Roll-off

Roll-off is the steepness of a transfer function with frequency, particularly in electrical network analysis, and most especially in connection with filter circuits in the transition between a passband and a stopband.

This can be shown to be so by considering the voltage transfer function, A, of the RC network:[1] Frequency scaling this to ωc = 1/RC = 1 and forming the power ratio gives, In decibels this becomes, or expressed as a loss, At frequencies well above ω=1, this simplifies to, Roll-off is given by, For a decade this is; and for an octave, A higher order network can be constructed by cascading first-order sections together.

In that circumstance, for n identical first-order sections in cascade, the voltage transfer function of the complete network is given by;[1] consequently, the total roll-off is given by, A similar effect can be achieved in the digital domain by repeatedly applying the same filtering algorithm to the signal.

[2] The calculation of transfer function becomes somewhat more complicated when the sections are not all identical, or when the popular ladder topology construction is used to realise the filter.

[3] Filters with a high roll-off were first developed to prevent crosstalk between adjacent channels on telephone FDM systems.