Прикладные математические методы в статистической радиотехнике. Федосов В.П. - 29 стр.

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ɧɢɣ ɬɨ, ɞɥɹ ɤɨɬɨɪɨɝɨ ɜɟɥɢɱɢɧɚ ɫɪɟɞɧɢɯ ɩɨɬɟɪɶ ɦɟɧɶɲɟ. ɉɨɞɫɱɢɬɚɟɦ ɜɟɥɢɱɢɧɵ
ɫɪɟɞɧɢɯ ɩɨɬɟɪɶ ɞɥɹ ɞɜɭɯ ɩɪɚɜɢɥ ɜɵɛɨɪɚ ɤɚɠɞɨɝɨ ɢɡ ɜɨɡɦɨɠɧɵɯ ɪɟɲɟɧɢɣ.
Ɉɞɧɨ ɩɪɚɜɢɥɨ G
0
ɦɨɠɟɬ ɛɵɬɶ ɫɮɨɪɦɭɥɢɪɨɜɚɧɨ ɬɚɤ: ɧɚɛɥɸɞɚɟɲɶ ɫɢɝɧɚɥ "0" -
ɩɪɢɧɢɦɚɣ ɪɟɲɟɧɢɟ J
0
, ɚ ɤɨɝɞɚ ɧɚɛɥɸɞɚɟɲɶ "1" - ɬɨ ɪɟɲɟɧɢɟ J
1
. ȼɷɬɨɦɫɥɭɱɚɟ
ɜɟɪɨɹɬɧɨɫɬɢ ɨɲɢɛɨɱɧɵɯ ɪɟɲɟɧɢɣ ɪɚɜɧɵ:
P{ɨɲɢɛɤɚ ɩɟɪɜɨɝɨ ɪɨɞɚ} = P{J
1
/s
0
} = p
0
P{1/s
0
};
P{ɨɲɢɛɤɚ ɜɬɨɪɨɝɨ ɪɨɞɚ} = P{J
0
/s
1
} = p
1
P{0/s
1
},
ɚ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ R
0
ɩɨɬɟɪɶ ɪɚɜɧɨ ɉ
01
p
0
p(1/s
0
)+ɉ
10
p
1
p(0/s
1
).
Ⱦɪɭɝɨɟ ɩɪɚɜɢɥɨ G
1
ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɬɚɤ: ɧɚɛɥɸɞɚɟɲɶ ɫɢɝɧɚɥ "0"- ɩɪɢɧɢɦɚɣ
ɪɟɲɟɧɢɟ J
1
ɢ ɧɚɨɛɨɪɨɬ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɜɟɪɨɹɬɧɨɫɬɢ ɨɲɢɛɨɱɧɵɯ ɪɟɲɟɧɢɣ ɪɚɜɧɵ:
P{ɨɲɢɛɤɚ ɩɟɪɜɨɝɨ ɪɨɞɚ} = p{J
1
/s
0
) = p
0
P{1/s
0
};
P{ɨɲɢɛɤɚ ɜɬɨɪɨɝɨ ɪɨɞɚ} = P{J
0
/s
1
} = p
1
P{0/s
1
},
ɚ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ R
1
ɩɨɬɟɪɶ ɪɚɜɧɨ ɉ
01
p
0
p(1/s
0
)+ɉ
10
p
1
p(0/s
1
).
ɉɪɢɧɹɬɵɣ ɤɪɢɬɟɪɢɣ ɤɚɱɟɫɬɜɚ ɨɬɞɚɟɬ ɩɪɟɞɩɨɱɬɟɧɢɟ ɩɪɚɜɢɥɭ G
0
, ɟɫɥɢ
R
0
/R
1
<1, ɬ.ɟ. ɤɨɝɞɚ R
0
<0,5(R
0
+R
1
). Ɍɚɤ ɤɚɤ R
0
+R
1
=ɉ
01
p
0
+ɉ
10
p
1
, ɬɨ ɩɪɢɯɨɞɢɦ ɤ
ɫɥɟɞɭɸɳɟɦɭ ɭɫɥɨɜɢɸ, ɩɪɢ ɜɵɩɨɥɧɟɧɢɢ ɤɨɬɨɪɨɝɨ ɩɪɢɧɢɦɚɟɬɫɹ ɩɪɚɜɢɥɨ ɩɪɢɧɹɬɢɹ
ɪɟɲɟɧɢɹ G
0
:
ɉ
01
p
0
p(1/s
0
)+ɉ
10
p
1
p(0/s
1
)<0,5(ɉ
01
p
0
+ɉ
10
p
1
). (8.1)
Ɏɨɪɦɭɥɚ (8.1), ɩɨɦɢɦɨ ɭɫɥɨɜɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ ɨɲɢɛɨɤ, ɨɩɪɟɞɟɥɹɟɦɵɯ ɜɟ-
ɪɨɹɬɧɨɫɬɧɵɦɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ ɩɨɦɟɯ ɜ ɤɚɧɚɥɟ ɫɜɹɡɢ, ɫɨɞɟɪɠɢɬ ɚɩɪɢɨɪɧɵɟ
ɜɟɪɨɹɬɧɨɫɬɢ ɫɨɨɛɳɟɧɢɣ ɢ ɜɟɥɢɱɢɧɵ ɩɨɬɟɪɶ ɨɬ ɧɟɩɪɚɜɢɥɶɧɵɯ ɪɟɲɟɧɢɣ. Ɉɩɪɟɞɟ-
ɥɟɧɢɟ ɢɥɢ ɧɚɡɧɚɱɟɧɢɟ ɜɟɥɢɱɢɧ p
0
, ɉ
10
ɢɉ
01
ɜ ɤɨɧɤɪɟɬɧɵɯ ɫɥɭɱɚɹɯ ɜɵɡɵɜɚɟɬ ɡɚ-
ɬɪɭɞɧɟɧɢɹ. Ʉɨɝɞɚ ɧɟɬ ɨɫɧɨɜɚɧɢɣ ɫɱɢɬɚɬɶ ɨɲɢɛɤɭ ɬɨɝɨ ɢɥɢ ɢɧɨɝɨ ɪɨɞɚ ɛɨɥɟɟ
ɨɩɚɫɧɨɣ, ɬɨ ɦɨɠɧɨ ɩɨɥɚɝɚɬɶ ɩɨɬɟɪɢ ɨɞɢɧɚɤɨɜɵɦɢ, ɢ ɬɨɝɞɚ ɜɟɥɢɱɢɧɚ ɫɪɟɞɧɢɯ
ɩɨɬɟɪɶ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɜɟɪɨɹɬɧɨɫɬɢ ɨɲɢɛɤɢ ɥɸɛɨɝɨ ɜɢɞɚ. Ʉɪɢɬɟɪɢɣ ɧɚɢɦɟɧɶ-
ɲɢɯ ɫɪɟɞɧɢɯ ɩɨɬɟɪɶ ɩɟɪɟɯɨɞɢɬ ɜ ɤɪɢɬɟɪɢɣ ɧɚɢɦɟɧɶɲɟɣ ɜɟɪɨɹɬɧɨɫɬɢ ɨɲɢɛɨɤ.
Ʉɨɝɞɚ ɠɟ ɧɢɱɟɝɨ "ɧɟɢɡɜɟɫɬɧɨ" ɨ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɫɬɪɭɤɬɭɪɟ ɢɫɬɨɱɧɢɤɚ ɫɨɨɛɳɟɧɢɣ
(ɬ.ɟ. ɜɟɪɨɹɬɧɨɫɬɢ p
0
ɢ p
1
ɧɟɢɡɜɟɫɬɧɵ), ɬɨ ɨɫɬɚɟɬɫɹ ɩɪɟɞɩɨɥɨɠɢɬɶ, ɱɬɨ ɫɨɨɛɳɟɧɢɹ
ɩɟɪɟɞɚɸɬɫɹ ɫ ɨɞɢɧɚɤɨɜɵɦɢ ɜɟɪɨɹɬɧɨɫɬɹɦɢ, ɬ.ɟ. p
1
= p
0
= 0,5.
ȿɫɥɢ ɩɥɚɬɚ ɡɚ ɨɲɢɛɤɢ ɨɞɢɧɚɤɨɜɚ ɢ ɪɚɜɧɵ ɦɟɠɞɭ ɫɨɛɨɣ ɜɟɪɨɹɬɧɨɫɬɢ ɩɟɪɟɞɚ-
ɜɚɟɦɵɯ ɫɨɨɛɳɟɧɢɣ, ɬɨ ɭɫɥɨɜɢɟ (8.1) ɫɬɚɧɨɜɢɬɫɹ ɩɪɨɫɬɵɦ:
p{1/s
0
)+p{0/s
0
}<1 ɢɥɢ p{1/s
0
)<p{0/s
0
}. (8.2)
Ɏɨɪɦɭɥɚ (8.2) ɨɡɧɚɱɚɟɬ, ɱɬɨ ɜɟɪɨɹɬɧɨɫɬɶ ɢɫɤɚɠɟɧɢɹ ɫɢɝɧɚɥɚ "0" ɦɟɧɶɲɟ ɜɟ-
ɪɨɹɬɧɨɫɬɢ ɢɫɤɚɠɟɧɢɹ ɫɢɝɧɚɥɚ "1" (ɬ.ɟ. ɜɟɪɨɹɬɧɨɫɬɶ ɥɨɠɧɨɝɨ ɫɢɝɧɚɥɚ "1" ɦɟɧɶɲɟ
ɜɟɪɨɹɬɧɨɫɬɢ ɧɟɩɨɞɚɜɥɟɧɢɹ ɢɫɬɢɧɧɨɝɨ ɫɢɝɧɚɥɚ "1"). ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɫɪɟɞɧɢɟ ɩɨɬɟ-
ɪɢ R
0
< 0,5, ɜ ɬɨ ɜɪɟɦɹ ɤɚɤ ɞɥɹ ɩɪɚɜɢɥɚ ɩɪɢ ɭɫɥɨɜɢɢ (8.2) ɫɪɟɞɧɢɟ ɩɨɬɟɪɢ R
1
>
0,5.
ɉɪɨɫɬɵɟ ɢ ɫɥɨɠɧɵɟ ɝɢɩɨɬɟɡɵ
Ɋɚɫɫɦɨɬɪɢɦ ɨɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ ɬɟɨɪɢɢ ɩɪɨɜɟɪɤɢ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɝɢɩɨɬɟɡ,
ɢɫɩɨɥɶɡɨɜɚɧɧɵɟ ɜ ɩɪɟɞɵɞɭɳɟɦ ɪɚɡɞɟɥɟ. ɇɚɩɨɦɧɢɦ, ɱɬɨ ɜɵɛɨɪ ɪɟɲɟɧɢɹ ɫɨɫɬɨɢɬ

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