Module theory 

The theory of chirp spread spectrum: error rates, cross-correlation, spectrum and same-SF interference.

Error rates over AWGN

After dechirping and the DFT, the bin of the transmitted symbol has the magnitude $M$ and every bin the noise $\mathcal{CN}(0, M/\rho)$, $\rho$ the SNR per sample. Normalised, the $M - 1$ other bins are Rayleigh and the bin of the symbol Rice with the location $v = \sqrt{2M\rho}$, so the noncoherent decision errs with the probability (O. Afisiadis, M. Cotting, A. Burg and A. Balatsoukas-Stimming, IEEE Trans. Wireless Commun. 19(2), 2020, eq. (16))

$$P_{s} = \int_{0}^{\infty} f_{\mathrm{Ri}}(y; v, 1)\,\left[1 - \left(1 - e^{-y^{2}/2}\right)^{M-1}\right] dy$$

(ser), that of $M$ orthogonal signals detected noncoherently; every wrong symbol being equally likely, the BER is $P_{s} M/(2(M - 1))$ (ber). ser_rice and ser_gumbel are the approximations of their eqs. (17)–(18), after T. Elshabrawy and J. Robert, IEEE Commun. Lett. 22(9), 2018.

Waveforms and spectrum

cross_correlation is the correlation of two continuous-time LoRa waveforms, eq. (7) of M. Chiani and A. Elzanaty, IEEE Internet Things J. 6(5), 2019: they are orthogonal only at one sample per chip. continuous_spectrum and spectral_lines are the two parts of the power spectrum of LoRa modulated by random symbols, their eqs. (20)–(25) in Fresnel functions; the lines hold exactly $1/M$ of the power (their Property 4) and occupied_bandwidth is the 99 % band of their Table I.

Same-SF interference

An interferer of power $P_{I}$ offset by $\tau$ chips leaves in bin $k$ the pattern $R_{k}$ (interference_pattern, the DFT of eq. (25) of Afisiadis et al.), of the magnitude of their eqs. (38)–(43) (interference_magnitudes). ser_interference and fer_interference are their approximations (61)–(63) and (64)–(67).

Functions

ber

The BER of the noncoherent receiver over AWGN at the SNR per sample snr_db (dB): ser M/(2(M - 1)), every wrong symbol being equally likely (comnum_mapping::theory::ber_orthogonal_noncoherent at $E_{s}/N_{0} = M\rho$).

continuous_spectrum

The continuous part of the power spectrum of LoRa modulated by independent uniform symbols, normalised as $G^{c}(f) B$, at the frequencies f in units of $B$ (Chiani and Elzanaty 2019, eqs. (20) and (25)); its integral is $1 - 1/M$.

cross_correlation

The cross-correlation $C_{\ell, m} = \frac{1}{T_{s}}\int_{0}^{T_{s}} x(t; \ell)\, x^{*}(t; m)\, dt$ of two continuous-time LoRa waveforms (Chiani and Elzanaty 2019, eq. (7)): $1$ for $\ell = m$, else $M \frac{e^{j2\pi\ell(m - \ell)/M} - e^{j2\pi m(m - \ell)/M}}{j2\pi(M - |m - \ell|)\,|m - \ell|}$.

fer_interference

The approximate frame error rate of frame uncoded symbols under AWGN and a same-SF interferer of power pi_db (dB, relative to the signal) whose frame of the same length overlaps a uniform number of symbols (Afisiadis et al. 2020, eqs. (64)–(67)), at the SNR per sample snr_db, the offset sampled every eps chips (as ser_interference).

interference_magnitudes

The magnitudes $|R_{k}|$ of the interference pattern in closed form, eqs. (39)–(43) of Afisiadis et al. (2020): $A_{k,1}$, $A_{k,2}$ (with their limits $\lceil\tau\rceil$ and $M - \lceil\tau\rceil$ when the sine of the denominator vanishes) and the phases $\theta_{k,1}$, $\theta_{k,2}$ as printed.

interference_pattern

The pattern $R_{k} = \mathrm{DFT}(x_{I} \odot x_{\mathrm{ref}}^{*})[k]$, $k = 0 \dots M - 1$, that an interferer of symbols s1, s2 offset by tau chips leaves after the dechirping (Afisiadis et al. 2020, eqs. (21) and (25)).

max_cross_correlation

The largest real cross-correlation $\max_{\ell \ne m} \Re C_{\ell, m}$ of the continuous-time waveforms (Chiani and Elzanaty 2019, Table I), which costs the coherent receiver at most $-10 \log_{10}(1 - \max \Re C)$ dB against orthogonal signals.

occupied_bandwidth

The band $[-z, z]$ (in units of $B$, returned as $2z$) of the complex envelope that holds the fraction fraction of the power, lines included (Chiani and Elzanaty 2019, Table I, $B_{99}$ for 0.99), from the spectrum on a grid of $B/(8M)$ up to span $B$ on each side.

ser

The exact SER of the noncoherent receiver over AWGN at the SNR per sample snr_db (dB), spreading factor sf (see the module): the SER of $M$ orthogonal signals (comnum_mapping::theory::ser_orthogonal_noncoherent) at $E_{s}/N_{0} = M\rho$.

ser_gumbel

The concise approximation of eq. (18) of Afisiadis et al. (2020), $Q(\sqrt{2M\rho} - \sqrt{2(\ln 2 \cdot \mathrm{SF} + \gamma_{\mathrm{EM}})})$, at the SNR per sample $\rho$ (snr_db), $\gamma_{\mathrm{EM}}$ Euler’s constant.

ser_interference

The approximate SER under AWGN and a same-SF interferer of power pi_db (dB, relative to the signal), at the SNR per sample snr_db, the offset sampled every eps chips over $[0, (M - 1)/2)$ (Afisiadis et al. 2020, eqs. (62)–(63); the paper takes eps = 1/5): a step below 10⁻³ chip is raised to it, a non-finite one is 1/5.

ser_rice

The approximation of the SER of eq. (17) of Afisiadis et al. (2020), after Elshabrawy and Robert (2018), the magnitudes taken as Gaussian, at the SNR per sample snr_db. As printed, the formula reads with the SNR per symbol $M\rho$ in place of its “SNR”; with it, it follows the figure, whose abscissa is the SNR per sample.

spectral_lines

The lines of the power spectrum, (f, P) at $f = nB/M$ for $|n| \le$ n_max (in units of $B$), $P = |\sum_{\ell} X(nB/M; \ell)|^{2}/(T_{s}^{2} M^{2})$ (Chiani and Elzanaty 2019, eq. (21)); they hold $1/M$ of the power (Property 4).