# Why should the cut off frequency and the transition bandwidth (BW) be expressed as a fraction of sampling rate?

I came across this confusion in Chapter 16,"Book - The Scientist and Engineers guide to DSP" and Topic is "Windowed Sinc-Filters". I am doing some calculations for designing the sinc filter and came across these doubts:

                               M = 4/BW


where "M" is the filter kernel length and "BW" is the transition bandwidth. The author says that "The BW must be expressed as a fraction of sampling rate with a value between 0 to 0.5 and same goes with the cut off frequency". For the values of M = 20, 40, 200, the BW is 0.2, 0.1, 0.02 and like we can see, it satisfies the condition of 0 to 0.5.

After I searched online, I understood that the value of cut off frequency and BW to exist between 0 to 0.5 has to do something with the power (dB) and presumed that 0 is 0db and 0.5 is -3 dB, since 10log(0.5) = -3dB.

My question is, what is the relationship between the cutoff of frequency and BW being a fraction of the sampling rate and between the values 0 to 0.5? What does it mean?. Is it something to do with the frequency components present beyond the cut off freq or BW being attenuated?

For example: If my sampling rate is 100sps, so the cut off frequency and BW must be a fraction of 100 sps with a value between 0 to 0.5?

• Check that book for 'Nyquist frequency' and 'sampling theorem', this should make it clear. Note that the cut-off frequency (and transition bandwidth) of a fixed discrete-time filter change if you change the sampling rate, so all frequencies are relative to the sampling rate. – Matt L. Feb 20 '15 at 14:02
• Oh I see, it has something to do with Sampling theorem. Ok I will check it out. Thanks! – PsychedGuy Feb 20 '15 at 14:15
• Or, from a different (plain DSP) perspective, the cut-off frequency (and transition bandwidth) is not related to the sampling rate at all. The filter operates on a sequence of samples, nothing in the filter description relates to the rate of these samples. It is only during the sampling and reconstruction steps that the sampling rate is a factor (and, of, course figuring out which fraction of the sample rate is required for the particular application). Hence, it makes sense to specify filters in relation to the sampling rate, 1, whatever the actual number may be. – Oscar Feb 20 '15 at 15:09
• Guys,the cutoff frequency and BW changes relative to the sampling rate, So, when I am designing the filter, should I keep the sampling rate as constant throughout? – PsychedGuy Feb 21 '15 at 11:59