A Comparative Study of Discrete Fourier Transform Based Detectors for LoRa SF=7 over AWGN and Multipath Channels with Cyclic Prefix Support

Long Range (LoRa) is a chirp spread spectrum (CSS) physical-layer technology that has become a leading candidate for low-power wide-area network (LPWAN) connectivity in the Internet of Things (IoT). At the receiver, the standard demodulator multiplies the incoming signal with a conjugate reference chirp and applies a one-dimensional discrete Fourier transform (DFT), reducing symbol detection to peak search in the frequency domain. While this non-coherent baseline is simple and robust under additive white Gaussian noise (AWGN), its symbol error rate (SER) degrades significantly in frequency-selective multipath channels, where parasitic spectral peaks distort the dominant tone. This paper presents a unified comparative study of seven LoRa detectors for spreading factor seven, six of which share a common one-dimensional DFT engine while a matched-filter bank operates directly in the time domain, with the six DFT detectors differing in their per-bin frequency-domain weighting and decision rule. The detector set spans the standard non-coherent DFT, a non-coherent matched filter bank, two coherent equalizers in the frequency domain (zero-forcing and minimum mean-square error), a phase-only equalizer, a maximal-ratio combiner with non-coherent decision, and an exhaustive maximum-likelihood detector that serves as a near‑optimal reference under the same preamble‑based CSI. To mitigate inter-symbol interference in the multipath case, every transmitted symbol is preceded by a cyclic prefix that converts the linear convolution with the channel into a circular convolution, enabling per-bin frequency-domain processing. Throughout the paper a deployment-realistic receiver model is adopted: the per-bin channel response is estimated by a frequency-domain least-squares estimator from a short preamble, and the noise variance is estimated blindly from the preamble residuals. The quality of the noise-variance estimator is reported separately as a diagnostic. Each detector is evaluated under both AWGN and a two-tap Rayleigh multipath channel through Monte Carlo simulation, and its execution time per call is recorded to provide a complementary view of computational cost. The framework introduced here clarifies how coherent processing, diversity combining, equalization, and exhaustive search trade detection performance against complexity within a single DFT-centric LoRa receiver architecture. The principal quantitative finding is that, under the two-tap Rayleigh multipath channel, the MMSE equalizer reaches SER ≈ 4.4×10⁻⁵ at SNR = −5 dB and tracks the exhaustive maximum-likelihood detector within 0.1 dB across the full SNR sweep, while costing only 1.26× the per-symbol time of the standard DFT receiver. Conversely, the standard non-coherent baseline hits an irreducible 16% error floor and the unregularized zero-forcing equalizer fails to reach the 10⁻² SER level at any SNR considered, isolating MMSE as the recommended choice in the multipath regime at every SNR for which a LoRa link is operationally viable.