Edit ECG article

_posts/2024-03-02-ecg.md

@@ -46,7 +46,7 @@ $$ V_\mathrm{out} + R I + R I + (2 R I + R_g I) - R I - R I = 0 \implies V_\math
 
 .
 
-Replacing $I$ with \eqref{eq_r_g}
+Replacing $I$ with equation \eqref{eq_r_g}
 
 $$ V_\mathrm{out} = -(2R + R_g)(V_1 - V_2)/R_g = (2R/R_1 + 1)(V_2 - V_1)$$
 
@@ -78,7 +78,7 @@ Another KVL also starting from $V_\mathrm{in}$ and passing through $R_1$, $C_1$,
 
 $$ V_\mathrm{in} - R_1 I - Z_{C_1} I - Z_2 I = V_\mathrm{out} \implies V_\mathrm{out} = - Z_2 I $$
 
-Replacing the current with the result found in \eqref{eq:band_filter_current}, we have:
+Replacing the current with the result found in equation \eqref{eq:band_filter_current}, we have:
 
 $$ V_\mathrm{out} = - V_\mathrm{in}\frac{Z_2}{R_1 + Z_{C_1}} = - V_\mathrm{in}\frac{1}{(1/R_2 + 1/Z_{C_2})(R_1 + Z_{C_1})} $$
 
@@ -128,43 +128,49 @@ For the case of the band pass filter, we will make use of the Op-Amp noise model
 {% endfigure %}
 {: refdef}
 
-The topology of the circuit is similar to the band pass filter presented in the {% figref fig_schematic %}, with the exception that some of the resistances will have to be replaced by impedances. A conversion table between the two is presented in {% figref tab_impedance %}.
+The topology of the circuit is similar to the band pass filter presented in the {% figref fig_schematic %}, with the exception that some of the resistances will have to be replaced by impedances. A conversion table between the two is presented in the table below ({% figref tab_impedance %}).
 
 {% figure caption: "Conversion table between the resistances of the band pass filter and their respective impedances." label:tab_impedance %}
+
 |*Noise Model*|*Band Pass Filter*|
 |---|---|
 |$R_1$|$Z_1 = R_1 + \frac{1}{s C_1}$|
 |$R_2$|$Z_2 = \left(\frac{1}{R_2} + s C_2\right)^{-1}$|
 |$R_3$|0|
+
 {% endfigure %}
 
 Making the appropriate variable changes in the total mean square noise voltage of the Op-Amp ($E_\mathrm{filter}^2$) as presented in {% cite texas_noise %}, we have:
 
-\begin{equation}\label{eq_noise_filter}
-\overline{E}^2_\mathrm{filter} = \frac{1}{2\pi} \mathrm{abs}\left(\int \mathrm{d} \omega \left( \overline{E}^2_\mathrm{no} \left( \frac{Z_2(i \omega)}{Z_1(i \omega)} \right)^2 + \overline{e_2}^2 + \overline{e_n}^2 \left(1 + \frac{Z_2(i \omega)}{Z_1(i \omega)} \right)^2 + \overline{I}_\mathrm{nn}(Z_2^2(i \omega)) \right)\right)
+\begin{equation}
+\label{eq_noise_filter}
 \end{equation}
 
+$$ \overline{E}^2_\mathrm{filter} = \frac{1}{2\pi} \mathrm{abs}\left(\int \mathrm{d} \omega \left( \overline{E}^2_\mathrm{no} \left( \frac{Z_2(i \omega)}{Z_1(i \omega)} \right)^2 + \overline{e_2}^2 + \overline{e_n}^2 \left(1 + \frac{Z_2(i \omega)}{Z_1(i \omega)} \right)^2 + \overline{I}_\mathrm{nn}(Z_2^2(i \omega)) \right)\right) $$
+
 .
 
-Where $Z_1$ and $Z_2$ are the respective impedances presented in {% figref tab_impedance %}. A little bit of algebraic manipulation reveals that the ratio $Z_2/Z_1$ is simply equation \eqref{eq:transfer_function_pretty} without the minus sign. So, we can further simplify \eqref{eq_noise_filter} as:
+Where $Z_1$ and $Z_2$ are the respective impedances presented in {% figref tab_impedance %}. A little bit of algebraic manipulation reveals that the ratio $Z_2/Z_1$ is simply equation \eqref{eq:transfer_function_pretty} without the minus sign. So, we can further simplify equation \eqref{eq_noise_filter} as:
 
-$${\overline{E}_\mathrm{filter}}^2 = 1/(2\pi)\left(\int \mathrm{d} \omega ({\overline{E}_\mathrm{no}}^2 H^2(i \omega) + \overline{e_2}^2 + \overline{e_n}^2 (1-H(i \omega))^2 + {\overline{I}_\mathrm{nn}} Z_2^2(i \omega))\right)$$
+$${\overline{E}_\mathrm{filter}}^2 = \frac{1}{2\pi}\left(\int \mathrm{d} \omega ({\overline{E}_\mathrm{no}}^2 H^2(i \omega) + \overline{e_2}^2 + \overline{e_n}^2 (1-H(i \omega))^2 + {\overline{I}_\mathrm{nn}} Z_2^2(i \omega))\right)$$
 
 Rearranging:
 
-$$ {\overline{E}_\mathrm{filter}}^2 = 1/(2\pi) \mathrm{abs}(\int \mathrm{d} \omega (({\overline{E}_\mathrm{no}}^2 +  \overline{e_n}^2) H^2(i \omega) + (\overline{e_2}^2 + \overline{e_n}^2) \-2\overline{e_n}^2H(i \omega) + {\overline{I}_\mathrm{nn}} Z_2^2(i \omega))) $$
+$$ {\overline{E}_\mathrm{filter}}^2 = \frac{1}{2\pi} \mathrm{abs}\left(\int \mathrm{d} \omega (({\overline{E}_\mathrm{no}}^2 +  \overline{e_n}^2) H^2(i \omega) + (\overline{e_2}^2 + \overline{e_n}^2) - 2\overline{e_n}^2H(i \omega) + {\overline{I}_\mathrm{nn}} Z_2^2(i \omega))\right) $$
 
 In the worst case scenario, this integral is bounded by the triangle inequality:
 
-$$ {\overline{E}_\mathrm{filter}}^2 \leq 1/(2\pi) \int \mathrm{d} \omega (({\overline{E}_\mathrm{no}}^2 +  \overline{e_n}^2) \mathrm{abs}(H(i \omega))^2 + (\overline{e_2}^2 + \overline{e_n}^2) + 2( \overline{e_n}^2) \mathrm{abs}(H(i \omega)) + {\overline{I}_\mathrm{nn}} \mathrm{abs}(Z_2(i \omega))^2) $$
+$$ {\overline{E}_\mathrm{filter}}^2 \leq \frac{1}{2\pi} \int \mathrm{d} \omega \left(({\overline{E}_\mathrm{no}}^2 +  \overline{e_n}^2) \mathrm{abs}(H(i \omega))^2 + (\overline{e_2}^2 + \overline{e_n}^2) + 2\overline{e_n}^2 \mathrm{abs}(H(i \omega)) + {\overline{I}_\mathrm{nn}} \mathrm{abs}(Z_2(i \omega))^2 \right) $$
 
 In which I have assumed that all the voltage and current noise sources aren't complex. Now, we will abuse the boundedness of the $\mathrm{abs}(H(i \omega))$ and $\mathrm{abs}(Z_2(i \omega))$ functions to use the ML inequality:
 
-$$ {\overline{E}_\mathrm{filter}}^2 \leq (\Delta \omega)/(2\pi) (({\overline{E}_\mathrm{no}}^2 +  \overline{e_n}^2)(2r \omega_2/(\omega_2 \- \omega_1))^2 \ + (\overline{e_2}^2 +  \overline{e_n}^2) \ + 2( \overline{e_n}^2) (2r \omega_2/(\omega_2 \-\omega_1)) + R_2 {\overline{I}_\mathrm{nn}} ) $$
+\begin{equation}\label{eq_noise_upper_bound}\end{equation}
+
+$$ {\overline{E}_\mathrm{filter}}^2 \leq \frac{\Delta \omega}{2\pi} \left(({\overline{E}_\mathrm{no}}^2 +  \overline{e_n}^2)\left(2r \frac{\omega_2}{\omega_2 - \omega_1}\right)^2 \ + (\overline{e_2}^2 +  \overline{e_n}^2) \ + 2( \overline{e_n}^2) \left(2r \frac{\omega_2}{\omega_2 -\omega_1}\right) + R_2 {\overline{I}_\mathrm{nn}} \right) $$
 
 .
 
-The term $\Delta \omega$, not to be confused with $\omega_2 \- \omega_1$, stands for the noise bandwidth of the Op-Amp in rad/s. Equation \eqref{eq:noise_upper_bound} with the definition of ${\overline{E}_\mathrm{no}}$ {% cite eq:ENO} provide a maximum upper bound to the output noise caused by the conjuction of the instrumentation amplifier and the band pass filter.
+The term $\Delta \omega$, not to be confused with $\omega_2 \- \omega_1$, stands for the noise bandwidth of the Op-Amp in rad/s. Equation \eqref{eq_noise_upper_bound} with the definition of ${\overline{E}_\mathrm{no}}$ equation \eqref{eq_ENO} provide a maximum upper bound to the output noise caused by the conjuction of the instrumentation amplifier and the band pass filter.
 
 Since all the $\overline{e_j}$ factors have a $k_B T$ term for noise due to thermal agitation, stabilizing the circuit's temperature with coolers will help with noise reduction.