The purpose of this problem is to illustrate the advantage of an nth-order low-pass Butter...

The purpose of this problem is to illustrate the advantage of an nth-order low-pass Butterwortk tiller over the cascade ot n identical low-pass sections by calculating the slope (in decibels per dccade) of each magnitude plot at (he corner frequency ωc. To facilitate the calculation, let y represent the magnitude of the plot (in decibels), and let x = log10ω. Then calculate dy/dx at ωc for each plot.

a) Show that at the corner frequency (ωc = 1 rad/s) of an nth-order low-pass prototype Butterworth filter,

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b) Sliuvv that for a cascade of n identical low-pass prototype sections, the slope at ωc is

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c) Compute dy/dx for each type of filter for n = 1, 2, 3, 4, and ∞.

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d) Discuss the significance of the results obtained in part (c).