Every bit that moves between two points, whether a Wi-Fi packet, a 5G voice call, a GPS pseudorange, satellite TV, deep-space telemetry from Voyager, or the SWIFT wire that just paid your rent, has been sampled, quantized, error-coded, modulated onto a carrier, dragged through a noisy channel, equalized, demodulated, decoded, and reassembled. This chapter is the reasoning chain behind that pipeline. We will derive the formulas, draw the constellations, and meet the algorithms that the entire 21st-century network runs on.
In Chapter 7 we put a voice on a carrier with AM and FM. The signal degraded a little with each repeater, a little more with distance, a little more with weather, and at some point it became unintelligible. There was no way to fix it, only to amplify it and accept the noise. Chapter 12 is the story of how engineers escaped that fate. By turning a continuous waveform into a stream of bits and treating those bits as the unit of communication, we gain three superpowers: lossless regeneration across thousands of repeaters, mathematical error correction that lets us run channels close to the theoretical limits Shannon proved in 1948, and a clean substrate for cryptography. Once you understand this chapter, you understand the physical layer of every modern radio, every fibre, every cable modem, and every satellite link on Earth.
We lean heavily on Chapter 3 throughout. Sampling, Fourier, convolution, cross-correlation, and noise are all tools we built there and now apply. We also build directly on Chapter 7, because every digital modulation scheme is just analog modulation with a finite alphabet of message values. Later chapters return the favour. Chapter 17 (DSP) zooms into the receiver internals, Chapter 23 (advanced communications) extends to OFDM and MIMO, and Chapter 24 (hardware security) turns power, EM, and timing into the very channels we will spend this chapter trying to make reliable.