0
$\begingroup$

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

enter image description here

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

|improve this question
$\endgroup$
  • 1
    $\begingroup$ 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) $\endgroup$ – Dan Boschen Jan 12 at 16:48
2
$\begingroup$

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 $H(f)=Y(f)/X(f)$.

|improve this answer
$\endgroup$
  • $\begingroup$ 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!! $\endgroup$ – Gibbs Jan 12 at 17:05
  • $\begingroup$ @Gibbs: that's right! $\endgroup$ – Matt L. Jan 12 at 17:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.