In this problem, we derive an empirical formula for the average signal-to-(quantization) noise ratio of a DM system with a sinusoidal signal of amplitude A and frequency fm as the test signal. Assume that the power spectral density of the granular noise generated by the system is governed by the formula S_N (ƒ) = ∆²/6ƒ_s where ƒ_s is the sampling rate and ∆ is the step size. The DM system is designed to handle analog message signals limited to bandwidth W.

(a) Show that the average quantization noise power produced by the system is N = 4π²A²ƒ²_mW/3ƒ³_s where it is assumed that the step size ∆ has been chosen in accordance with die formula used in Problem 3.27 so as to avoid slope overload.

(b) Hence determine the signal-to-(quantization) noise ratio of the DM system for a sinusoidal input.