10. I. T. Young, IEEE Trans. Inf. Theory IT-24, 773 (1978).
    [CrossRef]
  11. H. Hotelling, in Proceedings, Second Berkeley Symposium on Mathematical Statistics and Probability (U. of Calif. Press, Berkeley, 1951), pp. 23–42.
  12. T. W. Anderson, An Introduction to Multivariate Statistical Analysis (Wiley, New York, 1958).
  13. P. C. Mahalanobis, Proc. Natl. Inst. Sci. India 12, 49 (1936).
  14. C. Helstrom, Statistical Theory of Signal Detection (Pergamon, Oxford, England, 1960).
  15. N. J. Nilsson, Trans. IRE IT-7, 245 (1961).

1978 (1)

I. T. Young, IEEE Trans. Inf. Theory IT-24, 773 (1978).
[CrossRef]

1973 (1)

F. McNolty, Math. Comp. 27, 495 (1973).
[CrossRef]

1969 (1)

J. Salz, IEEE Trans. Inf. Theory IT-15, 644 (1969).
[CrossRef]

1967 (1)

B. Mandelbrot, IEEE Trans. Inf. Theory IT-13, 289 (1967).
[CrossRef]

1961 (1)

N. J. Nilsson, Trans. IRE IT-7, 245 (1961).

1960 (1)

G. L. Turin, IRE Trans. Inf. Theory IT-6, 311 (1960).
[CrossRef]

1959 (1)

R. C. Jones, Proc. IRE 47, 1481 (1959).
[CrossRef]

1936 (1)

P. C. Mahalanobis, Proc. Natl. Inst. Sci. India 12, 49 (1936).

Anderson, T. W.

T. W. Anderson, An Introduction to Multivariate Statistical Analysis (Wiley, New York, 1958).

Doob, J. L.

J. L. Doob, Stochastic Processes (Wiley, New York, 1953).

Helstrom, C.

C. Helstrom, Statistical Theory of Signal Detection (Pergamon, Oxford, England, 1960).

Hotelling, H.

H. Hotelling, in Proceedings, Second Berkeley Symposium on Mathematical Statistics and Probability (U. of Calif. Press, Berkeley, 1951), pp. 23–42.

Jones, R. C.

R. C. Jones, Proc. IRE 47, 1481 (1959).
[CrossRef]

Lathi, B. P.

B. P. Lathi, Signals, Systems and Communication (Wiley, New York, 1965).

Mahalanobis, P. C.

P. C. Mahalanobis, Proc. Natl. Inst. Sci. India 12, 49 (1936).

Mandelbrot, B.

B. Mandelbrot, IEEE Trans. Inf. Theory IT-13, 289 (1967).
[CrossRef]

McNolty, F.

F. McNolty, Math. Comp. 27, 495 (1973).
[CrossRef]

Nilsson, N. J.

N. J. Nilsson, Trans. IRE IT-7, 245 (1961).

Paley, R. E.

R. E. Paley, N. Wiener, Fourier Transforms in the Complex Domain (American Mathematical Society, Chicago, Colloquium Pub. 10, 1934).

Salz, J.

J. Salz, IEEE Trans. Inf. Theory IT-15, 644 (1969).
[CrossRef]

Turin, G. L.

G. L. Turin, IRE Trans. Inf. Theory IT-6, 311 (1960).
[CrossRef]

Wiener, N.

R. E. Paley, N. Wiener, Fourier Transforms in the Complex Domain (American Mathematical Society, Chicago, Colloquium Pub. 10, 1934).

N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series (Wiley, New York, 1949).

Young, I. T.

I. T. Young, IEEE Trans. Inf. Theory IT-24, 773 (1978).
[CrossRef]

IEEE Trans. Inf. Theory (3)

J. Salz, IEEE Trans. Inf. Theory IT-15, 644 (1969).
[CrossRef]

B. Mandelbrot, IEEE Trans. Inf. Theory IT-13, 289 (1967).
[CrossRef]

I. T. Young, IEEE Trans. Inf. Theory IT-24, 773 (1978).
[CrossRef]

IRE Trans. Inf. Theory (1)

G. L. Turin, IRE Trans. Inf. Theory IT-6, 311 (1960).
[CrossRef]

Math. Comp. (1)

F. McNolty, Math. Comp. 27, 495 (1973).
[CrossRef]

Proc. IRE (1)

R. C. Jones, Proc. IRE 47, 1481 (1959).
[CrossRef]

Proc. Natl. Inst. Sci. India (1)

P. C. Mahalanobis, Proc. Natl. Inst. Sci. India 12, 49 (1936).

Trans. IRE (1)

N. J. Nilsson, Trans. IRE IT-7, 245 (1961).

Other (7)

C. Helstrom, Statistical Theory of Signal Detection (Pergamon, Oxford, England, 1960).

H. Hotelling, in Proceedings, Second Berkeley Symposium on Mathematical Statistics and Probability (U. of Calif. Press, Berkeley, 1951), pp. 23–42.

T. W. Anderson, An Introduction to Multivariate Statistical Analysis (Wiley, New York, 1958).

R. E. Paley, N. Wiener, Fourier Transforms in the Complex Domain (American Mathematical Society, Chicago, Colloquium Pub. 10, 1934).

J. L. Doob, Stochastic Processes (Wiley, New York, 1953).

N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series (Wiley, New York, 1949).

B. P. Lathi, Signals, Systems and Communication (Wiley, New York, 1965).

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Figures (8)

Fig. 1
Fig. 1

Candidate filtering configurations.

Fig. 2
Fig. 2

Noise-whitening filter.

Fig. 3
Fig. 3

Matched filter.

Fig. 4
Fig. 4

Ninety-percent constant probability contours for a typical threat object.

Fig. 5
Fig. 5

Ninety-percent constant probability contours for a simplified threat containing only one real target and five imitators. Angular and linear scales are distorted.

Fig. 6
Fig. 6

Input-output for the ith matched filter.

Fig. 7
Fig. 7

Decomposition of inputs to Hw(ω) and Hi(ω)

Fig. 8
Fig. 8

Four-channel implementation of maximum likelihood estimation.

Tables (1)

Tables Icon

Table I Typical Signal-to-Noise Power Ratios; τ1 = 2 × 10−3 sec, τ2 = 2 × 10−2 sec, (S/N)P at a convenient scale

Equations (127)

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M ( f ) = N 1 / 2 π f b ,
K 1 = F ( 0 ) = ( 2 γ / ν ) π ( watt - seconds ) ,
h ^ ^ ( x , y ) = h ^ ( x , y ) / A col = h λ ( x , y ) / H λ · L ( λ ) · A col .
h λ ( u , v ) = L ( λ ) P col , λ · δ ( u , v ) = L ( λ ) H λ · A col δ ( u , v )
- h λ ( x , y ) d x d y = L ( λ ) A col H λ .
A d h λ ( x , y ) d x d y = L ( λ ) A col H λ
P in ( watts ) = λ 1 λ 2 P col , λ d λ ~ A col · L ( λ ) ¯ λ 1 λ 2 H λ d λ = H · A col · L ¯ .
S p = P in s ( Δ ) = H · A col · L ¯ · s ( Δ ) .
R 2 · H = R 2 λ 1 λ 2 H λ d λ .
β ( t ) = Δ λ · A d h ^ ^ ( x - x 0 - ν t , y - y 0 ) d x d y ~ exp ( - t 2 / 2 a ) ,
F ( f ) = K 1 exp [ - ( a / 2 ) ( 2 π f ) 2 ] .
W ( f ) = R E / ( 1 + j 2 π f τ 1 ) ; w ( t ) = ( R E / τ 1 ) · exp ( - t / τ 1 ) , t > 0.
N d ( ω ) = { N 1 + ω [ ( N p + N q ) R E 2 + N 01 ] + N 1 τ 1 2 ω 2 + N 01 τ 1 2 ω 3 } × [ ω ( 1 + τ 1 2 ω 2 ) ] - 1 , - < ω < .
s d ( t ) = ( R E K 1 / 2 τ 1 ) · exp ( - t / τ 1 + a / 2 τ 1 2 ) · { Φ [ t / ( 2 a ) 1 / 2 - a / τ 1 2 ] + 1 } ,
Φ ( x ) = erf ( x ) = ( 2 / π ) 0 x exp ( - t 2 ) d t ,
G ( f ) = j 2 π f τ 2 ( 1 + j 2 π f τ 2 ) - 1 ; g ( t ) = δ ( t ) - ( 1 / τ 2 ) exp ( - t / τ 2 ) , t > 0 ,
N c ( ω ) = ( N 02 + N 1 τ 2 2 ω + P ω 2 + Q ω 3 + R ω 4 ) × [ 1 + ( τ 1 2 + τ 2 2 ) ω 2 + τ 1 2 τ 2 2 ω 4 ] - 1 ,
c ( t ) = [ K 1 R E τ 2 / 2 ( τ 2 - τ 1 ) ] · [ l ( t , τ 1 ) m ( t , τ 1 ) - l ( t , τ 2 ) m ( t , τ 2 ) ] ,
l ( t , τ i ) = ( 1 / τ i ) exp ( - t / τ i + a / 2 τ i 2 ) ,
m ( t , τ i ) = Φ [ t / ( 2 a ) 1 / 2 - a / τ i 2 ] + 1
H A ( f ) = K A C * ( f ) exp ( - j 2 π f Δ ) / N c ( f ) ,
C ( f ) = - c ( t ) exp ( - j 2 π f t ) d t .
N m A ( ω ) = B A ω 2 exp ( - a ω 2 ) × ( N 02 + N 1 τ 2 2 ω + P ω 2 + Q ω 3 + R ω 4 ) - 1 , - < ω < ,
s A ( t ) = - S A ( f ) exp ( j 2 π f t ) d f = ( 1 / 2 π K A ) - N m A ( ω ) exp [ - j ω ( Δ - t ) ] d ω = - C ( f ) H A ( f ) exp ( j 2 π f t ) d f = K A - C ( f ) 2 [ N c ( f ) ] - 1 · exp [ - j 2 π f ( Δ - f ) ] d f , R m A ( τ ) = ( 1 / 2 π ) · - N m A ( ω ) exp ( j ω τ ) d ω = ( K A / 2 π ) · - S A ( ω ) exp [ j ω ( Δ + τ ) ] d ω = K A s A ( Δ + τ ) , H A ( f ) = [ K A C ( f ) ] - 1 N m A ( f ) exp ( - j 2 π f Δ ) , N m A ( f ) = K A 2 C ( f ) 2 / N c ( f ) = N c ( f ) H A ( f ) 2 , σ n 2 = R m A ( 0 ) = K A s A ( Δ ) = - N m A ( f ) d f = K A 2 ( S / N ) p = E [ n 2 ( t ) ] = K A 2 [ s A ( Δ ) ] 2 / R m A ( 0 ) .
H B ( f ) = K B S d * ( f ) exp ( - j 2 π f Δ ) / N d ( f ) ,
S d ( f ) = - s d ( t ) exp ( - j 2 π f t ) d t .
N m B ( ω ) = B B ω exp ( - a ω 2 ) × ( N 1 + E ω + N 1 τ 2 ω 2 + N 01 τ 1 2 ω 3 ) - 1 - < ω < ,
H p ( ω ) = j ω τ a [ ( 1 + j ω τ a ) ( 1 + j ω τ b ) ] - 1 = [ ( 1 + 1 / ω 2 τ a 2 ) ( 1 + ω 2 τ b 2 ) ] - 1 / 2 · exp { j [ arctan ( 1 / ω τ a ) - arctan ω τ b ] } ,
h p ( t ) = [ 2 π ( τ b - τ a ) ] - 1 exp ( - t / τ a ) + τ a [ 2 π ( τ a - τ b ) τ b ] - 1 · exp ( - t / τ b ) , t > 0.
N p ( ω ) = N c ( ω ) ( 1 + ω 2 τ b 2 ) - 1 + N 03 ,
s p ( t ) = ( K 1 R E τ a / 2 ) { [ ( τ a - τ 1 ) ( τ 1 - τ b ) ] - 1 τ 1 · l ( t , τ 1 ) m ( t , τ 1 ) - [ ( τ a - τ 1 ) ( τ a - τ b ) ] - 1 · τ a · l ( t , τ a ) m ( t , τ a ) + [ ( τ b - τ 1 ) ( τ a - τ b ) ] - 1 · τ b l ( t , τ b ) · m ( t , τ b ) } ,
H q ( f ) = K q C * ( f ) exp ( - j 2 π f Δ ) / Γ ( f ) ,
N q ( ω ) = B q ω 2 · exp ( - a ω 2 ) ( N 02 + P ω 2 + R ω 4 ) - 1 , - < ω < ,
s q ( t ) = B q [ 2 K q R ( r 1 - r 2 ) ] - 1 · [ z ( r 1 ) - z ( r 2 ) ] , t > 0 ,
R r 2 - P r + N 02 = 0 , r 1 r 2 .
s q ( t ) = B q [ 2 K q R ( N 2 - Q 2 ) ] - 1 · { g 1 ( N , t ) g 2 ( - N , t ) + g 1 ( - N , t ) g 2 ( N , t ) - g 1 ( Q , t ) g 2 ( - Q , t ) - g 1 ( - Q , t ) g 2 ( Q , t ) } ,
E [ n 2 ( t ) ] = B q [ 2 R ( N 2 - Q 2 ) ] - 1 · [ g 1 ( N , O ) g 2 ( - N , O ) + g 1 ( - N , O ) g 2 ( N , O ) - g 1 ( Q , O ) g 2 ( - Q , O ) - g 1 ( - Q , O ) g 2 ( Q , O ) ] ,
H I ( f ) = ( j 2 π f ) - 1 [ 1 - exp ( - j 2 π f T ] .
h I ( t ) = { 1 for 0 < t < T , 1 2 for t = 0 and t = T , 0 otherwise .
N I ( ω ) = 8 ( N 02 + N 1 τ 2 2 ω + P ω 2 + Q ω 3 + R ω 4 ) · [ sin ( ω T / 2 ) ] 2 · [ ω 2 ( 1 + τ 1 2 ω 2 ) ( 1 + τ 2 2 ω 2 ) ] - 1 , - < ω < .
s I ( t ) = K 1 R E τ 2 [ 2 ( τ 1 - τ 2 ) ] - 1 { τ 1 l ( τ , τ 1 ) · [ m ( t , τ 1 ) - exp ( T / τ 1 ) m ( t - T , τ 1 ) ] + τ 2 l ( t , τ 2 ) [ - m ( t , τ 2 ) + exp ( T / τ 2 ) m ( t - T , τ 2 ) ] } , t > 0 ,
E [ n 2 ( t ) ] = N 02 T + ( R - P τ 1 2 + N 02 τ 1 4 ) f 1 ( τ 1 ) ( 4 τ 1 τ 2 ) - 1 + ( P τ 2 2 - N 02 τ 2 4 - R ) f 1 ( τ 2 ) ( 4 τ 2 τ 2 ) - 1 + 2 N 1 τ 2 2 C / π + ( N 1 τ 2 2 τ 1 2 - Q ) [ f 2 ( - τ 1 ) f 3 * ( τ 1 ) + f 2 ( τ 1 ) f 3 ( - τ 1 ) ] ( π τ 2 ) - 1 + ( Q - N 1 τ 2 4 ) [ f 2 ( - τ 2 ) f 3 * ( τ 2 ) + f 2 ( τ 2 ) f 3 ( - τ 2 ) ] ( π τ 2 ) - 1 + ( 2 / π τ 2 ) [ ( N 1 τ 2 2 τ 1 2 - Q ) ln τ 1 + ( Q - N 1 τ 2 4 ) ln τ 2 ] ,
E i ( x ) = - - x ( 1 / t ) exp ( - t ) d t ,
H m ( ω ) = K S 0 * ( ω ) exp ( - j ω Δ ) = K S i * ( ω ) exp ( - j ω Δ ) / [ N i ( ω ) ] 1 / 2
N i ( λ ) = N i + ( λ ) N i - ( λ ) ,
- ln N i ( ω ) ( 1 + ω 2 ) - 1 d ω < ,
- ln H m ( ω ) ( 1 + ω 2 ) - 1 d ω < , H m ( ω ) L 2 ,
H m ( ω ) = [ 2 π N i + ( ω ) ] - 1 0 exp ( - j ω t ) × - S ( - u ) [ N i - ( u ) ] - 1 exp [ j u ( t - Δ ) ] d u d t .
H λ = watts / ( μ m · cm 2 ) = [ η ( λ ) A F 2 H λ , terr / π R 2 ] + [ η ( λ ) A F 1 H λ , solar / π R 2 ] + [ ( λ ) A W λ ( T ) / π R 2 ] ,
G ( T , λ 1 , λ 2 ) = G ( T ) = λ 1 λ 2 W λ ( T ) d λ = c 1 λ 1 λ 2 { λ 5 [ exp ( c 2 / λ T ) - 1 ] } - 1 d λ ,
Q terr = λ 1 λ 2 H λ , terr d λ , Q solar = λ 1 λ 2 H λ , solar d λ = 1.82 × 10 - 5 G ( 6000 K , λ 1 , λ 2 ) .
F ( T , λ 1 , λ 2 ) = F ( T ) = d G ( T , λ 1 , λ 2 ) / d T ,
M ( T , λ 1 , λ 2 ) = M ( T ) = d F ( T , λ 2 , λ 2 ) / d T .
H = η A F 2 Q terr / π R 2 + η A F 1 Q solar / π R 2 + A G ( T ) / π R 2 ,
H = f ( T , R , A , ) = K / π R 2 = A K / π R 2 = P in / A col · L ,
J ( λ 1 , λ 2 ) = J = R 2 · H = R 2 λ 1 λ 2 H λ d λ ,
d f = f R · Δ R + f A · Δ A + f T · Δ T + f · Δ + 1 2 · 2 f R 2 ( Δ R ) 2 + 1 2 · 2 f A 2 ( Δ A ) 2 + 1 2 · 2 f T 2 ( Δ T ) 2 + 1 2 · 2 f 2 · ( Δ ) 2 + 2 f R A ( Δ R ) ( Δ A ) + 2 f R T ( Δ R ) ( Δ T ) + 2 f R ( Δ R ) ( Δ ) + 2 f A T ( Δ A ) ( Δ T ) + 2 f A ( Δ A ) ( Δ ) + 2 f T ( Δ T ) ( Δ ) + ,
ν = R 2 · d f + γ = a x + a 2 y + a 1 z + ( d / π ) w + b x 2 + b 1 z 2 + c 1 x y + c 2 x z - ( 2 x / π R ) x w + c 3 y z + ( d 2 / π ) y w + ( d 1 / π ) z w + γ ,
u = a x + a 2 y + a 1 z + b x 2 + b 1 z 2 + c 1 x y + c 2 x z + c 3 y z + γ .
E ( ν ) = b σ R 2 + b 1 σ T 2 + ( d 1 / π ) σ T σ ρ T ,
E ( u ) = b σ R 2 + b 1 σ T 2 ,
σ ν 2 = a 2 σ R 2 + a 2 2 σ A 2 + a 1 2 σ T 2 + ( d 2 / π 2 ) σ 2 + 2 b 2 σ R 4 + 2 b 1 2 σ T 4 + c 1 2 σ R 2 σ A 2 + c 2 2 σ R 2 σ T 2 + ( 4 d 2 / π 2 R 2 ) σ R 2 σ 2 + c 3 2 σ A 2 σ T 2 + ( d 2 2 / π 2 ) σ A 2 σ 2 + ( d 1 2 / π 2 ) σ T 2 σ 2 + ( d 1 2 / π 2 ) σ 2 σ T 2 ρ T 2 + σ N 2 + 2 a 1 ( d / π ) ρ T σ σ T + 2 ( c 3 d 2 / π ) σ A 2 σ σ T ρ T - ( 4 c 2 d / π R ) σ R 2 σ σ T ρ T + ( 4 b 1 d 1 / π ) σ T 3 σ ρ T ,
J 1 ( λ 1 , λ 2 ) ν 1 = ( a ) 1 x + ( a 2 ) 1 y + ( a 1 ) 1 z + [ ( d ) 1 / π ] w + ,
J 2 ( λ 3 , λ 4 ) ν 2 = ( a ) 2 x + ( a 2 ) 2 y + ( a 1 ) 2 z + [ ( d ) 2 / π ] w + .
M 2 = E [ ( ν 1 - ν ¯ 1 ) ( ν 2 - ν ¯ 2 ) ] = ( a ) 1 ( a ) 2 σ R 2 + ( a 2 ) 1 ( a 2 ) 2 σ A 2 × ( a 1 ) 1 ( a 1 ) 2 σ T 2 + [ ( a 1 ) 1 ( d ) 2 / π ] ρ T σ σ T + [ ( d ) 1 ( a 1 ) 2 / π ] ρ T σ σ T + [ ( d 1 ) ( d ) 2 / π 2 ] σ 2 + 2 ( b ) 1 ( b ) 2 σ R 4 + 2 ( b 1 ) 1 [ ( d 1 ) 2 / π ] ρ T σ T 3 σ + 2 ( b 1 ) 1 ( b 1 ) 2 σ T 4 + ( c 1 ) 1 ( c 1 ) 2 σ R 2 σ A 2 + ( c 2 ) 1 ( c 2 ) 2 σ R 2 σ T 2 - [ 2 ( c 2 ) 1 ( d ) 2 / π R ] σ R 2 ρ T σ σ T - [ 2 ( c 2 ) 2 ( d ) 1 / π R ] σ R 2 ρ T σ σ T + [ 4 ( d ) 1 ( d ) 2 / π 2 R 2 ] σ R 2 σ 2 + ( c 3 ) 1 ( c 3 ) 2 σ A 2 σ T 2 + [ ( c 3 ) 1 ( d 2 ) 2 / π ] σ A 2 ρ T σ σ T + [ ( d 2 ) 1 ( c 3 ) 2 / π ] σ A 2 ρ T σ σ T + [ ( d 2 ) 1 ( d 2 ) 2 / π 2 ] σ A 2 σ 2 + 2 [ ( d 1 ) 1 ( b 1 ) 2 / π ] ρ T σ T 3 σ + [ ( d 1 ) 1 ( d 1 ) 2 / π 2 ] σ 2 σ T 2 + [ ( d 1 ) 1 ( d 1 ) 2 / π 2 ] ρ T 2 σ 2 σ T 2 .
α = ( x - μ ) Σ - 1 ( x - μ ) ,
x ¯ = ( 1 / n ) i = 1 n x i , s = [ 1 / ( n - 1 ) ] i = 1 n ( x i - x ¯ ) ( x i - x ¯ ) ,
β = ( x - x ¯ ) s - 1 ( x - x ¯ )
F = ( n - p ) n [ ( n 2 - 1 ) p ] - 1 · β
E ( β ) = ( n 2 - 1 ) p [ n ( n - p - 2 ) ] - 1 ,
Var ( β ) = 2 ( n - 2 ) p ( n 2 - 1 ) 2 × [ n 2 ( n - p - 2 ) 2 ( n - p - 4 ) ] - 1 = 2 ( n - 2 ) · [ p ( n - p - 4 ) ] - 1 · [ E ( β ) ] 2 .
γ = n ( x ¯ - μ ) Σ - 1 ( x ¯ - μ )
F = ( n - p ) [ p ( n - 1 ) ] - 1 · T 2
T 2 = n ( x ¯ - μ ) s - 1 ( x ¯ - μ )
U = D - L ,
D = x Σ - 1 ( μ T - μ I ) ,
L = ( ½ ) ( μ T + μ I ) Σ - 1 ( μ T - μ I ) ,
E ( U ) = d / 2             for x = x T = - d / 2             for x = x I
Var ( U ) = d             for x = x T or x I .
d = ( μ T - μ I ) Σ - 1 ( μ T - μ I ) ,
x ( Σ T - 1 - Σ I - 1 ) x + 2 ( μ I Σ I - 1 - μ T Σ T - 1 ) x + μ I Σ I - 1 μ I - μ T Σ T - 1 μ T + log [ Σ I / Σ T ] < 0
v 1 ( t ) = A 1 s 1 ( t - t 0 ) + A 2 s 2 ( t - t 0 ) + A 3 s 3 ( t - t 0 ) + A 4 s 4 ( t - t 0 ) + n ( t ) ,
v ( t ) = A 1 a 1 ( t - t 0 ) + A 2 a 2 ( t - t 0 ) + A 3 a 3 ( t - t 0 ) + A 4 a 4 ( t - t 0 ) + n 1 ( t ) ,
S i ( f ) = - s i ( t - t 0 ) exp ( - j 2 π f t ) d t , β i ( f ) = S i ( f ) / N + ( f ) = - a i ( t - t 0 ) exp ( - j 2 π f t ) d t ,
u i ( t ) = - U i ( f ) exp ( j 2 π f t ) d f = - β i ( f ) H i ( f ) exp ( j 2 π f t ) d f = ( K i / c i ) · - β i ( f ) 2 exp [ - j 2 π f ( Δ - t ) ] d f = ( c i / 2 π K i ) - N 2 ( ω ) exp [ - j ω ( Δ - t ) ] d ω .
c i 2 = 0 Δ a i 2 ( t - t 0 ) d t , λ i j = ( 1 / c i c j ) 0 Δ a i ( t - t 0 ) a j ( t - t o ) d t ,
R i ( Δ ) = R i ( t 0 ) = 0 Δ a i ( t - t 0 ) v ( t ) d t ,
R i ( t ) = 0 t a i ( Δ - τ - t 0 ) v ( t - τ ) d τ ,
h i ( τ ) = ( K i / c i ) a i ( Δ - τ - t 0 ) for τ 0 , = 0 for τ < 0 ,
A ^ 1 = ( 1 / c 1 ϕ ) [ R 1 ( t 0 ) x 1 / c 1 + R 2 ( t 0 ) x 2 / c 2 + R 3 ( t 0 ) x 3 / c 3 + R 4 ( t 0 ) x 4 / c 4 ] ,
A ^ 2 = ( 1 / c 2 ϕ ) [ R 1 x 2 / c 1 + R 2 x 5 / c 2 + R 3 x 6 / c 3 + R 4 x 7 / c 4 ] ,
A ^ 3 = ( 1 / c 3 ϕ ) [ R 1 x 3 / c 1 + R 2 x 6 / c 2 + R 3 x 8 / c 3 + R 4 x 9 / c 4 ] ,
A ^ 4 = ( 1 / c 4 ϕ ) [ R 1 x 4 / c 1 + R 2 x 7 / c 2 + R 3 x 9 / c 3 + R 4 x 10 / c 4 ] ,
x 1 = 1 + 2 λ 23 λ 24 λ 34 - λ 23 2 - λ 24 2 - λ 34 2 , x 2 = - λ 12 + λ 13 λ 23 + λ 14 λ 24 - λ 14 λ 23 λ 34 - λ 13 λ 24 λ 34 + λ 12 λ 34 2 , x 3 = - λ 13 + λ 12 λ 23 + λ 14 λ 34 - λ 12 λ 24 λ 34 - λ 14 λ 24 λ 23 + λ 13 λ 24 2 , x 4 = - λ 14 + λ 12 λ 24 + λ 13 λ 34 - λ 12 λ 23 λ 34 - λ 13 λ 23 λ 24 + λ 14 λ 23 2 , x 5 = 1 + 2 λ 13 λ 14 λ 34 - λ 13 2 - λ 14 2 - λ 34 2 , x 6 = - λ 23 + λ 12 λ 13 + λ 24 λ 34 - λ 12 λ 14 λ 34 - λ 13 λ 14 λ 24 + λ 14 2 λ 23 , x 7 = - λ 24 + λ 12 λ 14 + λ 23 λ 34 - λ 12 λ 13 λ 34 - λ 13 λ 14 λ 23 + λ 13 2 λ 24 , x 8 = 1 + 2 λ 12 λ 14 λ 24 - λ 12 2 - λ 14 2 - λ 24 2 , x 9 = - λ 34 + λ 13 λ 14 + λ 23 λ 24 - λ 12 λ 13 λ 24 - λ 12 λ 14 λ 23 + λ 12 2 λ 34 , x 10 = 1 + 2 λ 12 λ 13 λ 23 - λ 12 2 - λ 13 2 - λ 23 2 , ϕ = 1 - λ 12 2 - λ 13 2 - λ 14 2 - λ 23 2 - λ 24 2 - λ 34 2 + 2 λ 12 λ 13 λ 23 + 2 λ 12 λ 14 λ 24 + 2 λ 13 λ 14 λ 34 + 2 λ 23 λ 24 λ 34 + λ 12 2 λ 34 2 + λ 13 2 λ 24 2 + λ 14 2 λ 23 2 - 2 λ 12 λ 13 λ 24 λ 34 - 2 λ 13 λ 14 λ 23 λ 24 - 2 λ 12 λ 14 λ 23 λ 34 .
E ( A ^ i ) = A i when signal a i ( t ) is present , = 0 otherwise .
v ( t ) = A 1 a 1 ( t - t 0 ) + A 3 a 3 ( t - t 0 ) + n 1 ( t ) , E ( A ^ 3 ) = A 3 .
Var ( A ^ 1 ) = ( 1 / c 1 2 ϕ ) ( 1 - λ 23 2 - λ 24 2 - λ 34 2 + 2 λ 23 λ 24 λ 34 ) ,
Var ( A ^ 2 ) = ( 1 / c 2 2 ϕ ) ( 1 - λ 13 2 - λ 14 2 - λ 34 2 + 2 λ 13 λ 14 λ 34 ) ,
Var ( A ^ 3 ) = ( 1 / c 3 2 ϕ ) ( 1 - λ 12 2 - λ 14 2 - λ 24 2 + 2 λ 12 λ 14 λ 24 ) ,
Var ( A ^ 4 ) = ( 1 / c 4 2 ϕ ) ( 1 - λ 12 2 - λ 13 2 - λ 23 2 + 2 λ 12 λ 13 λ 23 )
cov ( A ^ 1 , A ^ 2 ) = x 2 / c 1 c 2 ϕ cov ( A ^ 1 , A ^ 3 ) = x 3 / c 1 c 3 ϕ , cov ( A ^ 1 , A ^ 4 ) = x 4 / c 1 c 4 ϕ , cov ( A ^ 2 , A ^ 3 ) = x 6 / c 2 c 3 ϕ , cov ( A ^ 2 , A ^ 1 ) = x 7 / c 2 c 4 ϕ cov ( A ^ 3 , A ^ 4 ) = x 9 / c 3 c 4 ϕ .
A ^ 1 = ( 1 / β c 1 ) [ R 1 ( 1 - λ 23 2 ) / c 1 - R 2 ( λ 12 - λ 13 λ 23 ) / c 2 - R 3 ( λ 13 - λ 12 λ 23 ) / c 3 ] ,
A ^ 2 = ( 1 / β c 2 ) [ R 2 ( 1 - λ 13 2 ) / c 2 - R 1 ( λ 12 - λ 23 ) / c 1 - R 3 ( λ 23 - λ 12 λ 13 ) / c 3 ] ,
A ^ 3 = ( 1 / β c 3 ) [ R 3 ( 1 - λ 12 2 ) / c 3 - R 1 ( λ 13 - λ 12 λ 23 ) / c 1 - R 2 ( λ 23 - λ 12 λ 13 ) / c 2 ] ,
E ( A ^ i ) = A i when signal a i ( t ) is present , = 0 otherwise .
Var ( A ^ j ) = ( 1 / β c j 2 ) ( 1 - λ i k 2 ) , j i , k ; i k ,
cov ( A ^ i , A ^ k ) = ( 1 / β c i c k ) ( λ i j λ j k - λ i k ) , j i , k ; i k
ρ 12 = ( λ 13 λ 23 - λ 12 ) [ ( 1 - λ 23 2 ) ( 1 - λ 13 2 ) ] - 1 / 2 .
v ( t ) = A 1 a 1 ( t - t 0 ) + A 2 a 2 ( t - t 0 - ) + n 1 ( t ) ,
E ( A ^ 1 ) = A 1 - A 2 · [ δ ( ) / c 1 - λ γ ( ) / c 2 ] · [ c 1 ( 1 - λ 2 ) ] - 1 ,
E ( A ^ 2 ) = A 2 - A 2 · [ γ ( ) / c 2 - λ δ ( ) / c 1 ] · [ c 2 ( 1 - λ 2 ) ] - 1 ,
E ( A ^ 1 ) = A 1 - A 1 · [ β ( ) / c 1 - λ δ ( ) / c 2 ] · [ c 1 ( 1 - λ 2 ) ] - 1 ,
E ( A ^ 2 ) = A 2 - A 1 · [ δ ( ) / c 2 - λ β ( ) / c 1 ] · [ c 2 ( 1 - λ 2 ) ] - 1 ,
δ ( ) = 0 Δ a 1 ( t - t 0 ) [ a 2 ( t - t 0 ) - a 2 ( t - t 0 - ) ] d t ,
β ( ) = 0 Δ a 1 ( t - t 0 ) [ a 1 ( t - t 0 ) - a 1 ( t - t 0 - ) ] d t ,
γ ( ) = 0 Δ + a 2 ( t - t 0 ) [ a 2 ( t - t 0 ) - a 2 ( t - t 0 - ) ] d t λ = λ 12 = ( 1 / c 1 c 2 ) · 0 Δ a 1 ( t - t 0 ) a 2 ( t - t 0 ) d t .
lim δ ( ) = λ c 1 c 2 ,             lim β ( ) = c 1 2 ,             lim γ ( ) = c 2 2 ,
lim E ( A ^ 1 ) = A 1 ,             lim E ( A ^ 2 ) = 0 ,
lim E ( A ^ 1 ) = 0 ,             lim E ( A ^ 2 ) = A 2 .
E ( A ^ 1 2 ) = A 1 2 + [ c 1 2 ( 1 - λ 2 ) ] - 1 + A 2 2 [ θ ( ) / c 1 - λ ϕ ( ) / c 2 ] 2 · [ c 1 2 ( 1 - λ 2 ) 2 ] - 1 + 2 A 1 A 2 [ θ ( ) - λ c 1 ϕ ( ) / c 2 ] · [ c 1 2 ( 1 - λ 2 ) ] - 1 ,
h 1 ( τ ) = ( K 1 / c 1 ) a 1 ( Δ - τ - t 0 ) h 2 ( τ ) = ( K 2 / c 2 ) a 2 ( Δ - τ - t 0 ) } ,
v ( t ) = A 1 a 1 ( t - t 0 ) + A 3 a 3 ( t - t 0 ) + n 1 ( t ) ,
E ( A ^ 1 ) = A 1 + A 3 c 3 ( λ 13 - λ λ 23 ) · [ c 1 ( 1 - λ 2 ) ] - 1 ,
E ( A ^ 2 ) = A 3 c 3 ( λ 23 - λ λ 13 ) · [ c 2 ( 1 - λ 2 ) ] - 1 ,
E ( A ^ 1 ) = A 1 - A 3 ( F / c 1 - λ G / c 2 ) · [ c 1 ( 1 - λ 2 ) ] - 1 ,
E ( A ^ 2 ) = A 3 + A 3 ( λ F / c 1 - G / c 2 ) · [ c 2 ( 1 - λ 2 ) ] - 1 ,
F = 0 Δ a 1 ( t - t 0 ) g ( t ) d t , G = 0 Δ a 2 ( t - t 0 ) g ( t ) d t , g ( t ) = a 2 ( t - t 0 ) - a 3 ( t - t 0 ) .

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