In this problem, we compare the performance of a DPCM system with that of an ordinary PCM system using companding. For a sufficiently large number of representation levels, the signal-to-(quantization) noise ratio of PCM systems, in general, is defined by 10 1og₁₀ (SNR) _O = α + 6n dB where 2ⁿ is the number of representation levels. For a companded PCM system using the μ-law, the constant α is itself defined by α = 4.77 — 20 1og₁₀ (log (1 + μ)) dB, for a DPCM system on the other hand, the constant a lies in the range —3 < α < 15 dBs. The formulas quoted herein apply to telephone-quality speech signals. Compare the performance of the DPCM system against that of the μ-companded PCM system with μ = 255 for each of the following scenarios:

(a) The improvement in (SNR) _O realized by DPCM over companded PCM for the same number of bits per sample.

(b) The reduction in the number of bits per sample required by DPCM, compared to the companded PCM for the same (SNR) _O.