# 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: $$H1(f)=\alpha_0^{-1},\ H2(f)=\alpha_0^{-1}\alpha_1e^{-j2\pi T_1f}+\alpha_0^{-1}\alpha_2e^{-j2\pi T_2f}$$.

• Hint: the response of feedback system is the forward gain divided by (1 + the loop gain) where the negative feedback is already implied. H1(f) is the forward rain and H1(f)H2(f) is the loop gain. You can derive that equation easily knowing Y(f)= H1(f)(X(f)-Y(f)H2(f))—- Solve that for Y(f)/X(f) – Dan Boschen Jan 12 at 16:48

$$U(f)= X(f)-H_2(f)Y(f)\tag{1}$$
$$Y(f)=U(f)H_1(f)\tag{2}$$
Now you can solve Eqs $$(1)$$ and $$(2)$$ to get the frequency response $$H(f)=Y(f)/X(f)$$.
• So I have that $Y=H1(X-YH2)=XH1-YH1H2$, then $Y(1+H1H2)=XH1$ so $\frac{Y}{X}=\frac{H1}{1+H1H2}=H$, and a possible solution could be $H1=1$ and $H2=H^{-1}-1$. The proposed solution in the exercise also works. Thanks for helping!! – Gibbs Jan 12 at 17:05