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 T. Such approximate synchronization allows for less precise timing control in mobile applications. Example waveforms for this type of application are those based on the zero autocorrelation zone (ZACZ) set of sequences ‎[13].

      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 Review

      The 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 (~ ), and hence this scheme does not retain the desirable constant envelope property of conventional chirps. This approach showed improvement in multi user system BER performance in multipath fading channels when compared to FSK frequency hopping code division multiple access (FH-CDMA) schemes. Their analysis and evaluation was based on a perfectly synchronized condition.

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 any nonlinear chirps, and validate with numerical and simulation results.

- 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 Signaling

In 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 T. In complex baseband form, the mth linear chirp waveform can be written as,

where N is the desired number of orthogonal (up-)chirp waveforms, m Î{0, 1, …N-1} is the user index, and T is the duration of the chirp waveform. The total bandwidth B that a set of N users occupies is B=2N/T, and each user signal occupies the same bandwidth. When perfectly synchronized, the waveforms in (1) are orthogonal. A completely analogous construction can be made with “downchirps” by using a negative sign on the mT/N term of the exponent of (1). In this paper we consider only upchirps, but all assumptions and results are analogous for downchirps. The instantaneous frequency of the signal in (1) can be written as,

B.    Synchronized Sinusoidal Chirp Signaling

Non-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  to find instantaneous frequency versus time. We propose two non-linear chirp signal sets which, qualitatively speaking, have more “spacing” between each signal’s time-frequency trace. This approach aims to fully use the available time-frequency “space” for signals in a set, and increase resilience to timing offsets for the practical QS case.

Case one uses a sinusoidal function for , with signal waveforms given by,

where  and  are selectable constants that can produce different time/frequency shapes. The instantaneous frequency can be shown to be,

We selected values for  and  as  and , respectively, as these qualitatively produce a larger “area coverage” in the TF plane than the linear set of signals. An example is plotted in ‎Fig. 1.

C.    Synchronized Quartic Chirp Signaling

In 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  was chosen as  This design yields a larger time/frequency coverage than the linear and sinusoidal nonlinear case. Note that although our chirp designs contain parameters that must be specified, for brevity we do not address optimal parameter selection here. Any such optimality would of course require specification of several assumptions or conditions (e.g., timing offset statistical distributions).

TF plots of both nonlinear waveforms using (4) and (7) are shown in ‎Fig. 1. Note that not all N waveforms are shown: specifically, only the two lowest and highest frequency signals are plotted to bound each signal type’s area. The nonlinear cases clearly occupy larger total areas in the TF plane. As ‎Fig. 1 depicts, the sinusoidal Case 1 signal set occupies a slightly larger TF area than the linear set but keeps the same starting and ending frequency and the same total bandwidth. The quartic Case 2 covers the largest area, with different starting and ending frequencies, but the same total bandwidth.

Fig. 1.    Time-frequency domain signal representation example showing upper-most and lower-most frequency signals for linear and two nonlinear signal sets.

                                                                                                                                                                       IV.        Quasi-synchronous Transmission

Many 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  in many communication systems. Asynchronism also of course arises from channel effects, primarily propagation delay. Delays are typically modeled as random for all these causes.

where is the delay associated with clock drift or uncompensated propagation delay for user m. Generally, these delays have value limited between 0 to T since other than packet transmission boundaries, effects of asynchronism recur over subsequent symbols (we assume delays are essentially constant over packet durations, and a given user signal uses the same chirp type for each symbols). A time/frequency domain representation of the set of quasi-synchronous signals of the form of (8) for only one asynchronous user (m=2) is depicted in ‎Fig. 2. We note that for certain values of timing offset , the non-synchronized TF signal can overlap another signal in the set nearly completely over a part of a symbol, and this yields relatively large MAI.

Fig. 2.    Time/frequency domain representation of quasi-synchronous transmission of linear chirps with one asynchronous user.

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  is the timing offset of user k, and we have used the unit-energy of each waveform.

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 , and via Euler’s identity and l’Hopital’s rule, we can find,