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For low-pass filters, we usually define the cutoff frequency as the frequency at which the gain drops by -3dB.

Another definition, however, comes up in filter design as one of the parameters that specify the filter (along with maximum passband attenuation, minimum stopband attenuation and the stopband frequency, see figure below.)

I fail to see how are these related as these two do not seem to be linked.

enter image description here

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  • \$\begingroup\$ You need to be clearer for instance what does one of the parameters that specify the filter particularly mean. Cut off is half power point which is 3.0103 dB or 3 dB approximately. \$\endgroup\$ – Andy aka Dec 20 '19 at 10:16
  • \$\begingroup\$ @Andyaka, Added illustrative figure for more clarity. In fact, my question is about how is the (cutoff) frequency defined at -3dB (the one you described) different from (or related to) the cutoff frequency we define in filter design. \$\endgroup\$ – Likely Dec 20 '19 at 10:30
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    \$\begingroup\$ OK, they are not linked and, in the picture the so-called cutoff frequency is incorrectly named (it should be called passband edge frequency for example. From where did you get the picture. Try this: zone.ni.com/images/reference/en-XX/help/371325F-01/lowpass.gif \$\endgroup\$ – Andy aka Dec 20 '19 at 10:57
  • \$\begingroup\$ @Andyaka, The picture comes from my understanding, which turned out to be wrong. \$\endgroup\$ – Likely Dec 20 '19 at 19:43
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I've taken your picture and corrected it showing the real 3 dB point at Fc: -

enter image description here

What you called the "cut-off frequency" is preferably called the lower pass band edge frequency or the lower transition band edge frequency. They are just "markers" that indicate where the real filter response cannot go beyond and are non mathematically related to the cut-off frequency - it is defined by the signal power falling to 50% commonly known as a half-power point.

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  • \$\begingroup\$ Your point is clear. Something bugs me though. Imagine we have ripples in the passband. How would one define cutoff frequency, i.e., what should we take as reference for the -3dB? is it the maximum/minimum ripple amplitude or mean amplitude perhaps? \$\endgroup\$ – Likely Dec 20 '19 at 19:49
  • \$\begingroup\$ It can be whatever you want it to be. I don’t believe there is any formulaic answer when it comes to complex filters. \$\endgroup\$ – Andy aka Dec 20 '19 at 22:46
  • \$\begingroup\$ Okay, thank you. \$\endgroup\$ – Likely Dec 21 '19 at 3:12
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Both definitions are not related to each other. Why should they? It really depends on the properties of the pass region and/or on special requirements.

  • If nothing else is mentioned, the passband edge (cut-off) for any first-order lowpass and for all BUTTERWORTH lowpass functions (second-order or higher) is specified at the -3dB point. However, this is NOT a fixed specification. Why shouldn`t we be open to require the passband edge for a specific application to be at the -1dB or -2dB point?

  • Example: Realize a second-order "maximally flat" (BUTTERWORTH) lowpass that deviates at f=fx from the value at f=0 by only 1.5 dB. In this case, the cut-off is specified at f=fx. The magnitude at the 3dB point is of less importance...

  • For all Chebyshev lowpass functions (with ripples in the passband) the passband edge - normally (!) - is specified according to the allowed ripple within the passband. That means: The cutoff is at a frequency where the magnitude leaves the ripple region (0.1 or 0.2 or 0.5 or 1.0 or....)dB.

  • For Bessel functions (where the frequency domain is less important because such a filter is used due to its time response properties) the passband - normally (!) - is specified at a frequency where the group delay is larger than a certain value - depending on a specific application.

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