Communication Basics #21 Worked Example: Nyquist Theorem & PCM

Instructor: Dr. Süleyman Burak ÇELİK

A worked example — the classic PCM calculation chain from analog signal to digital bit rate, solved step by step in light mode. Given a composite signal x(t) = 5[1 + 0.5 sin(2000 pi t) - cos(8000 pi t)], we expand and identify each frequency component, extract the maximum frequency (4 kHz from the cosine term), apply the Nyquist sampling theorem to get the minimum sampling rate (8000 samples per second), quantize with 256 levels (8 bits per sample), and compute the final bit rate: 8000 x 8 = 64 kbps — the standard voice channel. We then compare under-sampling and over-sampling scenarios to show why Nyquist is the critical threshold, and scale the same chain to a television signal (6 MHz bandwidth) to see why video needs 96 Mbps. Topics covered: - composite signal frequency decomposition - Nyquist sampling theorem: f_s ≥ 2 f_max - minimum sampling rate derivation - quantization levels and bits per sample (256 = 2^8) - PCM bit rate calculation: f_s x bits = 64 kbps - 64 kbps as the fundamental voice channel - under-sampling (aliasing) vs over-sampling trade-off - TV signal scaling: 6 MHz bandwidth, 96 Mbps bit rate Part of the AcEdumy Communication Basics series.