1
$\begingroup$

This question 10.21 is taken from GATE IN paper 2015.

10.21

I don't understand the answer given,

10.21 answer

I mean why is the frequency response at the pass band frequency considered to be $(1-\delta)$, isn't the ripple equally distributed on both sides, so shouldn't the average be One?

Hand drawn image, del1 passband and del2 stopband

Finally why is effective ripple calculated by subtracting from one?

$\endgroup$
2
$\begingroup$

The passband region of an equi-ripple filter is the region where its magnitude is in the interval $[1-\delta,1+\delta]$, where $\delta$ is the maximum approximation error. At the passband edge the value of the magnitude must equal $1-\delta$ (as shown in your drawing). The passband edge is the last frequency where the magnitude is still inside the interval $[1-\delta,1+\delta]$, and since the magnitude then falls off towards the stopband, the magnitude at the passband edge must be at the lower limit of the passband magnitude, which equals $1-\delta$.

The total magnitude response of a cascade of $M$ identical filters is

$$|H_{total}(\omega)|=|H(\omega)|^M$$

Consequently, the total magnitude at the passband edge is $(1-\delta)^M$. This value is then equal to $1-\delta_{total}$, which gives

$$\delta_{total}=1-(1-\delta)^M$$

$\endgroup$

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.