BPSK Demodulation

I did the LPFs that attenuate the f1+f2 components, and it filters perfectly (Using 200 taps x 2).

Now, I am confused with the loop filter (IIR filter). Many texts suggest using the proportional-integral control technique that I attached before.

I believe you are referring to the same thing except that yours has no proportional part (and you suggest to add it later if needed).

It has this transfer function: H(z) = c1 + c2/(1 - z^-1)

where c1 = 4*(theta)^2/(1+sqrt(2)*theta+theta^2)

and c2 = 2*sqrt(2)*theta/(1+sqrt(2)*theta+theta^2)

where theta = 2*pi*BW, BW is bandwidth of the filter in Hz.

Let me ask you some questions:

1- How to determine the cutoff frequency of this filter, regardless of the structure.

2- Is leaky integrator the same as PI technique? How different they are?

I think I have more questions but they relate to your answers of these.