LoRaWAN: A Mathematical Study of Simple-Signal Modulation Transforms and Realizing CSS (Chirp Spread Spectrum) by Means of FM

Abstract

LoRa, the physical layer beneath LoRaWAN, owes its long range and robustness to chirp spread spectrum (CSS) modulation, in which each symbol is a linear-frequency-modulated sweep. This article explores very basic modulation schemes mathematically to reveal the formal linkages among them and demonstrates that CSS is not an unusual method but a unique version of frequency modulation (FM) with a linear-frequency-ramp message. Beginning from the instantaneous-phase expression for a complex sinusoid, we develop the chirp signal, illustrate the up-chirp and down-chirp as a complex-conjugate pair, and demonstrate that using a linear modulating function into the FM formula produces the LoRa chirp phase. The cyclic frequency shift of a LoRa symbol's data bearing also equates to an offset or a phase break of the modulating ramp, which can be recovered as a single tone by a conjugate de-chirping receiver. Based on these foundations, we will describe how to create a CSS-like signal by driving an FM modulator with a chirp baseband, mapped to available hardware (the SX1276/SX1278/LR1276 transceivers and the STM32F103 host). Finally, we present an analogical hypothetical application that injects such a baseband into the FM channel of a standard UHF voice radio. We conclude with a symbolic, purely qualitative analysis of the distortion that occurs when a linear-frequency-modulated signal is created by means of an FM chain. The material presented here is entirely theoretical; it has no numerical measurement