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ɉɨɞɫɬɚɜɥɹɹ (1.48) ɜ (1.47), ɢɦɟɟɦ

«§

»Z 2k cos Zt dZ 1

S2 2k...(2k m 1)

2^ m 0^

¬©

2 ¹

¹ ¼

W2Zc >2(k 1@...>2(k 1) m 1@ `sin(Zt 1 mS)

2k m

(1.49)

m 1

( 1)kW2 >2(k 1)@!

sinZct

2k 3

ɉɨɞɫɬɚɜɥɹɹ (1.49) ɜ (1.46), ɩɨɥɭɱɢɦ:

| E2k | §W

g(t)

¦^

2k...(2k m 1)

2S W k 0 (2k)! © 2

m 0

W2Zc >2(k 1@...>2(k 1) m 1@ `sin (Zc t 1 mS)

2k m

(1.50)

m 1

( 1)kW2 >2(k 1)@!

sinZct

t2k 3

Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɭɫɬɨɣɱɢɜɨɫɬɢ ɫɢɫɬɟɦɵ ɮɢɥɶɬɪɨɜ ɫ ɩɨɥɭɱɟɧɧɨɣ ɢɦɩɭɥɶɫɧɨɣ ɩɟɪɟɯɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ ɪɚɫɫɦɨɬɪɢɦ ɩɨɜɟɞɟɧɢɟ g(t) ɩɪɢ t of. ȼɵ-

				\| E2k \| §W		·	2k
ɪɚɠɟɧɢɹ					¨	¸	ɹɜɥɹɸɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɪɚɡɥɨɠɟɧɢɹ ɮɭɧɤɰɢɢ
					¨	¸
				(2k)! ©2		¹
§	ZW ·		1	ɜ ɫɬɟɩɟɧɧɨɣ ɪɹɞ ɢ ɩɪɢ k o f ɛɵɫɬɪɨ ɭɛɵɜɚɸɬ, ɬ. ɤ. (2k)! ɜɨɡɪɚɫɬɚ-
¨cos		¸
¨cos		¸
©	2 ¹

ɸɬ ɝɨɪɚɡɞɨ ɛɵɫɬɪɟɟ, ɱɟɦ	\| E2k \| [4]. ɋɥɚɝɚɟɦɵɟ, ɫɬɨɹɳɢɟ ɩɨɞ ɜɬɨɪɵɦ ɡɧɚɤɨɦ
ɫɭɦɦɵ ɜ (4.19), ɢɦɟɸɬ ɜɢɞ	n!	sinZ t , ɚ ɤɨɷɮɮɢɰɢɟɧɬɵ	n!	o0 ɩɪɢ t of.

	tn		t n
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɢ	t of g o0 , ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɪɚɫɫɦɚɬɪɢɜɚɟɦɚɹ ɫɢ-

ɫɬɟɦɚ ɮɢɥɶɬɪɨɜ ɭɫɬɨɣɱɢɜɚ.

ɇɚɣɞɟɦ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɜɨɫɫɬɚɧɨɜɥɟɧɧɨɣ ɮɭɧɤɰɢɢ x(t). ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɞɥɹ ɷɬɨɝɨ ɜɵɪɚɠɟɧɢɟɦ (1.38), ɢɡ ɤɨɬɨɪɨɝɨ ɫɥɟɞɭɟɬ:

X ( jZ) Y ( jZ)	1	.	(1.51)

)( jZ)

ɋɩɟɤɬɪ Y ( jZ) ɧɚɯɨɞɢɬɫɹ ɢɡ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɜɨɫɫɬɚɧɨɜɥɟɧɧɨɣ ɮɭɧɤɰɢɢ

y(t):

			f	sinZc (t n't)
	y(t)		¦y(n't)			;	(1.52)
				Zc (t n't)
			n f
f	f				S	f
Y ( jZ) ³	¦y(n't)	sinZc (t n't)		e jZ t dt		¦y(n't) e jn'tZ .	(1.53)

fn f			Zc (t n't)	Zc n f

ɉɨɞɫɬɚɜɥɹɹ (1.53) ɜ (1.51) ɢ ɩɪɢɦɟɧɹɹ ɨɛɪɚɬɧɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ Ɏɭɪɶɟ, ɭɱɢɬɵɜɚɹ ɧɚɥɢɱɢɟ ɮɢɥɶɬɪɚ ɧɢɠɧɢɯ ɱɚɫɬɨɬ, ɩɨɥɭɱɢɦ:

		1		Zc				1	e jZ t dZ
x(t)		1		³ Y ( jZ)				1
x(t)			2S	³ Y ( jZ)			)( jZ)
			2S	Z			)( jZ)
				Z
					c
1			f			Zc		1		e jZ (t n't)dZ
1			¦y(n't)			Z³		1
			¦y(n't)
	2Zc n f						c	)( jZ)
							c
ɋɪɚɜɧɢɜɚɹ (1.54) ɫ (1.44), ɜɢɞɢɦ, ɱɬɨ
Zc		1			e jZ (t n't)dZ 2Sg(t n't) ,
³		1
³		)( jZ)
Z
	c

ɬɨɝɞɚ

	S	f
x(t)		¦y(n't)g(t n't) .
x(t)	Zc n	¦y(n't)g(t n't) .
	Zc n	f

(1.54)

(1.55)

ɉɨɞɫɬɚɜɥɹɹ (1.50) ɜ (1.55), ɢɦɟɟɦ

| E2k |

§W

x(t)

¦ y(n't)

2k...(2k m 1)

2SZ W

(2k)!

n f

k 0

m 0

W2Z

2(k 1 ... 2(k 1) m 1

Zc2k m

sin[Z (t n't) 1 mS] (1.56)

c >

@ `

(t n't)

m 1

( 1)

>2(k 1)@!

sinZc (t n't)

(t n't)2k 3

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɨɤɚɡɚɧɚ ɫɮɨɪɦɭɥɢɪɨɜɚɧɧɚɹ ɜ [5] ɬɟɨɪɟɦɚ ɨ ɬɨɦ, ɱɬɨ ɮɭɧɤɰɢɹ x(t) ɫɨ ɫɩɟɤɬɪɨɦ, ɨɝɪɚɧɢɱɟɧɧɵɦ ɱɚɫɬɨɬɨɣ Zc , ɩɨɥɧɨɫɬɶɸ ɨɩɪɟɞɟɥɹɟɬɫɹ ɞɢɫɤɪɟɬɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ y(n't) ɫɜɨɟɣ ɫɜɟɪɬɤɢ ɫ ɮɭɧɤɰɢɟɣ M (t) (1.37), ɨɬɫɬɨ-

ɹɳɢɦɢ ɞɪɭɝ ɨɬ ɞɪɭɝɚ ɧɟ ɛɨɥɟɟ, ɱɟɦ ɧɚ 't SZc ɫɟɤɭɧɞ. ȼɨɫɫɬɚɧɚɜɥɢɜɚɟɬɫɹ ɮɭɧɤɰɢɹ x(t) ɬɟɨɪɟɬɢɱɟɫɤɢ ɬɨɱɧɨ ɫ ɩɨɦɨɳɶɸ ɪɹɞɚ (1.56).

1.2.3. Ⱥɧɚɥɢɡ ɦɟɬɨɞɚ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɫ ɤɨɫɢɧɭɫɨɢɞɚɥɶɧɵɦɢ ɜɟɫɨɜɵɦɢ ɮɭɧɤɰɢɹɦɢ

Ɋɚɫɫɦɨɬɪɢɦ ɨɫɧɨɜɧɵɟ ɫɜɨɣɫɬɜɚ ɩɪɨɰɟɫɫɚ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɫɢɝɧɚɥɚ x(t) ɫ ɩɨɦɨɳɶɸ ɤɨɫɢɧɭɫɨɢɞɚɥɶɧɵɯ ɜɟɫɨɜɵɯ ɮɭɧɤɰɢɣ M(t) .

1. ɉɭɫɬɶ ɞɥɢɬɟɥɶɧɨɫɬɶ ɢɦɩɭɥɶɫɚ M(t) W o 0 , ɜ ɷɬɨɦ ɫɥɭɱɚɟ (1.56) ɢɦɟɟɬ

ɜɢɞ

x(t)

¦ y(n't) ®

>sinZc

(t n't)@¾

SZ W

t n't

n f

(1.57)

S f

y(n't)

sinZ

(t n't)

2 n¦f W

Zc (t n't)

ȼɵɪɚɠɟɧɢɟ (1.57) ɹɜɥɹɟɬɫɹ ɪɹɞɨɦ Ʉɨɬɟɥɶɧɢɤɨɜɚ, ɬɚɤ ɤɚɤ

§S

cos¨

n't ¸

lim

©W

(n't)

(1.58)

W o0

§S

x(t)cos¨

n't ¸dt

y n't

©W

lim

x(n't) .

(1.59)

W o0

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɩɨɫɨɛ ɤɨɫɢɧɭɫɨɢɞɚɥɶɧɨɣ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɩɪɢ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɣ ɞɥɢɬɟɥɶɧɨɫɬɢ ɜɟɫɨɜɨɣ ɮɭɧɤɰɢɢ M(t) ɫɜɨɞɢɬɫɹ ɤ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɩɨ Ʉɨ-

ɬɟɥɶɧɢɤɨɜɭ.

2. Ɋɟɡɭɥɶɬɚɬ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ:

f	f f	f	f
¦y(n't)	¦ ¦x(s)M(n't s)ds	³ x(s)	¦M(n't s)ds . (1.60)
n f	n f f	f	n f

ɂɡ (1.60) ɫɥɟɞɭɟɬ, ɱɬɨ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɢɧɬɟɝɪɚɥɶɧɵɯ ɨɬɫɱɟɬɨɜ y(n't)

ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɪɟɡɭɥɶɬɚɬ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɩɪɨɢɡɜɟɞɟɧɢɹ ɜɯɨɞɧɨɝɨ ɫɢɝɧɚɥɚ x(t) ɢ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɜɟɫɨɜɵɯ ɢɦɩɭɥɶɫɨɜ M(n't t) .

ȿɫɥɢ ɢɧɬɟɪɜɚɥ ɞɢɫɤɪɟɬɢɡɚɰɢɢ 't 2W, ɢɦɩɭɥɶɫɵ M(n't t) ɩɪɟɞɫɬɚɜɥɹ-

ɸɬ ɫɨɛɨɣ ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɩɨɥɭɜɨɥɧɵ ɨɞɧɨɣ ɤɨɫɢɧɭɫɨɢɞɵ, ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɛɵɬɶ

			§	2S	·	. ȿɫɥɢ
ɩɨɥɭɱɟɧɵ ɨɞɧɨɩɨɥɭɩɟɪɢɨɞɧɵɦ ɜɵɩɪɹɦɥɟɧɢɟɦ ɫɢɝɧɚɥɚ M	1	(t)	cos¨	2S	t ¸
ɩɨɥɭɱɟɧɵ ɨɞɧɨɩɨɥɭɩɟɪɢɨɞɧɵɦ ɜɵɩɪɹɦɥɟɧɢɟɦ ɫɢɝɧɚɥɚ M		(t)	cos¨		t ¸
			¨	't	¸
			©	't	¹

't W , ɬɨ M(n't t) ɹɜɥɹɸɬɫɹ ɪɟɡɭɥɶɬɚɬɨɦ ɞɜɭɯɩɨɥɭɩɟɪɢɨɞɧɨɝɨ ɜɵɩɪɹɦɥɟɧɢɹ ɫɢɝɧɚɥɚ M 1(t) . Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɩɨɫɨɛ ɩɨɥɭɱɟɧɢɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɜɟɫɨɜɵɯ ɮɭɧɤɰɢɣ M (t) ɜɟɫɶɦɚ ɩɪɨɫɬ.

3. Ɉɞɧɨ ɢɡ ɞɨɫɬɨɢɧɫɬɜ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɫ ɩɨɦɨɳɶɸ ɝɚɪɦɨɧɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ – ɛɨɥɶɲɚɹ ɞɥɢɬɟɥɶɧɨɫɬɶ ɢɫɩɨɥɶɡɭɟɦɵɯ ɢɦɩɭɥɶɫɨɜ ɢ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɩɨɧɢɠɟɧɧɚɹ ɱɚɫɬɨɬɚ ɩɪɨɢɡɜɨɞɹɳɟɣ ɮɭɧɤɰɢɢ M 1(t) . ɋɨɝɥɚɫɧɨ (1.43) ɞɥɢɬɟɥɶɧɨɫɬɶ ɢɦɩɭɥɶɫɚ M (t) ɨɝɪɚɧɢɱɟɧɚ ɫɜɟɪɯɭ ɭɫɥɨɜɢɟɦ W 3't , ɬ. ɟ. ɱɚɫɬɨɬɚ ɩɪɨɢɡɜɨɞɹɳɟɣ

ɮɭɧɤɰɢɢ ɞɨɥɠɧɚ ɛɵɬɶ
Zɉ	2S	!	2S		2	Zc .	(1.61)
Zɉ		!	3't	3		Zc .	(1.61)
	2W		3't	3
	24

ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɢɦɩɭɥɶɫɵ M (t) ɩɟɪɟɤɪɵɜɚɸɬɫɹ ɩɨ ɜɪɟɦɟɧɢ, ɩɨɷɬɨɦɭ ɞɥɹ ɪɟɚɥɢɡɚɰɢɢ ɬɚɤɨɝɨ ɪɟɠɢɦɚ ɪɚɛɨɬɵ ɧɟɨɛɯɨɞɢɦɨ 3-ɤɚɧɚɥɶɧɨɟ ɭɫɬɪɨɣɫɬɜɨ. ɇɚ ɢɧɮɨɪɦɚɰɢɨɧɧɵɟ ɜɯɨɞɵ ɜɫɟɯ 3-ɯ ɤɚɧɚɥɨɜ ɩɨɫɬɭɩɚɟɬ ɫɢɝɧɚɥ x(t), ɚ ɧɚ ɫɬɪɨɛɢɪɭɸɳɢɟ ɜɯɨɞɵ – ɫɞɜɢɧɭɬɵɟ ɩɨ ɜɪɟɦɟɧɢ ɧɚ 't ɢɦɩɭɥɶɫɵ M(n't t) . ȼɵɯɨɞɵ ɤɚɧɚ-

ɥɨɜ ɨɛɴɟɞɢɧɹɸɬɫɹ ɭɫɬɪɨɣɫɬɜɨɦ ɫɭɦɦɢɪɨɜɚɧɢɹ, ɧɚ ɜɵɯɨɞɟ ɤɨɬɨɪɨɝɨ ɮɨɪɦɢɪɭɟɬɫɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɨɬɫɱɟɬɨɜ y(n't) .

4. Ɏɨɪɦɭɥɚ (1.56) ɧɟ ɞɚɟɬ ɡɧɚɱɟɧɢɹ x(t) ɩɪɢ t=0. Ⱦɥɹ ɪɚɫɱɟɬɚ ɟɝɨ ɧɟɨɛɯɨɞɢɦɨ ɢɧɬɟɝɪɚɥ (1.48) ɜɡɹɬɶ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɩɪɢ t=0, ɬɨɝɞɚ ɜɦɟɫɬɨ (1.49) ɢɦɟɟɦ

ª§S

§ZW ·

2 º

2k 1

2 Z

2k 3

«¨

»Z2k dZ

(1.62)

4 2k 1

2 2k 3

«© 2

2 ¹ »

c ¬

ɉɨɞɫɬɚɜɥɹɹ (1.62) ɜ (1.46), ɩɨɥɭɱɢɦ:

| E2k | §W

2 ªS 2Z2k 1

2Z2k 3

g(t)

» .

(1.63)

2W k 0

(2k)!

2k 1

2k 3

ɉɪɢ W 't

g(t)

| E2k |

¨§

¸·2

(1.64)

2k 1 2k 2

W k¦0 (2k)!

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɩɨɫɨɛ ɤɨɫɢɧɭɫɨɢɞɚɥɶɧɨɣ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɨɬɥɢɱɚɟɬɫɹ ɛɨɥɶɲɟɣ ɞɥɢɬɟɥɶɧɨɫɬɶɸ ɜɟɫɨɜɵɯ ɢɦɩɭɥɶɫɨɜ, ɱɬɨ ɨɛɥɟɝɱɚɟɬ ɩɨɫɬɪɨɟɧɢɟ ɭɫɬɪɨɣɫɬɜ ɞɢɫɤɪɟɬɢɡɚɰɢɢ. ɉɪɢ ɩɨɥɭɱɟɧɢɢ ɜɟɫɨɜɵɯ ɢɦɩɭɥɶɫɨɜ ɩɭɬɟɦ ɜɵɩɪɹɦɥɟɧɢɹ ɩɪɨɢɡɜɨɞɹɳɟɣ ɝɚɪɦɨɧɢɱɟɫɤɨɣ ɮɭɧɤɰɢɢ ɟɟ ɱɚɫɬɨɬɚ ɨɝɪɚɧɢɱɟɧɚ ɫɧɢɡɭ ɜɟɥɢ-

ɱɢɧɨɣ 3Zc .

ȽɅȺȼȺ 2. ȺɇȺɅɂɁ ɉɈȽɊȿɒɇɈɋɌȿɃ ȾɂɋɄɊȿɌɂɁȺɐɂɂ

2.1. Ɇɟɬɨɞɢɱɟɫɤɚɹ ɩɨɝɪɟɲɧɨɫɬɶ ɞɢɫɤɪɟɬɢɡɚɰɢɢ

ɋɩɨɫɨɛ ɞɢɫɤɪɟɬɢɡɚɰɢɢ, ɛɚɡɢɪɭɸɳɢɣɫɹ ɧɚ ɬɟɨɪɟɦɟ Ʉɨɬɟɥɶɧɢɤɨɜɚ, ɨɛɥɚɞɚɟɬ ɪɹɞɨɦ ɧɟɞɨɫɬɚɬɤɨɜ. Ʉɚɤ ɜɢɞɧɨ ɢɡ ɜɵɪɚɠɟɧɢɹ (1.10), ɞɥɹ ɬɨɱɧɨɝɨ ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ ɢɫɯɨɞɧɨɣ ɮɭɧɤɰɢɢ ɧɟɨɛɯɨɞɢɦɨ ɩɨɥɭɱɢɬɶ ɢ ɩɪɨɫɭɦɦɢɪɨɜɚɬɶ ɪɟɚɤɰɢɢ ɮɢɥɶɬɪɚ ɧɢɠɧɢɯ ɱɚɫɬɨɬ ɧɚ ɜɯɨɞɧɵɟ ɢɦɩɭɥɶɫɵ ɩɨ ɜɫɟɣ ɨɫɢ ɜɪɟɦɟɧɢ ɨɬ f ɞɨ f ɢɥɢ ɯɨɬɹ ɛɵ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɨɝɨ ɤɨɥɢɱɟɫɬɜɚ ɢɦɩɭɥɶɫɨɜ ɞɨ ɢ ɩɨɫɥɟ ɚɩɩɪɨɤɫɢɦɢɪɭɟɦɨɝɨ ɭɱɚɫɬɤɚ ɮɭɧɤɰɢɢ. ɉɪɚɤɬɢɱɟɫɤɢ ɪɟɚɥɢɡɨɜɚɬɶ ɷɬɨ ɬɪɭɞɧɨ.

Ʉɪɨɦɟ ɬɨɝɨ, ɧɚ ɩɪɚɤɬɢɤɟ ɩɪɢɯɨɞɢɬɫɹ ɢɦɟɬɶ ɞɟɥɨ ɫ ɫɢɝɧɚɥɚɦɢ, ɨɝɪɚɧɢɱɟɧɧɵɦɢ ɜɨ ɜɪɟɦɟɧɢ ɢ ɨɛɥɚɞɚɸɳɢɦɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɛɟɫɤɨɧɟɱɧɨ ɲɢɪɨɤɢɦ ɫɩɟɤɬɪɨɦ, ɱɬɨ ɩɪɨɬɢɜɨɪɟɱɢɬ ɨɫɧɨɜɧɨɦɭ ɭɫɥɨɜɢɸ ɬɟɨɪɟɦɵ Ʉɨɬɟɥɶɧɢɤɨɜɚ.

Ɉɞɧɚɤɨ ɧɚ ɩɪɚɤɬɢɤɟ ɧɢɤɨɝɞɚ ɧɟ ɬɪɟɛɭɟɬɫɹ ɢɞɟɚɥɶɧɨ ɬɨɱɧɨɝɨ ɜɨɫɩɪɨɢɡɜɟɞɟɧɢɹ ɩɟɪɟɞɚɜɚɟɦɨɝɨ ɫɢɝɧɚɥɚ.

ɉɨɷɬɨɦɭ ɫ ɰɟɥɶɸ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɬɟɨɪɟɦɵ Ʉɨɬɟɥɶɧɢɤɨɜɚ ɞɥɹ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɫɢɝɧɚɥɨɜ ɪɟɚɥɶɧɵɣ ɫɩɟɤɬɪ ɫɢɝɧɚɥɚ, ɩɪɨɫɬɢɪɚɸɳɢɣɫɹ ɨɬ ɧɭɥɹ ɞɨ ɛɟɫɤɨɧɟɱɧɨɫɬɢ, ɭɫɥɨɜɧɨ ɨɝɪɚɧɢɱɢɜɚɸɬ ɧɟɤɨɬɨɪɵɦ ɞɢɚɩɚɡɨɧɨɦ ɱɚɫɬɨɬ ɨɬ ɧɭɥɹ ɞɨ Zc ,

ɜ ɤɨɬɨɪɨɦ ɫɨɫɪɟɞɨɬɨɱɟɧɚ ɨɫɧɨɜɧɚɹ ɱɚɫɬɶ ɷɧɟɪɝɢɢ ɫɩɟɤɬɪɚ. ɗɧɟɪɝɢɹ ɨɬɫɟɤɚɟɦɨɣ ɱɚɫɬɢ ɫɩɟɤɬɪɚ ɫɢɝɧɚɥɚ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɩɨɝɪɟɲɧɨɫɬɶ, ɜɨɡɧɢɤɚɸɳɭɸ ɡɚ ɫɱɟɬ ɨɝɪɚɧɢɱɟɧɢɣ ɫɩɟɤɬɪɚ ɫɢɝɧɚɥɚ ɱɚɫɬɨɬɨɣ Zc .

Ⱦɢɫɩɟɪɫɢɹ ɩɪɢɜɟɞɟɧɧɨɣ ɩɨɝɪɟɲɧɨɫɬɢ, ɜɨɡɧɢɤɚɸɳɟɣ ɜ ɪɟɡɭɥɶɬɚɬɟ ɨɝɪɚɧɢɱɟɧɢɹ ɱɚɫɬɨɬɧɨɝɨ ɫɩɟɤɬɪɚ ɫɢɝɧɚɥɚ,

S( jZ)2

DZc

c Zc

(2.1)

xmax xmin 2

S( jZ)2

dZ – ɫɪɟɞɧɹɹ ɦɨɳɧɨɫɬɶ ɨɬɫɟɤɚɟɦɨɣ ɱɚɫɬɢ ɱɚɫɬɨɬɧɨɝɨ ɫɩɟɤ-

ɝɞɟ

c Zc

ɬɪɚ ɮɭɧɤɰɢɢ x(t) ;

– ɞɥɢɬɟɥɶɧɨɫɬɶ ɮɭɧɤɰɢɢ x(t) ;

xmax , xmin – ɢɞɟɚɥɶɧɵɟ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ x(t) .

Ⱦɥɹ ɭɞɨɛɫɬɜɚ ɪɚɫɱɟɬɨɜ ɰɟɥɟɫɨɨɛɪɚɡɧɟɣ (2.1) ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ

DZc

QZc PT

(2.2)

xmax xmin 2

S( jZ)2

xmax

x2Z(x)dx –

ɫɢɝɧɚɥɚ;

ɝɞɟ

ɫɪɟɞɧɹɹ ɦɨɳɧɨɫɬɶ

c Zc

xmin

S( jZ)2

QZc

– ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɩɥɨɳɚɞɶ ɨɬɫɟɤɚɟɦɨɝɨ ɭɱɚɫɬɤɚ ɷɧɟɪɝɟɬɢ-

S( jZ)2

ɱɟɫɤɨɝɨ ɫɩɟɤɬɪɚ ɮɭɧɤɰɢɢ x(t) .

ȼɵɪɚɠɟɧɢɟ

(2.2) ɩɨ

ɡɚɞɚɧɧɨɣ ɜɟɥɢɱɢɧɟ DZc ɢ ɩɪɢ

ɢɡɜɟɫɬɧɵɯ

PT , xmax , xmin ɢ ɫɩɟɤɬɪɟ ɮɭɧɤɰɢɢ x(t) ɩɨɡɜɨɥɹɟɬ ɨɩɪɟɞɟɥɢɬɶ ɱɚɫɬɨɬɭ Zc , ɨɝɪɚ-

ɧɢɱɢɜɚɸɳɭɸ ɫɩɟɤɬɪ.

2.2. ɋɪɚɜɧɢɬɟɥɶɧɵɣ ɚɧɚɥɢɡ ɩɨɝɪɟɲɧɨɫɬɟɣ ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ ɫɢɝɧɚɥɨɜ ɩɨ ɢɧɬɟɝɪɚɥɶɧɵɦ ɨɬɫɱɟɬɚɦ

ɉɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɢɧɬɟɝɪɢɪɭɸɳɢɯ ɦɟɬɨɞɨɜ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɞɥɹ ɬɨɱɧɨɝɨ ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ ɫɢɝɧɚɥɚ ɩɨ ɢɧɬɟɝɪɚɥɶɧɵɦ ɨɬɫɱɟɬɚɦ ɧɟɨɛɯɨɞɢɦɨ ɪɟɚɥɢɡɨɜɚɬɶ ɤɨɪɪɟɤɬɢɪɭɸɳɢɣ ɮɢɥɶɬɪ ɫ ɢɦɩɭɥɶɫɧɨɣ ɩɟɪɟɯɨɞɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ (1.50), ɱɬɨ ɹɜɥɹɟɬɫɹ ɞɨɜɨɥɶɧɨ ɫɥɨɠɧɨɣ ɡɚɞɚɱɟɣ. ȼɫɥɟɞɫɬɜɢɟ ɷɬɨɝɨ ɩɪɟɞɫɬɚɜɥɹɟɬ ɢɧɬɟɪɟɫ ɨɰɟɧɤɚ ɩɨɝɪɟɲɧɨɫɬɢ, ɜɨɡɧɢɤɚɸɳɟɣ ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɜɦɟɫɬɨ ɫɩɟɰɢɚɥɶɧɨɝɨ ɜɨɫɫɬɚɧɚɜɥɢɜɚɸɳɟɝɨ ɮɢɥɶɬɪɚ ɢɞɟɚɥɶɧɨɝɨ ɮɢɥɶɬɪɚ ɧɢɠɧɢɯ ɱɚɫɬɨɬ. Ʉɪɨɦɟ ɬɨɝɨ, ɜ ɪɟɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ ɞɢɫɤɪɟɬɢɡɚɰɢɹ ɢ ɜɨɫɫɬɚɧɨɜɥɟɧɢɟ ɫɢɝɧɚɥɨɜ ɩɪɨɢɫɯɨɞɢɬ ɜ ɩɪɢɫɭɬɫɬɜɢɢ ɲɭɦɚ, ɩɪɢɧɰɢɩɢɚɥɶɧɨ ɨɝɪɚɧɢɱɢɜɚɸɳɟɝɨ ɞɨɫɬɢɠɢɦɭɸ ɬɨɱɧɨɫɬɶ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɢɧɮɨɪɦɚɰɢɢ.

Ɋɚɫɫɦɨɬɪɢɦ ɩɪɨɰɟɫɫ ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ ɫɢɝɧɚɥɚ x(t) ɩɨ ɢɧɬɟɝɪɚɥɶɧɵɦ ɨɬɫɱɟɬɚɦ, ɩɨɥɭɱɟɧɧɵɦ ɫɜɟɪɬɤɨɣ ɫ ɜɟɫɨɜɨɣ ɮɭɧɤɰɢɟɣ M (t) :

	n't W
y* (n't)	2	x*(t)M(n't t)dt ,
y* (n't)	³	x*(t)M(n't t)dt ,	(2.3)

n't W2

ɟɫɥɢ ɧɚ ɜɯɨɞɟ ɭɫɬɪɨɣɫɬɜɚ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɩɪɢɫɭɬɫɬɜɭɟɬ ɚɞɞɢɬɢɜɧɚɹ ɫɦɟɫɶ ɫɢɝɧɚɥɚ x(t) ɫ ɲɭɦɨɦ n(t):

x(t)* = x(t) + n(t),

(2.4)

ɝɞɟ n(t) – ɫɬɚɰɢɨɧɚɪɧɵɣ ɫɥɭɱɚɣɧɵɣ ɩɪɨɰɟɫɫ ɫ ɞɢɫɩɟɪɫɢɟɣ V 2 ɢ ɧɨɪɦɢɪɨɜɚɧɧɨɣ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɮɭɧɤɰɢɟɣ R(z). ɋɢɝɧɚɥ x(t) ɩɨ-ɩɪɟɠɧɟɦɭ ɢɦɟɟɬ ɫɩɟɤɬɪ X ( jZ) ,

ɨɝɪɚɧɢɱɟɧɧɵɣ ɱɚɫɬɨɬɨɣ Zc .

ȼ ɤɚɱɟɫɬɜɟ ɜɟɫɨɜɨɣ ɮɭɧɤɰɢɢ M (t) ɛɭɞɟɦ ɩɨɤɚ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɩɪɨɢɡɜɨɥɶ-

ɧɭɸ ɱɟɬɧɭɸ ɮɭɧɤɰɢɸ, ɨɩɪɟɞɟɥɟɧɧɭɸ ɧɚ ɢɧɬɟɪɜɚɥɟ		ª	W	,	W º		ɢ ɭɞɨɜɥɟɬɜɨɪɹɸ-
		«				»
		«	2		2	»
		¬	2		2	¼
ɳɭɸ ɭɫɥɨɜɢɸ ɧɨɪɦɢɪɨɜɤɢ
W

2
	³M(t)dt 1.						(2.5)

Ⱦɥɹ ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ ɢɫɯɨɞɧɨɝɨ ɫɢɝɧɚɥɚ ɧɚ ɜɯɨɞ ɮɢɥɶɬɪɚ ɧɢɠɧɢɯ ɱɚɫɬɨɬ ɫ ɩɟɪɟɞɚɬɨɱɧɨɣ ɮɭɧɤɰɢɟɣ

			e	jZ to	, \| Z \| Z
K( jZ)	°K	o	e		, \| Z \| Z	c	(2.6)
K( jZ)	®	o		\| Z \| !Zc		c	(2.6)
	¯0,			\| Z \| !Zc
	°

ɩɨɞɚɸɬɫɹ ɩɪɹɦɨɭɝɨɥɶɧɵɟ ɢɦɩɭɥɶɫɵ vn(t) ɞɥɢɬɟɥɶɧɨɫɬɢ p, ɫɮɨɪɦɢɪɨɜɚɧɧɵɟ ɢɡ ɨɬɫɱɟɬɨɜ y* (n't) ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:

°y* (n't)

| t n't |

vn (t)

°0,

| t n't |

ȼ ɪɟɡɭɥɶɬɚɬɟ ɨɛɪɚɛɨɬɤɢ 2N+1 ɨɬɫɱɟɬɚ ɩɨɥɭɱɢɦ ɫɢɝɧɚɥ

(t)

³K( jZ)Vn ( jZ) e jZ t dZ ¦un (t) ,

N 2S

n N

ɝɞɟ un(t) – ɪɟɚɤɰɢɹ ɮɢɥɶɬɪɚ ɧɚ ɢɦɩɭɥɶɫ vn(t);

Vn ( jZ) ³vn (t)e jZ t dt

– ɫɩɟɤɬɪ ɢɦɩɭɥɶɫɚ vn(t).

(2.7)

(2.8)

Ɏɭɧɤɰɢɹ x(t) , ɨɱɟɜɢɞɧɨ, ɫɨɜɩɚɞɚɟɬ ɫ ɜɯɨɞɧɵɦ ɫɢɝɧɚɥɨɦ x(t), ɟɫɥɢ

M(t) G(t),V 0, N f, p 0 .

ȼɪɟɚɥɶɧɨɣ ɫɢɬɭɚɰɢɢ ɧɢ ɨɞɧɨ ɢɡ ɷɬɢɯ ɭɫɥɨɜɢɣ ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ. ȼ ɫɜɹɡɢ

ɫɷɬɢɦ ɜɨɡɧɢɤɚɟɬ ɡɚɞɚɱɚ ɨɰɟɧɤɢ ɩɨɝɪɟɲɧɨɫɬɢ ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ ɫɢɝɧɚɥɚ ɩɪɢ ɤɨɧɟɱɧɨɦ N, p z 0 , ɜɵɛɪɚɧɧɨɣ ɜɟɫɨɜɨɣ ɮɭɧɤɰɢɢ M(t) ɢ ɡɚɞɚɧɧɨɣ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ

ɮɭɧɤɰɢɢ ɲɭɦɚ.

Ɋɚɫɫɦɨɬɪɢɦ ɨɬɞɟɥɶɧɨ ɮɭɧɤɰɢɸ un(t). ɉɨɥɚɝɚɹ Kɨ=1 ɢ ɩɨɞɫɬɚɜɥɹɹ ɜ (2.8)

sin

Z p

Vn ( jZ)

y * (n't)

(2.9)

Z p

ɩɨɥɭɱɢɦ:

(n't)

sin

Z p

un (t)

e jZ (t n't to)

dZ .

(2.10)

Z p

Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɩɪɢ M(t)

G (t)

ɢ p V

0 ɩɨɫɥɟɞɧɟɟ ɪɚɜɟɧɫɬɜɨ ɞɚɟɬ ɱɥɟ-

ɧɵ ɪɹɞɚ Ʉɨɬɟɥɶɧɢɤɨɜɚ uno (t) , ɞɥɹ ɤɨɬɨɪɵɯ

x(t to )

¦uno (t) .

(2.11)

n f

ɉɨɷɬɨɦɭ ɩɨɝɪɟɲɧɨɫɬɶ ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ ' ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɫɥɟ-

ɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:

¦uno (t)

¦un (t)

¦uno (t)

¦>un (t) uno (t)@.

(2.12)

n N

n f

|n|!N

n N

ɉɭɫɬɶ '

| u

(t) uo (t) |. ɇɚɣɞɟɦ ɞɥɹ ɜɟɥɢɱɢɧɵ '

ɨɰɟɧɤɭ ɫɜɟɪɯɭ. ɉɭ-

ɬɟɦ ɩɪɨɫɬɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ

' d ' 1 '

(2.13)

ɝɞɟ

sin

Z p

' 1n | y* (n't) x(n't) | | ³ e jZ

(t n't)

dZ |,

Z p

¨sin

jZ(t n't to )

| x(n't) |

³ e

' n

(2.14)

Z p

Ɋɚɫɫɦɨɬɪɢɦ ɤɚɠɞɭɸ ɢɡ ɜɟɥɢɱɢɧ

ɬɨɝɞɚ, ɫ ɭɱɟɬɨɦ ɧɟɪɚɜɟɧɫɬɜɚ

§Z p ·			1
Q¨		¸
	2
©		¹

' 1n,		' n2		ɨɬɞɟɥɶɧɨ. ɉɭɫɬɶ t n't to s ,
	Z p			§Z p ·3
				¨		¸


sin 2			d	©	2 ¹		,	(2.15)
Z p			d		6		,	(2.15)
Z p					6
	2

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