Frequency Domain Preamble-Based Channel Estimation and Equalization in LoRa

Abstract

This paper deals with multipath channel estimation and equalization in LoRa. It is suggested to take advantage of the cyclic property of the symbols in the LoRa frame preamble to obtain an interference-free version of the symbols in the frequency domain. Then, estimation methods used in multicarrier systems can be applied, such as the least square (LS), and the minimum mean square error (MMSE) estimators. It is shown that the cyclic property in LoRa is inherently independent of the length of the channel, making these estimation techniques robust to any frequency-selective channel. In addition the frequency domain zero-forcing (ZF) equalizer is used, and an original phase equalizer is introduced, taking advantage of the constant modulus property of LoRa symbols in the frequency domain. The performance of the investigated estimators and equalizers is shown through simulations, and applications to the presented results are further discussed.

An improved demodulation for LoRa through a threshold-based detection of error

“…In turn, we deduce from (13) that P E|h (β = 0) = 1 and P E|h (β = 1) = 0. This is consistent with the test in (12) Moreover, from ( 13), (15), and (17), we deduce that the overall detection rate in AWGN channel, denoted by P AW GN E , is given by…”

“…Finally, by substituting µ = √ N h, and since |µ| 2 = µ 2 R + µ 2 I , then P E1|h in (32) leads to (15), which concludes the proof. APPENDIX C PROOF OF (17) We keep the same definitions and assumptions as in Appendix B.…”

“…From ( 29) and (31), we find that APPENDIX B PROOF OF (15) Let X 0 , X 1 , .., X N −2 be N − 1 iid random variables obeying a χ 2 distribution with 2 degrees of freedom, and Z = |µ + X R + jX I | 2 a random variable independent of X k , k = 0,1,..,N −2, obeying a non-central χ 2 distribution with 2 degrees of freedom. Thus, µ ∈ C is defined as a (potentially non-zero) mean, and (X R +jX I ) ∼ N C(0,σ 2 ) (i.e.…”

“…The overall properties of CSS modulation have been analyzed in [6], [7]. Furthermore, signal processing algorithms have been suggested for time and frequency synchronization [8]- [13], multiple demodulations in presence of collision of LoRa signals [14], or multipath channel estimation [15]. Otherwise, numerous papers deal with performance analysis of the LoRa CSS waveform.…”

PAPR Reduction Techniques Based on Chirp Selection for Single and Multi-User Orthogonal Chirp Division Multiplexing System

“…We hereby mathematically formalize the principle of the OCDM modulation in [8]. First, it has been noticed in [19] that the DFT of a chirp g k,n is also a chirp in the frequency domain.…”