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I have seen frequency responses where amplitude response is constant( pure delay) and where phase response is constant(pure integrator).

In an integrator the constant phase is -90 deg. In a pure gain block the constant phase is 0 deg. Can we have a system which has a constant phase of say -45 deg?

I tried to find such a simple system analytically and ended up with an expression of system function as $$ H(j\omega)=e^{-j\pi/4} $$ Can we have a such a system in real life?

Background: I am trying to understand the importance of phase and amplitude plots individually. For eg: In the frequency response of pure delay systems, amplitude plot is constant and phase plot is a linear function of frequency. enter image description here

Similarly how will the system behave if

1) the phase is constant(other than -90 deg) and amplitude plot is a linearly decreasing function of frequency?

2) the phase plot is constant (say -90 deg, -45 deg)) and amplitude plot is also constant

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If you're talking about ideal conditions, then, sure, you can have exotic transfer functions with \$\tanh{s^\frac{2}{3}}\$, or what have you. I don't know where this one would be useful, but sure, you can.

If you're talking about real-life cases, even an integrator has a pole somewhere "palpable". With this in mind, filters with -3dB/oct exist (and others):

~3db/oct

These are mostly used in noise filtering, this one in particular serves for pink noise, typically filtering white noise (flat) to achieve -10dB/dec, the slope you see measured as ~-10.25dB/dec, because it's in the portion of the phase where it's not quite 45o. It's a lowpass due to the construction, but if you move further right on the slope, you have an integrator with -3dB/oct and 45o. Note that this is more of an idealistic approach, but the response is close to what you might get in real-life.

So, you can make filters that do not obey multiples of 20dB/dec, but the question is what for? That is up to the designer.

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