Performance Improvement of LoRa Modulation with Signal Combining and Semi-Coherent Detection
aa r X i v : . [ ee ss . SP ] F e b Performance Improvement of LoRa Modulation with Signal Combining
The Khai Nguyen, Ha H. Nguyen, and Ebrahim Bedeer
Abstract —Low-power long-range (LoRa) modulation has beenused to satisfy the low power and large coverage requirements ofInternet of Things (IoT) networks. In this paper, we investigateperformance improvements of LoRa modulation when a gatewayis equipped with multiple antennas. We derive the optimaldecision rules for both coherent and non-coherent detectionswhen combining signals received from multiple antennas. Wepresent expressions of the symbol/bit error probabilities ofboth the coherent and non-coherent detections in AWGN andRayleigh fading channels, respectively. Moreover, we also proposean iterative semi-coherent detection that does not require anyoverhead to estimate the channel-state-information (CSI) whileits performance can approach that of the coherent detection.Simulation and analytical results show very large power gainsprovided by the use of multiple antennas for all the detectionschemes considered.
Index Terms —Chirp-spread spectrum modulation, LoRa, Lo-RaWAN, non-coherent detection, signal combining.
I. I
NTRODUCTION
Internet of Things (IoT) networks aim to connect a massivenumber of end devices (EDs) that are typically battery poweredand expected to last for several years. Moreover, EDs canbe deployed in geographical areas extending to several tensof kilometers and served by a few gateways [1]. Existingcellular networks can provide long range coverage but usecomplex modulation, coding, and multiple access techniquesto deliver high data rates. As such, they are not suitable tosupport low power requirements of EDs that send informationat much lower data rates in many IoT applications. On theother hand, wireless local-area networks (WLANs), e.g., WiFi,cannot support very large coverage.Low-power wide-area networks (LPWANs) are emergingsolutions to balance the trade-off among coverage, powerrequirements and data rates of IoT networks [2]. Amongvarious solutions, low-power long-range (LoRa) technologyis currently one of the most widely deployed LPWAN tech-nologies around the world [1]. This technology is based ona proprietary chirp spread spectrum (CSS) modulation, alsoknown as LoRa modulation, in the PHY layer and LoRaWANprotocol in the MAC layer [2], [3].In the PHY layer, CSS operates in the ISM band (433 MHz,868 MHz, or 915 MHz, depending on which region in theworld) and encodes data symbols into chirp signals whosefrequency sweeps (either increasing or decreasing over time)the entire bandwidth once. There are three main parameters inCSS, namely, coding rate, bandwidth (125, 250 or 500 kHz),and spreading factor (from 7 to 12), that can be selected tobalance transmission rate, reception sensitivity, and coveragerange. In the MAC layer, the LoRaWAN protocol operates in a
The authors are with the Department of Electrical and Computer Engi-neering, University of Saskatchewan, Saskatoon, Canada S7N5A9. Emails: { khai.nguyen, ha.nguyen, e.bedeer } @usask.ca. star topology using ALOHA multiple access. In a typical LoRanetwork, each ED communicates with several single-antenna gateways in the uplink transmission. Then, the gatewaysforward the received signals for each ED, the received signalstrength indicator (RSSI) levels, and optional time stamps tothe LoRa network server (LNS) through IP backbone. TheLNS keeps the received message with the highest RSSI anddrops the rest. For downlink transmission to a specific ED, theLNS picks a gateway having the highest RSSI from that ED.It is pointed out that all research works on LoRa modulationconsider that gateways are equipped with a single antenna. Forexample, the authors in [4] analyze the bit error rate (BER)performance of LoRa modulation and provided tight closed-form approximations in both additive white Gaussian noise(AWGN) and Rayleigh fading channels. In [5], the authorsintroduce a slope-shift-keying LoRa scheme that increases theachievable bit rate of the conventional LoRa system by addinga down chirp and its cyclic shifts. The authors also developlow-complexity optimum coherent and non-coherent detectors,as well as, tight approximations for BER and symbol errorrate (SER) when the non-coherent detector is employed inthe presence of Rayleigh fading channels. In [6], the authorsinvestigate performance of LoRa in the presence of interfer-ence from another LoRa user that is neither chip nor phasealigned with the LoRa signal of the desired user. They deriveexpressions for SER and frame error rates. In [7], the authorsexploit index modulation to further improve the data ratesof conventional CSS systems and derive optimal detectionrules for both coherent and non-coherent detections. Further,they propose a low-complexity non-coherent detection schemewhose performance approaches the optimal performance.Given the widespread use of multiple antennas in wirelesscommunication systems, it is well expected that gateways inLoRa systems are fitted with multiple antennas. As such, it isnatural and useful to evaluate the BER performance of a LoRasystem equipped with multiple-antenna gateways and quantifythe performance gain over the system using single-antenna gateways. This is precisely the objective of this paper. Specif-ically, we first explain how multiple received signals at thegateway are combined under both cases of coherent and non-coherent detection. We then present the BER expressions forcoherent detection in AWGN channels and for non-coherentdetection in Rayleigh fading channels. Moreover, we proposea novel iterative semi-coherent detection that can blindlyestimate the CSI, i.e., without any training overhead, andsignificantly outperform the non-coherent detection, especiallywhen the number of antennas at the gateway increases. Theobtained results show that proper combining multiple receivedsignals at the gateway can significantly improve the LoRamodulation performance, which results in either power savingof EDs, or extended coverage. II. S
YSTEM M ODEL
We consider the uplink transmission of a LoRa networkwhere each ED communicates with a gateway equipped with L antennas . One LoRa symbol has a bandwidth B and duration T sym . Let T s = 1 /B be the sampling period. Then each LoRasymbol can be represented by M = T sym /T s = 2 SF samples,where SF ∈ { , , . . . , } is the spreading factor, which isalso the number of information bits encoded into one LoRasymbol. Let f c be the carrier frequency. Then the operatingfrequency range [ f c − B/ , f c + B/ is divided into M equally-spaced frequency bins, and each LoRa symbol (i.e., achirp signal) starts at an initial frequency belonging to one ofthe M frequency bins, and linearly increases/decreases until itreaches the maximum frequency f c + B/ . Then, it is folded to f c − B/ and continues to increase/decrease until the end of thesymbol duration T sym . It can be shown that LoRa modulationis M -ary orthogonal modulation [8].The baseband discrete-time basic up chirp (of length M samples) is given as [5]: x [ n ] = A exp (cid:18) j π (cid:18) n M − n (cid:19)(cid:19) , n = 0 , . . . , M − . (1)Then, the set of M orthogonal chirps can be simply con-structed from x [ n ] as x m [ n ] = x [ n + m ] , m = 0 , . . . , M .With the sampling period of T s = T sym /M , the equivalentanalog (continuous-time) chirps have an equal energy of E s = R T sym | x ( t ) | d t = (cid:16)P M − | x [ n ] | (cid:17) T s = A T sym .It also follows that the signal power is P signal = E s T sym = A .We consider a frequency-flat and slow Rayleigh fadingchannel between each ED and each receive antenna. With sucha channel model, the received signal at the ℓ th antenna of thegateway is given as y ℓ [ n ] = h ℓ x m [ n ] + w ℓ [ n ] = α ℓ exp ( jθ ℓ ) x m [ n ] + w ℓ [ n ] , (2)where w ℓ [ n ] ∼ CN (0 , σ ) is a zero-mean AWGN sample atthe ℓ th antenna, h ℓ = α ℓ exp ( jθ ℓ ) represents the complexchannel coefficient (in which α ℓ denotes the channel’s attenua-tion and θ ℓ represents the phase shift). For the Rayleigh fadingchannel, α ℓ follows a Rayleigh distribution with average powergain of 1, i.e., E (cid:8) | α ℓ | (cid:9) = 1 , whereas θ ℓ is uniformlydistributed over [0 , π ] . Note also that the case of an AWGNchannel corresponds to setting α ℓ = 1 and θ ℓ = 0 , ∀ ℓ .Before closing this section, it is important to commenton the signal-to-noise ratio (SNR) relevant to the consideredsystem model. Using the standard notation of N for theone-sided power spectral density of additive white Gaussiannoise, the noise power inside the signal bandwidth B is simply P noise = N B , which is also precisely the variance σ of theAWGN sample w ℓ [ n ] in (2). It then follows that the averageSNR at each receive antenna, denoted as SNR is given as
SNR = P signal P noise = E s /T sym N B = E s /MN . This SNR is exactly thesame as A /σ calculated based on the discrete-time basebandmodel in (2), which should be. In Section VI, all the analyticaland simulated BER curves are plotted versus this SNR . The techniques and results in this paper are also applicable when combin-ing signals received from different single-antenna gateways.
III. C
OHERENT D ETECTION OVER
AWGN C
HANNELS
In coherent detection, perfect knowledge of the CSI isassumed and used to perform the maximal ratio combining(MRC) of L received signal. Specifically, the received signalat the ℓ th antenna is first correlated with the complex conjugateof the respective channel coefficient as follows: v (coh) ℓ [ n ] = h ∗ ℓ y ℓ [ n ] = α ℓ x m [ n ] + ˆ w ℓ [ n ] , l = 1 , . . . , L, (3)where ˆ w ℓ [ n ] ∼ CN (cid:0) , α ℓ σ (cid:1) . The received signal after theMRC is given as r [ n ] = L X l =1 v (coh) ℓ [ n ] = L X l =1 α ℓ ! x m [ n ] + ˆˆ w [ n ] , (4)where ˆˆ w [ n ] ∼ CN (cid:16) , σ P Ll =1 α ℓ (cid:17) .To demodulate the combined signal r [ n ] , we perform nor-malizing and dechirping, i.e., multiplying r [ n ] with the scaledconjugate of the basic chirp x [ n ] as follows z [ n ] = r [ n ] qP Ll =1 α ℓ x ∗ [ n ] A = βA exp (cid:18) j π (cid:18) m M − m (cid:19)(cid:19)| {z } constant phase Ψ m exp (cid:18) j πmnM (cid:19)| {z } linear phase + ¯ w [ n ] , (5)where ¯ w [ n ] ∼ CN (0 , σ ) and β = qP Ll =1 α ℓ is the effectivegain due to MRC. Then, we perform the M -point FFT on thedechirped signal z [ n ] as Z (coh) [ k ] = 1 √ M M − X n =0 z [ n ] exp (cid:18) − j πnkM (cid:19) = 1 √ M M − X n =0 (cid:18) βA exp ( jπ Ψ m )exp (cid:18) j πmnM (cid:19) + ¯ w [ n ] (cid:19) = ( βA √ M exp ( jπ Ψ m ) + W [ m ] , k = mW [ k ] , k = 1 , . . . , M, k = m, (6)where W [ · ] ∼ CN (0 , σ ) is the noise sample after the FFT.One can see from (5) that the phase Ψ m is deterministic,hence, the equivalent decision variable can be re-written as Z (coh)R [ k ] = R n Z (coh) [ k ] exp ( − j Ψ m ) o = ( βA √ M + W R [ m ] , k = mW R [ k ] , k = 1 , . . . , M, k = m, (7)where R {·} returns the real value and W R [ · ] ∼ CN (0 , σ / .For ease in performance analysis, we scale the decision vari-able in (7) by √ B to obtain ˆ Z (coh)R [ k ] = Z (coh)R [ k ] / √ B . Usingthe relationships E s = A T sym = A M/B and σ = N B ,the scaled decision variable is expressed as ˆ Z (coh)R [ k ] = ( β √ E s + ˆ W R [ m ] , k = m ˆ W R [ k ] , k = 1 , . . . , M, k = m, (8)where ˆ W R [ · ] ∼ CN (0 , N / . Finally, the decision rule isgiven as ˆ m (coh) = arg max k =0 , ,...,M − ˆ Z (coh)R [ k ] . (9) Coherent detection is not very practical for fading channelssince it needs precise channel estimation (which typicallycomes with extra training overhead and complexity). As such,we complete this section by presenting the error probability ofLoRa modulation with MRC in AWGN channels, i.e., when α ℓ = 1 and θ ℓ = 0 , and β = √ L . One can observe that thedecision variable in (8) is the same as that for orthogonal M -ary FSK. Therefore, the error probability of LoRa modulationwith the maximum-ratio combining of L signals is the sameas in the single antenna case for orthogonal modulations,except with a power gain of L due to the MRC. Accordingly,the symbol error probability for the LoRa modulation canbe obtained by replacing E s with LE s in the symbol errorprobability of M -ary FSK using a single antenna [9, equation8.67]. This yields P (coh) s = 1 − √ π Z ∞−∞ (cid:18) √ π Z y −∞ exp (cid:18) − x (cid:19) dx (cid:19) M − × exp − y − r LE s N ! dy. (10)Finally, the bit error probability of LoRa modulation in AWGNchannels with MRC can be found as P (coh) b = M M − P (coh) s .IV. N ON -C OHERENT D ETECTION OVER F ADING C HANNELS
As mentioned before, precise knowledge of the CSI requirescomplex signal processing that may not be suitable for IoT net-works. As such, non-coherent detection of LoRa modulationis much more relevant and that’s why it is implemented inpractical systems. The first step in the non-coherent detectionof LoRa modulation is to perform normalizing and dechirpingof the received signal at the ℓ th antenna in (2) as follows v (non − coh) ℓ [ n ] = y ℓ [ n ] x ∗ [ n ] A , = α ℓ exp ( jθ ℓ ) A exp ( j Ψ m )exp (cid:18) j πmnM (cid:19) + ¯ w ℓ [ n ] , (11)where ¯ w ℓ [ n ] ∼ CN (0 , σ ) . Then by performing the M -pointFFT on the dechirped signal v (non − coh) ℓ [ n ] and dividing theresult by √ B , we obtain V ℓ [ k ] = ( √ E s α ℓ exp ( jθ ℓ ) exp ( jπ Ψ m ) + W ℓ [ m ] , k = mW ℓ [ k ] , k = 1 , . . . , M, k = m = ( √ E s a ℓ [ m ] + W ℓ [ m ] , k = 1 , . . . , M, k = mW ℓ [ k ] , otherwise , (12)where a ℓ [ k ] = α ℓ exp { jθ ℓ } exp { jπ Ψ k } = h ℓ exp { jπ Ψ k } ∼CN (0 , and W ℓ [ k ] ∼ CN (0 , N ) .Since CSI is not available in non-coherent detection, weapply the square-law combining to all the received signalsfrom the L antennas at bin k as Z (non − coh) [ k ] = L X l =1 | V ℓ [ k ] | = (P Ll =1 (cid:12)(cid:12) √ E s a ℓ [ m ] + W ℓ [ m ] (cid:12)(cid:12) , k = m P Ll =1 | W ℓ [ k ] | , otherwise . (13) Then, the final decision rule can be expressed as ˆ m (non − coh) = arg max k =0 , ,...,M − Z (non − coh) [ k ] . (14)To find the error probability for the non-coherent detectionof LoRa modulation, we follow a similar analysis as in [10].Without loss of generality, if we assume x [ n ] was transmit-ted, then the square-law combining output at the 0th bin is Λ = Z (non − coh) [0] = P Ll =1 (cid:12)(cid:12) √ E s a ℓ [0] + W ℓ [0] (cid:12)(cid:12) , whereasthe output at bin k , ∀ k , k = 0 is Λ k = Z (non − coh) [ k ] = P Ll =1 | W ℓ [ k ] | . To calculate the probability of error, we needfirst to find the probability that Λ > Λ k , ∀ k , k = 0 . Onecan show that Λ k , ∀ k , are mutually statistically independentrandom variables. Hence, P (Λ < Λ , Λ < Λ , . . . , Λ M − < Λ )= ( P (Λ < Λ )) M − = Z λ f ( λ ) dλ ! M − = − exp (cid:18) − λ σ (cid:19) L − X q =0 q ! (cid:18) λ σ (cid:19) q ! M − , (15)where f ( λ ) is the probability density function of Λ , givenas f ( λ ) = 1( σ ) L ( L − λ L − exp (cid:18) − λ σ (cid:19) , (16)and σ = E n | W ℓ [ k ] | o = N . The probability of a correctsymbol decision can be calculated by averaging (15) over theprobability density function of Λ , which is f ( λ ) = 1( σ ) L ( L − λ L − exp (cid:18) − λ σ (cid:19) , (17)where σ = E n(cid:12)(cid:12) √ E s a ℓ [ m ] + W ℓ [0] (cid:12)(cid:12) o = E s + N . Then,the probability of symbol error equals 1 minus the probabilityof a correct symbol decision and is given as [10, equation14.4–47] P (non − coh) s = 1 − Z ∞ σ ) L ( L − λ L − exp (cid:18) − λ σ (cid:19) × − exp (cid:18) − λ σ (cid:19) L − X q =0 q ! (cid:18) λ σ (cid:19) q ! M − dλ = 1 − Z ∞ γ c ) L ( L − λ L − exp (cid:18) − λ γ c (cid:19) × − exp ( − λ ) L − X q =0 λ q q ! ! M − dλ , (18)where ¯ γ c = E s /N = M · SNR . It is pointed out that (18)can be written in a closed-form finite sum as in [10, equation14.4–49]. However, such an expression is hard to compute forhigh values of M and L as in the case of LoRa modulationconsidered in this paper. In Section VI, we evaluate (18)directly using numerical integration.Finally, the bit error probability of non-coherent detec-tion of LoRa modulation with signal combining is given as P (non − coh) b = M M − P (non − coh) s V. P
ROPOSED S EMI -C OHERENT D ETECTION
In this section, we propose an iterative semi-coherent detec-tor for LoRa modulation when combining L received signalswithout needing extra overhead for CSI estimation. Section VIshows that the error performance of the proposed detector canclosely approach that of the coherent detector.Observe from (12) that if the detection of the firstLoRa symbol is correct, a least-square (LS) estimate ofthe channel coefficient h ℓ can be obtained as ˆ h ℓ = V ℓ [ ˆ m ] exp {− jπ Ψ ˆ m } / √ E s , where ˆ m is the detected valueof the first LoRa symbol. Then, for the detection of eachsubsequent LoRa symbol, instead of the square-law combiningin (13), we use ˆ a ℓ [ k ] = ˆ h ℓ exp { jπ Ψ k } to co-phase V ℓ [ k ] before coherently combining (adding) the signals receivedfrom all L antennas. Mathematically, the decision variable forthe proposed semi-coherent detection is Z (semi − coh) [ k ] = L X l =1 (ˆ a ℓ [ k ]) ∗ V ℓ [ k ] , (19)and the final decision rule can be expressed as ˆ m (semi − coh) = arg max k =0 , ,...,M − R { Z (semi − coh) [ k ] } . (20)There are two important observations regarding the abovesemi-coherent detection. First, since the decision rule in (20)is obtained by finding the maximum of R { Z (semi − coh) [ k ] } ,it is invariant to the scaling of the channel estimate ˆ h ℓ . Assuch, one can simply use ˆ h ℓ = V ℓ [ ˆ m ] exp {− jπ Ψ ˆ m } for thecombining operation in (19). Second, the performance of theabove semi-coherent is strongly influenced by the detectionof the first LoRa symbol, ˆ m . If the first LoRa symbol isnot detected correctly, the channel is estimated using a wrongbin, and hence contains only noise. To address this issue,we propose to perform average channel estimate and iterativeestimation and detection as follows.Assume that the fading channels stay constant for a co-herence time of τ c LoRa symbols. In the first stage, initialdetection of τ c LoRa symbols is carried out using the non-coherent detector in (14). Let ˆ m = [ ˆ m , . . . , ˆ m τ c ] be avector containing these initial detected LoRa symbols. For the i th symbol, let V ℓ,i [ ˆ m i ] denote the FFT value at the ˆ m i thbin. Then, the ℓ th channel coefficient is estimated as ˆ h ℓ,i = V ℓ [ ˆ m i ] exp {− jπ Ψ ˆ m i } and the average channel estimate isobtained as ˆ h (ave) ℓ = 1 τ c τ c X i =1 ˆ h ℓ,i = 1 τ c τ c X i =1 V ℓ [ ˆ m i ] exp {− jπ Ψ ˆ m i } . (21)Note that, because the noise W ℓ,i [ m ] are independent fromsymbol to symbol, when the coherent time goes to infin-ity, we can achieve the exact (scaled) channel value, i.e., ˆ h (ave) ℓ a . s −−−−→ τ c →∞ √ E s h ℓ .In the second stage, we use ˆ a (ave) ℓ [ k ] = ˆ h (ave) ℓ exp { jπ Ψ k } to combine L signals from the antenna array in a similarmanner as in (19) and perform new detection of all τ c LoRasymbols as in (20). After this new detection operation, a newaverage channel estimate is obtained, while the newly detected LoRa symbols are saved and compared to the previously-detected results. The iteration process between detection andestimation continues until convergence, i.e., when the detectedsymbols in two consecutive iterations are the same. Whenoperating at low SNRs, the iterative semi-coherent detectordoes not likely to converge because the initial non-coherentdetection is not reliable and the channel estimation is largelybased on the wrong frequency bins. In such a case, we use apredetermined number of iterations as a stopping criterion.The proposed iterative semi-coherent detection algorithm issummarized in Algorithm 1.
Algorithm 1
Iterative semi-coherent detection
Require:
Spreading factor (SF), τ c , N max . Initially detect τ c symbols ˆ m = [ ˆ m , . . . , ˆ m τ c ] with non-coherent detection. Estimate ˆ h ℓ,i corresponding to τ c detected LoRa symbols ( i =1 , , . . . , τ c ). Obtain the average channel estimate ˆ h (ave) ℓ as in (21). Save current decision as ˆ m prev = ˆ m . flag = 1 ; count = 0; while flag = 1 & count ≤ N max do count = count + 1; For the i th symbol, combine V ℓ,i [ k ] with ˆ a (ave) ℓ [ k ] =ˆ h (ave) ℓ exp { jπ Ψ k } as in (19) and perform a new detectionof the i th symbol ˆ m i as in (20), ∀ i = 1 , , . . . , τ c . Update ˆ h (ave) ℓ using newly detected symbols ˆ m . if ˆ m prev = ˆ m then flag = 1 else flag = 0 end if ˆ m prev = ˆ m end while return ˆ m . VI. R
ESULTS AND D ISCUSSION
In this section, we provide numerical results for the BERof both coherent detection in an AWGN channel and non-coherent detection in a Rayleigh fading channel to validateour theoretical analysis. We also show the BER performance ofthe proposed iterative semi-coherent detection. A minimum of Monte-Carlo trails were used to generate the BER curvesfor the detectors under study and the maximum number ofiterations of the semi-coherent detection is set to 50. For theRayleigh channel, it is assumed to be frequency flat and staysconstant over the duration of τ c = 10 LoRa symbols.Fig. 1 plots the simulated and theoretical BER curves ofboth coherent detection in AWGN channels and non-coherentdetection in Rayleigh fading channels versus the averagereceived signal-to-noise ratio per antenna, i.e.,
SNR (definedat the end of Section II) and for
SF = 10 . As one can observe,the simulated BER results of both detection techniques matchperfectly with the theoretical results. As expected, increasingthe number of antennas at the gateway helps to improvethe BER performance. In particular, for a BER of − ,increasing the number of antennas from L = 1 to 4 resultsin approximately 8 and 29 dB savings in the SNR (i.e., thetransmit power) for the coherent and non-coherent detectioncases, respectively. These numbers represent a tremendousperformance improvement of both detectors, especially thenon-coherent detector that does not require CSI knowledge. -30 -25 -20 -15 -10 -5 0 5 1010 -5 -4 -3 -2 -1 B E R Fig. 1. BERs of coherent detection in an AWGN channel and non-coherentdetection in a Rayleigh fading channel:
SF = 10 and L = 1 , , , . -15 -10 -5 0 5 10 15 2010 -4 -3 -2 -1 B E R Fig. 2. BER performance comparison among the coherent detection, non-coherent detection, and semi-coherent detection in Rayleigh fading channelsfor
SF = 7 , τ c = 10 , and L = 1 , , , . Fig. 2 compares the BER performance of the non-coherent(by theory), semi-coherent (by simulation), and coherent de-tection (by simulation) versus
SNR over Rayleigh fadingchannels for
SF = 7 and τ c = 10 . It is emphasized that theperformance of the coherent detection is obtained with perfectCSI knowledge, whereas CSI is not needed in the non-coherentdetection and it is blindly estimated (i.e., without any traininginformation) in the proposed iterative semi-coherent detection.One can see that the iterative semi-coherent detection performswithin a fraction of dB from the coherent detection. Also, onecan observe that properly combining the received signals frommultiple antennas provides similar SNR gains as in the casesof non-coherent and coherent detections. Such results clearlyshow the promise of using the proposed iterative semi-coherentdetection in LoRa modulation with signal combining.Finally, Fig. 3 depicts the influence of the coherence time onthe BER performance of the proposed semi-coherent detectionover a Rayleigh fading channel for SF = 7 , L = 4 and threedifferent values of coherence time, namely, τ c = 5 , , LoRa symbols. As expected, as the channels change slower,i.e., when the coherence time is larger, the blind channelestimate becomes more accurate, and hence the performance ofthe proposed semi-coherent detection approaches closer to thatof the coherent detection. For the worst situation considered,i.e., when τ c = 5 , the semi-coherent detection can still providean impressive SNR gain of more than 1 dB over the non-coherent detection. -16 -14 -12 -10 -8 -6 -410 -4 -3 -2 -1 B E R Fig. 3. Effect of coherence time on the performance of the proposed semi-coherent detection.
VII. C
ONCLUSION
We have investigated performance improvements of LoRamodulation when multiple antennas are employed at the gate-way and by combining signals received over these antennas.The optimal decision rules were established and the corre-sponding BER expressions were obtained for both coherentand non-coherent detections in AWGN and Rayleigh fadingchannels, respectively. More importantly, we also proposedan iterative semi-coherent detection whose performance ap-proaches that of the coherent detection without the need tospend extra resources for CSI training or estimation. Simu-lation results showed that increasing the number of antennasat the gateway from 1 to 4 results in approximately 8 and29 dB savings in the transmit power of the end deviceswhen the system operates over AWGN and Rayleigh fadingchannels, respectively. The results revealed that the semi-coherent detection performs within a fraction of dB fromthe coherent detection. As a future work, it would be usefulto perform theoretical performance analysis of the proposedsemi-coherent detection of LoRa signals.R
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