M-Ary Aggregate Spread Pulse Modulation in Lpwans for Iot Applications
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M-ary Aggregate Spread Pulse Modulation in LPWANs for IoT applications Alexei V. Nikitin Ruslan L. Davidchack Nonlinear LLC Sch. of Mathematics and Actuarial Sci., Wamego, Kansas, USA U. of Leicester, Leicester, UK E-mail: [email protected] E-mail: [email protected] Abstract—In low-power wide-area networks (LPWANs), various However, the designed pulse train Gˆ»:¼ given by (1) can be trade-offs among the bandwidth, data rates, and energy per bit “re-shaped” by linear filtering: have different effects on the quality of service under different ∑︁ propagation conditions (e.g. fading and multipath), interference G »:¼ = ¹Gˆ ∗ 6ˆº»:¼ = 퐴9 6ˆ»: −:9 ¼ , (3) scenarios, multi-user requirements, and design constraints. Such 9 compromises, and the manner in which they are implemented, fur- where 6ˆ»:¼ is the impulse response of the filter and the asterisk ther affect other technical aspects, such as system’s computational denotes convolution. The filter 6ˆ»:¼ can be, for example, a complexity and power efficiency. At the same time, this difference lowpass filter with a given bandwidth 퐵. If the filter 6ˆ»:¼ has a in trade-offs also adds to the technical flexibility in addressing a broader range of IoT applications. This paper addresses a sufficiently large time-bandwidth product (TBP) [4], [5], most of physical layer LPWAN approach based on the Aggregate Spread the samples in the reshaped train G »:¼ will have non-zero values, Pulse Modulation (ASPM) and provides a brief assessment of its and G »:¼ will have a much smaller PAPR than the designed properties in additive white Gaussian noise (AWGN) channel. In sequence Gˆ»:¼. Such low-PAPR signal can then be used for the binary ASPM the control of the quality of service is performed modulating a carrier. If the combination of the amplitude 퐴9 through the change in the spectral efficiency, i.e., the data rate : " at a given bandwidth. Implementing M-ary encoding in ASPM and the arrival time 9 of a pulse provides distinct “states,” " 5 further enables controlling service quality through changing the each pulse can encode log2 bits, and the raw bit rate b in 5 5 " 퐵 5 퐹 # " energy per bit (in about an order of magnitude range) as an such a train is b = p log2 . When b = ¹ s/ pº log2 , it additional trade-off parameter. Such encoding is especially useful results in a low-rate message encoded in a wideband waveform. for improving the ASPM’s energy per bit performance, thus For example, for the arrival times in (1) one can use increasing its range and overall energy efficiency, and making it more attractive for use in LPWANs for IoT applications. :9 = 9#p ¸ Δ: »<9 ¼ , (4) Index Terms—Aggregate spread pulse modulation (ASPM), : : < # : < intermittently nonlinear filtering (INF), Internet of things (IoT), where Δ is a positive integer, 0 ≤ Δ » 9 ¼ < p, and Δ » ¼ < LoRa, low-power wide-area network (LPWAN), M-ary ASPM (M- Δ: »;¼ for < < ;. Then for <9 = 1, 2,...," and 퐴9 = const the " ASPM), physical layer (PHY), spread spectrum. pulse train given by (1) encodes log2 bits per pulse. We will refer to such M-ary encoding with 퐴9 = const as “unipolar.” 0 I.I NTRODUCTION Another bit can be added by using 퐴9 = ¹−1º 9 , where 09 is In the Aggregate Spread Pulse Modulation (ASPM) [1], [2] either “0” or “1,” and we will refer to such signaling as “bipolar.” the information is encoded in the amplitudes 퐴9 and/or the Then for bipolar M-ary signaling equation (1) can be rewritten “arrival times” :9 of the pulses in a digital “pulse train” Gˆ»:¼ as ∑︁ G »:¼ = È: = 9# ¸ : »< ¼É ¹− º09 , with only relatively small fraction of samples having non-zero ˆ p Δ 9 1 (5) values: 9 < , ,...," 0 G : ∑︁ : : 퐴 , where 9 = 1 2 /2 and 9 is either “0” or “1.” arXiv:2106.10179v1 [eess.SP] 18 Jun 2021 ˆ» ¼ = È = É (1) 9 9 G »:¼ 9 For a given designed pulse sequence ˆ the spectral, temporal and amplitude structures of the reshaped train G »:¼ where : is the sample index of the 9-th pulse, 퐴 is its amplitude, 9 9 will be determined by the choice of 6ˆ»:¼. In particular, it may and the double square brackets denote the Iverson bracket [3] be desirable to select a filter 6ˆ»:¼ that minimizes the PAPR 1 if % is true G : 6 : È%É = , (2) of » ¼. Note that if the time duration of ˆ» ¼ extends over 0 otherwise multiple interpulse intervals, the instantaneous amplitudes and/or where % is a statement that can be true or false. The average phases [6] of the resulting waveform are no longer representative “pulse rate” 5p in such a train is 5p = 퐹s/#p, where 퐹s is the of individual pulses. Instead, they are a “piled-up” aggregate of sample rate, and #p = h:9 − :9−1i is the average interpulse the contributions from multiple “stretched” pulses. interval. Note that for #p 1 the pulse rate is much smaller than The key property of the large-TBP pulse shaping filter the Nyquist rate. Also note that for #p 1 this train has a large (PSF) 6ˆ»:¼ is that its autocorrelation function (ACF), i.e., the peak-to-average power ratio (PAPR) even when j퐴9 j = const, convolution of 6ˆ»:¼ with its matched filter 6»:¼ = 6ˆ»−:¼, has a and is generally unsuitable for use as a modulating signal. much smaller TBP, in particular, sufficiently smaller than the Nikitin and Davidchack M-ary Aggregate Spread Pulse Modulation in LPWANs for IoT applications ratio 퐵/5p. Then, after demodulation and analog-to-digital (A/D) II.M- ARY VARIANTS OF ASPM conversion in the receiver, the encoded binary sequence can be In the binary ASPM, each pulse encodes one bit, hence the recovered by filtering with 6»:¼ and sampling the resulting pulse energy per bit 퐸b and the energy per pulse 퐸p are equal to each train at : = 9#p¸Δ: »<¼ (i.e., using 6»:¼ as a decimation filter). 퐸 퐸 " other, b = p. By encoding log2 bits per pulse with the same A good choice for the PSF would be a pulse that combines a 퐸 퐸 " energy, the energy per bit is reduced to b = p/log2 . Such small TBP of its ACF (e.g., close to that of a Gaussian pulse) encoding is especially useful for improving the ASPM’s energy with ACF’s compact frequency support. An example would be per bit performance, thus increasing its range and overall energy a raised-cosine (RC) filter [7, e.g] with unity roll-off factor. efficiency, and making it more attractive for use in LPWANs for The minimum required (Nyquist) sample rate for such a filter IoT applications. will be double its (baseband) physical bandwidth 퐵, and the sample rate 퐹s can be expressed as 퐹s = 2#s퐵, where #s ≥ 1 is A. Single-sideband M-ary ASPM with constant-envelope pulses the oversampling factor. To minimize the power consumption, For example, Fig. 1 illustrates a single-sideband M-ary ASPM the memory usage, and the computational complexity of the link which uses constant-envelope transmitted pulses and is digital processing, it is beneficial to keep the sample rate in the suitable for both coherent and noncoherent detection. transceivers designed for IoT applications as low as possible, In Fig. 1(I), the designed pulse train Gˆ»:¼ according i.e., to use #s = 1. Through the rest of the paper, we will assume to (5) is filtered with 6ˆ»:¼ and ℎˆ»:¼ to form the shaped 퐹 퐵 sampling with the Nyquist rate s = 2 . trains G6 »:¼ and Gℎ »:¼. After digital-to-analog (D/A) conversion, G »:¼ Since for a given designed pulse sequence ˆ the temporal G6 ¹Cº and Gℎ ¹Cº are used for quadrature amplitude modulation of G »:¼ and amplitude structures of the reshaped train are a carrier with frequency 5c, providing the transmitted waveform 6»:¼ ˆ determined by the PSF ˆ , these structures can be substantially G6 ¹Cº sin¹2c 5cCº ¸Gℎ ¹Cº cos¹2c 5cCº (Fig. 1(II)). If 6ˆ»:¼ and ℎ»:¼ different even for the pulse shaping filters with the same are, say, the real and imaginary parts, respectively, of a nonlinear ACF. As discussed in [1], one can construct a great variety of chirp with the desired ACF, e.g. large-TBP pulse shaping filters 6ˆ »:¼ with the same small-TBP 8 6ˆ»:¼ ¸ 8 ℎˆ»:¼ = È0 ≤: <=É exp ¹8 Φ»:¼º , (8) ACF F »:¼, so that ¹6ˆ8 ∗ 68 º»:¼ = F »:¼ for any 8, while the : convolutions of any 6ˆ8 ¹Cº with 69 ¹Cº for 8 < 9 (cross-correlations) where Φ» ¼ is the phase, then this waveform will occupy only have large TBPs. Further, this property will also effectively a single sideband with the physical bandwidth 퐵 equal to the hold for the PSFs ℎˆ8 »:¼ such that ℎˆ8 »:¼ is the discrete Hilbert baseband bandwidth of the chirp. In addition, if the pulses do # = : < transform of 6ˆ8 »:¼, i.e., ℎˆ8 »:¼ = 퐻 f6ˆ8 »:¼g [8], [9]. Therefore, not overlap (e.g., p ¡ ¸ max< ¹Δ » ¼º), this waveform will using various PSFs combinations we can design different coherent consist of constant-envelope pulses. and noncoherent modulation schemes with emphasis on particular For noncoherent detection (Fig. 1(III)), in the receiver’s (Rx) spectral and/or temporal properties of the modulated signal. quadrature demodulator the noisy passband signal is multiplied by the orthogonal sinusoidal signals from a local oscillator, A. Binary (“one bit per pulse”) encoding lowpassed, and converted to the in-phase and quadrature digital For example, in [10] we describe single-sideband, constant- signals 퐼 »:¼ and & »:¼. Filtering 퐼 »:¼ and & »:¼ with the pairs of envelope coherent and noncoherent ASPM configurations that the filters 6»:¼ and ℎ»:¼, as shown in Fig.