# How does cascading low pass modules affect pass band ripples?

This question 10.21 is taken from GATE IN paper 2015.

I don't understand the answer given,

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?

Finally why is effective ripple calculated by subtracting from one?

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