# reconstructing a frequency response function from a factor

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

$$\frac{-0.191714-0.786113i}{1+0i}=-0.191714-0.786113i$$

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

$$\phi=\tan^{-1}(\frac{-0.786113}{-0.191714})=76,2°$$

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 and calculate the frequency response function via FFT 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 ...]