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