Efficient Design Of Chirp Spread Spectrum Modulation For Low-Power Wide .

NGUYEN et al.: EFFICIENT DESIGN OF CSS MODULATION FOR LPWANsFig. 1.Block diagram of an LUT-based CSS transmitter.samples of xm (t) taken at rate Fsamp BW over the symbol duration. In particular, the baseband discrete-time (digital)samples of the basic chirp are given as x0 [n] x0 (t) t nT samp μnTsampBW exp j2πnTsamp22 2nn ; n 0, 1, . . . , M 1. (6) exp j2π2M2On the other hand, it is simple to show that digital samplesof the mth chirp can be obtained by cyclically shifting thedigital basic chirp by m samples. Furthermore, the digital basicchirp repeats itself after every M samples, i.e., x0 [n M] x0 [n], n. Therefore the mth digital chirp at baseband can alsobe mathematically written as xm [n] x0 [n m].The above properties of the digital chirps are useful forimplementing modulation and demodulation of CSS signals. In particular, a conventional LUT-based CSS transmitterdesign [16], [17] is shown in Fig. 1. The most important component of the transmitter is a ROM that stores M samples ofthe basic chirp as described in (6). CSS modulator starts with achannel encoding block that converts binary data into a streamof SF-bit symbols. Symbol mapping is then performed bycyclically shifting the basic chirp by an amount equivalent tothe symbol value, producing a corresponding digital basebandCSS symbol. Each digital baseband symbol is then up-sampledbefore being converted to an analog baseband signal. There isa pair of low-pass filters (LPFs) to remove spectral aliasescaused by up-sampling. Conversion of the signal from analogbaseband to a desired carrier frequency, e.g., 902–928 MHzfor LoRa in North America, is done by mixing the signal witha digitally controlled oscillator (DCO) that provides in-phaseand quadrature components of the carrier frequency. Note thatit is possible to store the preupconverted basic chirp to eliminate the need for the up-conversion filters, but it comes at thecost of increasing the ROM size.are not phase-locked, and w[n] is AWGN noise sample withzero mean and variance N0 . For such an input/output model,it is simple to see that the signal-to-noise ratio (SNR) isSNR (1/N0 ).De-chirping is performed on every received symbol by multiplying with the complex conjugate of the basic chirp. If thebasic chirp is an up-chirp, its conjugation is a down-chirp withthe same chirp rate, and vice versa. In essence, de-chirpingunwinds the second-order phase function applied on the transmitted signal, leaving only the constant and linear phase terms.This is shown as follows:vm [n] ym [n]x0 [n] Demodulation of CSS signals requires a ROM that storesthe complex conjugate of the digital basic chirp. Since LoRasignals have a narrow BW (500 kHz or less), the channelcan be considered as having a constant power gain across theBW (i.e., a flat channel) and the received signal after beingdown-converted to digital baseband can be expressed asn 0, 1, . . . , M 1. (7)In the above expression, ψ is a random phase rotation causedby the fact that the oscillators at the transmitter and receiver n m(n m)2 exp(jψ) exp j2π w[n]2M2 2nn exp j2π2M2 2nm m2m ŵ[n] exp(jψ) exp j2π 2M2 2 mmj2π nm exp jψ j2πexp ŵ[n]2M2M constant phaselinear phase(8)where ŵ[n] w[n]x0 [n] is also an AWGN noise samplewith zero mean and variance N0 . Thus, in the absence ofnoise, the CSS signal after de-chirping is a pure sinusoidwith a frequency of m/M (cycles/sample). Denote the constantphase term as ψm ψ 2π([m2 /2M] [m/2]), then symboldemodulation can be done by performing M-point DFT on thede-chirped signal to obtain M 11 j2π nkvm [n] expVm [k] MM n 0 M 1j2π nm j2π nk1 exp{jψm } expexpMMM n 0 M 11 j2π nk ŵ[n] expMM n 0 W[k] exp{jψm }j2π n(m k) W[k]exp MM n 0 M exp{jψm } W[m], if k m W[k], otherwise. B. CSS Demodulationym [n] exp(jψ)xm [n] w[n],9505M 1 (9)C. BER Performance AnalysisThe above analysis shows that all the power of a CSS symbol is concentrated at a single frequency bin, namely the mthbin if the transmitted CSS symbol is m, whereas all the otherM 1 bins contain only noise. At this point, it is useful to

9506IEEE INTERNET OF THINGS JOURNAL, VOL. 6, NO. 6, DECEMBER 2019define a peak SNR (PSNR) as 2 E M exp{jψm } PSNR W[k] 2 MN0(10)which indicates the difference in terms of power between the“power” bin versus the rest (“noise” bins). It is obvious thata higher PSNR would lead to a better chance of detecting thetransmitted symbol correctly. The ratio between PSNR andSNR, which is M in linear scale, is called the processing gain.Using a higher SF would lead to a higher processing gain andthus improve the performance of the transmission, but at thecost of a lower transmission rate.The transmitted symbol can be detected coherently or noncoherently. Coherent detection requires the receiver to bephase-locked to the transmitter in order to demodulate thesignal. The coherent receiver has a phase detector that workstogether with a phase-locked loop (PLL) to synchronize thephases between the receiver and the transmitter. As such,coherent detection is often referred to as synchronous detection. Noncoherent detection, on the other hand, does notexploit the phase information and the decision is made solelybased on the signal envelope. While it is less complex thancoherent detection, the performance is not as good.In particular, if the transmitter and the receiver are fullysynchronized, i.e., the phase term ψm is known at the receiver,coherent detection can be carried out asdcoh argmax Vm [k] exp{ jψm }.(11)k 0,.,M 1Otherwise, a simpler noncoherent detection is dnon-coh argmax Vm [k] .(12)k 0,.,M 1The above description and analysis clearly shows thatCSS is an M-ary orthogonal modulation. As such, its errorperformance in an AWGN channel is exactly the same as thatof M-ary FSK (M-FSK). In particular, BER of the coherentdetection can be expressed as [18]Pb,coh M2(M 1) 2 M 1 y 11 xdx1 exp 22π 2π 2 12 log2 MEb exp y dy(13) 2 N0where the SNR per bit, (Eb /N0 ), is related to the SNR as (notethat each orthogonal symbol has M samples while carrying SFdata bits)EbM SNR .N0SF(14)Fig. 2. BER performance of 64-ary CSS (i.e., SF 6) with coherent andnoncoherent detection over an AWGN channel.On the other hand, for noncoherent detection, the BER is [19]M 1 M 1 ( 1)m 1MPb,non-coh m2(M 1)m 1m 1 m log2 MEb.(15)exp m 1N0Fig. 2 plots the BER performance of a CSS system withSF 6 (i.e., M 64) obtained with both coherent and noncoherent detection. As can be seen, simulation results matchexactly with the theoretical results in (13) and (15). Observethat, for this particular case of M 64, coherent detectionyields a performance gain of approximately 0.6 dB over thesimpler noncoherent detection.Before closing this section, we point out that a similardescription of LoRa modulation and noncoherent detectionis presented in [20]. Reference [20], however, does not discuss coherent detection nor does it provide theoretical BERexpressions of the two detection methods.III. P HASE C ONTINUITY OF CSS S IGNALSAs explained in the previous section, data is encoded intothe phase function of a CSS symbol, which is a quadraticfunction of time. Since the phase determines the instantaneoussignal amplitude, phase discontinuity would lead to amplitudediscontinuity. Since the phase of a particular CSS symbol isalso a cyclically shifted version of the basic chirp’s phase,there are inherent phase jumps occurring at symbols boundaries. The amount of phase jump depends on the modulateddata, i.e., randomly. The discontinuities at symbol boundariesspread the signal power outside its BW. As an example, Fig. 3plots (in dashed lines) both phase function (top panel) and realpart of the amplitude (bottom panel) of three CSS symbolscreated by the cyclically shifting method with SF 4. Thefirst symbol is the basic chirp, i.e., m 0, followed by twosymbols corresponding to m 2 and 4, respectively. Notethat each CSS symbol starts and ends at the same phase. Ascan be seen, there are phase jumps at symbol boundaries anddiscontinuities occur in the signal amplitude.

NGUYEN et al.: EFFICIENT DESIGN OF CSS MODULATION FOR LPWANsFig. 3.9507Illustration of the phase property of CSS signals.Fig. 4.We point out that the CSS symbols shown in Fig. 3 areup-sampled to show the smooth phase function over time. Inparticular, the basic chirp is taken by sampling (2) at Fsamp L BW, which gives the up-converted basic chirp as 2nn(L), n 0, . . . , ML 1x0 [n] exp j2π 2L2ML2(16)where L is the up-sampling factor. The signals in Fig. 3 areplotted with L 32. The symbols produced by the cyclic shiftmethod can be expressed as(L)[n] xm(L)x0 [mod(n mL, ML)](17)which has the initial phase of(L)(L)[0] φ0 [0] xmmL(mL)2 22L2MLmm2 (18)( 2π rads).2M2It is not hard to see that the noncontinuous phase can beturned into continuous by forcing all CSS symbols to start atthe zero phase. In particular, the modulated CSS symbol withzero-starting phase can be expressed as 2mm(L)(L)[n]. xm(19)x̂m [n] exp j2π2M2 The continuous-phase CSS signal is plotted in Fig. 3 in solidlines. It is obvious that there are no more phase jumps noramplitude discontinuities at the symbol boundaries.Fig. 4 compares the power spectrum density (PSD) of twoCSS signals, with and without phase correction. Aside fromthe phase offset correction, the two signals are generated withthe same parameters, i.e., SF 10, BW 125 kHz, andL 8, and using the same set of random data symbols. Asexpected, the phase-corrected signal has much better out-ofband suppression. The difference is about 10 dB measured at500 kHz away from the carrier frequency.It is also noted that there are spectrum spikes occurred withthe CSS signal without phase correction. This is because, sincePower spectra of CSS signals with an without phase correction.its samples do not add to zero, the basic chirp contains a dccomponent. As a consequence, modulation by cyclic-shiftingaligns the dc component of all the transmitted symbols inphase, causing spectrum spikes at Nyquist frequencies. Onthe other hand, the phase correction not only produces smoothphase transition between symbols but also randomly rotates thedc component of each symbol based on the data it modulates,thus effectively eliminating the spectrum spikes.IV. E FFICIENT C HIRP T RANSMITTER D ESIGNWhile there are many papers investigating different aspectsof CSS modulation [13], [14], [21], [22], research work onefficient implementation of a CSS system at the physical layeris missing. This is quite surprising given the fact that theimplementation aspect of CSS is particularly important dueto low-cost and low-power requirements in the majority ofIoT applications. It appears that the common method to generate a CSS signal is LUT-based (as shown in Fig. 1), whichis the method patented and commercialized by Semtech [15].The LUT-based design is fairly simple and effective, presenting a challenge for Semtech’s competitors to come up with adifferent design that is as simple and yet effective for LoRa’sphysical layer.One of the disadvantages of the LUT-based design, as shownin Fig. 1, is scalability. In particular, in order to supportmultiple SFs, the LUT size must be large enough to hold allpossible basic chirps, each of which corresponds to one particular SF. Of particular interest is the LoRa gateway design,which should be able to handle multiple nodes at various BWand SF simultaneously. The gateway typically employs DACsand ADCs running at a fixed rate of about 2 Msps.2 Therefore,in order to support multiple BWs, the gateway’s transceiverwould need to employ variable-rate up/down conversion modules or storing chirp samples at a higher rate, either of whichwould increase the cost of the transceiver considerably. For2 The spec is taken from Semtech SX1257, which is the RF front-endtransceiver employed in most of the LoRa gateways [23].

9508Fig. 5.IEEE INTERNET OF THINGS JOURNAL, VOL. 6, NO. 6, DECEMBER 2019OCG.example, given the DAC sampling rate of 2 Msps, in order tosupport a minimum LoRa BW of 7.8125 kHz [24], the upconversion module must support an up-conversion rate of upto L 256. Assuming the dual DAC resolution is 8-bit, whichmeans each complex sample consumes two bytes, the size ofthe LUT must be at least212 2SF 256 4 161 536 bytes(20)SF 6which is a little more than 4 MB. Note that the size wouldgrow exponentially, to 67 MB, if LoRa standard is extendedto support up to SF 16.This section proposes an efficient design of the chirp transmitter, called OCG. The OCG can support a wide range of SFand BW at low cost. Moreover, the OCG uses just accumulators and adders to generate the continuous phase function of(L)(L)the chirp signal over time, i.e., φ̂m [n] [( x̂m [n])/2π ]. Inparticular, by substituting (16) and (17) into (19), the phasefunction can be expressed asφ̂m(L) [n] 22 m m (n mL), n mL ML n mL22M22L2ML2 2 m m (n mL ML), otherwise n mL ML2M22L2ML2 2nnmn2 ML 2L , n mL ML 2ML(21)2nnmnn ,otherwise.2ML2LL2MLIt should be noted that the term (n/2L) shifts the frequencydown by an amount of (BW/2) so that the CSS spectrumis centered at baseband. Moreover, the term (n/L) shifts thefrequency down by an amount equivalent to BW, providingthe frequency “wrap around” effect of the chirp signal.Since n2 expressed as!n 1k 0 (2k 1),the phase function can be further !n 1 k 0 (2k 1 2mL ML) , if n mL ML2ML2φ̂m(L) [n] !n 1 (2k 1 2mL ML 2ML) k 0, otherwise.(22)2ML2Of particular interest is the phase function when n ML mL,which can be written as φ̂m(L) [n] n ML mL!ML mL 1(2k 1 2mL ML 2ML) k 02ML2!n 1 2mL ML 2ML)(2kk ML mL 2ML2!ML mL 1(2k 1 2mL ML) k 02ML22ML(ML mL) 2!n 2MLk ML mL (mod(2k 1 2mL, 2ML) ML) 2ML2!n(mod(2k 1 2mL, 2ML) ML) k ML mL2ML2 (M m).(23)Since the integer term (M m) can be ignored due to thephase wrap-around effect, combining (22) and (23) yields anequivalent phase function for a chirp signal as!n(mod(2k 1 2mL, 2ML) ML)φm(L) [n] k 02ML2n 0, 1, . . . , ML 1.(24)The above expression of the phase function allows thechirp signal to be generated efficiently in the proposed OCGdesign as shown in Fig. 5. The OCG comprises of four

NGUYEN et al.: EFFICIENT DESIGN OF CSS MODULATION FOR LPWANsmain components: the frequency accumulator, the frequencymanipulator, the phase accumulator, and a vector rotator.The first three components create the phase function asin (24). In particular, the frequency accumulator 100 creates a frequency that is linearly increasing over time, i.e., aramp. The width of the accumulator has to be large enoughto accommodate the combination of the largest SF with thelowest BW, i.e., Fsys 1 (bits) (25)freq acc width SFmax log2BWminwhere Fsys , SFmax , and BWmin are, respectively, the systemclock rate, the highest SF, and the minimum BW supported.There is one extra bit added to support the half-step frequencycorrection 101.The frequency manipulator introduces jumps into thefrequency ramp at the symbol boundaries according to inputsymbol symbol in, selected BW and SF. It comprises ofmultiple logics, labeled from 101 to 108.Since phase is the integration of frequency over time,there is a half-step frequency correction 101 applied to theoutput of the frequency accumulator to correct for the difference between discrete and continuous integration of thelinearly changing frequency over time. The output of thehalf-step frequency correction is the term 2k 1 describedin (24).Symbol modulation is performed by adding a frequency offset to the output of the frequency accumulator 102. The offsetis obtained by performing bit shifting 103 on the input symbolto make sure symbol in has a proper format before addingit to the frequency accumulator’s output. The amount of leftshifting depends on the system clock rate and the desired BWof the generated chirp signal and is given as Fsys 1symbol in left shift log2BW log2 (L) 1 (bits)(26)i.e., the bit-shifting operation generates the term 2mL describedin (24).SF of the chirp signal is controlled by performing bit-wiseAND 105 of a mask 104 and the modulated frequency sample.The mask, which is created based on SF and BW, controls howfast the frequency rolls over. In particular, the mask has thesame width as the frequency accumulator. The log2 (ML) 1least significant bits of the mask are all ones, while the remaining bits are all zeros. The mask performs the modulo-by-2MLoperation described in (24). For example, with the same BWand if SF is reduced by one, the mask will mask off one morebit at the top of the frequency sample, effectively making thefrequency rolls over twice as fast. The opposite can be said ifBW is reduced by one.The frequency spectrum is centered at zero by invertingthe most significant bit of the masked frequency sample 106,then change the number format from unsigned to signed. Inparticular, before inverting, the frequency rolls over at 0 andBW, but after inverting the frequency rolls over at (BW/2)and (BW/2), respectively. Mathematically, spectrum centeringis represented as the term ML in (24).9509Fig. 6.SSB up-conversion of a chirp signal using a quadrature modulator.The actual center frequency of the chirp can be addeddirectly to the inverted frequency sample 108. The center frequency word has the same width as the frequencyaccumulator, as such, the center frequency resolution isFsys /freq acc width (Hz).The phase accumulator 110 performs discrete integration ofthe frequency samples to provide phase samples for the generated chirp symbol. There is a selective inverter 109, located atthe input to the phase accumulator, to control the chirp direction, i.e., up-chirp or down-chirp. Mathematically, the phaseaccumulator performs the main summation in (24).Finally, the vector rotator 111 performs phase rotation by(L)an amount of φm [n] on an input vector as"#(L)(27)y(L)m [n] x[n] exp j2π φm [n]where x[n] and y[n, m] are the input and output vectors,respectively.To generate a continuous-phase chirp signal, the input vectorx[n] is set to a scalar, which is determined by the magnitudeof the chirp signal. Alternatively, when the continuous phaseis not required between two adjacent chirp symbols, extra datainformation can be encoded as the phase of the input vector.The vector rotator can be built effectively using a pipelinedcoordinate rotation d

Efficient Design of Chirp Spread Spectrum Modulation for Low-Power Wide-Area Networks Tung T. Nguyen , Ha H. Nguyen , Senior Member, IEEE, Robert Barton, and Patrick Grossetete Abstract—LoRa is an abbreviation for low power and long range and it refers to a communication technology developed for low-power wide-area networks (LPWANs). Based .