Full and final version is available at: https://ieeexplore.ieee.org/document/9340403
Multi user orthogonal chirp spread spectrum (OCSS) can improve the spectral inefficiency of chirp spread spectrum (CSS) but is only feasible with perfect synchronism and without any channel dispersion. Asynchronism, channel dispersion, or unexpectedly large Doppler shifts can cause multiple access interference (MAI), which degrades performance. Conditions with small timing offsets we term “quasi-synchronous” (QS). In this paper, we propose two new sets of nonlinear chirps to improve CSS system performance in QS conditions. We analytically and numerically evaluate cross-correlation distributions. We also derive the bit error probability for Binary CSS analytically and validate our theoretical result with both numerical and simulation results; our error probability expression is applicable to any binary time-frequency (TF) chirp waveform. Finally, we show that in QS conditions our two new nonlinear chirp designs outperform the classical linear chirp and all existing nonlinear chirps from the literature. To complete our analysis, we demonstrate that our nonlinear CSS designs outperform existing chirps in two realistic (empirically modeled) dispersive air to ground channels.
Nonlinear
Quasi-Synchronous Multi User Chirp Spread
Spectrum Signaling Nozhan Hosseini,
Index Terms—chirp spread spectrum; multiple access communication system; quasi-synchronous transmission; ## I. IntroductionMany wireless communication systems will need to accommodate a larger number of users in the future. One application in particular in which this is critical is low data rate, long range communication links with very large numbers of nodes, such as the internet of things (IoT), internet of flying things (IoFT), etc. These systems demand advanced multi-access techniques with minimal multiple access interference (MAI). They should also be robust to multiple impairments, including multipath channel distortion, Doppler spreading, and interference.
These time frequency (TF) waveforms have several useful properties including energy efficiency, and if wideband enough, robustness to interference, non-linear distortion, multipath fading, and eavesdropping. They can also be used for high resolution ranging and channel estimation. Underwater acoustic wireless communication systems [3], [4] can also use chirp waveforms to advantage in the presence of very rapid underwater acoustic channel fading. The “long-range” (LoRa) technology developed for IoT applications uses a proprietary CSS modulation scheme that aims to provide wide-area, low power and low cost IoT communications [5], [6]. In the literature, different chirp waveforms have been categorized: linear, various types of nonlinear, amplitude variant as well as constant amplitude forms. Modulation can be accomplished in several ways, one of the simplest being binary chirps that sweep either up or down in frequency over a bit period. Chirps can of course be used in on-off signaling or as basic waveforms for frequency shift keying (FSK). Higher-order modulation can be attained with chirps in a number of ways, e.g., by using multiple sub-bands, different start/stop frequencies (somewhat akin to pulse position modulation, PPM), and via distinct chirp waveforms within a given band. A disadvantage of CSS signaling is
spectral inefficiency. This can be addressed by accommodating multiple users
with a set of properly designed chirps in the available bandwidth. For multiple
access, a set of chirp signals is required, and all waveforms in this set would
ideally be orthogonal, without MAI. Achieving orthogonality is easy enough with
synchronized waveforms [7]-[10],
but in many practical cases, e.g., with mobile platforms, attaining and
maintaining synchronism is challenging. Waveforms designed to be orthogonal
when synchronized typically exhibit large inter-signal cross correlation when
asynchronous [11],
[12],
and this of course induces MAI and degrades performance. Thus finding a set of
waveforms that achieves low cross correlation among signals when asynchronous
is desirable. In general this is very difficult in practice, so researchers
have focused on quasi-synchronous (QS) conditions. This refers to the case when
synchronization is approximate, typically limited to some small fraction of a
symbol time In this paper, we explore the analysis and design of CSS waveforms for quasi-synchronous operation in multiple access systems using both linear and nonlinear TF functions. We quantify MAI and error probability analytically in asynchronous conditions, and provide two new nonlinear chirp designs that yield better performance under modest asynchronism (i.e., QS). The remainder of this paper is organized as follows: Section II provides a brief literature review on chirp signaling and CSS, and in Section III we introduce linear and two proposed nonlinear chirp designs. Section IV describes the quasi-synchronous condition and provides analytical and numerical cross correlation results for linear and any nonlinear chirp signal sets; we compare cross correlation values for a full range of delays. Section V addresses analytical binary CSS (BCSS) bit error ratio (BER) performance evaluation and simulation validation. In Section VI, we evaluate QS performance of our proposed waveforms in comparison to linear and other chirp waveforms in the literature, and over an empirical air-ground channel model. Section VII concludes the paper. ## II. Literature ReviewThe literature on the general use of chirps is fairly extensive (radar, channel modeling, etc.), so we only provide highlights. We focus primarily on communication aspects. The chirp technique proposal made by S. Darlington in 1947 was related to waveguide transmission for pulsed radar systems with long range performance and high range resolution [14]. B. M Oliver first used “chirp” in his memorandum entitled “not with a Bang, but a Chirp,” and 6 years later, acoustic chirp devices were developed at Bell Labs. Hardware constraints were a limiting factor for their development. In [15], the authors described an experimental communication system employing chirp modulation in the HF band for air-ground communication. In [7],
the authors proposed an orthogonal linear amplitude-variant chirp modulation
scheme where each user employs a unique frequency modulated chirp rate. The
scheme defines orthogonal linear chirps with different chirp rates or TF
slopes. To satisfy orthogonality with their design, they impose amplitude
variation (~ The authors of [8] used a set of orthogonal linear chirped waveforms based on the Fresnel transform and its convolution theorem to design an orthogonal chirp division multiplexing (OCDM) system. They compared this to orthogonal frequency division multiplexing (OFDM) and showed that their OCDM system outperformed the conventional OFDM system by exhibiting greater resilience to inter symbol interference when the OFDM system had an insufficient guard interval. Compared to OFDM the OCDM scheme had identical PAPR performance and only slightly higher complexity. Discrete Fourier transform-precoded-OFDM (DFT-P-OFDM) outperformed OCDM in terms of PAPR and had identical BER performance. In this work, the authors also assumed perfect synchronization between all transmitters and receivers. In [9], the authors presented their orthogonal quadratic and exponential non-linear chirp designs. Users are assigned unique chirp rates that vary either quadratically or exponentially versus time (yielding different signal bandwidths among users). These designs also required amplitude variation to maintain orthogonality. A similar approach was followed in [10] for nonlinear trigonometric and hyperbolic CSS waveforms, again assuming full synchronization. The authors in [16] presented another set of orthogonal chirps by exploiting the advantages of the fractional Fourier transform (FrFT) adopted from [17]. They claimed that the proposed method has lower MAI than the conventional method in [17] and should yield better system performance. Their signal amplitude is constant over the chirp duration, but again, a fully synchronous system was assumed. The authors in [18] proposed an iterative receiver to improve BER performance in frequency-selective fading channels and opened the possibility of space-time coding multiple input multiple output (MIMO) schemes for orthogonal code division chirps (OCDM). Finally, in [19] and [20] we discussed the implementation of a low complexity transceiver based on discrete Fourier transform spread orthogonal frequency division multiplexing (DFT-s-OFDM). In this work, we provided insight into how chirp waveforms for radar and communication can be synthesized without major modifications to the physical layer of today’s OFDM based wireless communication systems. In [21] we investigated air to ground channel fading effects on the BER performance of CSS systems. Specifically, we simulated performance in some “canonical” Ricean fading channels, and over realistic aeronautical channels based on extensive measurements. Our approach for CSS here enforces constant signal envelope and nearly equal signal bandwidths for all users. Primarily, we relax the perfect synchronization constraint and find designs that can yield better multi-user performance when quasi-synchronous. To the best of our knowledge, this is the first appearance of chirp schemes designed for practical QS operation. The main contributions of this paper can be listed as follows: - Our proposed approach improves the spectral inefficiency of chirp spread spectrum (CSS) in asynchronous or quasi-synchronous conditions, via introduction of two new nonlinear chirp signals sets that cover a larger area in the time-frequency plane than existing chirps. - We provide analytical results for
cross-correlations of linear chirps and an algorithmic method to compute
cross-correlations for - We derive the bit error probability for binary CSS for any chirp waveforms, for arbitrary received signal energies, and validate our theoretical result with both numerical and simulation results. - We show that in QS conditions our two new nonlinear chirp designs outperform the classical linear chirp and all existing nonlinear chirps from the literature, on the additive white Gaussian noise channel and an example practical air-ground (AG) channel. ## III. Chirp Signal Designs## A. Synchronized Linear Chirp SignalingIn this paper, the core
formula for generating frequency-modulated (chirp) waveforms is adopted from
the kernel Fresnel transform theorem method, related to the Talbot effect,
where the discrete Fresnel transform (DFnT) provides the coefficients of the
optical field of an image, first observed by Talbot [22];
this is discussed in lightwave and optical communication applications [8].
Using the continuous Fresnel transform provided in [23]
and expressed in the form of convolution as noted in [24],
we obtain the formula to generate orthogonal linear “up-chirps” (low to high
frequency) and “down chirps” (high to low frequency) with symbol duration
where
## B. Synchronized Sinusoidal Chirp SignalingNon-linear chirp waveforms can easily be generated with arbitrary shapes in the time-frequency plane. The most well-known examples are exponential, quadratic, and sawtooth [9], [10]. Here we propose a mathematical derivation for generating two specific nonlinear chirp waveforms with no amplitude variation (with the aim of keeping PAPR low). A nonlinear phase function Ψ(t) is employed as in (3),
This
phase function can modify the instantaneous frequency to any desired nonlinear
TF shape. One can find the chirp signal’s time-frequency shape via the time
derivative Case
one uses a sinusoidal function for
where
We selected values for ## C. Synchronized Quartic Chirp SignalingIn order to further increase spacing between each signal’s time/frequency trace, we constructed another nonlinear signal set with the following instantaneous frequency:
The corresponding phase functions are,
where TF plots of
both nonlinear waveforms using (4)
and (7)
are shown in Fig. 1.
Note that not all ## IV. Quasi-synchronous TransmissionMany
modern communication systems have been developed assuming quasi-synchronous
conditions, where clocks of different user terminals (or, nodes) are not
perfectly synchronized, but are “close” to synchronized. Their mean clock
frequencies may be essentially identical, but drift and jitter cause clocks to
deviate from this mean over the long and short terms. This asynchronism is
usually bounded (a small portion of a symbol duration
where Multiple access interference (MAI) is quantified by the cross correlation between signals of the form of (1) and (8). Computing the cross correlation values requires an integration, which can be written as follows:
where for the linear chirp case we have,
where
again By dividing
the integral into two parts as indicated in Fig. 2,
we computed each integral based on the signal within the corresponding time
segment. this integral has a closed form solution for any arbitrary offset
where The
integral for nonlinear chirps has no closed form solution in general, yet for
arbitrary non-linear chirp waveforms one can obtain a very good approximation
by modeling any nonlinear TF trajectory as a set of Specifically,
in Fig. 3, we show a symbol duration divided into
where
where
with
where In general
mentioned delays can be well modeled as random, and we can assess the quality
of any chirp signal set statistically, by considering cross correlation to be
conditioned upon delay, then averaging that over the probability density
function of delay. An example set of This figure
shows results for a set of (a) (b) We observe
that beyond a certain small value of delay, the quartic nonlinear signals yield
a smaller average correlation value for nearly the entire range of timing
offset for the two smaller values of Note that
correlation plots are symmetric around 0.5 For a more complete representation of the cross correlation distributions for these chirp types, we provide histograms of all cross correlation values for all offsets for our three signal sets in Fig. 7 (a) to (c). The histograms show that the largest correlation values, which cause the most severe MAI, are less likely for the nonlinear cases than the linear set. ## V. Analytical Performance EvaluationFor a
multiuser
where ku
is an equiprobable M-ary symbol , T is
the symbol duration and function p(t) is the unit rectangular pulse equal to one
for J
symbols.
M sub-bands, and within each sub-band, a set of N chirp waveforms is used to accommodate
the N users.Each sub-band
has bandwidth We
first assume an additive white Gaussian noise (AWGN) channel, and hence can
consider detection during a single symbol interval. Performance is evaluated
for user
where At the receiver, matched filters convolve the received signal with a bank of time-reversed versions of the transmitted chirps. An alternative heterodyne detector (correlator) can also be used instead of matched filter detectors, as explained in [26]. Decision circuits complete the receiver symbol detection. Assuming
user
Analysis
is analogous for the transmission of “1.” Based upon the transmitted symbols,
and using the expression derived for cross-correlations, we find that the
correlator outputs are Gaussian with variance |