Problem $5$

In Lab $7$ you created a circuit to detect a heartbeat. The second stage of the circuit was, in fact, a filter designed to attenuate unwanted low$-$frequency and high$-$frequency interference. That filter stage is shown below.

a$)$ Using the parameter values you used in Lab $7,$ find an expression for the transfer function $\backslash (\backslash$mathbf$\{$H$\}(\backslash$omega$)=\backslash )\backslash (\backslash$mathbf$\{$V$\}\_\{\backslash$mathbf$\{0\}\}/\backslash$mathbf$\{$V$\}\_\{\backslash$mathbf$\{$i$\}\}\backslash ).$ You should be able to express $\backslash (\backslash$mathbf$\{$H$\}(\backslash$omega$)\backslash )$ as a product of three terms: a gain term $\backslash (-\backslash$mathrm$\{$R$\}\_\{\backslash$mathrm$\{$f$\}\}/\backslash$mathrm$\{$R$\}\_\{\backslash$mathrm$\{$i$\}\}\backslash ),$ a first$-$order low$-$pass filter, and a first$-$order high$-$pass filter.

b$)$ For this architecture, the low and high cut$-$off $($corner$)$ frequencies can be approximated as the corner frequencies of the high$-$pass and low$-$pass terms of the transfer function. Compute the low and high cut$-$off $($corner$)$ frequencies in Hertz $($ Hz $).$

$($Recall that cycle frequency $\backslash ($ f$=\backslash$omega $/2\backslash$pi $\backslash )).$