52 lines
1.6 KiB
Markdown
52 lines
1.6 KiB
Markdown
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## Frequency simulation
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Model is based on https://www.dsprelated.com/showarticle/973.php
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### Time domain block diagram
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### Z domain block diagram
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## DDMTD Phase detector gain ($K_{PD}$)
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$$
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\begin{align*}
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\Delta tag[n] &= \dfrac{N}{2\pi} \cdot \Delta\theta[n] \\[10pt]
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\Delta tag(z) &= \dfrac{N}{2\pi} \cdot \Delta\theta(z) \\[10pt]
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\dfrac{\Delta tag(z)}{\Delta \theta(z)} &= \dfrac{N}{2\pi} \\[10pt]
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K_{PD} &= \dfrac{N}{2\pi} \\[10pt]
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\end{align*}
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$$
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## Loop Filter
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Low pass biquad filter calculator: https://www.earlevel.com/main/2021/09/02/biquad-calculator-v3/
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## Si549 DCXO gain ($K_{DCXO}$)
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Consider Si549 as a phase accumulator:
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$$
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\begin{align*}
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\theta_{DCXO}[n] &= 2\pi f[n] \cdot T + \theta_{DCXO}[n-1] \\[10pt]
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\theta_{DCXO}[n] &= 2\pi \bigg( f_{center} + 0.0001164 \cdot (\text{BASEADPLL} + ADPLL[n]) \cdot \dfrac{f_{center}}{10^6} \bigg) \cdot T_s + \theta_{DCXO}[n-1] \\[10pt]
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\theta_{DCXO}[n] &= \underbrace{ 2\pi \bigg(0.0001164 \cdot \dfrac{f_{center}}{10^6} \cdot T_s \bigg)}_{K_{DCXO}} \cdot ADPLL[n] +
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\underbrace{ 2\pi \bigg( f_{center} + 0.0001164 \cdot \text{BASEADPLL} \cdot \dfrac{f_{center}}{10^6} \bigg) }_{f} \cdot T_s + \theta_{DCXO}[n-1] \\[20pt]
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K_{DCXO} &= 2\pi 0.0001164 \cdot \dfrac{f_{center}}{10^6} \cdot T_s \\[10pt]
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K_{DCXO} &= 2\pi * 0.0001164 \cdot \dfrac{f_{center}}{10^6} \cdot \dfrac{1}{f_s} \\[10pt]
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K_{DCXO} &= 2\pi * 0.0001164 \cdot \dfrac{f_{center}}{10^6} \cdot \dfrac{N}{f_{center}} \\[10pt]
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K_{DCXO} &= \dfrac{0.0001164\cdot N\cdot 2\pi}{10^6} \\[10pt]
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\end{align*}
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$$
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