Frequency response of each component of a system given its global response Given the following block diagram, find the frequency responses $H1(f)$ and $H2(f)$. The frequency response of the whole system has to be $H(f)=(\alpha_0+\alpha_1e^{-j2\pi T_1f}+\alpha_2e^{-j2\pi T_2f})^{-1}$




The fact that there's a loop confuses me. I would express $Y(f)$ as $Y(f)=X(f)H1(f)+(X(f)-X(f)H1(f)H2(f))\ H1(f)+...$, but it doesn't seem to be correct. Could you give me some hints? Thanks in advance!


  Solution: $H1(f)=\alpha_0^{-1},\ H2(f)=\alpha_0^{-1}\alpha_1e^{-j2\pi T_1f}+\alpha_0^{-1}\alpha_2e^{-j2\pi T_2f}$.
With feedback systems such as the one given it's often easy to define an additional signal at the output of the adder. This gives the following equations:

$$U(f)= X(f)-H_2(f)Y(f)\tag{1}$$

and

$$Y(f)=U(f)H_1(f)\tag{2}$$

Now you can solve Eqs $(1)$ and $(2)$ to get the frequency response $H(f)=Y(f)/X(f)$.