On the pole locations of sharp cutoff low-pass filters

It is shown that the poles of sharp cutoff low-pass filters, Butterworth- or Chebyshev-derived, can be obtained analytically for orders 3, 4, 6, and 8, through the solution of quadratic, cubic, or quartic algebraic equations. For the Butterworth case, higher order filters having a value of 3, 4, 6, 8 for the ratio (order of filter)/(order of zero) are also shown to have analytical solutions for the poles. Explicit solutions for the Butterworth-derived filter are given for third and fourth orders.