Дискретизация сигналов. Учебное пособие
.pdfȺ. Ʌ. Ɍɢɦɨɮɟɟɜ, Ⱥ. ɏ. ɋɭɥɬɚɧɨɜ
ȾɂɋɄɊȿɌɂɁȺɐɂə ɋɂȽɇȺɅɈȼ
ɍɱɟɛɧɨɟ ɩɨɫɨɛɢɟ
Ɇɨɫɤɜɚ ȼɨɥɨɝɞɚ
«ɂɧɮɪɚ-ɂɧɠɟɧɟɪɢɹ»
2024
ɍȾɄ 621.391 ȻȻɄ 32.88
Ɍ41
Ɋɟɰɟɧɡɟɧɬ:
ɤ. ɬ. ɧ., ɞɨɰɟɧɬ ɍɮɢɦɫɤɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ ɧɚɭɤɢ ɢ ɬɟɯɧɨɥɨɝɢɣ (ɍɍɇɢɌ)
Ɇɟɲɤɨɜ ɂ. Ʉ.
Ɍɢɦɨɮɟɟɜ, Ⱥ. Ʌ.
Ɍ41 Ⱦɢɫɤɪɟɬɢɡɚɰɢɹ ɫɢɝɧɚɥɨɜ : ɭɱɟɛɧɨɟ ɩɨɫɨɛɢɟ / Ⱥ. Ʌ. Ɍɢɦɨɮɟɟɜ, Ⱥ. ɏ. ɋɭɥɬɚɧɨɜ. – Ɇɨɫɤɜɚ ; ȼɨɥɨɝɞɚ : ɂɧɮɪɚ-ɂɧɠɟɧɟɪɢɹ, 2024. – 64 ɫ. : ɢɥ.
ISBN 978-5-9729-1797-6
ɉɪɢɜɟɞɟɧɵ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɢ ɜɚɪɢɚɧɬɵ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɚɧɚɥɨɝɨɜ ɬɟɨɪɟɦɵ Ʉɨɬɟɥɶɧɢɤɨɜɚ ɞɥɹ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɢ ɤɨɫɢɧɭɫɨɢɞɚɥɶɧɨɣ ɜɟɫɨɜɵɯ ɮɭɧɤɰɢɣ. ɂɡɥɨɠɟɧɵ ɫɩɟɰɢɚɥɶɧɵɟ ɫɩɨɫɨɛɵ ɞɢɫɤɪɟɬɢɡɚɰɢɢ, ɩɨɡɜɨɥɹɸɳɢɟ ɭɦɟɧɶɲɢɬɶ ɞɢɧɚɦɢɱɟɫɤɭɸ ɩɨɝɪɟɲɧɨɫɬɶ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɲɢɪɨɤɨɩɨɥɨɫɧɵɯ ɫɢɝɧɚɥɨɜ.
Ⱦɥɹ ɩɨɞɝɨɬɨɜɤɢ ɛɚɤɚɥɚɜɪɨɜ ɢ ɦɚɝɢɫɬɪɨɜ ɩɨ ɝɪɭɩɩɟ ɧɚɩɪɚɜɥɟɧɢɣ ɢ ɫɩɟɰɢɚɥɶɧɨɫɬɟɣ «ɗɥɟɤɬɪɨɧɢɤɚ, ɪɚɞɢɨɬɟɯɧɢɤɚ ɢ ɫɢɫɬɟɦɵ ɫɜɹɡɢ».
ɍȾɄ 621.391 ȻȻɄ 32.88
ISBN 978-5-9729-1797-6 |
Ɍɢɦɨɮɟɟɜ Ⱥ. Ʌ., ɋɭɥɬɚɧɨɜ Ⱥ. ɏ., 2024 |
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ɂɡɞɚɬɟɥɶɫɬɜɨ «ɂɧɮɪɚ-ɂɧɠɟɧɟɪɢɹ», 2024 |
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Ɉɮɨɪɦɥɟɧɢɟ. ɂɡɞɚɬɟɥɶɫɬɜɨ «ɂɧɮɪɚ-ɂɧɠɟɧɟɪɢɹ», 2024 |
ɋɈȾȿɊɀȺɇɂȿ |
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ȼɜɟɞɟɧɢɟ...................................................................................................................... |
4 |
Ƚɥɚɜɚ 1. Ɍɟɨɪɟɬɢɱɟɫɤɢɟ ɨɫɧɨɜɵ ɞɢɫɤɪɟɬɢɡɚɰɢɢ................................................. |
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1.1. Ⱦɢɫɤɪɟɬɢɡɚɰɢɹ ɩɨ Ʉɨɬɟɥɶɧɢɤɨɜɭ........................................................................ |
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1.1.1. Ɍɟɨɪɟɦɚ Ʉɨɬɟɥɶɧɢɤɨɜɚ ɜɨ ɜɪɟɦɟɧɧɨɣ ɨɛɥɚɫɬɢ............................................... |
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1.1.2. Ɍɟɨɪɟɦɚ Ʉɨɬɟɥɶɧɢɤɨɜɚ ɜ ɱɚɫɬɨɬɧɨɣ ɨɛɥɚɫɬɢ.................................................. |
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1.2. ɂɧɬɟɝɪɢɪɭɸɳɢɟ ɦɟɬɨɞɵ ɞɢɫɤɪɟɬɢɡɚɰɢɢ......................................................... |
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1.2.1. Ⱥɧɚɥɨɝ ɬɟɨɪɟɦɵ Ʉɨɬɟɥɶɧɢɤɨɜɚ ɞɥɹ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɜɟɫɨɜɨɣ |
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ɮɭɧɤɰɢɢ...................................................................................................................... |
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1.2.2. Ⱥɧɚɥɨɝ ɬɟɨɪɟɦɵ Ʉɨɬɟɥɶɧɢɤɨɜɚ ɞɥɹ ɤɨɫɢɧɭɫɨɢɞɚɥɶɧɨɣ ɜɟɫɨɜɨɣ |
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ɮɭɧɤɰɢɢ...................................................................................................................... |
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1.2.3. Ⱥɧɚɥɢɡ ɦɟɬɨɞɚ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɫ ɤɨɫɢɧɭɫɨɢɞɚɥɶɧɵɦɢ ɜɟɫɨɜɵɦɢ |
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ɮɭɧɤɰɢɹɦɢ.................................................................................................................. |
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Ƚɥɚɜɚ 2. Ⱥɧɚɥɢɡ ɩɨɝɪɟɲɧɨɫɬɟɣ ɞɢɫɤɪɟɬɢɡɚɰɢɢ................................................. |
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2.1. Ɇɟɬɨɞɢɱɟɫɤɚɹ ɩɨɝɪɟɲɧɨɫɬɶ ɞɢɫɤɪɟɬɢɡɚɰɢɢ.................................................... |
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2.2. ɋɪɚɜɧɢɬɟɥɶɧɵɣ ɚɧɚɥɢɡ ɩɨɝɪɟɲɧɨɫɬɟɣ ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ ɫɢɝɧɚɥɨɜ |
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ɩɨ ɢɧɬɟɝɪɚɥɶɧɵɦ ɨɬɫɱɟɬɚɦ....................................................................................... |
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2.3. ɂɧɫɬɪɭɦɟɧɬɚɥɶɧɚɹ ɫɨɫɬɚɜɥɹɸɳɚɹ ɩɨɝɪɟɲɧɨɫɬɢ ɞɢɫɤɪɟɬɢɡɚɰɢɢ |
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(ɞɢɧɚɦɢɱɟɫɤɚɹ ɩɨɝɪɟɲɧɨɫɬɶ) ................................................................................... |
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2.3.1. ɋɨɫɬɚɜɥɹɸɳɢɟ ɞɢɧɚɦɢɱɟɫɤɨɣ ɩɨɝɪɟɲɧɨɫɬɢ................................................ |
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2.3.2. Ɋɚɫɱɟɬ ɨɩɬɢɦɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɞɢɫɤɪɟɬɢɡɚɰɢɢ............................................. |
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(ɜɪɟɦɟɧɢ ɞɚɬɢɪɨɜɤɢ ɨɬɫɱɟɬɚ).................................................................................... |
41 |
2.3.3. Ⱥɧɚɥɢɡ ɞɢɧɚɦɢɱɟɫɤɨɣ ɩɨɝɪɟɲɧɨɫɬɢ ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ |
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ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɜɟɫɨɜɨɣ ɮɭɧɤɰɢɢ............................................................................ |
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2.3.4. ɋɪɚɜɧɢɬɟɥɶɧɵɣ ɚɧɚɥɢɡ ɞɢɧɚɦɢɱɟɫɤɨɣ ɩɨɝɪɟɲɧɨɫɬɢ ɢɧɬɟɝɪɢɪɭɸɳɢɯ |
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ɭɫɬɪɨɣɫɬɜ ɞɢɫɤɪɟɬɢɡɚɰɢɢ.......................................................................................... |
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Ƚɥɚɜɚ 3. Ⱦɢɫɤɪɟɬɢɡɚɰɢɹ ɜ ɩɪɢɫɭɬɫɬɜɢɢ ɲɭɦɚ................................................... |
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3.1. Ɉɫɨɛɟɧɧɨɫɬɢ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɢɡɨɛɪɚɠɟɧɢɣ...................................................... |
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3.2. ȼɵɛɨɪ ɱɚɫɬɨɬɵ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ......................................................................... |
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ɋɩɢɫɨɤ ɥɢɬɟɪɚɬɭɪɵ................................................................................................. |
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3
ȼȼȿȾȿɇɂȿ
Ɋɚɡɜɢɬɢɟ ɷɥɟɤɬɪɨɧɢɤɢ, ɢɧɮɨɪɦɚɬɢɤɢ, ɢɧɮɨɤɨɦɦɭɧɢɤɚɰɢɣ ɩɪɢɜɟɥɨ ɤ ɬɨɦɭ, ɱɬɨ ɲɢɪɨɱɚɣɲɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɟ ɩɨɥɭɱɢɥɚ ɰɢɮɪɨɜɚɹ ɨɛɪɚɛɨɬɤɚ ɢɧɮɨɪɦɚɰɢɢ – ɨɧɚ ɢɫɩɨɥɶɡɭɟɬɫɹ ɜɟɡɞɟ, ɨɬ ɰɢɮɪɨɜɨɝɨ ɬɟɪɦɨɦɟɬɪɚ ɞɨ ɫɢɫɬɟɦɵ ɪɚɫɩɨɡɧɚɜɚɧɢɹ ɥɢɰ ɩɪɢ ɜɢɞɟɨɫɴɟɦɤɟ. ɉɪɢ ɷɬɨɦ ɜɨ ɦɧɨɝɢɯ ɫɥɭɱɚɹɯ ɢɫɯɨɞɧɚɹ ɢɧɮɨɪɦɚɰɢɹ ɩɨɫɬɭɩɚɟɬ ɜ ɚɧɚɥɨɝɨɜɨɦ ɜɢɞɟ, ɬɨ ɟɫɬɶ ɜ ɜɢɞɟ ɧɟɩɪɟɪɵɜɧɵɯ ɫɢɝɧɚɥɨɜ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɤɚɤɢɦ-ɥɢɛɨ ɩɪɨɰɟɫɫɚɦ ɨɤɪɭɠɚɸɳɟɝɨ ɦɢɪɚ. ɂ ɩɟɪɜɨɟ, ɱɬɨ ɧɟɨɛɯɨɞɢɦɨ ɫɞɟɥɚɬɶ – ɷɬɨ ɩɟɪɟɜɟɫɬɢ ɢɧɮɨɪɦɚɰɢɸ ɜ ɰɢɮɪɨɜɭɸ ɮɨɪɦɭ. ɉɪɨɰɟɫɫ «ɨɰɢɮɪɨɜɤɢ» ɢɧɮɨɪɦɚɰɢɢ, ɢɥɢ ɚɧɚɥɨɝɨ-ɰɢɮɪɨɜɨɝɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɫɢɝɧɚɥɨɜ, ɫɨɫɬɨɢɬ ɢɯ ɞɜɭɯ ɷɬɚɩɨɜ: ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɢ ɤɜɚɧɬɨɜɚɧɢɹ.
ɉɪɢ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɜɯɨɞɧɨɣ ɫɢɝɧɚɥ ɡɚɦɟɧɹɟɬɫɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶɸ ɫɜɨɢɯ ɦɝɧɨɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ, ɜɡɹɬɵɯ ɜ ɨɩɪɟɞɟɥɟɧɧɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ. ɉɪɢ ɤɜɚɧɬɨɜɚɧɢɢ ɤɚɠɞɨɦɭ ɡɧɚɱɟɧɢɸ ɫɢɝɧɚɥɚ ɫɬɚɜɢɬɫɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɟ ɧɟɤɨɬɨɪɨɟ ɡɧɚɱɟɧɢɟ ɟɞɢɧɢɰ ɲɤɚɥɵ ɤɜɚɧɬɨɜɚɧɢɹ (ɤɜɚɧɬɨɜ), ɤɨɬɨɪɨɟ ɢ ɫɬɚɧɟɬ ɰɢɮɪɨɜɵɦ ɡɧɚɱɟɧɢɟɦ ɫɢɝɧɚɥɚ.
ȼɥɢɬɟɪɚɬɭɪɟ ɨɱɟɧɶ ɯɨɪɨɲɨ ɢ ɩɨɞɪɨɛɧɨ ɨɩɢɫɚɧɨ ɦɧɨɠɟɫɬɜɨ ɦɟɬɨɞɨɜ ɚɧɚɥɨɝɨ-ɰɢɮɪɨɜɨɝɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ, ɢɫɫɥɟɞɨɜɚɧɵ ɢɯ ɜɨɡɦɨɠɧɨɫɬɢ ɢ ɨɫɨɛɟɧɧɨɫɬɢ. ɇɨ ɜ ɪɹɞɟ ɫɥɭɱɚɟɜ, ɨɛɵɱɧɨ, ɤɨɝɞɚ ɛɵɫɬɪɨɞɟɣɫɬɜɢɹ ɢɦɟɸɳɟɣɫɹ ɷɥɟɦɟɧɬɧɨɣ ɛɚɡɵ ɢ ɚɩɩɚɪɚɬɭɪɵ ɧɟ ɯɜɚɬɚɟɬ ɞɥɹ ɩɨɥɭɱɟɧɢɹ ɬɪɟɛɭɟɦɨɣ ɬɨɱɧɨɫɬɢ, ɜɨɡɧɢɤɚɟɬ ɩɨɬɪɟɛɧɨɫɬɶ ɜ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɦɟɧɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɧɵɯ ɦɟɬɨɞɨɜ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɢ, ɜ ɩɟɪɜɭɸ ɨɱɟɪɟɞɶ, ɷɬɨ ɤɚɫɚɟɬɫɹ ɫɩɨɫɨɛɨɜ ɞɢɫɤɪɟɬɢɡɚɰɢɢ.
ȼɞɚɧɧɨɦ ɩɨɫɨɛɢɢ ɪɚɫɫɦɨɬɪɟɧɵ ɫɩɟɰɢɚɥɶɧɵɟ ɦɟɬɨɞɵ ɞɢɫɤɪɟɬɢɡɚɰɢɢ, ɩɨɡɜɨɥɹɸɳɢɟ ɩɨɜɵɫɢɬɶ ɬɨɱɧɨɫɬɶ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɛɵɫɬɪɨɢɡɦɟɧɹɸɳɢɯɫɹ ɫɢɝɧɚɥɨɜ.
4
ȽɅȺȼȺ 1. ɌȿɈɊȿɌɂɑȿɋɄɂȿ ɈɋɇɈȼɕ ȾɂɋɄɊȿɌɂɁȺɐɂɂ
1.1.Ⱦɢɫɤɪɟɬɢɡɚɰɢɹ ɩɨ Ʉɨɬɟɥɶɧɢɤɨɜɭ
1.1.1.Ɍɟɨɪɟɦɚ Ʉɨɬɟɥɶɧɢɤɨɜɚ ɜɨ ɜɪɟɦɟɧɧɨɣ ɨɛɥɚɫɬɢ
ɉɪɢ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɫɢɝɧɚɥɨɜ ɧɟɩɪɟɪɵɜɧɚɹ ɩɨ ɜɪɟɦɟɧɢ ɮɭɧɤɰɢɹ x(t) ɩɪɟ-
ɨɛɪɚɡɭɟɬɫɹ ɜ ɮɭɧɤɰɢɸ xɞ(t) ɞɢɫɤɪɟɬɧɨɝɨ ɚɪɝɭɦɟɧɬɚ. Ɍɚɤɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɦɨ-
ɠɟɬ ɛɵɬɶ ɜɵɩɨɥɧɟɧɨ ɩɭɬɟɦ ɜɡɹɬɢɹ ɨɬɫɱɟɬɨɜ ɮɭɧɤɰɢɢ x(t) ɜ ɨɩɪɟɞɟɥɟɧɧɵɟ ɞɢɫ-
ɤɪɟɬɧɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ t0 ,t1,t2 ,...tn . ȼ ɪɟɡɭɥɶɬɚɬɟ ɮɭɧɤɰɢɹ x(t) ɡɚɦɟɧɹɟɬɫɹ ɫɨɜɨɤɭɩɧɨɫɬɶɸ ɦɝɧɨɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ x(ti ), i 0,1, 2,...n .
ȼɪɟɦɟɧɧɨɣ ɢɧɬɟɪɜɚɥ T ti ti 1 ɦɟɠɞɭ ɞɜɭɦɹ ɫɨɫɟɞɧɢɦɢ ɮɢɤɫɢɪɨɜɚɧ-
ɧɵɦɢ ɦɨɦɟɧɬɚɦɢ ɜɪɟɦɟɧɢ, ɜ ɤɨɬɨɪɵɯ ɡɚɞɚɟɬɫɹ ɞɢɫɤɪɟɬɧɚɹ ɮɭɧɤɰɢɹ, ɧɚɡɵɜɚɟɬɫɹ ɢɧɬɟɪɜɚɥɨɦ ɞɢɫɤɪɟɬɢɡɚɰɢɢ. ȼɟɥɢɱɢɧɚ, ɨɛɪɚɬɧɚɹ ɢɧɬɟɪɜɚɥɭ ɞɢɫɤɪɟɬɢɡɚɰɢɢ
fɞ |
1 |
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(1.1) |
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T |
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ɧɚɡɵɜɚɟɬɫɹ ɱɚɫɬɨɬɨɣ ɞɢɫɤɪɟɬɢɡɚɰɢɢ.
ɑɚɫɬɨɬɚ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɞɨɥɠɧɚ ɜɵɛɢɪɚɬɶɫɹ ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɬɨɛɵ ɩɨ ɨɬɫɱɟɬɚɦ x(ti ) ɦɨɠɧɨ ɛɵɥɨ ɫ ɡɚɞɚɧɧɨɣ ɬɨɱɧɨɫɬɶɸ ɩɨɥɭɱɢɬɶ ɢɫɯɨɞɧɭɸ ɮɭɧɤɰɢɸ.
ɂɡɜɟɫɬɧɨ ɧɟɫɤɨɥɶɤɨ ɤɪɢɬɟɪɢɟɜ ɜɵɛɨɪɚ ɱɚɫɬɨɬɵ ɞɢɫɤɪɟɬɢɡɚɰɢɢ. ɇɚɢɛɨɥɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɧɵɦ ɹɜɥɹɟɬɫɹ ɱɚɫɬɨɬɧɵɣ, ɩɨɥɭɱɢɜɲɢɣ ɧɚɡɜɚɧɢɟ ɬɟɨɪɟɦɵ Ʉɨɬɟɥɶɧɢɤɨɜɚ [1] (ɢɫɩɨɥɶɡɭɸɬɫɹ ɬɚɤɠɟ ɧɚɡɜɚɧɢɹ ɬɟɨɪɟɦɚ ɨɬɫɱɟɬɨɜ, ɬɟɨɪɟɦɚ ɇɚɣɤɜɢɫɬɚ).
Ɍɟɨɪɟɦɚ Ʉɨɬɟɥɶɧɢɤɨɜɚ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: ɟɫɥɢ ɧɟɩɪɟɪɵɜɧɚɹ ɮɭɧɤɰɢɹ x(t) ɢɦɟɟɬ ɫɩɟɤɬɪ, ɨɝɪɚɧɢɱɟɧɧɵɣ ɱɚɫɬɨɬɨɣ fc , ɬɨ ɨɧɚ ɩɨɥɧɨ-
ɫɬɶɸ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶɸ ɫɜɨɢɯ ɡɧɚɱɟɧɢɣ ɜ ɬɨɱɤɚɯ, ɨɬɫɬɨɹɳɢɯ
ɧɚ ɪɚɫɫɬɨɹɧɢɢ T |
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Ⱦɥɹ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɬɟɨɪɟɦɵ ɪɚɫɫɦɨɬɪɢɦ ɜɵɪɚɠɟɧɢɹ ɩɪɹɦɨɝɨ ɢ ɨɛɪɚɬɧɨɝɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ɏɭɪɶɟ ɧɟɩɪɟɪɵɜɧɨɣ ɮɭɧɤɰɢɢ x(t)
5
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ȼ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ ɮɭɧɤɰɢɢ ɫ ɨɝɪɚɧɢɱɟɧɧɵɦ ɫɩɟɤɬɪɨɦ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ
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Ⱦɨɩɨɥɧɢɦ ɮɭɧɤɰɢɸ S( jZ) |
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ɞɨ ɩɟɪɢɨɞɢɱɟɫɤɨɣ ɫ ɩɟɪɢɨɞɨɦ, ɪɚɜɧɵɦ 2Zc , |
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ɢ ɪɚɡɥɨɠɢɦ ɟɟ ɜ ɪɹɞ Ɏɭɪɶɟ |
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ɋɪɚɜɧɢɜɚɹ ɜɵɪɚɠɟɧɢɹ (1.4) ɢ (1.6), ɡɚɦɟɱɚɟɦ, ɱɬɨ ɨɧɢ ɫɨɜɩɚɞɚɸɬ ɫ ɬɨɱɧɨ- |
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ɫɬɶɸ ɞɨ ɩɨɫɬɨɹɧɧɨɝɨ ɦɧɨɠɢɬɟɥɹ |
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ɬɟɥɶɧɨ, |
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ɉɨɞɫɬɚɜɢɜ ɧɚɣɞɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ Ck ɜ (1.5), ɩɨɥɭɱɚɟɦ
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ɉɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ (1.7) ɜ (1.4), ɡɚɦɟɧɵ ɡɧɚɤɚ ɩɪɢ k (ɬɚɤ ɤɚɤ ɫɭɦɦɢɪɨɜɚɧɢɟ ɩɪɨɢɡɜɨɞɢɬɫɹ ɩɨ ɜɫɟɦ ɩɨɥɨɠɢɬɟɥɶɧɵɦ ɢ ɨɬɪɢɰɚɬɟɥɶɧɵɦ ɡɧɚɱɟɧɢɹɦ k) ɢ ɩɟɪɟɫɬɚɧɨɜɤɢ ɨɩɟɪɚɰɢɣ ɫɭɦɦɢɪɨɜɚɧɢɹ ɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɩɨɥɭɱɢɦ
ȼɵɱɢɫɥɢɦ ɢɧɬɟɝɪɚɥ
ɬɚɤ ɤɚɤ
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ɉɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ (1.9) ɜ (1.8) ɨɤɨɧɱɚɬɟɥɶɧɨ ɢɦɟɟɦ
(1.8)
(1.9)
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x(t) ¦ x(k't) |
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ɉɨɥɭɱɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɚɧɚɥɢɬɢɱɟɫɤɢ ɬɟɨɪɟɦɭ Ʉɨɬɟɥɶɧɢɤɨ-
ɜɚ.
7
ɂɡ (1.10) ɜɢɞɧɨ, ɱɬɨ ɧɟɩɪɟɪɵɜɧɚɹ ɮɭɧɤɰɢɹ, ɨɛɥɚɞɚɸɳɚɹ ɨɝɪɚɧɢɱɟɧɧɵɦ ɫɩɟɤɬɪɨɦ, ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɪɚɡɥɨɠɟɧɢɟɦ ɜ ɪɹɞ, ɤɚɠɞɵɣ ɱɥɟɧ ɤɨɬɨɪɨɝɨ
ɜɵɪɚɠɚɟɬɫɹ ɨɞɢɧɚɤɨɜɨɣ ɮɭɧɤɰɢɟɣ ɜɢɞɚ (ɮɭɧɤɰɢɹ ɨɬɫɱɟɬɨɜ), ɧɨ ɫ ɪɚɡɥɢɱ-
ɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ x(k't) .
Ɋɹɞ (1.10) ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɤɚɧɨɧɢɱɟɫɤɨɟ ɪɚɡɥɨɠɟɧɢɟ ɫɥɭɱɚɣɧɨɝɨ ɩɪɨɰɟɫɫɚ ɫ ɤɨɨɪɞɢɧɚɬɧɵɦɢ ɮɭɧɤɰɢɹɦɢ (ɞɟɬɟɪɦɢɧɢɪɨɜɚɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ ɜɪɟɦɟɧɢ) ɢ ɜɟɫɨɜɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ x(k't) , ɹɜɥɹɸɳɢɦɢɫɹ ɫɥɭɱɚɣɧɵɦɢ ɜɟɥɢɱɢ-
ɧɚɦɢ, ɪɚɜɧɵɦɢ ɦɝɧɨɜɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɫɢɝɧɚɥɚ ɜ ɬɨɱɤɚɯ k't .
Ʉɚɤ ɢɡɜɟɫɬɧɨ, ɮɭɧɤɰɢɹ ɜɢɞɚ siny y ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɪɟɚɤɰɢɸ ɢɞɟɚɥɶɧɨ-
ɝɨ ɮɢɥɶɬɪɚ ɧɢɠɧɢɯ ɱɚɫɬɨɬ ɫ ɝɪɚɧɢɱɧɨɣ ɱɚɫɬɨɬɨɣ Zc ɧɚ ɞɟɥɶɬɚ-ɮɭɧɤɰɢɸ.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɟɫɥɢ ɜ ɩɪɢɟɦɧɨɦ ɭɫɬɪɨɣɫɬɜɟ ɩɨɦɟɫɬɢɬɶ ɬɚɤɨɣ ɮɢɥɶɬɪ ɢ ɩɪɨɩɭɫɬɢɬɶ ɱɟɪɟɡ ɧɟɝɨ ɞɢɫɤɪɟɬɢɡɢɪɨɜɚɧɧɵɣ ɫɢɝɧɚɥ, ɩɪɟɞɫɬɚɜɥɹɸɳɢɣ ɫɨɛɨɣ ɩɨ-
ɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɫ ɱɚɫɬɨɬɨɣ fɞ 2 fc |
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ɜɟɫɶɦɚ ɤɪɚɬɤɨɜɪɟɦɟɧɧɵɯ ɢɦɩɭɥɶ- |
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ɫɨɜ, ɚɦɩɥɢɬɭɞɵ ɤɨɬɨɪɵɯ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɵ ɨɬɫɱɟɬɚɦ ɢɫɯɨɞɧɨɣ ɧɟɩɪɟɪɵɜɧɨɣ ɮɭɧɤɰɢɢ, ɬɨ, ɫɭɦɦɢɪɭɹ ɜɵɯɨɞɧɵɟ ɫɢɝɧɚɥɵ ɮɢɥɶɬɪɚ, ɦɨɠɧɨ ɜɨɫɩɪɨɢɡɜɟɫɬɢ ɫ ɞɨɫɬɚɬɨɱɧɨ ɜɵɫɨɤɨɣ ɫɬɟɩɟɧɶɸ ɬɨɱɧɨɫɬɢ ɢɫɯɨɞɧɵɣ ɧɟɩɪɟɪɵɜɧɵɣ ɫɢɝɧɚɥ.
1.1.2. Ɍɟɨɪɟɦɚ Ʉɨɬɟɥɶɧɢɤɨɜɚ ɜ ɱɚɫɬɨɬɧɨɣ ɨɛɥɚɫɬɢ
ɂɧɨɝɞɚ ɫɢɝɧɚɥ ɧɟɨɛɯɨɞɢɦɨ ɩɪɟɞɫɬɚɜɢɬɶ ɫ ɩɨɦɨɳɶ ɱɚɫɬɨɬɧɵɯ ɜɵɛɨɪɨɤ ɫɩɟɤɬɪɚɥɶɧɨɣ ɮɭɧɤɰɢɢ S(Z) , ɚ ɧɟ ɜɪɟɦɟɧɧɵɯ ɜɵɛɨɪɨɤ ɮɭɧɤɰɢɢ s(t) . Ⱦɥɹ ɮɭɧɤɰɢɢ S(Z) ɦɨɠɧɨ ɫɨɫɬɚɜɢɬɶ ɪɹɞ, ɚɧɚɥɨɝɢɱɧɵɣ ɜɵɪɚɠɟɧɢɸ (1.10). ɗɬɨ ɧɟɬɪɭɞɧɨ
ɫɞɟɥɚɬɶ ɧɚ ɨɫɧɨɜɚɧɢɢ ɜɡɚɢɦɧɨɣ ɡɚɦɟɧɹɟɦɨɫɬɢ ɩɟɪɟɦɟɧɧɵɯ t |
ɢ Z ɜ ɩɪɟɨɛɪɚɡɨ- |
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ɜɚɧɢɹɯ Ɏɭɪɶɟ. ɉɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɜɵɪɚɠɟɧɢɸ (1.10) |
ɷɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ t ɫɥɟɞɭɟɬ |
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ɩɨɦɟɧɹɬɶ ɧɚ Z , 2Zc ɧɚ Tc , 2 fc ɧɚ Tc |
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8
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨɥɭɱɚɟɦ
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k f cTc |
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ȿɫɥɢ ɪɚɧɟɟ ɜɪɟɦɟɧɧɨɣ ɢɧɬɟɪɜɚɥ ɦɟɠɞɭ ɞɜɭɦɹ ɫɨɫɟɞɧɢɦɢ ɜɵɛɨɪɤɚɦɢ ɧɟ ɞɨɥɠɟɧ ɛɵɥ ɩɪɟɜɵɲɚɬɶ 2S 2Zc , ɬɨ ɬɟɩɟɪɶ ɱɚɫɬɨɬɧɵɣ ɢɧɬɟɪɜɚɥ ɧɟ ɞɨɥɠɟɧ ɩɪɟ-
ɜɵɲɚɬɶ 2S |
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. ɉɪɢ ɲɢɪɢɧɟ |
ɫɩɟɤɬɪɚ |
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2Zc , |
ɨɯɜɚɬɵɜɚɸɳɟɣ ɨɛɥɚɫɬɶ ɱɚɫɬɨɬ |
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, ɱɢɫɥɨ ɜɵɛɨɪɨɤ ɪɚɜɧɨ 2Zc |
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ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɜɵɛɨɪɤɢ |
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ɹɜɥɹɸɬɫɹ ɤɨɦɩɥɟɤɫɧɵɦɢ ɱɢɫɥɚɦɢ ɢ ɜ |
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ɤɚɠɞɨɣ ɨɬɫɱɟɬɧɨɣ ɬɨɱɤɟ ɧɚ ɨɫɢ ɱɚɫɬɨɬ ɞɨɥɠɧɵ ɛɵɬɶ ɡɚɞɚɧɵ ɞɜɚ ɩɚɪɚɦɟɬɪɚ –
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k2S |
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(ɢɥɢ ɦɨɞɭɥɶ ɢ ɚɪɝɭɦɟɧɬ). Ɍɚɤɢɦ ɨɛɪɚ- |
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ɞɟɣɫɬɜɢɬɟɥɶɧɚɹ ɢ ɦɧɢɦɚɹ ɱɚɫɬɢ S¨ |
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ɡɨɦ, ɨɛɳɟɟ ɱɢɫɥɨ ɩɚɪɚɦɟɬɪɨɜ ɩɨɥɭɱɚɟɬɫɹ ɜɞɜɨɟ ɛɨɥɶɲɢɦ, ɱɟɦ ɩɪɢ ɜɪɟɦɟɧɧɨɦ
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ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɫɢɝɧɚɥɚ, ɤɨɝɞɚ ɜɵɛɨɪɤɢ s¨ |
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– ɞɟɣɫɬɜɢɬɟɥɶɧɵɟ ɱɢɫɥɚ. ɂɡɛɵ- |
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2 fc ¹ |
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ɬɨɱɧɨɫɬɶ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɫɢɝɧɚɥɚ ɜ ɱɚɫɬɨɬɧɨɣ ɨɛɥɚɫɬɢ ɥɟɝɤɨ ɭɫɬɪɚɧɹɟɬɫɹ, ɟɫɥɢ
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k2S |
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ɹɜɥɹɸɬɫɹ ɤɨɦɩɥɟɤɫɧɨ-ɫɨɩɪɹɠɟɧɧɵɦɢ ɮɭɧɤɰɢ- |
ɭɱɟɫɬɶ, ɱɬɨ S¨ |
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ɹɦɢ, ɬɚɤ ɱɬɨ ɡɚɞɚɧɢɟ ɨɞɧɨɣ ɢɡ ɧɢɯ ɨɞɧɨɡɧɚɱɧɨ ɨɩɪɟɞɟɥɹɟɬ ɞɪɭɝɭɸ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɩɟɤɬɪ ɫɢɝɧɚɥɚ ɩɨɥɧɨɫɬɶɸ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɫɨɜɨɤɭɩɧɨɫɬɶɸ ɤɨɦɩɥɟɤɫɧɵɯ ɜɵɛɨɪɨɤ, ɜɡɹɬɵɯ ɬɨɥɶɤɨ ɜ ɨɛɥɚɫɬɢ ɩɨɥɨɠɢɬɟɥɶɧɵɯ ɱɚɫɬɨɬ, ɢ ɱɢɫɥɨ ɧɟɡɚɜɢɫɢɦɵɯ ɩɚɪɚɦɟɬɪɨɜ ɢɥɢ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ ɫɢɝɧɚɥɚ ɪɚɜɧɨ 2 fcTc , ɤɚɤ ɢ ɩɪɢ ɩɪɟɞɫɬɚɜɥɟ-
ɧɢɢ ɫɢɝɧɚɥɚ ɜɨ ɜɪɟɦɟɧɧɨɣ ɨɛɥɚɫɬɢ.
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1.2.ɂɧɬɟɝɪɢɪɭɸɳɢɟ ɦɟɬɨɞɵ ɞɢɫɤɪɟɬɢɡɚɰɢɢ
ȼɨɛɳɟɦ ɫɥɭɱɚɟ ɞɢɫɤɪɟɬɢɡɚɰɢɹ ɫɨɫɬɨɢɬ ɜ ɡɚɦɟɧɟ ɧɟɩɪɟɪɵɜɧɨɝɨ ɫɢɝɧɚɥɚ x(t) ɧɚɛɨɪɨɦ ɞɢɫɤɪɟɬɧɵɯ ɡɧɚɱɟɧɢɣ y(tn ) , ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɵ
ɤɚɤ ɪɟɡɭɥɶɬɚɬ ɫɜɟɪɬɤɢ ɫɢɝɧɚɥɚ x(t) |
ɫ ɜɟɫɨɜɨɣ ɮɭɧɤɰɢɟɣ M(t) : |
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³x(t)M(t tn ) dt , |
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ɝɞɟ y(t n ) – ɮɢɤɫɢɪɨɜɚɧɧɵɟ ɜ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ tn , n |
1,2,... , ɡɧɚɱɟɧɢɹ ɜɵɯɨɞ- |
ɧɨɝɨ ɫɢɝɧɚɥɚ. |
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ȼ ɢɞɟɚɥɶɧɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɜ ɤɚɱɟɫɬɜɟ ɮɭɧɤɰɢɢ M(t) ɢɫɩɨɥɶɡɭɟɬɫɹ |
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G -ɮɭɧɤɰɢɹ, ɪɟɡɭɥɶɬɚɬɨɦ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɹɜɥɹɸɬɫɹ ɦɝɧɨɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɜɯɨɞ- |
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ɧɨɝɨ ɫɢɝɧɚɥɚ: |
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³x(t)G(t tn ) dt x(tn ) , |
(1.12) |
f
ɤɨɬɨɪɵɟ ɢɫɩɨɥɶɡɭɸɬɫɹ ɩɪɢ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɩɨ Ʉɨɬɟɥɶɧɢɤɨɜɭ.
ȼɨ ɜɫɟɯ ɪɟɚɥɶɧɵɯ ɫɥɭɱɚɹɯ ɮɭɧɤɰɢɹ M(t) ɢɦɟɟɬ ɤɨɧɟɱɧɭɸ ɞɥɢɬɟɥɶɧɨɫɬɶ,
ɱɬɨ ɩɪɢɜɨɞɢɬ ɤ ɩɨɹɜɥɟɧɢɸ ɩɨɝɪɟɲɧɨɫɬɢ ɞɢɫɤɪɟɬɢɡɚɰɢɢ. ȼ ɨɬɥɚɠɟɧɧɨɣ ɢ ɧɚɫɬɪɨɟɧɧɨɣ ɚɩɩɚɪɚɬɭɪɟ ɜɵɩɨɥɧɹɟɬɫɹ ɭɫɥɨɜɢɟ ɧɨɪɦɢɪɨɜɤɢ
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³ M(t) dt 1, |
(1.13) |
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ɭɫɬɪɚɧɹɸɳɟɟ ɫɬɚɬɢɱɟɫɤɭɸ ɩɨɝɪɟɲɧɨɫɬɶ ɞɢɫɤɪɟɬɢɡɚɰɢɢ: ɩɪɢ x(t) a |
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y(tn ) ³aM(t tn )dt a . |
(1.14) |
f
Ⱦɢɫɤɪɟɬɢɡɚɰɢɹ ɢɡɦɟɧɹɸɳɢɯɫɹ ɜɨ ɜɪɟɦɟɧɢ ɫɢɝɧɚɥɨɜ ɩɪɢ M(t) z G(t) ɩɪɢ-
ɜɨɞɢɬ ɤ ɩɨɹɜɥɟɧɢɸ ɞɢɧɚɦɢɱɟɫɤɨɣ ɩɨɝɪɟɲɧɨɫɬɢ.
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