Why do we choose cutoff frequency to be taken for "1/sqrt(2) magnitude reduce"?

Question

Why don’t we instead choose (calculate) our bandwidth to be between two purple “dot’s”? Why don’t those purple dot’s represent points for “cutoff frequency”? Looking at this picture “frequency components” after the right purple dot (or before the left purple dot) would be attenuated (yellow lines + after cutoff frequency). I presume that red line I drew represents that components of a signal would neither increase or decrease in magnitude. Following that logic, if I want to send signal and (theoretically) receive same signal through this channel (having this response) than it’s better for signal to have only components between those two purple dots. Any components of signal after right purple dot or left purple dot would be attenuated and hence we won’t receive same signal (signal with same shape)?

Answer 1

I presume that red line I drew represents that components of a signal would neither increase or decrease in magnitude.

That would be a wrong presumption. The magnitude of the frequency response is (almost) never truly flat, and you would see this the more you zoom in. That's why you need to pick a convention of what exactly the "band edge" is.

As with most conventions, it's somewhat arbitrary. Half power has the advantage that if adjacent filters have the same cutoff frequency the total power will be flat. So if you add the output of an 1 kHz high pass and a 1 kHz low pass, the energy sum of the outputs will be flat with no dips or peaks in the cross over region.

-3dB is also not the only convention either: a Linkwitz Riley filter typically uses -6dB at the nominal cutoff. -3dB also doesn't make a lot of sense for parametric EQ filters and for high and low shelve filter.

So it all depends on context and one should never automatically assume -3dB.

Answer 2

Fundamentally it's an accepted convention. The most typical attribute could be stated as that the power at the output of the channel (filter) drops to half value at the cutoff frequencies compared to the center (or passband) frequency.

But then you would ask why not one forth? So somewhat half power is more meaningful and useful than quarter power. There could be some obscure (or forgotten) merits of selecting the half power. Probably and historically related to telephony industry as the definition most typically based on communications industry of a century ago.

Note that half power cutoff definition does not apply for ideally frequency selective filters. Furthermore, analog and digital filter design techniques rely on specifying passband edge and stopband corner frequencies as well instead of just a single cutoff frequency statement.