Given I have coefficients a0, a1, a2, b1, and b2, defining the difference equation for a digital filter as:
y[n] = a0 * x[n] + a1 * x[n - 1] + a2 * x[n - 2] - b1 * y[n - 1] - b2 * y[n - 2]
Which defines a low-pass filter with particular cutoff frequency, how can I obtain the coefficients A0, A1, A2, B1, B2, which similarly define a high-pass filter with the same cutoff frequency? I'm aware there are so-called "bandform transformations" for converting a prototype low-pass into a high-pass, but to my knowledge, these are not directly applicable to discrete-time/digital filters, so I am unaware of any way to apply them to this problem.
If these coefficients are derived from a complex-conjugate pair of zeros and/or of poles given the discrete transfer function for the low-pass filter would be:
H(z) = (z - Zero) * (Z - Zero)/[(Z - Pole) * (Z - Pole)]
Is there then a way to transform this function to the corresponding high-pass filter I'm looking for to get the poles and zeros from the new transfer function?