Direct form expansion of the transfer function for a digital Butterworth low-pass filter

For a digital Butterworth low-pass general-order filter, generalized expressions for the filter coefficients of the transfer function are derived, with representation in direct form as a single high-order filter, rather than as a combination of second-order and first-order subfilters. By using double precision to overcome the usual problems of numerical accuracy with the direct form, the advantages of simplicity and economy of programming effort are realized.