  In vibration-based condition monitoring, optimal filter design improves fault
detection by enhancing weak fault signatures within vibration signals. This
process involves optimising a derived objective function from a defined
objective. The objectives are often based on proxy health indicators to
determine the filter's parameters. However, these indicators can be compromised
by irrelevant extraneous signal components and fluctuating operational
conditions, affecting the filter's efficacy. Fault detection primarily uses the
fault component's prominence in the squared envelope spectrum, quantified by a
squared envelope spectrum-based signal-to-noise ratio. New optimal filter
objective functions are derived from the proposed generalised envelope
spectrum-based signal-to-noise objective for machines operating under variable
speed conditions. Instead of optimising proxy health indicators, the optimal
filter coefficients of the formulation directly maximise the squared envelope
spectrum-based signal-to-noise ratio over targeted frequency bands using
standard gradient-based optimisers. Four derived objective functions from the
proposed objective effectively outperform five prominent methods in tests on
three experimental datasets.
