# Frequency analysis

What's the difference between Constant Bandwidth Filter and Constant percentage bandwidth filter? I mean not just the basic difference but a little detailed one, couldn't find much on the internet!

Consider a band-pass filter with centre frequency $$f_0$$ and bandwidth $$\Delta f$$, determined by a specified attenuation with respect to the response at the centre frequency.

Consider the ratio

$$Q = \frac{f_0}{\Delta f}$$

defining a quality factor for the filter (the quality factor can be defined in different ways, more or less equivalent, but this is the most useful for your question).

Now imagine to have a filter with variable $$f_0$$ or several filters with different $$f_0$$'s. You can easily imagine two limiting cases: the variable filter has bandwidth independent of the centre frequency; or the variable filter has also variable $$\Delta f$$ in such a way that the ratio $$Q$$ remains constant, independent of the centre frequency. The former type of variable filter is a constant bandwidth filter; the latter a constant percentage bandwidth filter.

For instance, when you analyse a signal with the fast-Fourier transform (FFT), it's like having many constant bandwidth filters, one for each bin. The same type of filter is present in analogue RF spectrum analyser.

The constant percentage bandwidth filter is more typical of spectral analysis in acoustic applications, for instance when you consider octave or third-octave filters. What do they mean these names?

An octave corresponds to a frequency doubling: so in octave filters the higher endpoint of the bandwidth has frequency equal to twice that of the lower endpoint. Let's denote by $$f_\mathrm{L}$$ the lower endpoint and by $$f_\mathrm{H}=2f_\mathrm{L}$$ the higher one. The centre frequency is $$f_0 = (f_\mathrm{L}+f_\mathrm{H})/2 = 3f_\mathrm{L}/2$$ and the bandwidth is $$\Delta f = f_\mathrm{H}-f_\mathrm{L} = f_\mathrm{L}$$. The quality factor is then

$$Q = \frac{3f_\mathrm{L}/2}{f_\mathrm{L}} = \frac{3}{2}$$

The quality factor is independent of the centre frequency and the filter is indeed a constant percentage bandwidth filter. If you take the reciprocal of $$Q$$, you get a ratio, typically less than 1, which can be expressed in percentage of the bandwidth with respect to the center frequency: hence the name constant percentage bandwidth filter.