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On the LoRa Modulation for IoT: Waveform Properties and Spectral Analysis
Marco Chiani, Ahmed Elzanaty
TL;DR
This paper addresses incomplete and potentially inaccurate characterizations of LoRa modulation, including analyses that assume orthogonality. It develops time- and frequency-domain models, derives waveform cross-correlation and Fresnel-function spectral expressions, and finds non-orthogonal waveforms, continuous and discrete spectra, and a discrete-spectrum power fraction of exactly 1/M.
Problem
Prior LoRa analyses used limited time-domain descriptions or treated the modulation as orthogonal, motivating a complete signal characterization.
Method
The paper mathematically characterizes M-ary LoRa in the time and frequency domains, deriving waveform cross-correlation and continuous and discrete power spectra in closed form.
Results
LoRa waveforms are non-orthogonal and become asymptotically orthogonal only as M increases; its discrete spectrum contains exactly 1/M of the overall signal power.
Takeaways & Limitations
LoRa spectrum analysis must account for discrete spectral lines and an occupied bandwidth generally larger than the deviation B, while orthogonality is an approximation limited to large M.
Abstract
from arXiv · showhide
An important modulation technique for Internet of Things (IoT) is the one proposed by the LoRa allianceTM. In this paper we analyze the M-ary LoRa modulation in the time and frequency domains. First, we provide the signal description in the time domain, and show that LoRa is a memoryless continuous phase modulation. The cross-correlation between the transmitted waveforms is determined, proving that LoRa can be considered approximately an orthogonal modulation only for large M. Then, we investigate the spectral characteristics of the signal modulated by random data, obtaining a closed-form expression of the spectrum in terms of Fresnel functions. Quite surprisingly, we found that LoRa has both continuous and discrete spectra, with the discrete spectrum containing exactly a fraction 1/M of the total signal power.
I. INTRODUCTION
The paper develops a complete time- and frequency-domain characterization of M-ary LoRa modulation, addressing gaps in waveform modeling and spectral analysis. It shows that LoRa waveforms are non-orthogonal except asymptotically, and that the spectrum includes a discrete component with power fraction 1/M.
- Motivation: The paper addresses limited LoRa waveform descriptions and the absence of spectral characterization in prior literature.Earlier analyses considered performance by simulation or by treating LoRa as orthogonal, while spectral characteristics had not been addressed.
- Contributions: The authors provide analytical expressions for the M-ary LoRa signal in continuous-time and discrete-time domains.The time-domain model characterizes the modulated waveform before the frequency-domain analysis.
- Contributions: LoRa waveforms are non-orthogonal, but become asymptotically orthogonal as M increases.The paper derives their cross-correlation and quantifies performance loss relative to orthogonal modulation.
- Contributions: Closed-form continuous and discrete spectra are derived in terms of Fresnel functions.The analytical spectrum is also compared with experimental data from commercial LoRa devices.
- Contributions: The discrete spectrum contains exactly 1/M of the overall signal power.The analysis also supports investigation of compliance with spectral masks governing out-of-band emissions and power spectral density.
- Implications: The characterization helps select spreading factor, maximum frequency deviation, and transmitted power to meet system requirements.These parameters determine bandwidth, spectrum shape, spectral efficiency, discrete-spectrum power, cross-correlation, and SNR penalty relative to orthogonal modulation.
II. LORA SIGNAL MODEL
The LoRa signal model describes M-ary chirp waveforms whose instantaneous frequency sweeps linearly across a frequency interval and wraps modulo B. LoRa uses M = 2^SF and BT_s = M, linking spreading factor, bandwidth, and symbol duration.
- Signal model: LoRa is an M-ary chirp spread-spectrum modulation with M waveforms having different initial frequencies.Each waveform is swept over the frequency interval (f0 − B/2, f0 + B/2).
- Signal model: The instantaneous frequency increases linearly and wraps to the lower frequency boundary when reaching the maximum.Mathematically, the wrap operation is a reduction modulo B.
- Signal model: A frequency sweep over B does not imply that the signal bandwidth equals B.The paper defers the bandwidth characterization to its spectral analysis.
- Signal parameters: M = 2^SF and BT_s = M, where T_s is the symbol interval.These parameter relations define the LoRa configuration used in the signal model.
- Signal parameters: The reciprocal of the modulation bit-rate ratio is interpreted as spectral efficiency in bit/s/Hz.The paper reports spectral-efficiency values for M ranging from 2^3 to 2^12.
A. Continuous-time description
The continuous-time LoRa signal is an M-ary chirp modulation whose waveforms are memoryless continuous-phase signals, but are not generally orthogonal. Their cross-correlation decreases asymptotically with increasing M, while correlation also produces an SNR penalty relative to orthogonal modulation.
- Signal model: LoRa uses M chirp waveforms with different initial frequencies, linearly sweeping across a frequency interval and wrapping modulo B at the upper limit.The instantaneous frequency continues increasing after reaching the maximum, reduced modulo B.
- Model consistency: A missing 1/2 factor in prior quadratic phase definitions makes their instantaneous frequency inconsistent with the original LoRa signal model.The discrepancy also affects the corresponding discrete-time analyses.
- Phase properties: The LoRa modulation is memoryless and continuous-phase, with each symbol waveform depending only on its current symbol and having equal initial and final phase.The phase diagram over consecutive symbols visualizes this continuity.
- Signal model: The continuous-time signal is represented by a constant-amplitude complex envelope x(t; a) = γ exp{j φ(t; a)} over each symbol interval.The amplitude parameter γ accounts for passband signal power, and the normalized analysis assumes γ = 1 unless otherwise stated.
- Cross-correlation: Continuous-time LoRa waveforms are orthogonal only for specific symbol separations, so the modulation is generally non-orthogonal.The zero-correlation conditions depend on the symbol difference and the spreading factor.
- Cross-correlation: As M increases, the LoRa waveforms become asymptotically orthogonal.The cross-correlation analysis also determines the maximum correlation and its associated SNR penalty relative to orthogonal modulation.
B. Discrete-time description
The discrete-time description samples LoRa at chip rate, dechirps the samples, and uses a DFT to identify the transmitted symbol. Although the discrete-time waveforms are orthogonal algebraically, bandwidth filtering can undermine that property in practice.
- Sampling: Sampling at chip rate uses Tc = Ts/M = 1/B seconds over each LoRa symbol interval.This produces M samples per symbol.
- Orthogonality: The discrete-time LoRa waveforms are orthogonal under the stated discrete-time inner-product condition.This property follows directly from the sampled signal representation.
- Demodulation: Dechirping converts the sampled waveform into a complex sinusoid whose frequency identifies the modulating symbol.The DFT then concentrates the signal at the corresponding symbol index.
- Demodulation: The DFT of the dechirped signal has one non-zero element at the position of symbol a, with value M.A receiver can therefore decide from the maximum DFT component.
- Practical limitation: Filtering to bandwidth B distorts the signal because its actual bandwidth is larger than B, so the filtered samples need not remain orthogonal.For large M, the signal bandwidth is approximately B, supporting rate-B sampling with dechirping and DFT detection.
III. SPECTRAL ANALYSIS OF THE LORA MODULATION
The paper derives LoRa’s power spectrum analytically and identifies both continuous and discrete spectral components. The discrete component contains exactly 1/M of the total signal power.
- Spectral analysis: The LoRa spectrum is derived in closed form using Fresnel functions and can also be evaluated through a discrete Fourier transform.The analysis addresses the spectrum of LoRa modulation in the frequency domain.
- Spectral components: Exactly 1/M of the overall signal power lies in the discrete spectrum.The remaining signal power belongs to the continuous spectral component.
A. Power Spectrum of LoRa Modulated Signals
For random i.i.d. symbol sequences, the paper models LoRa as a randomly modulated process and decomposes its power spectral density into continuous and discrete parts. The waveform transforms are obtained analytically with Fresnel functions or numerically through the DFT.
- Random signal model: The source is modeled as a sequence of independent, identically distributed discrete random variables.The resulting random signal takes values from the finite set of LoRa symbol waveforms.
- Power spectral density: The power spectral density of the randomly modulated LoRa process is the sum of continuous and discrete components.The component expressions are obtained using frequency-domain analysis of randomly modulated signals.
- Analytical spectrum: The Fourier transforms of the LoRa waveforms can be expressed analytically in terms of Fresnel functions.The paper introduces an auxiliary function represented through Fresnel functions to obtain the spectrum.
- DFT spectrum: A DFT of N uniformly spaced samples provides spectrum samples with frequency spacing Δf = 1/Ts = B/M.Zero-padding reduces the frequency step to Δf = 1/(kTs) = B/(kM) when finer resolution is needed.
B. Total Power of the Discrete spectrum
The LoRa modulation has spectral lines because its mean value is nonzero, and those discrete components carry exactly 1/M of the overall signal power.
- 1/M of the overall signal power is contained in LoRa’s discrete spectrum.This result is stated as an exact fraction of total power.
- A nonzero mean value of the signal produces lines in the LoRa spectrum.The discrete spectrum is attributed to the signal’s mean component.
- The discrete-spectrum power is obtained analytically from the mean-value derivation for the memoryless modulation.The derivation uses the chip-rate relation Tc = 1/B and an integral for integer m.
IV. NUMERICAL RESULTS
The numerical results characterize LoRa’s continuous and discrete spectra across spreading factors, compare analytical spectra with commercial-device measurements, and assess regulatory compliance.
- Spectrum versus spreading factor: For SF ∈ {3, 7, 10, 12}, Fig. 3 displays LoRa’s continuous and discrete complex-envelope spectrum versus normalized frequency f/B.The discrete spectrum is represented by spectral lines alongside the continuous component.
- Spectrum versus spreading factor: 1/M of the total signal power lies in the discrete spectrum, corresponding to 12.5% for SF = 3 and M = 8.The total power of all discrete spectral lines equals 1/M.
- Bandwidth: Increasing M makes the spectrum more compact, with most complex-envelope power contained between −B/2 and B/2 for large M.For smaller spreading factors, restricting the bandwidth to B can omit signal power and distort the signal.
- Bandwidth: For M ≥ 27, almost all signal power is contained within bandwidth B, according to the 99%-power bandwidth B99 analysis.The bandwidth B99 is centered on f0 and contains 99% of the power.
- Experimental comparison: Analytical power spectra are compared with Welch estimates from 16-byte random-payload IQ waveforms recorded from a commercial LoRa transceiver.The comparison uses B = 125 kHz, sample rate fs = 4B, and frequency bins of width Δf = B/256.
- Experimental comparison: The numerical spectrum analysis is limited by frequency aliasing because the experimental samples use fs = 4B.This sampling rate is not large enough to completely eliminate frequency aliasing.
- Regulatory compliance: For the G1 sub-band, the analytically calculated passband spectrum meets adjacent-band maximum-power limits under the considered one- and three-channel configurations.The cases use Ps = 14 dBm and Δf = 1 kHz, with B = 250 kHz for one channel and B = 125 kHz for three channels.
V. CONCLUSIONS
The paper derives LoRa’s analytical spectrum and compares it with experimental data and ISM-band masks, finding discrete spectral lines and generally wider occupied bandwidth than B.
- V. CONCLUSIONS: The paper derives analytical expressions for the continuous and discrete spectra of M-ary LoRa modulation and compares them with measurements and ISM-band limits.The conclusions summarize both spectral derivation and validation against experiments and regulatory masks.
- V. CONCLUSIONS: The discrete spectral lines contain exactly 1/M of the overall signal power.This is a paper-level conclusion about LoRa’s spectrum.
- V. CONCLUSIONS: The occupied bandwidth is generally larger than the frequency deviation B.The conclusion distinguishes occupied bandwidth from the modulation’s deviation parameter.