I am looking at a stationary monopole source with a frequency of 80 Hz with the driving data $1+0i$. It creates an output $-0.191714-0.786113i$ at a certain point $p$. I calculate the "transfer factor" from the origin of the monopole to point $p$ by


The factor tells me that the resulting amplitude at point p is

$$A=\sqrt{(-0.191714)^2+(-0.786113)^2}=0.809$$ and the phase is


Now I create a cosinus signal with an amplitude of 1, frequency of 80Hz that is 100 samples long. I try reconstruct the amplitude and phase change via a convolution of the signal with a impulse response (IFFT of the frequency response function). The result should be that my cosinussignal is shifted by $\phi$ with an amplitude $A$.

My question is: What is the correct frequency response function in this case (complex, also 100 samples long)?

Edit: a cumbersome way to solve the problem is to generate a impulse response
with the given amplitude and delay by phase

enter image description here

and calculate the frequency response function via FFT

enter image description here

the frequency response function generated by the FFT looks like

[0.8091 + 0i , -0.7948 + 0.1516i , 0.7523 - 0.2978i, -0.6831 + 0.4335i ...]

which looks like it can be reconstructed by something like $$F(i\omega)=A e^{i \phi n}$$ where I dont know what n is.


In the meantime I have solved the problem. If anybody is interested: The answer is actually easier than first thought. Because the monopole is stationary it sends out a simple sine/cos wave and the frequency response function from the origin of the monopole to point $p$ simply contains the complex factor in every single entry:

[−0.191714−0.786113i , −0.191714−0.786113i , −0.191714−0.786113i ...]
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