# 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. It is most typically applied to the insertion loss of the network, but can, in principle, be applied to any relevant function of frequency, and any technology, not just electronics. It is usual to measure roll-off as a function of logarithmic frequency; consequently, the units of roll-off are either decibels per decade (dB/decade), where a decade is a tenfold increase in frequency, or decibels per octave (dB/8ve), where an octave is a twofold increase in frequency.

The concept of roll-off stems from the fact that in many networks roll-off tends towards a constant gradient at frequencies well away from the cut-off point of the frequency curve. Roll-off enables the cut-off performance of such a filter network to be reduced to a single number. Note that roll-off can occur with decreasing frequency as well as increasing frequency, depending on the bandform of the filter being considered: for instance a low-pass filter will roll-off with increasing frequency, but a high-pass filter or the lower stopband of a band-pass filter will roll-off with decreasing frequency. For brevity, this article describes only low-pass filters. This is to be taken in the spirit of prototype filters; the same principles may be applied to high-pass filters by interchanging phrases such as "above cut-off frequency" and "below cut-off frequency".

## First-order roll-off

A simple first-order network such as a RC circuit will have a roll-off of 20 dB/decade. This is approximately equal (to within normal engineering required accuracy) to 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:

$A={\frac {V_{o}}{V_{i}}}={\frac {1}{1+i\omega RC}}$ Frequency scaling this to ωc = 1/RC = 1 and forming the power ratio gives,

$|A|^{2}={\frac {1}{1+\left({\omega \over \omega _{c}}\right)^{2}}}={\frac {1}{1+\omega ^{2}}}$ In decibels this becomes,

$10\log \left({\frac {1}{1+\omega ^{2}}}\right)$ or expressed as a loss,

$L=10\log \left({1+\omega ^{2}}\right)\ \mathrm {dB}$ At frequencies well above ω=1, this simplifies to,

$L\approx 10\log \left(\omega ^{2}\right)=20\log \omega \ \mathrm {dB}$ Roll-off is given by,

$\Delta L=20\log \left({\omega _{2} \over \omega _{1}}\right)\ \mathrm {dB/interval_{2,1}}$ $\Delta L=20\log 10=20\ \mathrm {dB/decade}$ and for an octave,

$\Delta L=20\log 2\approx 20\times 0.3=6\ \mathrm {dB/8ve}$ ## Higher order networks

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;

$A_{\mathrm {T} }=A^{n}\$ consequently, the total roll-off is given by,

$\Delta L_{\text{T}}=n\,\Delta L=6n{\text{ dB/8ve}}$ A similar effect can be achieved in the digital domain by repeatedly applying the same filtering algorithm to the signal.