Transient analysis of filters inside allowed band


I understand that a zone pass filter will cut off portions of the signal in the cutoff zone, and allow or amplify the frequency parts of the signal in the allowed zone.

Now lets assume a signal that is in the allowed zone. If we exclude minor amplifications, will the signal change in time due to all the components of the filter? I mean resistors, capacitors or inductors.

So, if we have for example a sine signal in the allowed zone, will it change (excluding the amplitude due to minor amplifications)?
Or will there be some modifications? For example getting triangular a bit?

Does it make sense to try to analyze filters in the time domain? Or we just accept them as they are (black box approach) and just work with them only in the frequency domain?

Best Answer

Or will there be some modifications? For example getting triangular a bit?

No, not if the filter is linear. By more or less definition, the output signal from a linear filter does not contain any frequency that isn't present in the input signal.

If the input is a sinusoidal signal and the output is not a sinusoidal signal, even if by just a bit, the filter is not linear since, as Fourier analysis shows, a non-sinusoidal signal necessarily has multiple sinusoidal components of different, related frequencies.

Thus, to make the sinusoid triangular a bit requires adding frequency components that are not present in the input signal, i.e., adding harmonic distortion.

In summary, if the filter is linear, a sinusoidal input of (angular) frequency \$\omega\$ will result in a sinusoidal output of frequency \$\omega \$ with, at most, a modified amplitude and phase.

$$v_I(t) = \cos\omega t $$

$$v_O(t) = |H(\omega)|\cos[\omega t + \phi(\omega)] $$

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