Chapter 15, Problem 38P

(a)

To determine

Prove that an nth-order low-pass Butterworth filter is $\frac{dy}{dx}=-10n\frac{\text{dB}}{\text{dec}}$ at the corner frequency ${\omega }_c=1\frac{\text{rad}}{\text{s}}$.

(b)

To determine

Prove that a cascade of n identical low-pass prototype sections is $\frac{dy}{dx}=\frac{-20n\left(2^{\frac{1}{n}}-1\right)}{2^{\frac{1}{n}}}\frac{\text{dB}}{\text{dec}}$ at the corner frequency ${\omega }_c$.

(c)

To determine

Find $\frac{dy}{dx}$ for each type of filter for $n=1,2,3,4,\text{and}\infty .$

(d)

To determine

Explain the significance of the results obtained in Part (c).