I have two cascaded one-pole sections with the same coefficient b for both sections. $$ y_A[n]=x_A[n]+b\cdot(y_A[n−1]−x_A[n])\\ y_B[n]=y_A[n]+b\cdot(y_B[n−1]−y_A[n]) $$
The final output of the system is $y_B[n]$.
I would like to re-implement the system as one biquad section of the form:
$$ y[n]=a_0x[n]+a_1x[n-1]+a_2x[n-2]-b_1y[n-1]-b_2y[n-2] $$
The final output of the system is $y[n]$.
I would like to know what calculations I have to perform in order to make the output of the biquad exactly match the output of the one-pole cascaded system. $$ y_B[n] = y[n] $$ Obviously the 5 biquad coefficients $a_0\; a_1\; a_2\; b_1\; b_2$ will have to be determined as a function of the $b$ coefficient as found in the cascaded system above.
Can someone lead me through the math?