Roll-off

A simple first-order network such as a RC circuit will have a roll-off of 20 dB/decade. This is a little over 6 dB/octave and is the more usual description given for this roll-off. This can be shown to be so by considering the voltage transfer function, A, of the RC network:

Frequency scaling this to ω_c = 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. If a unity gain buffer amplifier is placed between each section (or some other active topology is used) there is no interaction between the stages. In that circumstance, for n identical first-order sections in cascade, the voltage transfer function of the complete network is given by;

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.

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. In a ladder filter each section of the filter has an effect on its immediate neighbours and a lesser effect on more remote sections so the response is not a simple Aⁿ even when all the sections are identical. For some filter classes, such as the Butterworth filter, the insertion loss is still monotonically increasing with frequency and quickly asymptotically converges to a roll-off of 20n dB/decade, but in others, such as the Chebyshev or elliptic filter the roll-off near the cut-off frequency is much faster and elsewhere the response is anything but monotonic. Nevertheless, all filter classes eventually converge to a roll-off of 20n dB/decade theoretically at some arbitrarily high frequency, but in many applications this will occur in a frequency band of no interest to the application and parasitic effects may well start to dominate long before this happens.

Filters with a high roll-off were first developed to prevent crosstalk between adjacent channels on telephone FDM systems. Roll-off is also significant on audio loudspeaker crossover filters: here the need is not so much for a high roll-off but that the roll-offs of the high frequency and low-frequency sections are symmetrical and complementary. An unusual need for high roll-off arises in EEG machines. Here the filters mostly make do with a basic 20 dB/decade roll-off, however, some instruments provide a switchable 35 Hz filter at the high frequency end with a faster roll-off to help filter out noise generated by muscle activity.