Дискретизация сигналов. Учебное пособие
.pdfȽɪɭɛɭɸ ɨɰɟɧɤɭ ɯɚɪɚɤɬɟɪɚ ɡɚɜɢɫɢɦɨɫɬɢ ɞɢɧɚɦɢɱɟɫɤɨɣ ɩɨɝɪɟɲɧɨɫɬɢ ɨɬ ɜɢɞɚ ɜɟɫɨɜɨɣ ɮɭɧɤɰɢɢ ɢ ɩɚɪɚɦɟɬɪɨɜ ɫɢɝɧɚɥɚ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ. Ⱦɥɢɬɟɥɶɧɨɫɬɶ ɜɟɫɨɜɨɣ ɮɭɧɤɰɢɢ M(t) WM ɫɜɹɡɚɧɚ ɫ ɬɚɤɢɦ ɩɚɪɚɦɟɬɪɨɦ
ɭɫɬɪɨɣɫɬɜɚ ɞɢɫɤɪɟɬɢɡɚɰɢɢ, ɤɚɤ ɚɩɟɪɬɭɪɧɵɣ ɫɞɜɢɝ, ɢɥɢ ɫɢɫɬɟɦɚɬɢɱɟɫɤɚɹ ɫɨɫɬɚɜɥɹɸɳɚɹ ɜɪɟɦɟɧɢ ɡɚɞɟɪɠɤɢ ɨɬɫɱɟɬɚ tɡɫ , ɚ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɷɬɨɣ ɞɥɢɬɟɥɶɧɨɫɬɢ
'WM – ɫ ɚɩɟɪɬɭɪɧɵɦ ɜɪɟɦɟɧɟɦ ta . ɏɚɪɚɤɬɟɪ ɢɡɦɟɧɟɧɢɹ ɫɢɝɧɚɥɚ ɜɨ ɜɪɟɦɟɧɢ ɭɞɨɛɧɨ ɨɰɟɧɢɜɚɬɶ ɫ ɩɨɦɨɳɶɸ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɮɭɧɤɰɢɢ B(W) , ɨɩɪɟɞɟɥɹɟɦɨɣ, ɜ
ɫɜɨɸ ɨɱɟɪɟɞɶ, ɩɨ ɚɦɩɥɢɬɭɞɧɨɦɭ ɫɩɟɤɬɪɭ ɢ ɧɟ ɡɚɜɢɫɹɳɟɣ ɨɬ ɮɨɪɦɵ ɫɢɝɧɚɥɚ. ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɞɥɹ ɫɨɩɨɫɬɚɜɥɟɧɢɹ ɫ ɞɥɢɬɟɥɶɧɨɫɬɶɸ ɜɟɫɨɜɨɣ ɮɭɧɤɰɢɢ WM ɢɧɬɟɪ-
ɜɚɥɨɦ ɤɨɪɪɟɥɹɰɢɢ Wɤ (ɪɢɫ. 1.1). Ⱦɢɧɚɦɢɱɟɫɤɚɹ ɩɨɝɪɟɲɧɨɫɬɶ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɦɨɠɟɬ ɛɵɬɶ ɨɰɟɧɟɧɚ ɜɟɥɢɱɢɧɨɣ
WM 'WM |
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Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɩɪɢ M(t) G (t) ɚɩɟɪɬɭɪɧɨɟ ɜɪɟɦɹ ɢ ɚɩɟɪɬɭɪɧɵɣ ɫɞɜɢɝ ɭɫɬɪɨɣɫɬɜɚ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɪɚɜɧɵ ɧɭɥɸ ɢ 'ɞ 0 ɩɪɢ ɥɸɛɨɣ ɫɤɨɪɨɫɬɢ ɢɡɦɟɧɟɧɢɹ ɜɯɨɞɧɨɝɨ ɫɢɝɧɚɥɚ. Ⱦɥɹ ɪɟɚɥɶɧɵɯ ɜɟɫɨɜɵɯ ɮɭɧɤɰɢɣ, ɤɨɝɞɚ WM z 0, ɞɢɧɚɦɢɱɟɫɤɚɹ ɩɨɝɪɟɲɧɨɫɬɶ ɨɬɫɭɬɫɬɜɭɟɬ ɬɨɥɶɤɨ ɩɪɢ Wɤ o f, ɬ. ɟ. ɩɪɢ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɩɨɫɬɨɹɧ-
ɧɵɯ ɫɢɝɧɚɥɨɜ.
Ɍɚɤɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɦɟɯɚɧɢɡɦɚ ɜɨɡɧɢɤɧɨɜɟɧɢɹ ɞɢɧɚɦɢɱɟɫɤɨɣ ɩɨɝɪɟɲɧɨɫɬɢ ɩɨɡɜɨɥɹɟɬ ɪɚɫɫɦɨɬɪɟɬɶ ɦɟɬɨɞɵ ɩɨɜɵɲɟɧɢɹ ɬɨɱɧɨɫɬɢ ɩɪɨɰɟɫɫɚ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɩɭɬɟɦ ɭɱɟɬɚ ɮɨɪɦɵ ɢ ɞɥɢɬɟɥɶɧɨɫɬɢ ɜɟɫɨɜɨɣ ɮɭɧɤɰɢɢ M(t) ɩɪɢ ɩɨɥɭɱɟɧɢɢ ɨɬɫɱɟɬɨɜ ɫɢɝɧɚɥɚ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɩɪɨɰɟɫɫ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɤɚɤ ɩɪɨɯɨɠɞɟɧɢɟ ɜɯɨɞɧɨɝɨ ɫɢɝɧɚɥɚ ɱɟɪɟɡ ɡɜɟɧɨ ɫ ɢɦɩɭɥɶɫɧɨɣ ɩɟɪɟɯɨɞɧɨɣ ɮɭɧɤɰɢɟɣ M(t) . Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɬɨɱɧɨɝɨ ɨɬɫɱɟɬɚ ɜɯɨɞɧɨɝɨ ɫɢɝɧɚɥɚ ɧɟɨɛɯɨɞɢɦ ɰɢɮɪɨɜɨɣ ɮɢɥɶɬɪ, ɪɟɚɥɢɡɭɸɳɢɣ ɨɛɪɚɬɧɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ. Ⱦɢɧɚɦɢɱɟɫɤɚɹ ɩɨɝɪɟɲɧɨɫɬɶ ɩɪɢ ɬɚɤɨɦ ɩɨɞɯɨɞɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɧɟ ɲɢɪɢɧɨɣ ɢ ɮɨɪɦɨɣ ɜɟɫɨɜɨɣ ɮɭɧɤɰɢɢ, ɚ ɧɟɫɬɚɛɢɥɶɧɨɫɬɶɸ ɷɬɢɯ ɩɚɪɚɦɟɬɪɨɜ ɢ ɩɨɝɪɟɲɧɨɫɬɶɸ ɢɯ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɩɪɢ ɪɟɚɥɢɡɚɰɢɢ ɜɨɫɫɬɚɧɚɜɥɢɜɚɸɳɟɝɨ ɮɢɥɶɬɪɚ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɢɫɤɪɟɬɧɵɟ ɨɬɫɱɟɬɵ, ɩɨɥɭɱɚɟɦɵɟ ɜ ɩɪɨɰɟɫɫɟ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɪɟɚɥɶɧɵɯ ɫɢɝɧɚɥɨɜ, ɹɜɥɹɸɬɫɹ ɧɟ ɦɝɧɨɜɟɧɧɵɦɢ, ɚ ɢɧɬɟɝɪɚɥɶɧɵɦɢ ɜɵɛɨɪɤɚɦɢ ɫɢɝɧɚɥɚ. ɗɬɢ ɜɵɛɨɪɤɢ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɪɟɡɭɥɶɬɚɬ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɩɪɨɢɡ-
11
ɜɟɞɟɧɢɹ ɜɯɨɞɧɨɝɨ ɫɢɝɧɚɥɚ ɢ ɜɟɫɨɜɨɣ ɮɭɧɤɰɢɢ. ɉɨɷɬɨɦɭ ɦɟɬɨɞɵ ɞɢɫɤɪɟɬɢɡɚɰɢɢ, ɭɱɢɬɵɜɚɸɳɢɟ ɮɨɪɦɭ ɢ ɞɥɢɬɟɥɶɧɨɫɬɶ ɜɟɫɨɜɨɣ ɮɭɧɤɰɢɢ M(t) , ɧɚɡɵɜɚɸɬɫɹ ɢɧɬɟ-
ɝɪɢɪɭɸɳɢɦɢ ɦɟɬɨɞɚɦɢ ɞɢɫɤɪɟɬɢɡɚɰɢɢ. ɇɢɠɟ ɪɚɫɫɦɨɬɪɟɧɵ ɞɜɚ ɦɟɬɨɞɚ ɢɧɬɟɝɪɢɪɭɸɳɟɣ ɞɢɫɤɪɟɬɢɡɚɰɢɢ: ɫ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɢ ɤɨɫɢɧɭɫɨɢɞɚɥɶɧɨɣ ɜɟɫɨɜɨɣ ɮɭɧɤɰɢɟɣ.
B(t),M(t)
M(t tn )
B( |
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tn ) |
W |
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tn |
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WM |
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Ɋɢɫ. 1.1. ɋɨɩɨɫɬɚɜɥɟɧɢɟ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɮɭɧɤɰɢɢ ɫɢɝɧɚɥɚ
ɢɜɟɫɨɜɨɣ ɞɢɫɤɪɟɬɢɡɢɪɭɸɳɟɣ ɮɭɧɤɰɢɢ
1.2.1.Ⱥɧɚɥɨɝ ɬɟɨɪɟɦɵ Ʉɨɬɟɥɶɧɢɤɨɜɚ ɞɥɹ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɜɟɫɨɜɨɣ ɮɭɧɤɰɢɢ
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɨɰɟɫɫ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɫɢɝɧɚɥɚ x(t) , ɨɩɢɫɵɜɚɟɦɵɣ ɮɨɪɦɭ-
ɥɨɣ (1.11), ɤɨɝɞɚ ɜ ɤɚɱɟɫɬɜɟ M(t) ɢɫɩɨɥɶɡɭɟɬɫɹ ɩɪɹɦɨɭɝɨɥɶɧɵɣ ɢɦɩɭɥɶɫ ɤɨɧɟɱ-
ɧɨɣ ɞɥɢɬɟɥɶɧɨɫɬɢ: |
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M(t) |
1, W 2 dt dW |
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ɋɩɟɤɬɪɚɥɶɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ X ( jZ) |
ɢ Yn ( jZ) ɫɢɝɧɚɥɨɜ x(t) |
ɢ y(tn ) |
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ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɩɨɫɬɨɹɧɧɨɝɨ ɦɧɨɠɢɬɟɥɹ ɫɜɹɡɚɧɵ ɫɨɨɬɧɨɲɟɧɢɟɦ [2] |
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Yn ( jZ) |
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jZ |
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ɢɡ ɤɨɬɨɪɨɝɨ ɫɥɟɞɭɟɬ |
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X ( jZ) |
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jZ |
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Yn ( jZ) . |
(1.18) |
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1 e jZW |
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ɉɨɥɶɡɭɹɫɶ ɮɨɪɦɭɥɨɣ ɗɣɥɟɪɚ, ɩɨɥɭɱɢɦ: |
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X ( jZ) |
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0,5Z |
e j0,5ZW Yn ( jZ) . |
(1.19) |
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sin 0,5ZW |
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ɉɪɟɨɛɪɚɡɨɜɚɧɢɟ (1.11) ɹɜɥɹɟɬɫɹ ɥɢɧɟɣɧɵɦ, ɩɨɷɬɨɦɭ ɫɩɟɤɬɪ ɫɢɝɧɚɥɚ y(tn ) |
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ɨɫɬɚɟɬɫɹ ɨɝɪɚɧɢɱɟɧɧɵɦ ɡɧɚɱɟɧɢɹɦɢ Z c |
ɢ Z c . ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɫɨɝɥɚɫɧɨ ɬɟɨ- |
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ɪɟɦɟ Ʉɨɬɟɥɶɧɢɤɨɜɚ ɩɨ ɞɢɫɤɪɟɬɧɵɦ ɡɧɚɱɟɧɢɹɦ y(tn ) , ɨɬɫɱɢɬɵɜɚɟɦɵɦ ɱɟɪɟɡ ɢɧ-
ɬɟɪɜɚɥ Ɍ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɣ ɭɫɥɨɜɢɸ T |
1 |
(ɝɞɟ fc |
Zc |
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2 fc |
2S |
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ɫɬɚɧɨɜɥɟɧɚ ɫɩɟɤɬɪɚɥɶɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ Yn ( jZ) . ɉɨɫɥɟɞɧɹɹ, ɜ ɫɜɨɸ |
ɨɱɟɪɟɞɶ, |
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ɨɞɧɨɡɧɚɱɧɨ ɨɩɪɟɞɟɥɹɟɬ ɫɩɟɤɬɪɚɥɶɧɭɸ ɯɚɪɚɤɬɟɪɢɫɬɢɤɭ X ( jZ) |
ɫɢɝɧɚɥɚ |
x(t) , ɟɫ- |
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ɥɢ ɬɨɥɶɤɨ, ɢɧɬɟɪɜɚɥ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɜɵɛɪɚɧ ɢɡ ɭɫɥɨɜɢɹ |
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W |
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(1.20) |
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ɗɬɨ ɭɫɥɨɜɢɟ ɫɥɟɞɭɟɬ ɢɡ ɜɵɪɚɠɟɧɢɹ (1.19) – ɢɧɬɟɪɜɚɥ W ɞɨɥɠɟɧ ɛɵɬɶ ɬɚɤɢɦ, ɱɬɨɛɵ ɞɨ ɱɚɫɬɨɬɵ Zc ɮɭɧɤɰɢɹ sin 0,5ZW ɧɟ ɩɪɢɧɢɦɚɥɚ ɧɭɥɟɜɨɝɨ ɡɧɚɱɟɧɢɹ.
ɋɥɟɞɭɟɬ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ, ɬ. ɟ. ɩɪɢ ɩɪɨɢɡɜɨɥɶɧɨɦ ɜɢɞɟ ɫɢɝɧɚɥɚ x(t) , ɜɨɫɫɬɚɧɨɜɥɟɧɢɟ ɫɩɟɤɬɪɚɥɶɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɢɫɯɨɞɧɨɝɨ ɫɢɝɧɚɥɚ ɩɨ ɫɩɟɤɬɪɚɥɶɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɟ ɩɪɟɨɛɪɚɡɨɜɚɧɧɨɝɨ ɫɢɝɧɚɥɚ ɧɟɜɨɡɦɨɠɧɨ. Ɉɞɧɚɤɨ ɩɪɢ ɨɝɨɜɨɪɟɧɧɨɦ ɭɫɥɨɜɢɢ – ɨɝɪɚɧɢɱɟɧɧɨɫɬɢ ɫɩɟɤɬɪɚ ɜɯɨɞɧɨɣ ɮɭɧɤɰɢɢ – ɞɚɧɧɚɹ ɡɚɞɚɱɚ ɹɜɥɹɟɬɫɹ ɤɨɪɪɟɤɬɧɨɣ. Ɍɟɦ ɫɚɦɵɦ ɞɨɤɚɡɵɜɚɟɬɫɹ ɩɟɪɜɚɹ ɱɚɫɬɶ ɚɧɚɥɨɝɚ ɬɟɨɪɟɦɵ Ʉɨɬɟɥɶɧɢɤɨɜɚ ɞɥɹ ɢɧɬɟɝɪɢɪɭɸɳɟɣ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɫ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɜɟɫɨɜɨɣ ɮɭɧɤɰɢɢ.
13
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɜɬɨɪɨɣ ɱɚɫɬɢ, ɬ. ɟ. ɫɢɧɬɟɡ ɪɹɞɚ ɞɥɹ ɬɟɨɪɟɬɢɱɟɫɤɢ ɬɨɱɧɨɝɨ ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ ɮɭɧɤɰɢɢ ɫ ɨɝɪɚɧɢɱɟɧɧɵɦ ɫɩɟɤɬɪɨɦ ɩɨ ɟɟ ɢɧɬɟɝɪɚɥɶɧɵɦ ɜɵɛɨɪɤɚɦ, ɩɨɥɭɱɟɧɨ ɜ ɪɚɛɨɬɟ [3]. ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɫɨɫɬɨɢɬ ɜ ɫɥɟɞɭɸɳɟɦ.
ȼɵɪɚɠɚɹ ɮɭɧɤɰɢɸ x(t) ɱɟɪɟɡ ɟɟ ɫɩɟɤɬɪɚɥɶɧɭɸ ɯɚɪɚɤɬɟɪɢɫɬɢɤɭ (1.19), ɩɨ-
ɥɭɱɢɦ:
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Zc |
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e jZ(t 0,5W)dZ. |
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x(t) |
³Yn ( jZ) |
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ɋɩɟɤɬɪɚɥɶɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ Yn ( jZ) ɦɨɠɟɬ ɛɵɬɶ ɨɩɪɟɞɟɥɟɧɚ: |
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ɉɨɞɫɬɚɜɥɹɹ (1.22) ɜ (1.21), ɧɚɣɞɟɦ |
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ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɮɨɪɦɭɥɨɣ ɗɣɥɟɪɚ, ɜɵɪɚɠɟɧɢɟ (1.23) ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɜ ɜɢɞɟ
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Ɏɭɧɤɰɢɸ ɜɢɞɚ z / sin z ɩɪɟɨɛɪɚɡɭɟɦ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
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14
ɝɞɟ Bm – ɱɢɫɥɚ Ȼɟɪɧɭɥɥɢ, ɩɪɢɱɟɦ
Bm |
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ɋ ɭɱɟɬɨɦ (1.25) ɜɵɪɚɠɟɧɢɟ (1.24) ɡɚɩɢɲɟɦ ɜ ɜɢɞɟ
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(ɩɨɫɤɨɥɶɤɭ ɫɭɦɦɢɪɨɜɚɧɢɟ ɜɟɞɟɬɫɹ ɩɨ ɨɬɪɢɰɚɬɟɥɶɧɵɦ ɢ ɩɨɥɨɠɢɬɟɥɶɧɵɦ n , ɡɧɚɤ ɩɟɪɟɞ n ɡɚɦɟɧɟɧ ɧɚ ɨɛɪɚɬɧɵɣ).
ɉɭɬɟɦ ɩɨɞɫɬɚɧɨɜɤɢ
z Z (t nT 0,5W) |
(1.28) |
ɢɧɬɟɝɪɚɥɵ ɩɨɞ ɡɧɚɤɨɦ ɜɧɭɬɪɟɧɧɟɣ ɫɭɦɦɵ ɩɪɢɜɨɞɹɬɫɹ ɤ ɜɢɞɭ
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Ⱦɥɹ ɬɚɤɢɯ ɢɧɬɟɝɪɚɥɨɜ ɢɡɜɟɫɬɧɨ ɪɚɡɥɨɠɟɧɢɟ:
³z2m cos z dz z2m sin z 2m³z2m 1 sin z dz ; |
(1.29) |
15
³z2m 1 sin z dz z2m 1 cos z (2m 1)³z2m 2 cos z dz . |
(1.30) |
ɋ ɭɱɟɬɨɦ (1.29) ɢ (1.30) ɜɵɪɚɠɟɧɢɟ (1.27) ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
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(nT ) sinZ |
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cosZc (t nT |
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ɝɞɟ Q2m 2 (T) ɢ P2m (T) – ɩɨɥɢɧɨɦɵ ɩɨ ɱɟɬɧɵɦ ɫɬɟɩɟɧɹɦ ɚɪɝɭɦɟɧɬɚ
T1/(t nT 0,5W) , ɩɪɢɱɟɦ m o f. ȼɵɪɚɠɟɧɢɟ ɜ ɮɢɝɭɪɧɵɯ ɫɤɨɛɤɚɯ ɩɨ ɚɧɚɥɨ-
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ɪɹɞɚ Ʉɨɬɟɥɶɧɢɤɨɜɚ ɦɨɠɧɨ ɧɚɡɜɚɬɶ ɮɭɧɤ- |
ɝɢɢ ɫ ɮɭɧɤɰɢɟɣ |
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ɰɢɟɣ ɢɧɬɟɝɪɚɥɶɧɵɯ ɨɬɫɱɟɬɨɜ. Ɉɛɨɡɧɚɱɢɜ ɟɟ ɤɚɤ g(t nT 0,5W) , ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɜɵɪɚɠɟɧɢɟ (1.31) ɜ ɫɥɟɞɭɸɳɟɣ ɤɨɦɩɚɤɬɧɨɣ ɮɨɪɦɟ:
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Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨɞɨɛɧɨ ɬɨɦɭ, ɤɚɤ ɫ ɩɨɦɨɳɶɸ ɪɹɞɚ Ʉɨɬɟɥɶɧɢɤɨɜɚ ɦɨɠɧɨ ɬɟɨɪɟɬɢɱɟɫɤɢ ɬɨɱɧɨ ɜɨɫɫɬɚɧɨɜɢɬɶ ɮɭɧɤɰɢɸ ɫ ɨɝɪɚɧɢɱɟɧɧɵɦ ɫɩɟɤɬɪɨɦ, ɫ ɩɨɦɨɳɶɸ ɪɹɞɚ (1.31) ɦɨɠɧɨ ɬɟɨɪɟɬɢɱɟɫɤɢ ɬɨɱɧɨ ɜɨɫɫɬɚɧɨɜɢɬɶ ɮɭɧɤɰɢɸ ɫ ɨɝɪɚɧɢɱɟɧɧɵɦ ɫɩɟɤɬɪɨɦ ɩɨ ɟɟ ɢɧɬɟɝɪɚɥɶɧɵɦ ɜɵɛɨɪɤɚɦ. ɉɪɢ W o 0 ɪɹɞ (1.31) ɨɛɪɚɳɚɟɬɫɹ ɜ ɪɹɞ Ʉɨɬɟɥɶɧɢɤɨɜɚ, ɩɨɫɤɨɥɶɤɭ lim >yn (nT ) /W@ ɩɪɢ W o 0 ɪɚɜɟɧ x(nT ) .
Ⱥɧɚɥɢɡ ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɩɨɥɢɧɨɦɵ P2m (T) ɢ Q2m 2 (T) ɛɵɫɬɪɨ ɫɯɨɞɹɬɫɹ,
ɬɚɤ ɱɬɨ ɞɥɹ ɩɨɥɭɱɟɧɢɹ ɩɪɢɟɦɥɟɦɨɣ ɧɚ ɩɪɚɤɬɢɤɟ ɬɨɱɧɨɫɬɢ ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ ɦɨɠɧɨ ɨɝɪɚɧɢɱɢɬɶɫɹ ɡɧɚɱɟɧɢɹɦɢ m ɩɨɪɹɞɤɚ 2 ɢɥɢ 3. ɇɚɩɪɢɦɟɪ, ɩɪɢ m 2 ɢɦɟɟɦ ɫɥɟɞɭɸɳɟɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɮɭɧɤɰɢɢ ɢɧɬɟɝɪɚɥɶɧɵɯ ɨɬɫɱɟɬɨɜ:
16
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ucosZc (t nT 0,5W)
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¹
ɂɡ ɮɨɪɦɭɥɵ (1.33) ɧɟɜɨɡɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ ɢɧɬɟɝɪɚɥɶɧɵɯ ɨɬɫɱɟɬɨɜ ɜ ɬɨɱɤɟ t (nT 0,5W) , ɬɚɤ ɤɚɤ ɨɧɚ ɢɦɟɟɬ ɜ ɷɬɨɣ ɬɨɱɤɟ ɪɚɡɪɵɜ ɧɟ-
ɩɪɟɪɵɜɧɨɫɬɢ. ɑɬɨɛɵ ɨɛɨɣɬɢ ɷɬɭ ɬɪɭɞɧɨɫɬɶ, ɨɛɪɚɬɢɦɫɹ ɤ ɢɫɯɨɞɧɨɣ ɮɨɪɦɭɥɟ (1.24), ɢɡ ɤɨɬɨɪɨɣ ɫɥɟɞɭɟɬ, ɱɬɨ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ ɢɧɬɟɝɪɚɥɶɧɵɯ ɨɬɫɱɟɬɨɜ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɬɨɱɤɟ ɪɚɜɧɨ
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ɢɥɢ ɫ ɭɱɟɬɨɦ (1.25) |
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1.2.2. Ⱥɧɚɥɨɝ ɬɟɨɪɟɦɵ Ʉɨɬɟɥɶɧɢɤɨɜɚ ɞɥɹ ɤɨɫɢɧɭɫɨɢɞɚɥɶɧɨɣ ɜɟɫɨɜɨɣ ɮɭɧɤɰɢɢ
ɋɪɟɞɢ ɲɢɪɨɤɨɝɨ ɤɥɚɫɫɚ ɜɟɫɨɜɵɯ ɮɭɧɤɰɢɣ, ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɛɵɬɶ ɢɫɩɨɥɶɡɨɜɚɧɵ ɞɥɹ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɫɢɝɧɚɥɨɜ, ɛɨɥɶɲɨɣ ɢɧɬɟɪɟɫ ɫ ɬɨɱɤɢ ɡɪɟɧɢɹ ɩɪɚɤɬɢɱɟɫɤɨɣ ɪɟɚɥɢɡɚɰɢɢ ɜɵɡɵɜɚɸɬ ɝɚɪɦɨɧɢɱɟɫɤɢɟ ɮɭɧɤɰɢɢ. ɂɫɫɥɟɞɭɟɦ ɜɨɡɦɨɠɧɨɫɬɶ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɜ ɤɚɱɟɫɬɜɟ ɜɟɫɨɜɨɣ ɮɭɧɤɰɢɢ ɤɨɫɢɧɭɫɨɢɞɚɥɶɧɨɝɨ ɢɦɩɭɥɶɫɚ
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17
Ɋɚɫɫɦɨɬɪɢɦ ɮɭɧɤɰɢɸ
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f
ɦɝɧɨɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɤɨɬɨɪɨɣ ɜ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ tn n't ɫɨɜɩɚɞɚɸɬ ɫ ɮɭɧɤɰɢɟɣ (1.11). ȿɫɥɢ ɮɭɧɤɰɢɢ x(t) ɢ M(t) ɢɦɟɸɬ ɫɩɟɤɬɪɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ X ( jZ)
ɢ )( jZ) , ɬɨ ɩɨ ɫɜɨɣɫɬɜɭ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ɏɭɪɶɟ ɫɩɟɤɬɪ ɮɭɧɤɰɢɢ y(t) ɜɵɱɢɫɥɹ-
ɟɬɫɹ ɤɚɤ
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ɉɭɫɬɶ |
ɫɩɟɤɬɪ ɜɯɨɞɧɨɝɨ ɫɢɝɧɚɥɚ x(t) ɨɝɪɚɧɢɱɟɧ |
ɱɚɫɬɨɬɨɣ |
Zc , ɬ. ɟ. |
X ( jZ) 0 |
ɩɪɢ | Z |! Zc . ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɪɢ ɭɫɥɨɜɢɢ |
ɤɨɧɟɱɧɨɫɬɢ |
ɮɭɧɤɰɢɢ |
)( jZ) ɧɚ ɜɫɟɣ ɨɫɢ ɱɚɫɬɨɬ ( f,f) ɫɩɟɤɬɪY ( jZ) ɬɚɤɠɟ ɧɟ ɛɭɞɟɬ ɫɨɞɟɪɠɚɬɶ ɫɨ-
ɫɬɚɜɥɹɸɳɢɯ ɫ ɱɚɫɬɨɬɨɣ ɜɵɲɟ Zc , ɬ. ɟ. Y ( jZ) = 0 ɩɪɢ | Z |! Zc .
Ɏɭɧɤɰɢɹ M(t) ɢɦɟɟɬ ɫɩɟɤɬɪ
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Ɏɭɧɤɰɢɹ (1.39) ɜ ɬɨɱɤɟ | |
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. ɉɨɫɥɟ ɪɚɫɤɪɵɬɢɹ ɟɟ ɩɨ ɩɪɚɜɢɥɭ Ʌɨɩɢɬɚɥɹ ɢɦɟɟɦ: |
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Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɮɭɧɤɰɢɹ ɤɨɧɟɱɧɚ ɧɚ ɜɫɟɣ ɱɚɫɬɨɬɧɨɣ ɨɫɢ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɮɭɧɤɰɢɹ y(t) ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɫɥɨɜɢɹɦ ɬɟɨɪɟɦɵ Ʉɨɬɟɥɶɧɢ-
ɤɨɜɚ ɢ ɦɨɠɟɬ ɛɵɬɶ ɞɢɫɤɪɟɬɢɡɢɪɨɜɚɧɚ ɫ ɱɚɫɬɨɬɨɣ 2Zc , ɜ ɪɟɡɭɥɶɬɚɬɟ ɱɟɝɨ ɩɨɥɭ-
ɱɢɦ ɫɨɜɨɤɭɩɧɨɫɬɶ ɨɬɫɱɟɬɨɜ y (n't) , ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (1.37) ɪɚɜɧɵɯ
18
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f
Ⱦɥɹ ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ ɮɭɧɤɰɢɢ y(t) ɩɨ ɨɬɫɱɟɬɚɦ y (n't) ɧɟɨɛɯɨɞɢɦ ɢɞɟɚɥɶ-
ɧɵɣ ɮɢɥɶɬɪ ɧɢɠɧɢɯ ɱɚɫɬɨɬ ɫ ɝɪɚɧɢɱɧɨɣ ɱɚɫɬɨɬɨɣ Zc .
ɂɫɯɨɞɹ ɢɡ (1.38), ɮɭɧɤɰɢɸ y(t) ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ ɪɟɡɭɥɶɬɚɬ ɩɪɨɯɨɠɞɟɧɢɹ ɮɭɧɤɰɢɢ x(t) ɱɟɪɟɡ ɭɫɬɪɨɣɫɬɜɨ ɫ ɩɟɪɟɞɚɬɨɱɧɨɣ ɮɭɧɤɰɢɟɣ W ( jZ) )( jZ) .
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɥɹ ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ ɮɭɧɤɰɢɢ x(t) ɩɨ ɮɭɧɤɰɢɢ y(t) ɧɟɨɛɯɨɞɢɦ ɤɨɪɪɟɤɬɢɪɭɸɳɢɣ ɮɢɥɶɬɪ ɫ ɩɟɪɟɞɚɬɨɱɧɨɣ ɮɭɧɤɰɢɟɣ
Wɤ( jZ) |
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Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ ɫɢɝɧɚɥɚ x(t) ɩɨ ɞɢɫɤɪɟɬɧɵɦ ɡɧɚɱɟɧɢɹɦ y(n't) ɧɟɨɛɯɨɞɢɦɚ ɫɢɫɬɟɦɚ ɮɢɥɶɬɪɨɜ, ɫɨɫɬɨɹɳɚɹ ɢɡ ɢɞɟɚɥɶɧɨɝɨ ɮɢɥɶɬɪɚ ɧɢɠɧɢɯ ɱɚɫɬɨɬ ɫ ɝɪɚɧɢɱɧɨɣ ɱɚɫɬɨɬɨɣ Zc ɢ ɤɨɪɪɟɤɬɢɪɭɸɳɟɝɨ ɮɢɥɶɬɪɚ ɫ ɩɟɪɟɞɚ-
ɬɨɱɧɨɣ ɮɭɧɤɰɢɟɣ (1.42).
ȼɵɪɚɠɟɧɢɟ (1.42) ɧɚɤɥɚɞɵɜɚɟɬ ɧɚ ɮɭɧɤɰɢɸ )( jZ) ɬɪɟɛɨɜɚɧɢɟ ɨɬɫɭɬ-
ɫɬɜɢɹ ɧɭɥɟɣ ɜ ɩɨɥɨɫɟ ɩɪɨɩɭɫɤɚɧɢɹ ɮɢɥɶɬɪɚ ɧɢɠɧɢɯ ɱɚɫɬɨɬ. ɂɡ (1.39) ɫɥɟɞɭɟɬ,
ɱɬɨ ɩɟɪɜɵɣ ɧɭɥɶ )( jZ) ɢɦɟɟɬ ɜ ɬɨɱɤɟ Z |
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, ɬ. ɟ. ɞɨɥɠɧɨ ɜɵɩɨɥɧɹɬɶɫɹ ɭɫɥɨ- |
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ɜɢɟ Z1 ! Zc , ɨɬɤɭɞɚ |
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ɂɦɩɭɥɶɫɧɭɸ ɩɟɪɟɯɨɞɧɭɸ ɯɚɪɚɤɬɟɪɢɫɬɢɤɭ ɫɢɫɬɟɦɵ ɜɨɫɫɬɚɧɚɜɥɢɜɚɸɳɢɯ ɮɢɥɶɬɪɨɜ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ, ɢɫɩɨɥɶɡɭɹ ɫɜɹɡɶ ɜɪɟɦɟɧɧɵɯ ɢ ɱɚɫɬɨɬɧɵɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ:
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cos Zt dZ . (1.44) |
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Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɢɧɬɟɝɪɚɥɚ ɜ (1.44) ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɩɪɢɜɟɞɟɧɧɵɦ ɜ [4] ɪɚɡɥɨɠɟɧɢɟɦ ɜ ɪɹɞ
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ɝɞɟ E2k – ɱɢɫɥɚ ɗɣɥɟɪɚ.
ɉɨɞɫɬɚɜɥɹɹ (1.45) ɜ (1.44) ɢ ɜɵɧɨɫɹ ɡɧɚɤ ɫɭɦɦɵ ɡɚ ɡɧɚɤ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ, ɩɨɥɭɱɢɦ
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Ɋɚɫɫɦɨɬɪɢɦ ɨɬɞɟɥɶɧɨ ɢɧɬɟɝɪɚɥ |
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Ⱦɥɹ ɩɨɥɭɱɟɧɧɵɯ ɢɧɬɟɝɪɚɥɨɜ ɜ [4] ɞɚɟɬɫɹ ɪɚɡɥɨɠɟɧɢɟ |
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sin(Zt |
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