Прикладные математические методы в статистической радиотехнике. Федосов В.П. - 47 стр.

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9.2.Ɉɩɬɢɦɚɥɶɧɚɹ ɨɰɟɧɤɚ ɚɦɩɥɢɬɭɞɵ ɫɢɝɧɚɥɚ
ɂɫɩɨɥɶɡɭɟɦ ɫɥɟɞɭɸɳɭɸ ɦɨɞɟɥɶ ɫɢɝɧɚɥɚ:
s(t) = a c(t), ɝɞɟ ɚ - ɚɦɩɥɢɬɭɞɚ,
ɩɨɞɥɟɠɚɳɚɹ ɨɰɟɧɤɟ;
ɫ(t) - ɡɚɞɚɟɬ ɮɨɪɦɭ ɫɢɝɧɚɥɚ ɫ ɟɞɢɧɢɱɧɨɣ ɚɦɩɥɢɬɭɞɨɣ. Ɍɨɝɞɚ
ɢɫɯɨɞɧɚɹ ɜɵɛɨɪɤɚ ɫɨɞɟɪɠɢɬ ɞɜɚ ɫɥɚɝɚɟɦɵɯ:
x(t)=s(t)+[(t); ɜɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ ɟɫɬɶ
ɲɭɦ. Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɚɥɝɨɪɢɬɦɚ ɨɰɟɧɤɢ ɚɦɩɥɢɬɭɞɵ ɨɩɪɟɞɟɥɢɦ ɮɭɧɤɰɢɨɧɚɥ
ɩɪɚɜɞɨɩɨɞɨɛɢɹ L
(ɚ), ɤɨɬɨɪɵɣ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɨɜɦɟɫɬɧɭɸ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨ-
ɹɬɧɨɫɬɢ ɟɞɢɧɢɱɧɵɯ ɢɡɦɟɪɟɧɢɣ X, ɩɨɥɭɱɟɧɧɵɯ ɧɚ ɨɫɧɨɜɟ ɞɢɫɤɪɟɬɢɡɚɰɢɢ
x(t) ɩɨ
ɬɟɨɪɟɦɟ Ʉɨɬɟɥɶɧɢɤɨɜɚ. Ɍɚɤɢɟ ɢɡɦɟɪɟɧɢɹ, ɟɫɥɢ ɲɭɦ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɧɨɪ-
ɦɚɥɶɧɵɣ ɫɥɭɱɚɣɧɵɣ ɩɪɨɰɟɫɫ, ɛɭɞɭɬ ɧɟɡɚɜɢɫɢɦɵɦɢ, ɢ ɢɯ ɫɨɜɦɟɫɬɧɚɹ ɩɥɨɬɧɨɫɬɶ
ɜɟɪɨɹɬɧɨɫɬɟɣ ɪɚɜɧɚ ɩɪɨɢɡɜɟɞɟɧɢɸ ɨɞɧɨɦɟɪɧɵɯ ɩɥɨɬɧɨɫɬɟɣ ɜɟɪɨɹɬɧɨɫɬɟɣ:
L(
a)=f(X/a)=(2S)
-n/2
V
-n
exp{-0,5V
-2
¦
n
i 1
[x
i
-ac(t
i
)]
2
}
ɢɥɢ
l(a)=(2S)
n/2
V
n
ln[L(a) 2V
2
] =
¦
n
i 1
[x
i
-ac(t
i
)]
2
.
Ɉɩɪɟɞɟɥɢɦ ɷɤɫɬɪɟɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɷɬɨɝɨ ɮɭɧɤɰɢɨɧɚɥɚ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ
ɬɪɟɬɶɢɦ ɤɪɢɬɟɪɢɟɦ ɢɡ ɩɪɢɜɟɞɟɧɧɨɝɨ ɪɚɧɟɟ ɫɩɢɫɤɚ. Ⱦɥɹ ɷɬɨɝɨ ɩɪɨɞɢɮɮɟɪɟɧɰɢɪɭ-
ɟɦ ɩɨɥɭɱɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɩɨ
ɚ ɢ ɩɪɢɪɚɜɧɹɟɦ ɩɨɥɭɱɟɧɧɭɸ ɩɪɨɢɡɜɨɞɧɭɸ ɤ ɧɭɥɸ,
ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɪɚɫɤɪɵɜ ɤɜɚɞɪɚɬ ɪɚɡɧɨɫɬɢ ɜ ɫɤɨɛɤɚɯ:
wl(a)/wa = 2
¦
n
i 1
x
i
c(t
i
)+2a*
¦
n
i 1
c
2
(t
i
) = 0,
ɨɬɤɭɞɚ ɧɚɣɞɟɦ ɨɰɟɧɢɜɚɟɦɭɸ ɚɦɩɥɢɬɭɞɭ:
a* =
¦
n
i 1
x
i
c(t
i
)/
¦
n
i 1
c
2
(t
i
)=
=(1/
E
0
)
¦
n
i 1
x
i
c(t
i
), ɝɞɟ E =
¦
n
i 1
c
2
(t
i
) - ɷɧɟɪɝɢɹ ɟɞɢɧɢɱɧɨɝɨ ɫɢɝɧɚɥɚ. ɉɨɫɥɟ ɩɟɪɟ-
ɯɨɞɚ ɤ ɧɟɩɪɟɪɵɜɧɨɣ ɪɟɚɥɢɡɚɰɢɢ
x(t) ɩɪɢ nof ɩɨɥɭɱɢɦ a*= =
³
T
0
x(t)c(t)dt, ɫɥɟɞɨ-
ɜɚɬɟɥɶɧɨ, ɚɥɝɨɪɢɬɦ ɨɰɟɧɤɢ ɚɦɩɥɢɬɭɞɵ ɨɫɧɨɜɚɧ ɧɚ ɜɡɚɢɦɨɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɨɛɪɚ-
ɛɨɬɤɟ ɪɟɚɥɢɡɚɰɢɢ x(t) ɢ ɢɡɜɟɫɬɧɨɝɨ ɟɞɢɧɢɱɧɨɝɨ ɫɢɝɧɚɥɚ c(t). Ɉɩɪɟɞɟɥɢɦ ɤɚɱɟɫɬɜɨ
ɩɨɥɭɱɟɧɧɨɣ ɨɰɟɧɤɢ, ɬ.ɟ. ɫɦɟɳɟɧɢɟ ɢ ɞɢɫɩɟɪɫɢɸ. ɋɦɟɳɟɧɢɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɦɚɬɟ-
ɦɚɬɢɱɟɫɤɢɦ ɨɠɢɞɚɧɢɟɦ ɨɰɟɧɤɢ a*:
¢a*²=(1/E
0
)¢
³
T
0
[[(t)+a
0
c(t)]c(t)dt² = ¢(a
0
/E
0
)
³
T
0
c
2
(t)dt² = a
0
,

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