смод
.pdf(x,y) ɩɥɨɫɤɨɫɬɶɸ, ɩɚɪɚɥɥɟɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ x0y. ɋɟɱɟɧɢɟ ɩɪɨɣɞɟɬ ɩɨ ɤɪɢɜɨɣ Ʉ. ɋɩɪɨɟɰɢɪɭɟɦ ɷɬɭ ɤɪɢɜɭɸ ɧɚ ɩɥɨɫɤɨɫɬɶ x0y. ɉɪɨɟɤɰɢɹ, ɨɩɪɟɞɟɥɹɟɦɚɹ ɭɪɚɜɧɟɧɢɟɦ M(x,y) = z, ɪɚɡɞɟɥɢɬ ɩɥɨɫɤɨɫɬɶ x0y ɧɚ ɞɜɟ ɨɛɥɚɫɬɢ. Ⱦɥɹ ɨɞɧɨɣ ɢɡ ɧɢɯ ɜɵɩɨɥɧɹɟɬɫɹ ɭɫɥɨɜɢɟ Z < z. Ɉɛɨɡɧɚɱɢɦ ɷɬɭ ɩɥɨɫɤɨɫɬɶ ɱɟɪɟɡ [. ȼɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɜ ɷɬɭ ɩɥɨɫɤɨɫɬɶ ɢ ɟɫɬɶ ɢɧɬɟɝɪɚɥɶɧɚɹ ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, G(z) =
P((X,Y) [)= ³³f(x,y)dxdy. Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ G(z) ɩɨ z, ɩɨɥɭɱɢɦ ɢɫɤɨɦɭɸ ɮɭɧɤɰɢɸ
[
g(z). Ɉɫɬɚɥɨɫɶ ɜɵɪɚɡɢɬɶ ɱɟɪɟɡ z ɩɪɟɞɟɥɵ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɢ ɧɚɩɢɫɚɬɶ ɜɵɪɚɠɟɧɢɟ ɞɥɹ g(z).
ɉɊɂɆȿɊ. ɇɚɣɬɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫɭɦɦɵ ɞɜɭɯ ɋȼ X ɢ Y, ɢɦɟɸɳɢɯ ɫɨɜɦɟɫɬɧɭɸ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ f(x,y). ɋɭɦɦɚ Z = X + Y. ɇɚɣɬɢ g(z).
ɇɚ ɩɥɨɫɤɨɫɬɢ x0y ɢɦɟɟɬɫɹ ɥɢɧɢɹ, ɭɪɚɜɧɟɧɢɟ ɤɨɬɨɪɨɣ x+y=z ɟɫɬɶ ɩɪɹɦɚɹ, ɨɬɫɟɤɚɸɳɚɹ ɧɚ ɨɫɹɯ ɨɬɪɟɡɤɢ, ɪɚɜɧɵɟ z. ɗɬɚ ɩɪɹɦɚɹ ɞɟɥɢɬ ɜɫɸ ɩɥɨɫɤɨɫɬɶ ɧɚ ɞɜɟ ɱɚɫɬɢ. Ɍɚ ɱɚɫɬɶ, ɱɬɨ ɥɟɜɟɟ ɢ ɧɢɠɟ ɷɬɨɣ ɩɪɹɦɨɣ, ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɭɫɥɨɜɢɸ X+Y<Z. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɮɨɪɦɭɥɨɣ (5.1), ɝɞɟ ɩɥɨɫɤɨɫɬɶ [ ɟɫɬɶ ɩɥɨɫɤɨɫɬɶ, ɧɚ ɤɨɬɨɪɨɣ ɜɵɩɨɥɧɹɟɬɫɹ ɷɬɨ ɭɫɥɨɜɢɟ, ɢɦɟɟɦ:
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G(z)= ³³f(x,y)dxdy = ³ ³f(x, y)dxdy |
= ³f(x,z - x)dx. |
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ȿɫɥɢ ɋȼ X ɢ Y ɧɟɡɚɜɢɫɢɦɵ, ɬɨ ɧɟɬɪɭɞɧɨ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɢɧɬɟɝɪɚɥ ɩɪɢ ɪɚɜɟɧɫɬɜɟ f(x,y) = f(x) f(y) ɩɪɟɜɪɚɳɚɟɬɫɹ ɜ ɢɧɬɟɝɪɚɥ ɫɜɟɪɬɤɢ. Ɍɚɤ, ɫɜɟɪɬɤɚ ɞɜɭɯ ɩɪɹɦɨɭɝɨɥɶɧɵɯ ɡɚɤɨɧɨɜ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ ɞɚɟɬ ɡɚɤɨɧ ɋɢɦɩɫɨɧɚ.
5.4. ɋɭɦɦɚ ɞɜɭɯ ɧɨɪɦɚɥɶɧɵɯ ɋȼ
ȿɫɥɢ ɩɨɞɫɬɚɜɢɬɶ ɜ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɢɧɬɟɝɪɚɥɚ ɫɜɟɪɬɤɢ ɞɜɚ ɧɨɪɦɚɥɶɧɵɯ ɨɞɧɨɦɟɪɧɵɯ ɡɚɤɨɧɚ ɞɥɹ ɋȼ X ɢ ɋȼ Y, ɬɨ ɩɨɫɥɟ ɧɟɫɥɨɠɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɩɨɥɭɱɢɦ ɫɧɨɜɚ ɧɨɪɦɚɥɶɧɭɸ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɋȼ Z, ɭ ɤɨɬɨɪɨɣ ɆɈ ɪɚɜɧɨ ɫɭɦɦɟ ɆɈ ɤɚɠɞɨɣ ɋȼ ɢ ɞɢɫɩɟɪɫɢɹ ɬɚɤɠɟ ɪɚɜɧɚ ɫɭɦɦɟ ɞɢɫɩɟɪɫɢɣ, ɟɫɥɢ ɋȼ ɧɟɤɨɪɪɟɥɢɪɨɜɚɧɵ, ɚ ɟɫɥɢ ɤɨɪɪɟɥɢɪɨɜɚɧɵ, ɬɨ Vz2=Vx2+Vy2r2RVxVy (+, ɟɫɥɢ ɤɨɷɮɮɢɰɢɟɧɬ ɤɨɪɪɟɥɹɰɢɢ ɛɨɥɶɲɟ ɧɭɥɹ ɢ ɧɚɨɛɨɪɨɬ).
Ɂɚɞɚɧɵ: ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ f(x) ɋȼ-ɚɪɝɭɦɟɧɬɚ, ɮɭɧɤɰɢɹ y = M(x), ɫɜɹɡɵɜɚɸɳɚɹ ɋȼ-ɚɪɝɭɦɟɧɬ ɢ ɋȼ-ɮɭɧɤɰɢɸ. Ɉɩɪɟɞɟɥɢɬɶ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ f(y) ɋȼ-ɮɭɧɤɰɢɢ. Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ f(y) ɜ ɨɛɳɟɦ ɜɢɞɟ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɪɚɜɟɧɫɬɜɨ ɷɥɟɦɟɧɬɚɪɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ, ɧɟɢɡɦɟɧɹɸɳɢɯɫɹ ɩɪɢ ɥɸɛɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɯ: f(x)dx = f(y)dy. Ɉɬɫɸɞɚ f(y) = f(x)/(dy/dx).
Ⱦɥɹ ɪɟɲɟɧɢɹ ɫɮɨɪɦɭɥɢɪɨɜɚɧɧɨɣ ɡɚɞɚɱɢ ɧɟɨɛɯɨɞɢɦɨ ɧɚɣɬɢ ɮɭɧɤɰɢɸ, ɨɛɪɚɬɧɭɸ M(x). Ɂɚɬɟɦ ɨɩɪɟɞɟɥɢɬɶ ɹɤɨɛɢɚɧ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ J = |dM(x)/dx| ɢ ɩɨɞɫɬɚɜɢɬɶ ɜɦɟɫɬɨ y ɨɛɪɚɬɧɭɸ ɮɭɧɤɰɢɸ x=M-1(y) ɞɥɹ ɜɫɟɯ ɡɧɚɱɟɧɢɣ x, ɨɩɪɟɞɟɥɹɟɦɵɯ ɧɟɨɞɧɨɡɧɚɱɧɨɫɬɶɸ ɨɛɪɚɬɧɨɣ ɮɭɧɤɰɢɢ M-1(y). ɉɨɬɨɦ ɜɫɟ ɩɨɥɭɱɟɧɧɵɟ ɜɵɪɚɠɟɧɢɹ ɫɥɨɠɢɬɶ.
ȿɫɥɢ ɮɭɧɤɰɢɹ y = M(x) ɢɦɟɟɬ ɤɭɫɨɱɧɨ-ɥɨɦɚɧɵɣ ɜɢɞ, ɬɨ ɞɚɧɧɭɸ ɡɚɞɚɱɭ ɦɨɠɧɨ ɪɟɲɚɬɶ ɢ ɝɪɚɮɢɱɟɫɤɢ.
6. ɋɅɍɑȺɃɇɕȿ ɉɊɈɐȿɋɋɕ (ɎɍɇɄɐɂɂ)
6.1. Ɉɛɳɢɟ ɩɨɧɹɬɢɹ
ɋɜɟɞɟɧɢɹ ɨ ɫɥɭɱɚɣɧɵɯ ɩɪɨɰɟɫɫɚɯ, ɢɯ ɨɫɨɛɟɧɧɨɫɬɹɯ, ɦɟɬɨɞɚɯ ɨɩɢɫɚɧɢɹ, ɩɚɪɚɦɟɬɪɚɯ ɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɯ ɦɨɠɧɨ ɧɚɣɬɢ ɜ ɭɱɟɛɧɨɦ ɩɨɫɨɛɢɢ [5, ɪɚɡɞ. 1].
6.2.ɍɡɤɨɩɨɥɨɫɧɵɟ ɫɥɭɱɚɣɧɵɟ ɩɪɨɰɟɫɫɵ (ɍɋɉ)
ȼɚɠɧɨɫɬɶ ɷɬɢɯ ɩɪɨɰɟɫɫɨɜ ɞɥɹ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɪɚɞɢɨɬɟɯɧɢɤɢ ɬɪɟɛɭɟɬ ɛɨɥɟɟ ɩɨɞɪɨɛɧɨɝɨ ɢɯ ɪɚɫɫɦɨɬɪɟɧɢɹ.
Ⱦɥɹ ɛɨɥɟɟ ɩɨɞɪɨɛɧɨɝɨ ɚɧɚɥɢɡɚ ɨɩɪɟɞɟɥɢɦ ɨɝɢɛɚɸɳɭɸ Um(t) ɢ ɮɚɡɭ M(t) ɍɋɉ. ɑɚɫɬɨ ɨɝɢɛɚɸɳɭɸ ɨɩɪɟɞɟɥɹɸɬ ɩɨ ɮɨɪɦɭɥɟ
Um(t)=[u2(t)+u12(t)]1/2, (6.1)
ɝɞɟ u1(t) - ɫɨɩɪɹɠɟɧɧɵɣ ɩɨ Ƚɢɥɶɛɟɪɬɭ ɩɪɨɰɟɫɫ. ɉɪɢɦɟɧɹɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ Ƚɢɥɶɛɟɪɬɚ ɤ ɢɫɯɨɞɧɨɦɭ ɜɵɪɚɠɟɧɢɸ, ɩɨɥɭɱɚɟɦ u1(t)=-Um(t)sin[Z0t--M(t)]. Ɍɨɱɧɨɫɬɶ ɜɵɪɚɠɟɧɢɹ ɞɥɹ Um(t) ɢɧɨɝɞɚ ɦɨɠɟɬ ɜɵɡɵɜɚɬɶ ɫɨɦɧɟɧɢɟ, ɩɨɫɤɨɥɶɤɭ ɬɨɥɶɤɨ ɞɥɹ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɤɨɥɟɛɚɧɢɣ ɪɚɜɟɧɫɬɜɨ (6.1) ɧɟɫɨɦɧɟɧɧɨ. Ɉɩɪɟɞɟɥɢɦ, ɧɚɫɤɨɥɶɤɨ ɩɚɪɚɦɟɬɪɵ ɍɋɉ ɜɥɢɹɸɬ ɧɚ ɬɨɱɧɨɫɬɶ ɷɬɨɣ ɮɨɪɦɭɥɵ.
ɂɫɩɨɥɶɡɭɹ ɢɡɜɟɫɬɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ ɞɥɹ ɫɨɫɬɚɜɥɹɸɳɢɯ ɤɨɦɩɥɟɤɫɧɨɣ ɚɦɩɥɢɬɭɞɵ ɚɧɚɥɢɬɢɱɟɫɤɨɝɨ ɫɢɝɧɚɥɚ z(t), ɩɨɥɭɱɢɦ
Uc(t)=Um(t)cosM(t) ɢ Us(t)=Um(t)sinM(t).
ɉɪɢɦɟɧɹɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ Ƚɢɥɶɛɟɪɬɚ ɤ ɢɫɯɨɞɧɨɦɭ ɜɵɪɚɠɟɧɢɸ ɞɥɹ ɍɋɉ ɢ ɢɫɩɨɥɶɡɭɹ ɫɨɫɬɚɜɥɹɸɳɢɟ ɤɨɦɩɥɟɤɫɧɨɣ ɨɝɢɛɚɸɳɟɣ, ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ
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Ɋɚɡɥɨɠɢɦ ɮɭɧɤɰɢɢ Uc(t) ɢ Us(t) ɜ ɩɨɞɵɧɬɟɝɪɚɥɶɧɵɯ ɜɵɪɚɠɟɧɢɹɯ ɜ ɪɹɞ Ɍɟɣɥɨɪɚ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ x = t ɢ ɩɨɱɥɟɧɧɨ ɩɪɨɢɧɬɟɝɪɢɪɭɟɦ. ɉɨɥɭɱɢɦ
u1 t
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= Uc(t)sinZ0t +Us(t)cosZ0t +Q(t), |
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ɝɞɟ Q(t) - ɨɫɬɚɬɨɱɧɨɟ ɫɥɚɝɚɟɦɨɟ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɟɟ ɨɬɛɪɨɲɟɧɧɭɸ ɱɚɫɬɶ ɫɭɦɦɵ. ɉɨɞɫɬɚɜɢɜ ɜ ɜɵɪɚɠɟɧɢɟ (6.2) Uc(t) ɢ Us(t), ɩɨɥɭɱɢɦ
u1(t)=-Um(t)sin[Z0t-M(t)]+Q(t). (6.3)
ɂɡ ɮɨɪɦɭɥɵ (6.3) ɜɢɞɧɨ, ɱɬɨ ɟɫɥɢ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɮɭɧɤɰɢɟɣ Q(t), ɬɨ ɫɨɩɪɹɠɟɧɧɵɣ ɩɨ Ƚɢɥɶɛɟɪɬɭ ɍɋɉ ɢɦɟɟɬ ɬɚɤɭɸ ɠɟ ɨɝɢɛɚɸɳɭɸ, ɱɬɨ ɢ ɢɫɯɨɞɧɵɣ ɍɋɉ.
ɂɡ ɬɚɛɥɢɰ ɨɩɪɟɞɟɥɟɧɧɵɯ ɢɧɬɟɝɪɚɥɨɜ ɢɡɜɟɫɬɧɨ:
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ɋ ɭɱɟɬɨɦ ɷɬɢɯ ɜɵɪɚɠɟɧɢɣ ɮɨɪɦɭɥɭɞɥɹ Q(t) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ: |
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0,5Q(t)=1/Z0(Uc'(t)cosZ0t -Us'(t)sinZ0t)- |
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2(U |
''(t)sinZ t -Us''(t)cosZ t)+... |
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ɋɱɢɬɚɟɦ, ɱɬɨ ɩɨɥɨɫɚ ɨɝɢɛɚɸɳɟɣ ɪɚɜɧɚ 'Z, ɩɨɷɬɨɦɭ ɜɬɨɪɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ ɫɜɨɢɦ ɡɧɚɱɟɧɢɹɦ ɧɟ ɩɪɟɜɨɫɯɨɞɹɬ 'ZUm(t). ɉɨɷɬɨɦɭ ɦɨɠɧɨ ɩɨɥɚɝɚɬɶ, ɱɬɨ Q(t) = '
ZUm(t)/ Z0.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ:
u1(t)= -Um(t)sin[Z0t-M(t)]+'ZUm(t)/Z0.
Ɉɬɫɸɞɚ ɜɢɞɧɨ, ɱɬɨ ɞɥɹ ɍɋɉ ɮɭɧɤɰɢɢ u(t) ɢ u1(t) ɢɦɟɸɬ ɨɞɢɧɚɤɨɜɭɸ ɨɝɢɛɚɸɳɭɸ ɫ ɩɨɝɪɟɲɧɨɫɬɶɸ, ɡɚɜɢɫɹɳɟɣ ɨɬ ɨɬɧɨɲɟɧɢɹ ɲɢɪɢɧɵ ɫɩɟɤɬɪɚ ɤ ɟɝɨ ɫɪɟɞɧɟɣ ɱɚɫɬɨɬɟ. Ⱦɥɹ ɭɡɤɨɩɨɥɨɫɧɵɯ ɫɥɭɱɚɣɧɵɯ ɩɪɨɰɟɫɫɨɜ ɨɛɹɡɚɬɟɥɶɧɵɦ ɹɜɥɹɟɬɫɹ ɜɵɪɚɠɟɧɢɟ 'Z/Z0<<1, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɨɝɢɛɚɸɳɚɹ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɬɪɟɛɨɜɚɧɢɹɦ, ɤɨɬɨɪɵɟ ɤ ɧɟɣ ɩɪɟɞɴɹɜɥɹɸɬɫɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɨɩɪɟɞɟɥɟɧɢɟɦ ɍɋɉ, ɬ.ɟ. ɹɜɥɹɟɬɫɹ ɤɚɫɚɬɟɥɶɧɨɣ ɜ ɬɨɱɤɚɯ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɦɚɤɫɢɦɚɥɶɧɵɦ ɡɧɚɱɟɧɢɹɦ ɍɋɉ ɢ ɢɦɟɟɬ ɨɛɳɢɟ ɡɧɚɱɟɧɢɹ ɫ ɧɢɦ ɜ ɬɨɱɤɚɯ ɤɚɫɚɧɢɹ.
Ɏɚɡɚ M(t) ɨɞɧɨɡɧɚɱɧɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɡɜɟɫɬɧɵɦɢ ɫɨɨɬɧɨɲɟɧɢɹɦɢ ɞɥɹ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɤɨɦɩɥɟɤɫɧɨɝɨ ɱɢɫɥɚ ɜ ɩɨɤɚɡɚɬɟɥɶɧɨɣ ɮɨɪɦɟ.
Ƚɪɚɮɢɱɟɫɤɢ ɍɋɉ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ ɜɟɤɬɨɪɚ, ɜɪɚɳɚɸɳɟɝɨɫɹ ɫ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ Z0, ɞɥɢɧɚ ɜɟɤɬɨɪɚ ɦɟɞɥɟɧɧɨ ɦɟɧɹɟɬɫɹ ɜɨ ɜɪɟɦɟɧɢ ɬɚɤ ɠɟ, ɤɚɤ ɢ ɮɚɡɨɜɵɣ ɭɝɨɥ M(t). ɂɫɯɨɞɧɵɣ ɍɋɉ ɹɜɥɹɟɬɫɹ ɩɪɨɟɤɰɢɟɣ ɜɟɤɬɨɪɚ ɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɭɸ ɨɫɶ. ȿɫɥɢ ɜɫɸ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ ɡɚɫɬɚɜɢɬɶ ɜɪɚɳɚɬɶɫɹ ɫ ɬɨɣ ɠɟ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ, ɧɨ ɜ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ, ɬɨ ɬɚ ɠɟ ɩɪɨɟɤɰɢɹ ɛɭɞɟɬ ɨɝɢɛɚɸɳɟɣ Uc(t).
ȿɫɥɢ ɢɫɯɨɞɧɵɣ ɍɋɉ ɹɜɥɹɟɬɫɹ ɧɨɪɦɚɥɶɧɵɦ, ɬɨ Uc(t) ɢ Us(t) ɬɚɤɠɟ ɹɜɥɹɸɬɫɹ ɧɨɪɦɚɥɶɧɵɦɢ ɫɥɭɱɚɣɧɵɦɢ ɩɪɨɰɟɫɫɚɦɢ. ȿɫɥɢ ɍɋɉ u(t) ɧɨɪɦɚɥɟɧ, ɫɬɚɰɢɨɧɚɪɟɧ, ɢɦɟɟɬ ɧɭɥɟɜɨɟ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɢ ɮɭɧɤɰɢɸ ɤɨɪɪɟɥɹɰɢɢ K(W)=V2U(W) cosZ0W ɬɨ Uc(t) ɢ Us(t) ɬɚɤɠɟ ɢɦɟɸɬ ɧɭɥɟɜɵɟ ɫɪɟɞɧɢɟ ɡɧɚɱɟɧɢɹ ɢ ɤɨɪɪɟɥɹɰɢɨɧɧɭɸ ɮɭɧɤɰɢɸ K(W). ȼ ɬɨ ɠɟ ɜɪɟɦɹ Uc(t) ɢ Us(t) ɜɡɚɢɦɧɨ ɧɟɤɨɪɪɟɥɢɪɨɜɚɧɵ, ɚ ɬɚɤ ɤɚɤ ɨɧɢ ɧɨɪɦɚɥɶɧɵ, ɬɨ ɢ ɜɡɚɢɦɧɨ ɧɟɡɚɜɢɫɢɦɵ. ɋɨɦɧɨɠɢɬɟɥɶ U(W) ɹɜɥɹɟɬɫɹ ɨɝɢɛɚɸɳɟɣ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɮɭɧɤɰɢɢ R(W).
6.3.Ɉɝɢɛɚɸɳɚɹ ɢ ɮɚɡɚ ɭɡɤɨɩɨɥɨɫɧɨɝɨ ɋɉ
ɉɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ ɨɝɢɛɚɸɳɟɣ ɢ ɮɚɡɵ ɍɋɉ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ, ɫɨɜɟɪɲɚɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ, ɤɨɬɨɪɵɟ ɛɵɥɢ ɢɫɩɨɥɶɡɨɜɚɧɵ ɞɥɹ ɢɯ ɩɨɥɭɱɟɧɢɹ. ɗɬɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɩɨɤɚɡɵɜɚɸɬ, ɱɬɨ ɨɝɢɛɚɸɳɚɹ ɢ ɮɚɡɚ ɹɜɥɹɸɬɫɹ ɧɟɡɚɜɢɫɢɦɵɦɢ ɋȼ ɤɚɤ ɜ ɫɨɜɩɚɞɚɸɳɢɟ, ɬɚɤ ɢ ɜ ɧɟɫɨɜɩɚɞɚɸɳɢɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ. Ɉɞɧɨɦɟɪɧɚɹ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɨɝɢɛɚɸɳɟɣ (ɜ ɨɞɢɧ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ) ɩɨɞɱɢɧɹɟɬɫɹ ɡɚɤɨɧɭ Ɋɟɥɟɹ, ɚ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɮɚɡɵ ɪɚɜɧɨɦɟɪɧɚ ɜ ɩɪɟɞɟɥɚɯ ɨɬ -S ɞɨ S.
ɋɥɨɠɧɵɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɩɨɤɚɡɵɜɚɸɬ, ɱɬɨ ɰɟɧɬɪɢɪɨɜɚɧɧɚɹ ɤɨɪɪɟɥɹɰɢɨɧɧɚɹ ɮɭɧɤɰɢɹ ɨɝɢɛɚɸɳɟɣ ɩɪɢɛɥɢɠɟɧɧɨ ɪɚɜɧɚ ɤɜɚɞɪɚɬɭ ɨɝɢɛɚɸɳɟɣ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɮɭɧɤɰɢɢ ɢɫɯɨɞɧɨɝɨ ɍɋɉ. ɋɩɟɤɬɪɚɥɶɧɚɹ ɩɥɨɬɧɨɫɬɶ ɦɨɳɧɨɫɬɢ ɨɝɢɛɚɸɳɟɣ ɢɦɟɟɬ ɞɜɚ ɫɥɚɝɚɟɦɵɯ: ɞɟɥɶɬɚ-ɮɭɧɤɰɢɸ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɭɸ ɩɨɫɬɨɹɧɧɨɣ ɫɨɫɬɚɜɥɹɸɳɟɣ ɨɝɢɛɚɸɳɟɣ, ɢ ɫɩɟɤɬɪɚɥɶɧɭɸ ɩɥɨɬɧɨɫɬɶ ɮɥɸɤɬɭɚɰɢɨɧɧɨɣ ɫɨɫɬɚɜɥɹɸɳɟɣ, ɤɨɬɨɪɚɹ ɹɜɥɹɟɬɫɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟɦ Ɏɭɪɶɟ ɨɬ ɤɜɚɞɪɚɬɚ ɨɝɢɛɚɸɳɟɣ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɮɭɧɤɰɢɢ ɢɫɯɨɞɧɨɝɨ ɍɋɉ.
ȿɫɥɢ ɋɉ ɹɜɥɹɟɬɫɹ ɫɭɦɦɨɣ ɭɡɤɨɩɨɥɨɫɧɨɝɨ ɧɨɪɦɚɥɶɧɨɝɨ ɩɪɨɰɟɫɫɚ ɢ ɫɢɧɭɫɨɢɞɵ ɫɨ ɫɥɭɱɚɣɧɨɣ ɧɚɱɚɥɶɧɨɣ ɮɚɡɨɣ, ɬɨ ɦɝɧɨɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɫɢɧɭɫɨɢɞɵ ɪɚɫɩɪɟɞɟɥɟɧɵ ɩɨ ɡɚɤɨɧɭ ɚɪɤɫɢɧɭɫɚ, ɫɭɦɦɚ - ɩɨ ɞɜɭɦɨɞɚɥɶɧɨɦɭ ɡɚɤɨɧɭ. ɉɨɫɥɟ ɩɪɢɦɟɧɟɧɢɹ ɬɟɯ ɠɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ, ɱɬɨ ɢ ɞɥɹ ɭɡɤɨɩɨɥɨɫɧɨɝɨ ɧɨɪɦɚɥɶɧɨɝɨ ɋɉ, ɩɨɥɭɱɢɦ ɞɥɹ ɨɝɢɛɚɸɳɟɣ ɪɚɫɩɪɟɞɟɥɟɧɢɟ Ɋɚɣɫɚ
f(A)=Aexp[-(A2-A02)/2]I0(AA0), A>0,
ɝɞɟ A0=Am/V, Am- ɚɦɩɥɢɬɭɞɚ ɫɢɧɭɫɨɢɞɚɥɶɧɨɝɨ ɫɢɝɧɚɥɚ, V- ɫɪɟɞɧɟɤɜɚɞɪɚɬɢɱɟɫɤɨɟ ɨɬɤɥɨɧɟɧɢɟ ɲɭɦɚ.
ɉɪɢ Amo0 ɪɚɫɩɪɟɞɟɥɟɧɢɟ Ɋɚɣɫɚ ɩɟɪɟɯɨɞɢɬ ɜ ɪɚɫɩɪɟɞɟɥɟɧɢɟ Ɋɟɥɟɹ. ɉɪɢ ɛɨɥɶɲɢɯ ɨɬɧɨɲɟɧɢɹɯ Am/V, ɬ.ɟ. ɩɪɢ A0>>1(ɨɬɧɨɲɟɧɢɟ ɫɢɝɧɚɥ/ɲɭɦ),
ɪɚɫɩɪɟɞɟɥɟɧɢɟ Ɋɚɣɫɚ ɦɨɠɟɬ ɛɵɬɶ ɚɩɩɪɨɤɫɢɦɢɪɨɜɚɧɨ ɧɨɪɦɚɥɶɧɵɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ ɫ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦ ɨɠɢɞɚɧɢɟɦ, ɪɚɜɧɵɦ A0.
7.ɅɂɇȿɃɇɕȿ ɉɊȿɈȻɊȺɁɈȼȺɇɂə ɋɅɍɑȺɃɇɕɏ ɉɊɈɐȿɋɋɈȼ
ȼ ɪɚɞɢɨɬɟɯɧɢɤɟ ɬɚɤɢɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɜɵɩɨɥɧɹɸɬɫɹ ɫ ɩɨɦɨɳɶɸ ɥɢɧɟɣɧɵɯ ɪɚɞɢɨɬɟɯɧɢɱɟɫɤɢɯ ɰɟɩɟɣ ɢ ɫɢɫɬɟɦ. Ʉ ɥɢɧɟɣɧɵɦ ɰɟɩɹɦ ɦɨɠɧɨ ɨɬɧɟɫɬɢ ɭɫɢɥɢɬɟɥɢ, ɮɢɥɶɬɪɵ, ɞɥɢɧɧɵɟ ɥɢɧɢɢ ɢ ɞɪ. ɉɨɥɧɨɟ ɨɩɢɫɚɧɢɟ ɥɢɧɟɣɧɨɣ ɰɟɩɢ ɦɨɠɧɨ ɜɵɩɨɥɧɢɬɶ ɪɟɲɟɧɢɟɦ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɥɢɧɟɣɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ, ɨɞɧɚɤɨ ɞɥɹ ɷɬɨɝɨ ɦɨɠɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɩɟɪɟɞɚɬɨɱɧɵɦɢ ɮɭɧɤɰɢɹɦɢ ɢ ɢɦɩɭɥɶɫɧɵɦɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ. Ɏɢɡɢɱɟɫɤɚɹ ɨɫɭɳɟɫɬɜɢɦɨɫɬɶ ɨɛɟɫɩɟɱɢɜɚɟɬɫɹ ɜɵɩɨɥɧɟɧɢɟɦ ɧɟɤɨɬɨɪɵɯ ɭɫɥɨɜɢɣ. Ɇɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɷɬɢ ɭɫɥɨɜɢɹ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ:
|
f |
|
g(t)= 0 ɩɪɢ t<0; lim g(t)= 0; |
³ |
|log|K(jZ)|2/(1+Z2)|dZ<f. |
t of |
|
|
|
f |
|
ɉɟɪɟɞɚɬɨɱɧɚɹ ɮɭɧɤɰɢɹ ɢ ɢɦɩɭɥɶɫɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɫɜɹɡɚɧɵ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟɦ Ɏɭɪɶɟ.
ȿɫɥɢ ɢɡɜɟɫɬɧɚ ɫɩɟɤɬɪɚɥɶɧɚɹ ɩɥɨɬɧɨɫɬɶ ɦɨɳɧɨɫɬɢ ɋɉ ɧɚ ɜɯɨɞɟ ɥɢɧɟɣɧɨɣ ɫɢɫɬɟɦɵ, ɬɨ ɫɩɟɤɬɪɚɥɶɧɚɹ ɩɥɨɬɧɨɫɬɶ ɦɨɳɧɨɫɬɢ ɜɵɯɨɞɧɨɝɨ ɋɉ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ Wy(Z)=Wx(Z)|K(jZ)|2. ȿɫɥɢ ɜɵɩɨɥɧɢɬɶ ɨɛɪɚɬɧɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ Ɏɭɪɶɟ ɨɬ ɤɚɠɞɨɝɨ ɢɡ ɫɨɦɧɨɠɢɬɟɥɟɣ ɢ ɨɬ ɪɟɡɭɥɶɬɚɬɚ ɩɟɪɟɦɧɨɠɟɧɢɹ, ɬɨ ɩɨɥɭɱɢɦ ɢɧɬɟɝɪɚɥ ɫɜɟɪɬɤɢ ɦɟɠɞɭ ɤɨɪɪɟɥɹɰɢɨɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ (ɄɎ) ɜɯɨɞɧɨɝɨ ɋɉ ɢ ɢɦɩɭɥɶɫɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ, ɚ ɜ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ ɄɎ ɜɵɯɨɞɧɨɝɨ ɋɉ. ȼ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɩɪɢɦɟɧɟɧɢɹ ɷɬɢɯ ɞɜɭɯ ɮɨɪɦɭɥ ɢ ɪɚɡɥɢɱɚɸɬ ɫɩɟɤɬɪɚɥɶɧɵɣ ɢ ɜɪɟɦɟɧɧɨɣ ɦɟɬɨɞɵ ɚɧɚɥɢɡɚ ɥɢɧɟɣɧɵɯ ɫɢɫɬɟɦ. ɇɨ ɫɥɟɞɭɟɬ ɢɦɟɬɶ ɜ ɜɢɞɭ, ɱɬɨ ɷɬɢɦ ɚɩɩɚɪɚɬɨɦ ɦɨɠɧɨ ɩɨɥɶɡɨɜɚɬɶɫɹ, ɟɫɥɢ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɧɭɥɟɜɵɟ. Ⱥ ɟɫɥɢ ɩɪɢɦɟɧɹɬɶ ɫɩɟɤɬɪɚɥɶɧɵɣ ɦɟɬɨɞ, ɬɨ ɪɟɡɭɥɶɬɚɬ ɛɭɞɟɬ ɫɨɨɬɜɟɬɫɬɜɨɜɚɬɶ ɫɬɚɰɢɨɧɚɪɧɨɦɭ ɪɟɠɢɦɭɫɢɫɬɟɦɵ.
7.1. Ⱦɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ ɋɉ
ɉɭɫɬɶ ɢɦɟɟɬɫɹ ɋɉ X(t), ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɣ ɭɫɥɨɜɢɹɦ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɨɫɬɢ, ɫ ɆɈ Ɇ(t) ɢ ɄɎ Ʉ(W). Ɉɩɪɟɞɟɥɢɦ ɆɈ ɢ ɄɎ ɋɉ ɞɥɹ ɩɪɨɢɡɜɨɞɧɨɣ Y(t)=dX(t)/dt ɢ ɜ ɤɚɤɢɯ ɰɟɩɹɯ ɷɬɨ ɜɵɩɨɥɧɹɟɬɫɹ.
ɉɪɟɞɫɬɚɜɢɦ ɩɪɨɢɡɜɨɞɧɭɸ ɜ ɜɢɞɟ ɩɪɟɞɟɥɚ ɢ ɭɫɪɟɞɧɢɦ ɨɛɟ ɱɚɫɬɢ ɪɚɜɟɧɫɬɜɚ:
M (t)=<Y(t)>= |
lim |
X t 't X t |
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't o0 |
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't o0 |
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.= lim |
M x t 't M x t |
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't o0 |
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Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɄɎ ɩɟɪɟɣɞɟɦ ɤ ɰɟɧɬɪɢɪɨɜɚɧɧɵɦ ɋɉ. ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ, ɄɎ ɢɦɟɟɬ ɜɢɞ
K(W)= Y t W Y t |
o |
o |
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d X t d X t W |
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o |
o |
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dt |
d t W |
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w |
2 |
ª o o |
º |
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« |
X t X t W |
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¼ |
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wtw t W |
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ɉɨɫɤɨɥɶɤɭ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ ɟɫɬɶ ɥɢɧɟɣɧɚɹ ɨɩɟɪɚɰɢɹ, ɬɨ ɭɫɪɟɞɧɟɧɢɟ ɢ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ ɦɨɠɧɨ ɩɨɦɟɧɹɬɶ ɦɟɫɬɚɦɢ, ɬɨɝɞɚ ɩɨɥɭɱɢɦ Ky(W) = -w2Kx(W)/w W2. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɪɨɰɟɫɫ, ɞɥɹ ɤɨɬɨɪɨɝɨ ɫɭɳɟɫɬɜɭɟɬ ɜɬɨɪɚɹ ɫɦɟɲɚɧɧɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ɨɬ ɄɎ ɩɨ ɟɟ ɚɪɝɭɦɟɧɬɭ, ɹɜɥɹɟɬɫɹ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɵɦ ɋɉ. ȼ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɦ ɫɥɭɱɚɟ ɷɬɨɬ ɋɉ ɧɟ ɹɜɥɹɟɬɫɹ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɵɦ, ɢ, ɟɫɥɢ ɧɟɨɛɯɨɞɢɦɨ ɧɚɣɬɢ ɄɎ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ ɢɫɯɨɞɧɨɝɨ ɋɉ, ɧɚɞɨ ɞɜɚɠɞɵ ɩɪɨɞɢɮɮɟɪɟɧɰɢɪɨɜɚɬɶ ɟɝɨ ɄɎ. Ɍɚɤɢɟ ɠɟ ɫɨɨɬɧɨɲɟɧɢɹ ɫɩɪɚɜɟɞɥɢɜɵ ɢ ɞɥɹ ɦɧɨɝɨɤɪɚɬɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ.
ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɫɜɨɣɫɬɜɚɦɢ ɄɎ, ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ |Kx''(W)|d-Kx''(0) = Vy2. Ɂɧɚɱɢɬ, ɜɬɨɪɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ɫɭɳɟɫɬɜɭɟɬ ɩɪɢ ɥɸɛɨɦ W, ɟɫɥɢ ɬɨɥɶɤɨ ɨɧɚ ɫɭɳɟɫɬɜɭɟɬ ɩɪɢ W=0, ɢɥɢ ɫɭɳɟɫɬɜɭɟɬ ɤɨɧɟɱɧɚɹ ɞɢɫɩɟɪɫɢɹ ɞɥɹ ɩɪɨɢɡɜɨɞɧɨɣ (ɫɤɨɪɨɫɬɢ). ɉɨɷɬɨɦɭ ɥɟɝɤɨ ɩɪɨɜɟɪɢɬɶ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɨɫɬɶ ɋɉ ɩɨ ɜɟɥɢɱɢɧɟ ɜɬɨɪɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɢɫɯɨɞɧɨɝɨ ɋɉ ɩɪɢ W = 0. ɇɚɩɪɢɦɟɪ, ɞɥɹ ɧɟɞɢɮɮɟɪɟɧɰɢɪɭɟɦɨɝɨ ɋɉ K ''(0) = f ɢɥɢ
K ''(+0) z K ''(-0).
ɉɪɢɦɟɧɹɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ Ɏɭɪɶɟ ɤ ɜɬɨɪɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ ɄɎ ɢ ɧɚɯɨɞɹ ɟɟ ɞɢɫɩɟɪɫɢɸ, ɩɨɥɭɱɢɦ
f
Wy(Z)=Z2Wx(Z); Vy2= -K ''x(W)=(2S)-1/2 ³ Z2Wx(Z)dZ,
f
ɨɬɤɭɞɚ ɫɥɟɞɭɟɬ, ɱɬɨ ɞɥɹ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɨɫɬɢ ɋɉ ɟɝɨ ɫɩɟɤɬɪɚɥɶɧɚɹ ɩɥɨɬɧɨɫɬɶ ɞɨɥɠɧɚ ɭɛɵɜɚɬɶ ɩɪɢ ɜɵɫɨɤɢɯ ɱɚɫɬɨɬɚɯ ɛɵɫɬɪɟɟ, ɱɟɦ Z-3. ȼ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ɩɪɨɰɟɫɫ ɛɭɞɟɬ ɧɟɞɢɮɮɟɪɟɧɰɢɪɭɟɦɵɦ.
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɢɦɟɪɵ ɚɧɚɥɢɡɚ ɥɢɧɟɣɧɵɯ ɰɟɩɟɣ ɩɪɢ ɜɨɡɞɟɣɫɬɜɢɢ ɧɚ ɢɯ ɜɯɨɞɵ ɛɟɥɨɝɨ ɲɭɦɚ.
7.2. Ⱦɟɣɫɬɜɢɟ ɛɟɥɨɝɨ ɲɭɦɚ ɧɚ RC-ɮɢɥɶɬɪ ɧɢɠɧɢɯ ɱɚɫɬɨɬ
Ⱦɥɹ ɬɚɤɨɣ ɰɟɩɢ ɩɟɪɟɞɚɬɨɱɧɚɹ ɮɭɧɤɰɢɹ ɟɫɬɶ K(jZ)=1/(1+jZRC), ɚ ɢɦɩɭɥɶɫɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɢɦɟɟɬ ɜɢɞ g(t)=exp(-t/RC)/(RC). Ɉɩɪɟɞɟɥɢɦ ɫɩɟɤɬɪɚɥɶɧɭɸ ɩɥɨɬɧɨɫɬɶ ɦɨɳɧɨɫɬɢ ɢ ɄɎ ɋɉ ɧɚ ɜɵɯɨɞɟ ɬɚɤɨɣ ɰɟɩɢ, ɟɫɥɢ ɧɚ ɜɯɨɞ ɩɨɞɤɥɸɱɟɧ ɢɫɬɨɱɧɢɤ ɛɟɥɨɝɨ ɲɭɦɚ. Ⱦɥɹ ɷɬɨɝɨ ɧɟɨɛɯɨɞɢɦɨ ɭɦɧɨɠɢɬɶ ɜɯɨɞɧɭɸ ɫɩɟɤɬɪɚɥɶɧɭɸ ɩɥɨɬɧɨɫɬɶ ɧɚ ɤɜɚɞɪɚɬ ɦɨɞɭɥɹ ɩɟɪɟɞɚɬɨɱɧɨɣ ɮɭɧɤɰɢɢ ɰɟɩɢ. ɉɨɫɤɨɥɶɤɭ ɫɩɟɤɬɪɚɥɶɧɚɹ ɩɥɨɬɧɨɫɬɶ ɦɨɳɧɨɫɬɢ ɛɟɥɨɝɨ ɜɯɨɞɧɨɝɨ ɲɭɦɚ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɱɚɫɬɨɬɵ, ɬɨ ɫɩɟɤɬɪɚɥɶɧɚɹ ɩɥɨɬɧɨɫɬɶ ɦɨɳɧɨɫɬɢ ɜɵɯɨɞɧɨɝɨ ɩɪɨɰɟɫɫɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɨɣ ɤɜɚɞɪɚɬɚ ɦɨɞɭɥɹ ɚɦɩɥɢɬɭɞɧɨ-ɱɚɫɬɨɬɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɰɟɩɢ ɢ ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɟɟ ɩɨɫɬɨɹɧɧɨɣ ɜɪɟɦɟɧɢ. Ʉɨɪɪɟɥɹɰɢɨɧɧɚɹ ɮɭɧɤɰɢɹ ɜɵɯɨɞɧɨɝɨ ɋɉ ɫɜɹɡɚɧɚ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟɦ Ɏɭɪɶɟ ɫɨ ɫɩɟɤɬɪɚɥɶɧɨɣ ɩɥɨɬɧɨɫɬɶɸ, ɹɜɥɹɟɬɫɹ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨɣ ɢ ɱɟɦ ɛɨɥɶɲɟ ɩɨɫɬɨɹɧɧɚɹ ɜɪɟɦɟɧɢ ɰɟɩɢ, ɬɟɦ ɦɟɞɥɟɧɧɟɟ ɫɧɢɠɚɟɬɫɹ ɤɨɪɪɟɥɹɰɢɹ ɡɧɚɱɟɧɢɣ ɜɵɯɨɞɧɨɝɨ ɋɉ ɩɪɢ ɭɜɟɥɢɱɟɧɢɢ ɡɚɞɟɪɠɤɢ ɦɟɠɞɭ ɧɢɦɢ. ȼɨɡɦɨɠɧɚɹ ɮɨɪɦɚ ɪɟɚɥɢɡɚɰɢɢ ɜɵɯɨɞɧɨɝɨ ɋɉ ɬɚɤɠɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɟɥɢɱɢɧɨɣ RC ɢ ɱɟɦ ɛɨɥɶɲɟ RC, ɬɟɦ ɦɟɞɥɟɧɧɟɟ ɢɡɦɟɧɟɧɢɹ ɜɨ ɜɪɟɦɟɧɢ ɦɝɧɨɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɢ ɦɟɧɶɲɟ ɪɚɡɦɚɯ ɪɟɚɥɢɡɚɰɢɢ (ɦɟɧɶɲɟ ɞɢɫɩɟɪɫɢɹ ɜɵɯɨɞɧɨɝɨ ɋɉ).
7.3. Ⱦɟɣɫɬɜɢɟ ɛɟɥɨɝɨ ɲɭɦɚ ɧɚ ɤɨɥɟɛɚɬɟɥɶɧɵɣ ɤɨɧɬɭɪ
Ⱥɧɚɥɨɝɢɱɧɨ ɦɨɠɧɨ ɪɚɫɫɦɨɬɪɟɬɶ ɮɨɪɦɭɥɵ ɢ ɝɪɚɮɢɤɢ ɞɥɹ ɞɜɭɯ ɞɨɛɪɨɬɧɨɫɬɟɣ. ɉɪɢ ɷɬɨɦ ɢɧɟɪɰɢɨɧɧɵɟ ɫɜɨɣɫɬɜɚ ɷɬɨɣ ɰɟɩɢ ɡɚɞɚɸɬɫɹ ɧɟ ɞɨɛɪɨɬɧɨɫɬɶɸ, ɚ ɩɨɥɨɜɢɧɨɣ ɩɨɥɨɫɵ ɩɪɨɩɭɫɤɚɧɢɹ, ɩɨɫɤɨɥɶɤɭɩɟɪɟɯɨɞɧɵɣ ɩɪɨɰɟɫɫ ɨɩɪɟɞɟɥɹɟɬɫɹ ɧɟ ɢɡɦɟɧɟɧɢɟɦ ɦɝɧɨɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɜɵɯɨɞɟ ɷɬɨɣ ɰɟɩɢ, ɚ ɢɡɦɟɧɟ-
ɧɢɟɦ ɜɨ ɜɪɟɦɟɧɢ ɨɝɢɛɚɸɳɟɣ. Ɋɟɚɥɢɡɚɰɢɹ ɭɡɤɨɩɨɥɨɫɧɨɝɨ ɫɥɭɱɚɣɧɨɝɨ ɩɪɨɰɟɫɫɚ ɧɚ ɜɵɯɨɞɟ ɤɨɥɟɛɚɬɟɥɶɧɨɝɨ ɤɨɧɬɭɪɚ ɢɦɟɟɬ ɨɝɢɛɚɸɳɭɸ ɢ ɡɚɩɨɥɧɟɧɢɟ ɫ ɱɚɫɬɨɬɨɣ Z0, ɛɥɢɡɤɨɣ ɤ ɟɝɨ ɪɟɡɨɧɚɧɫɧɨɣ ɱɚɫɬɨɬɟ.
8.ɌȿɈɊɂə ɉɊɈȼȿɊɄɂ ɋɌȺɌɂɋɌɂɑȿɋɄɂɏ ȽɂɉɈɌȿɁ
8.1.ɉɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ ɫɢɧɬɟɡɚ ɚɥɝɨɪɢɬɦɨɜ ɨɛɪɚɛɨɬɤɢ ɫɢɝɧɚɥɨɜ ɧɚ ɮɨɧɟ ɩɨɦɟɯ
ɉɪɢ ɧɚɥɢɱɢɢ ɩɨɦɟɯ ɜ ɫɦɟɫɢ ɫ ɫɢɝɧɚɥɨɦ ɧɚ ɜɯɨɞɟ ɭɫɬɪɨɣɫɬɜɚ ɨɛɪɚɛɨɬɤɢ ɜɨɡɧɢɤɚɸɬ ɨɲɢɛɤɢ ɜɨɫɩɪɨɢɡɜɟɞɟɧɢɹ ɩɟɪɟɞɚɜɚɟɦɨɝɨ ɫɨɨɛɳɟɧɢɹ. ɍɫɬɪɨɣɫɬɜɨ ɨɛɪɚɛɨɬɤɢ, ɢɥɢ ɩɪɢɟɦɧɢɤ, ɨɛɟɫɩɟɱɢɜɚɸɳɢɣ ɦɢɧɢɦɭɦ ɢɫɤɚɠɟɧɢɣ ɩɟɪɟɞɚɜɚɟɦɨɝɨ ɫɨɨɛɳɟɧɢɹ, ɧɚɡɵɜɚɟɬɫɹ ɨɩɬɢɦɚɥɶɧɵɦ ɢɥɢ ɢɞɟɚɥɶɧɵɦ, ɬ.ɟ. ɧɚɢɥɭɱɲɢɦ. ȼ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɡɚɞɚɱɢ, ɪɟɲɚɟɦɨɣ ɭɫɬɪɨɣɫɬɜɨɦ ɨɛɪɚɛɨɬɤɢ ɫɢɝɧɚɥɨɜ, ɢɫɩɨɥɶɡɭɸɬɫɹ ɪɚɡɥɢɱɧɵɟ ɤɨɥɢɱɟɫɬɜɟɧɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɢɫɤɚɠɟɧɢɣ ɩɟɪɟɞɚɜɚɟɦɨɝɨ ɫɨɨɛɳɟɧɢɹ ɢɥɢ ɤɪɢɬɟɪɢɢ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɢ ɜɵɛɪɚɧɧɨɦ ɤɪɢɬɟɪɢɢ ɢ ɡɚɞɚɧɧɵɯ ɭɫɥɨɜɢɹɯ ɩɪɢɟɦɚ ɨɩɬɢɦɚɥɶɧɵɣ ɩɪɢɟɦɧɢɤ ɨɛɟɫɩɟɱɢɜɚɟɬ ɦɢɧɢɦɚɥɶɧɵɟ ɢɫɤɚɠɟɧɢɹ ɫɨɨɛɳɟɧɢɹ. ɗɬɨɬ ɦɢɧɢɦɚɥɶɧɵɣ ɭɪɨɜɟɧɶ ɢɫɤɚɠɟɧɢɣ ɱɚɫɬɨ ɧɚɡɵɜɚɸɬ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɩɨɦɟɯɨɭɫɬɨɣɱɢɜɨɫɬɶɸ. ɉɪɢ ɡɚɞɚɧɧɵɯ ɭɫɥɨɜɢɹɯ ɩɪɢɟɦɚ ɨɩɬɢɦɚɥɶɧɵɣ ɩɪɢɟɦɧɢɤ ɨɩɪɟɞɟɥɹɟɬ ɜɟɪɯɧɢɣ ɩɪɟɞɟɥ ɩɨɦɟɯɨɭɫɬɨɣɱɢɜɨɫɬɢ, ɧɟɞɨɫɬɢɠɢɦɵɣ ɪɟɚɥɶɧɵɦ ɪɚɞɢɨɩɪɢɟɦɧɢɤɨɦ. ɑɚɫɬɨ ɨɩɬɢɦɚɥɶɧɵɣ ɩɪɢɟɦɧɢɤ ɧɟ ɹɜɥɹɟɬɫɹ ɩɪɚɤɬɢɱɟɫɤɢ ɪɟɚɥɢɡɭɟɦɵɦ, ɧɨ ɩɨɡɜɨɥɹɟɬ ɫɪɚɜɧɢɬɶ ɪɚɡɥɢɱɧɵɟ ɮɢɡɢɱɟɫɤɢ ɪɟɚɥɢɡɭɟɦɵɟ ɭɫɬɪɨɣɫɬɜɚ ɨɛɪɚɛɨɬɤɢ ɫɢɝɧɚɥɨɜ ɢ ɜɵɛɪɚɬɶ ɫɪɟɞɢ ɧɢɯ ɧɚɢɛɨɥɟɟ ɩɨɞɯɨɞɹɳɢɣ.
Ɏɨɪɦɭɥɢɪɨɜɤɚ ɩɪɨɛɥɟɦɵ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɫɥɟɞɭɸɳɟɦ. ɂɦɟɟɬɫɹ ɫɨɜɨɤɭɩɧɨɫɬɶ ɜɨɡɦɨɠɧɵɯ ɫɨɫɬɨɹɧɢɣ s0,...,sm (ɹɜɥɟɧɢɣ ɩɪɢɪɨɞɵ, ɩɪɢɱɢɧ ɩɨɹɜɥɟɧɢɹ ɫɨɛɵɬɢɣ ɢ ɬ.ɞ.), ɤɨɬɨɪɵɟ ɩɪɟɞɫɬɚɜɥɹɸɬ ɩɨɥɧɭɸ ɝɪɭɩɩɭ ɫɨɛɵɬɢɣ, ɬ.ɟ. ɫɭɦɦɚ ɜɟɪɨɹɬɧɨɫɬɟɣ ɢɯ ɩɨɹɜɥɟɧɢɹ ɪɚɜɧɚ ɟɞɢɧɢɰɟ (p0+...+pm=1). Ɋɚɫɫɦɨɬɪɢɦ ɞɚɥɟɟ ɫɨɜɨɤɭɩɧɨɫɬɶ ɪɟɡɭɥɶɬɚɬɨɜ ɧɚɛɥɸɞɟɧɢɣ x1,...,xn (ɜɵɛɨɪɨɱɧɵɯ ɡɧɚɱɟɧɢɣ), ɡɚɜɢɫɹɳɢɯ ɨɬ ɬɨɝɨ, ɤɚɤɨɟ ɢɡ ɭɩɨɦɹɧɭɬɵɯ ɫɨɫɬɨɹɧɢɣ ɜ ɞɟɣɫɬɜɢɬɟɥɶɧɨɫɬɢ ɢɦɟɟɬ ɦɟɫɬɨ. ɉɭɫɬɶ ɬɚɤɠɟ ɭɫɥɨɜɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɩɨɥɭɱɟɧɧɨɣ ɜ ɪɟɡɭɥɶɬɚɬɟ ɨɩɵɬɚ ɜɵɛɨɪɤɢ ɢɦɟɟɬ ɜɢɞ fn(x1,...,xn/sk) ɢ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɫɨɫɬɨɹɧɢɸ sk, k = 0,1,..., m.
ɂɦɟɸɬɫɹ ɬɚɤɠɟ: ɧɚɛɨɪ ɪɟɲɟɧɢɣ J0,...,Jm ɨɬɧɨɫɢɬɟɥɶɧɨ ɢɫɬɢɧɧɨɫɬɢ ɫɨɫɬɨɹɧɢɣ; ɩɪɚɜɢɥɚ ɜɵɛɨɪɚ ɪɟɲɟɧɢɣ G(Ji/x1,...,xn), ɩɪɢɩɢɫɵɜɚɸɳɢɟ ɤɚɠɞɨɦɭ ɜɨɡɦɨɠɧɨɦɭ ɪɟɡɭɥɶɬɚɬɭ ɧɚɛɥɸɞɟɧɢɣ x1,...,xn ɨɞɧɨ ɢɡ ɪɟɲɟɧɢɣ Ji, i=0,1,..., m, ɚ ɬɚɤɠɟ ɮɭɧɤɰɢɹ ɩɨɬɟɪɶ ɉ(si,Jk), ɭɱɢɬɵɜɚɸɳɚɹ ɩɨɫɥɟɞɫɬɜɢɹ ɜɵɛɨɪɚ ɪɟɲɟɧɢɣ ɢ ɤɪɢɬɟɪɢɣ ɤɚɱɟɫɬɜɚ f(ɉ) ɜɵɛɨɪɚ ɪɟɲɟɧɢɣ, ɫɜɹɡɚɧɧɵɯ ɫ ɮɭɧɤɰɢɟɣ ɩɨɬɟɪɶ.
ɉɪɨɛɥɟɦɚ ɦɨɠɟɬ ɛɵɬɶ ɫɮɨɪɦɭɥɢɪɨɜɚɧɚ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: ɩɪɢ ɡɚɞɚɧɧɵɯ ɪɚɫɩɪɟɞɟɥɟɧɢɹɯ pi, fn(x1,...,xn/si), ɧɚɛɨɪɟ ɪɟɲɟɧɢɣ Ji, ɮɭɧɤɰɢɢ ɩɨɬɟɪɶ ɉ ɢ ɤɪɢɬɟɪɢɢ ɤɚɱɟɫɬɜɚ f ɬɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɧɚɢɥɭɱɲɟɟ ɜ ɫɦɵɫɥɟ ɩɪɢɧɹɬɨɝɨ ɤɪɢɬɟɪɢɹ ɩɪɚɜɢɥɨ G ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɪɟɡɭɥɶɬɚɬɨɜ ɧɚɛɥɸɞɟɧɢɣ ɞɥɹ ɜɵɛɨɪɚ ɪɟɲɟɧɢɹ. ɗɬɨ ɩɪɚɜɢɥɨ ɹɜɥɹɟɬɫɹ ɪɚɡɧɨɜɢɞɧɨɫɬɶɸ ɫɬɚɬɢɫɬɢɱɟɫɤɨɝɨ ɜɵɜɨɞɚ, ɩɨɥɭɱɚɟɦɨɝɨ ɩɨ ɪɟ-
ɡɭɥɶɬɚɬɚɦ ɧɚɛɥɸɞɟɧɢɣ, ɨ ɧɟɢɡɜɟɫɬɧɵɯ ɫɬɨɪɨɧɚɯ ɢɡɭɱɚɟɦɨɝɨ ɹɜɥɟɧɢɹ, ɢɥɢ ɬɨɱɧɟɟ, ɨ ɩɪɢɧɹɬɨɣ ɦɨɞɟɥɢ ɜ ɭɫɥɨɜɢɹɯ ɧɟɩɨɥɧɨɣ ɚɩɪɢɨɪɧɨɣ ɢɧɮɨɪɦɚɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɷɬɨɣ ɦɨɞɟɥɢ.
ɉɪɨɢɥɥɸɫɬɪɢɪɭɟɦ ɫɤɚɡɚɧɧɨɟ ɧɚ ɩɪɨɫɬɨɦ ɩɪɢɦɟɪɟ.
ɉɭɫɬɶ ɫɢɫɬɟɦɚ ɩɟɪɟɞɚɱɢ ɢɧɮɨɪɦɚɰɢɢ, ɧɚɩɪɢɦɟɪ, ɫɢɫɬɟɦɚ ɫɜɹɡɢ, ɫɨɫɬɨɢɬ ɢɡ ɩɟɪɟɞɚɬɱɢɤɚ, ɜɵɪɚɛɚɬɵɜɚɸɳɟɝɨ ɞɜɚ ɫɨɨɛɳɟɧɢɹ: "ɞɚ" ɢ "ɧɟɬ", ɤɨɬɨɪɵɟ ɩɪɟɨɛɪɚɡɭɸɬɫɹ ɜ ɤɨɞɢɪɭɸɳɟɦ ɭɫɬɪɨɣɫɬɜɟ ɜ ɫɢɝɧɚɥɵ "1" ɢ "0" ɢ ɩɟɪɟɞɚɟɬɫɹ ɱɟɪɟɡ ɤɚɧɚɥ ɫɜɹɡɢ ɫ ɩɨɦɟɯɚɦɢ, ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɩɪɟɨɛɪɚɡɨɜɵɜɚɬɶ ɫɢɝɧɚɥ "1' ɜ "0" ɢ ɧɚɨɛɨɪɨɬ. ɂɦɟɟɬɫɹ ɧɚ ɞɪɭɝɨɦ ɤɨɧɰɟ ɤɚɧɚɥɚ ɫɜɹɡɢ ɩɪɢɟɦɧɨɟ ɭɫɬɪɨɣɫɬɜɨ, ɤɨɬɨɪɨɟ ɩɪɟɨɛɪɚɡɨɜɵɜɚɟɬ ɫɢɝɧɚɥɵ ɜ ɫɨɨɛɳɟɧɢɹ ɜ ɢɫɤɚɠɟɧɧɨɦ ɢɥɢ ɧɟɢɫɤɚɠɟɧɧɨɦ ɜɢɞɟ. Ɉɩɟɪɚɬɨɪ (ɚɜɬɨɦɚɬɢɱɟɫɤɨɟ ɭɫɬɪɨɣɫɬɜɨ), ɧɚɛɥɸɞɚɸɳɢɣ (ɪɟɝɢɫɬɪɢɪɭɸɳɟɟ) ɫɢɝɧɚɥ ɧɚ ɜɵɯɨɞɟ ɩɪɢɟɦɧɨɝɨ ɭɫɬɪɨɣɫɬɜɚ, ɞɨɥɠɟɧ ɪɚɫɲɢɮɪɨɜɚɬɶ ɩɟɪɟɞɚɱɭ, ɬ.ɟ. ɨɩɪɟɞɟɥɢɬɶ ɩɨ ɩɪɢɧɹɬɨɦɭɫɢɝɧɚɥɭɩɟɪɟɞɚɧɧɨɟ ɫɨɨɛɳɟɧɢɟ.
ȿɫɥɢ ɛɵ ɧɟ ɛɵɥɨ ɩɨɦɟɯ, ɬɨ ɧɟ ɛɵɥɨ ɛɵ ɢ ɩɪɨɛɥɟɦ, ɫɜɹɡɚɧɧɵɯ ɫ ɨɬɜɟɬɨɦ ɧɚ ɜɨɩɪɨɫ, ɤɚɤɨɟ ɫɨɨɛɳɟɧɢɟ ɛɵɥɨ ɩɨɫɥɚɧɨ. Ɉɞɧɚɤɨ ɢɡ-ɡɚ ɢɫɤɚɠɟɧɢɣ ɩɨɦɟɯɚɦɢ ɩɪɢɧɹɬɵɣ ɫɢɝɧɚɥ ɧɟ ɜɫɟɝɞɚ ɛɭɞɟɬ ɞɨɫɬɨɜɟɪɧɨ ɭɤɚɡɵɜɚɬɶ ɧɚ ɬɨ, ɤɚɤɨɟ ɫɨɨɛɳɟɧɢɟ ɛɵɥɨ ɩɟɪɟɞɚɧɨ, ɬ. ɟ. ɛɭɞɭɬ ɫɥɭɱɚɢ, ɤɨɝɞɚ ɩɪɢɧɢɦɚɟɬɫɹ ɫɢɝɧɚɥ "1" ɩɪɢ ɩɟɪɟɞɚɱɟ ɫɨɨɛɳɟɧɢɹ "ɧɟɬ", ɢ ɧɚɨɛɨɪɨɬ. ȼɨɡɧɢɤɚɟɬ ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɞɚɬɶ ɨɩɟɪɚɬɨɪɭ ɡɚɪɚɧɟɟ ɩɪɚɜɢɥɨ ɩɨɜɟɞɟɧɢɹ, ɧɟ ɩɨɥɚɝɚɹɫɶ ɧɚ ɟɝɨ ɢɧɬɭɢɰɢɸ ɢ ɫɭɛɴɟɤɬɢɜɧɵɟ ɫɭɠɞɟɧɢɹ, ɡɚɜɢɫɹɳɢɟ ɨɬ ɫɥɭɱɚɣɧɨɫɬɟɣ, ɧɚɩɪɢɦɟɪ, ɫɟɦɟɣɧɨɝɨ ɯɚɪɚɤɬɟɪɚ. ɗɬɚ ɫɢɬɭɚɰɢɹ ɢɦɟɟɬ ɜɫɟ ɷɥɟɦɟɧɬɵ ɩɪɢɜɟɞɟɧɧɨɣ ɩɨɫɬɚɧɨɜɤɢ ɩɪɨɛɥɟɦɵ. Ɍɚɤ, ɫɨɨɛɳɟɧɢɹ "ɧɟɬ" ɢ "ɞɚ" ɹɜɥɹɸɬɫɹ ɜɡɚɢɦɧɨ ɧɟɫɨɜɦɟɫɬɢɦɵɦɢ ɫɨɫɬɨɹɧɢɹɦɢ ɩɟɪɟɞɚɬɱɢɤɚ s0 ɢ s1. Ⱥɩɪɢɨɪɧɵɟ ɜɟɪɨɹɬɧɨɫɬɢ ɷɬɢɯ ɫɨɫɬɨɹɧɢɣ p0 ɢ p1=1-p0 ɞɚɸɬ ɫɬɚɬɢɫɬɢɱɟɫɤɭɸ ɫɬɪɭɤɬɭɪɭ ɢɫɬɨɱɧɢɤɚ ɫɨɨɛɳɟɧɢɣ. Ɋɟɡɭɥɶɬɚɬɚɦɢ ɧɚɛɥɸɞɟɧɢɣ ɹɜɥɹɸɬɫɹ ɫɢɝɧɚɥɵ "1" ɢ "0". ɍɫɥɨɜɧɵɟ ɜɟɪɨɹɬɧɨɫɬɢ ɷɬɢɯ ɫɢɝɧɚɥɨɜ p{0/s0}=1-p{1/s0}, p{1/s1}=1-p{0/s1} ɨɩɪɟɞɟɥɹɸɬɫɹ ɜɟɪɨɹɬɧɨɫɬɧɵɦɢ ɫɜɨɣɫɬɜɚɦɢ ɩɨɦɟɯ ɜ ɤɚɧɚɥɟ. ȼɟɥɢɱɢɧɵ p{0/s0} ɢ p{1/s1} ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɜɟɪɨɹɬɧɨɫɬɢ ɬɨɝɨ, ɱɬɨ ɫɢɝɧɚɥɵ "0" ɢ "1" ɩɪɢɧɢɦɚɸɬɫɹ ɧɚ ɩɪɢɟɦɧɨɦ ɤɨɧɰɟ ɥɢɧɢɢ ɫɜɹɡɢ ɛɟɡ ɢɫɤɚɠɟɧɢɣ ɩɨɦɟɯɚɦɢ, ɚ p{1/s0} ɢ p{0/s1} - ɜɟɪɨɹɬɧɨɫɬɢ ɨɲɢɛɨɤ ɞɜɭɯ ɬɢɩɨɜ: ɩɟɪɟɯɨɞɚ "0" ɜ "1" ɢ ɧɚɨɛɨɪɨɬ. ɇɚɛɨɪ ɪɟɲɟɧɢɣ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɫɨɫɬɨɢɬ ɢɡ J0 (ɩɟɪɟɞɚɧɨ ɫɨɨɛɳɟɧɢɟ "ɧɟɬ") ɢ J1 (ɩɟɪɟɞɚɧɨ ɫɨɨɛɳɟɧɢɟ "ɞɚ"), ɚ ɩɪɚɜɢɥɚ ɪɟɲɟɧɢɹ ɩɪɟɞɩɢɫɵɜɚɸɬ ɨɩɟɪɚɬɨɪɭ, ɤɚɤɨɟ ɢɡ ɷɬɢɯ ɞɜɭɯ ɪɟɲɟɧɢɣ ɨɧ ɞɨɥɠɟɧ ɩɪɢɧɢɦɚɬɶ, ɟɫɥɢ ɨɧ ɧɚɛɥɸɞɚɟɬ ɫɢɝɧɚɥ "1" ɢɥɢ ɫɢɝɧɚɥ "0". Ɏɭɧɤɰɢɹ ɩɨɬɟɪɶ ɜ ɷɬɨɦ ɩɪɢɦɟɪɟ ɞɨɥɠɧɚ ɭɱɢɬɵɜɚɬɶ ɩɨɫɥɟɞɫɬɜɢɹ ɨɲɢɛɨɱɧɵɯ ɪɟɲɟɧɢɣ ɨɩɟɪɚɬɨɪɚ ɢ ɧɚɡɧɚɱɚɟɬ "ɩɥɚɬɭ" ɉ01>0 ɡɚ ɨɲɢɛɤɭ ɩɟɪɜɨɝɨ ɪɨɞɚ (ɩɪɢɧɹɬɨ ɪɟɲɟɧɢɟ, ɱɬɨ ɛɵɥɨ ɩɟɪɟɞɚɧɨ ɫɨɨɛɳɟɧɢɟ "ɞɚ", ɤɨɝɞɚ ɧɚ ɫɚɦɨɦ ɞɟɥɟ ɩɟɪɟɞɚɜɚɥɨɫɶ "ɧɟɬ") ɢ "ɩɥɚɬɭ" ɉ10>0 ɡɚ ɨɲɢɛɤɭ ɜɬɨɪɨɝɨ ɪɨɞɚ. Ɇɨɠɧɨ ɜɜɟɫɬɢ ɧɚɪɹɞɭ ɫ ɩɨɬɟɪɹɦɢ ɡɚ ɧɟɩɪɚɜɢɥɶɧɵɟ ɪɟɲɟɧɢɹ ɜɵɢɝɪɵɲɢ ɡɚ ɩɪɚɜɢɥɶɧɵɟ ɪɟɲɟɧɢɹ (ɨɬɪɢɰɚɬɟɥɶɧɵɟ ɪɟɲɟɧɢɹ), ɨɞɧɚɤɨ ɨɝɪɚɧɢɱɢɜɚɸɬɫɹ ɜɜɟɞɟɧɢɟɦ ɬɨɥɶɤɨ ɩɨɬɟɪɶ ɢɡ-ɡɚ ɧɟɩɪɚɜɢɥɶɧɵɯ ɪɟɲɟɧɢɣ, ɩɪɢɧɢɦɚɹ "ɩɥɚɬɭ" ɡɚ ɩɪɚɜɢɥɶɧɵɟ ɪɟɲɟɧɢɹ ɪɚɜɧɨɣ ɧɭɥɸ.
Ʉɪɢɬɟɪɢɟɦ ɤɚɱɟɫɬɜɚ ɜɵɛɨɪɚ ɪɟɲɟɧɢɹ ɦɨɠɟɬ ɫɥɭɠɢɬɶ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɩɨɬɟɪɶ ɢɡ-ɡɚ ɨɲɢɛɨɱɧɵɯ ɪɟɲɟɧɢɣ, ɜɡɜɟɲɟɧɧɨɟ ɫ ɜɟɪɨɹɬɧɨɫɬɹɦɢ ɢɯ ɩɨɹɜɥɟɧɢɹ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɫɨɝɥɚɫɧɨ ɷɬɨɦɭ ɤɪɢɬɟɪɢɸ ɜɵɛɢɪɚɟɬɫɹ ɢɡ ɞɜɭɯ ɜɨɡɦɨɠɧɵɯ ɪɟɲɟ-
ɧɢɣ ɬɨ, ɞɥɹ ɤɨɬɨɪɨɝɨ ɜɟɥɢɱɢɧɚ ɫɪɟɞɧɢɯ ɩɨɬɟɪɶ ɦɟɧɶɲɟ. ɉɨɞɫɱɢɬɚɟɦ ɜɟɥɢɱɢɧɵ ɫɪɟɞɧɢɯ ɩɨɬɟɪɶ ɞɥɹ ɞɜɭɯ ɩɪɚɜɢɥ ɜɵɛɨɪɚ ɤɚɠɞɨɝɨ ɢɡ ɜɨɡɦɨɠɧɵɯ ɪɟɲɟɧɢɣ.
Ɉɞɧɨ ɩɪɚɜɢɥɨ G0 ɦɨɠɟɬ ɛɵɬɶ ɫɮɨɪɦɭɥɢɪɨɜɚɧɨ ɬɚɤ: ɧɚɛɥɸɞɚɟɲɶ ɫɢɝɧɚɥ "0" - ɩɪɢɧɢɦɚɣ ɪɟɲɟɧɢɟ J0, ɚ ɤɨɝɞɚ ɧɚɛɥɸɞɚɟɲɶ "1" - ɬɨ ɪɟɲɟɧɢɟ J1. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɜɟɪɨɹɬɧɨɫɬɢ ɨɲɢɛɨɱɧɵɯ ɪɟɲɟɧɢɣ ɪɚɜɧɵ:
P{ɨɲɢɛɤɚ ɩɟɪɜɨɝɨ ɪɨɞɚ} = P{J1/s0} = p0P{1/s0}; P{ɨɲɢɛɤɚ ɜɬɨɪɨɝɨ ɪɨɞɚ} = P{J0/s1} = p1P{0/s1}, ɚ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ R0 ɩɨɬɟɪɶ ɪɚɜɧɨ ɉ01p0p(1/s0)+ɉ10p1p(0/s1).
Ⱦɪɭɝɨɟ ɩɪɚɜɢɥɨ G1 ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɬɚɤ: ɧɚɛɥɸɞɚɟɲɶ ɫɢɝɧɚɥ "0"- ɩɪɢɧɢɦɚɣ ɪɟɲɟɧɢɟ J1 ɢ ɧɚɨɛɨɪɨɬ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɜɟɪɨɹɬɧɨɫɬɢ ɨɲɢɛɨɱɧɵɯ ɪɟɲɟɧɢɣ ɪɚɜɧɵ:
P{ɨɲɢɛɤɚ ɩɟɪɜɨɝɨ ɪɨɞɚ} = p{J1/s0) = p0P{1/s0}; P{ɨɲɢɛɤɚ ɜɬɨɪɨɝɨ ɪɨɞɚ} = P{J0/s1} = p1P{0/s1}, ɚ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ R1 ɩɨɬɟɪɶ ɪɚɜɧɨ ɉ01p0p(1/s0)+ɉ10p1p(0/s1).
ɉɪɢɧɹɬɵɣ ɤɪɢɬɟɪɢɣ ɤɚɱɟɫɬɜɚ ɨɬɞɚɟɬ ɩɪɟɞɩɨɱɬɟɧɢɟ ɩɪɚɜɢɥɭ G0, ɟɫɥɢ
R0/R1<1, ɬ.ɟ. ɤɨɝɞɚ R0<0,5(R0+R1). Ɍɚɤ ɤɚɤ R0+R1=ɉ01p0+ɉ10p1, ɬɨ ɩɪɢɯɨɞɢɦ ɤ ɫɥɟɞɭɸɳɟɦɭ ɭɫɥɨɜɢɸ, ɩɪɢ ɜɵɩɨɥɧɟɧɢɢ ɤɨɬɨɪɨɝɨ ɩɪɢɧɢɦɚɟɬɫɹ ɩɪɚɜɢɥɨ ɩɪɢɧɹɬɢɹ
ɪɟɲɟɧɢɹ G0:
ɉ01p0p(1/s0)+ɉ10p1p(0/s1)<0,5(ɉ01p0+ɉ10p1). (8.1)
Ɏɨɪɦɭɥɚ (8.1), ɩɨɦɢɦɨ ɭɫɥɨɜɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ ɨɲɢɛɨɤ, ɨɩɪɟɞɟɥɹɟɦɵɯ ɜɟɪɨɹɬɧɨɫɬɧɵɦɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ ɩɨɦɟɯ ɜ ɤɚɧɚɥɟ ɫɜɹɡɢ, ɫɨɞɟɪɠɢɬ ɚɩɪɢɨɪɧɵɟ ɜɟɪɨɹɬɧɨɫɬɢ ɫɨɨɛɳɟɧɢɣ ɢ ɜɟɥɢɱɢɧɵ ɩɨɬɟɪɶ ɨɬ ɧɟɩɪɚɜɢɥɶɧɵɯ ɪɟɲɟɧɢɣ. Ɉɩɪɟɞɟɥɟɧɢɟ ɢɥɢ ɧɚɡɧɚɱɟɧɢɟ ɜɟɥɢɱɢɧ p0, ɉ10 ɢ ɉ01 ɜ ɤɨɧɤɪɟɬɧɵɯ ɫɥɭɱɚɹɯ ɜɵɡɵɜɚɟɬ ɡɚɬɪɭɞɧɟɧɢɹ. 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 ɧɚɢɦɟɧɶɲɢɯ ɫɪɟɞɧɢɯ ɩɨɬɟɪɶ ɩɟɪɟɯɨɞɢɬ ɜ ɤɪɢɬɟɪɢɣ ɧɚɢɦɟɧɶɲɟɣ ɜɟɪɨɹɬɧɨɫɬɢ ɨɲɢɛɨɤ. Ʉɨɝɞɚ ɠɟ ɧɢɱɟɝɨ "ɧɟɢɡɜɟɫɬɧɨ" ɨ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɫɬɪɭɤɬɭɪɟ ɢɫɬɨɱɧɢɤɚ ɫɨɨɛɳɟɧɢɣ (ɬ.ɟ. ɜɟɪɨɹɬɧɨɫɬɢ p0 ɢ p1 ɧɟɢɡɜɟɫɬɧɵ), ɬɨ ɨɫɬɚɟɬɫɹ ɩɪɟɞɩɨɥɨɠɢɬɶ, ɱɬɨ ɫɨɨɛɳɟɧɢɹ ɩɟɪɟɞɚɸɬɫɹ ɫ ɨɞɢɧɚɤɨɜɵɦɢ ɜɟɪɨɹɬɧɨɫɬɹɦɢ, ɬ.ɟ. p1= p0 = 0,5.
ȿɫɥɢ ɩɥɚɬɚ ɡɚ ɨɲɢɛɤɢ ɨɞɢɧɚɤɨɜɚ ɢ ɪɚɜɧɵ ɦɟɠɞɭ ɫɨɛɨɣ ɜɟɪɨɹɬɧɨɫɬɢ ɩɟɪɟɞɚɜɚɟɦɵɯ ɫɨɨɛɳɟɧɢɣ, ɬɨ ɭɫɥɨɜɢɟ (8.1) ɫɬɚɧɨɜɢɬɫɹ ɩɪɨɫɬɵɦ:
p{1/s0)+p{0/s0}<1 ɢɥɢ p{1/s0)<p{0/s0}. (8.2)
Ɏɨɪɦɭɥɚ (8.2) ɨɡɧɚɱɚɟɬ, ɱɬɨ ɜɟɪɨɹɬɧɨɫɬɶ ɢɫɤɚɠɟɧɢɹ ɫɢɝɧɚɥɚ "0" ɦɟɧɶɲɟ ɜɟɪɨɹɬɧɨɫɬɢ ɢɫɤɚɠɟɧɢɹ ɫɢɝɧɚɥɚ "1" (ɬ.ɟ. ɜɟɪɨɹɬɧɨɫɬɶ ɥɨɠɧɨɝɨ ɫɢɝɧɚɥɚ "1" ɦɟɧɶɲɟ ɜɟɪɨɹɬɧɨɫɬɢ ɧɟɩɨɞɚɜɥɟɧɢɹ ɢɫɬɢɧɧɨɝɨ ɫɢɝɧɚɥɚ "1"). ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɫɪɟɞɧɢɟ ɩɨɬɟ- 0,5.ɪɢ R0 < 0,5, ɜ ɬɨ ɜɪɟɦɹ ɤɚɤ ɞɥɹ ɩɪɚɜɢɥɚ ɩɪɢ ɭɫɥɨɜɢɢ (8.2) ɫɪɟɞɧɢɟ ɩɨɬɟɪɢ R1 >
ɉɪɨɫɬɵɟ ɢ ɫɥɨɠɧɵɟ ɝɢɩɨɬɟɡɵ
Ɋɚɫɫɦɨɬɪɢɦ ɨɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ ɬɟɨɪɢɢ ɩɪɨɜɟɪɤɢ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɝɢɩɨɬɟɡ, ɢɫɩɨɥɶɡɨɜɚɧɧɵɟ ɜ ɩɪɟɞɵɞɭɳɟɦ ɪɚɡɞɟɥɟ. ɇɚɩɨɦɧɢɦ, ɱɬɨ ɜɵɛɨɪ ɪɟɲɟɧɢɹ ɫɨɫɬɨɢɬ
ɜ ɩɪɢɧɹɬɢɢ ɢɥɢ ɨɬɤɥɨɧɟɧɢɢ ɝɢɩɨɬɟɡɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɨɡɦɨɠɧɵɯ ɫɨɫɬɨɹɧɢɣ ɢɡɭɱɚɟɦɨɝɨ ɹɜɥɟɧɢɹ ɩɨ ɪɟɡɭɥɶɬɚɬɚɦ ɧɚɛɥɸɞɟɧɢɹ. ɇɚɩɪɢɦɟɪ, ɩɪɢ ɨɛɧɚɪɭɠɟɧɢɢ ɫɢɝɧɚɥɚ ɧɚ ɮɨɧɟ ɲɭɦɚ ɪɟɡɭɥɶɬɚɬ ɧɚɛɥɸɞɟɧɢɹ ɜɵɯɨɞɧɨɝɨ ɷɮɮɟɤɬɚ ɩɪɢɟɦɧɨɝɨ ɭɫɬɪɨɣɫɬɜɚ ɦɨɠɟɬ ɨɬɧɨɫɢɬɶɫɹ ɬɨɥɶɤɨ ɤ ɲɭɦɭ (s0) ɢɥɢ ɤ ɫɦɟɫɢ ɫɢɝɧɚɥɚ ɢ ɲɭɦɚ (s1). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜɵɛɨɪ ɪɟɲɟɧɢɹ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɩɪɢɧɹɬɢɢ ɢɥɢ ɨɬɤɥɨɧɟɧɢɢ ɝɢɩɨɬɟɡɵ H0. ɉɪɢɧɹɬɢɟ ɝɢɩɨɬɟɡɵ H0 ɩɪɢɜɨɞɢɬ ɤ ɜɵɜɨɞɭ ɨ ɬɨɦ, ɱɬɨ ɧɚɛɥɸɞɚɜɲɢɣɫɹ ɜɵɯɨɞɧɨɣ ɷɮɮɟɤɬ ɨɬɧɨɫɢɬɫɹ ɬɨɥɶɤɨ ɤ ɲɭɦɭ. ɉɪɨɬɢɜɨɩɨɥɨɠɧɚɹ ɝɢɩɨɬɟɡɚ H1 ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɬɨɦ, ɱɬɨ ɷɮɮɟɤɬ ɨɬɧɨɫɢɬɫɹ ɤ ɫɦɟɫɢ ɫɢɝɧɚɥɚ ɢ ɲɭɦɚ, ɢ ɧɚɡɵɜɚɟɬɫɹ ɚɥɶɬɟɪɧɚɬɢɜɨɣ.
Ʉɥɚɫɫ ɝɢɩɨɬɟɡ ɧɚɡɵɜɚɟɬɫɹ ɩɪɨɫɬɵɦ, ɟɫɥɢ ɨɧ ɫɨɞɟɪɠɢɬ ɬɨɥɶɤɨ ɨɞɧɭɝɢɩɨɬɟɡɭ, ɢ ɫɥɨɠɧɵɦ, ɟɫɥɢ ɝɢɩɨɬɟɡ ɧɟ ɦɟɧɶɲɟ ɞɜɭɯ. ȼ ɩɪɢɜɟɞɟɧɧɨɦ ɩɪɢɦɟɪɟ ɩɪɨɜɟɪɹɟɦɚɹ ɝɢɩɨɬɟɡɚ ɹɜɥɹɟɬɫɹ ɩɪɨɫɬɨɣ. ȿɫɥɢ ɨɛɧɚɪɭɠɟɧɢɟ ɦɨɠɟɬ ɩɪɨɢɡɜɨɞɢɬɶɫɹ ɧɚ ɮɨɧɟ ɪɚɡɥɢɱɧɵɯ ɫɦɟɫɟɣ ɲɭɦɚ ɢ ɩɨɦɟɯ (ɧɚɩɪɢɦɟɪ, ɜɧɭɬɪɢɩɪɢɟɦɧɨɝɨ ɲɭɦɚ ɢɥɢ ɬɨɝɨ ɠɟ ɲɭɦɚ ɢ ɜɧɟɲɧɟɣ ɩɨɦɟɯɢ), ɬɨ ɝɢɩɨɬɟɡɚ ɫɥɨɠɧɚɹ. ɂ ɟɫɥɢ ɩɪɨɢɡɜɨɞɢɬɫɹ ɨɛɧɚɪɭɠɟɧɢɟ ɨɞɧɨɝɨ ɫɢɝɧɚɥɚ ɢɡ ɢɯ ɫɨɜɨɤɭɩɧɨɫɬɢ, ɬɨ ɤɥɚɫɫ ɚɥɶɬɟɪɧɚɬɢɜ ɹɜɥɹɟɬɫɹ ɫɥɨɠɧɵɦ.
ȼɵɛɨɪɤɚ
ɋ ɤɚɠɞɵɦ ɫɥɭɱɚɣɧɵɦ ɷɤɫɩɟɪɢɦɟɧɬɨɦ ɫɜɹɡɚɧɚ ɧɟɤɨɬɨɪɚɹ ɫɥɭɱɚɣɧɚɹ ɜɟɥɢɱɢɧɚ [, ɡɧɚɱɟɧɢɹ ɤɨɬɨɪɨɣ ɟɫɬɶ ɪɟɡɭɥɶɬɚɬɵ ɧɚɛɥɸɞɟɧɢɹ. Ɋɟɡɭɥɶɬɚɬ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ n ɫɥɭɱɚɣɧɵɯ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɷɬɨɣ ɋȼ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ n ɜɨɡɦɨɠɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ x1,..., xn ɋȼ [. Ʉɚɠɞɨɟ ɢɡ ɷɬɢɯ ɡɧɚɱɟɧɢɣ ɧɚɡɵɜɚɟɬɫɹ ɜɵɛɨɪɨɱɧɵɦ ɢɥɢ ɟɞɢɧɢɱɧɵɦ ɢɡɦɟɪɟɧɢɟɦ, ɚ ɢɯ ɫɨɜɨɤɭɩɧɨɫɬɶ - ɜɵɛɨɪɤɨɣ. ɉɭɫɬɶ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜɟɪɨɹɬɧɨɫɬɟɣ ɋȼ [ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɢɧɬɟɝɪɚɥɶɧɨɣ ɮɭɧɤɰɢɟɣ F1(x) ɢɥɢ ɩɥɨɬɧɨɫɬɶɸ ɜɟɪɨɹɬɧɨɫɬɟɣ f1(x). Ɍɨɝɞɚ ɜɵɛɨɪɤɚ x1,..., xn ɩɨɥɭɱɟɧɚ ɢɡ ɪɚɫɩɪɟɞɟɥɟɧɢɹ F1(x) ɢɥɢ f1(x). ȼɢɞ ɮɭɧɤɰɢɢ F1(x) ɢɥɢ f1(x) ɦɨɠɟɬ ɡɚɜɢɫɟɬɶ ɨɬ ɫɨɫɬɨɹɧɢɹ sk ɢɡɭɱɚɟɦɨɝɨ ɹɜɥɟɧɢɹ. ɑɬɨɛɵ ɷɬɨ ɩɨɞɱɟɪɤɧɭɬɶ, ɛɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ ɷɬɢ ɮɭɧɤɰɢɢ F1(x/sk) ɢɥɢ f1(x/sk). Ʉɚɠɞɚɹ ɜɵɛɨɪɤɚ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɬɨɱɤɭ ɜ n-ɦɟɪɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɢɥɢ ɜɟɤɬɨɪ X=(x1,..., xn). ɗɬɚ ɨɛɥɚɫɬɶ ɧɚɡɵɜɚɟɬɫɹ ɜɵɛɨɪɨɱɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ ɢɥɢ ɩɪɨɫɬɪɚɧɫɬɜɨɦ ɜɵɛɨɪɨɤ. ȼɟɪɨɹɬɧɨɫɬɧɵɟ ɫɜɨɣɫɬɜɚ ɜɵɛɨɪɤɢ X ɨɩɢɫɵɜɚɸɬɫɹ n-ɦɟɪɧɨɣ ɩɥɨɬɧɨɫɬɶɸ ɜɟɪɨɹɬɧɨɫɬɢ. ȿɫɥɢ ɟɞɢɧɢɱɧɵɟ ɢɡɦɟɪɟɧɢɹ ɜɵɛɨɪɤɢ ɧɟɡɚɜɢɫɢɦɵ, ɬɨ ɷɬɚ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɩɪɨɢɡɜɟɞɟɧɢɟɦ ɨɞɧɨɦɟɪɧɵɯ ɩɥɨɬɧɨɫɬɟɣ ɜɟɪɨɹɬɧɨɫɬɟɣ ɤɚɠɞɨɝɨ ɜɵɛɨɪɨɱɧɨɝɨ ɡɧɚɱɟɧɢɹ. ȿɫɥɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜɵɛɨɪɤɢ ɞɢɫɤɪɟɬɧɨ, ɬɨ ɫɨɜɦɟɫɬɧɨɟ ɫɨɛɵɬɢɟ - ɩɨɥɭɱɟɧɢɟ n ɢɡɦɟɪɟɧɢɣ ɟɫɬɶ ɩɪɨɢɡɜɟɞɟɧɢɟ ɜɟɪɨɹɬɧɨɫɬɟɣ ɤɚɠɞɨɝɨ ɜɵɛɨɪɨɱɧɨɝɨ ɡɧɚɱɟɧɢɹ.
ɋɨɜɦɟɫɬɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜɵɛɨɪɨɱɧɵɯ ɡɧɚɱɟɧɢɣ ɧɚɡɵɜɚɸɬ ɮɭɧɤɰɢɟɣ ɩɪɚɜɞɨɩɨɞɨɛɢɹ ɜɵɛɨɪɤɢ X ɪɚɡɦɟɪɚ n.
ɇɚɛɨɪ ɪɟɲɟɧɢɣ ɢ ɩɪɚɜɢɥɨ ɜɵɛɨɪɚ ɪɟɲɟɧɢɣ
ɇɚɛɨɪ ɪɟɲɟɧɢɣ J0,J1,...,Jm ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɪɹɞ ɥɨɝɢɱɟɫɤɢɯ ɭɬɜɟɪɠɞɟɧɢɣ ɨ ɬɨɦ, ɤɚɤɚɹ ɢɡ ɝɢɩɨɬɟɡ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɨɫɬɨɹɧɢɣ s0,...,sm ɢɡɭɱɚɟɦɨɝɨ ɹɜɥɟɧɢɹ ɢɫ-