Nonlinear Quasi-Synchronous Multi User Chirp Spread Spectrum Signaling

Nozhan Hosseini, Member, IEEE, David W. Matolak, Senior Member, IEEE

Abstract— 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.

Index Terms—chirp spread spectrum; multiple access communication system; quasi-synchronous transmission;

                                                                                                                                                                                                      I.     Introduction

      Many 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.

where (i) denotes (k-m)T/N and (ii) denotes otherwise. This expression has the smallest value (0) when  or . Correlation is of course one when  and .

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  linear segments of very small duration. We do not address the mathematical intricacies here, but as  gets large, for continuous TF functions our approximation should converge to the exact cross correlation integral result. The total cross correlation for any two nonlinear TF functions (integral of (9)) is then the summation of the  small segments of linear cross correlations. Figure 3 illustrates the approximation method, where for each small segment, a specific linear equation is used to approximate the TF function. Naturally, as the segment length decreases (and  increases), the approximation improves.

Fig. 3.    Method used to find cross correlation for nonlinear TF shapes.

Specifically, in Fig. 3, we show a symbol duration divided into  equal segments. By taking the derivative of the nonlinear TF function at each segment of width we can find the slope ( ) and intercept ( ) of each line segment  where  , , and . The variables  and  are constant frequency and time values, respectively, of the center of the line segment,  is the index on  and  is the instantaneous frequency as defined previously in (2) and (5). We can then write the nonlinear chirp signal cross correlation for the mth and kth signals as,

where  and  can be written as,

where  and  are the phases of the segment, and  can be written as,

with  defined analogously. Therefore, using (13) and (14) in (12), and making use of the online Wolfram Alpha intelligence computational integral engine ‎[25] the cross correlation of the two nonlinear chirps can be approximated as,

where  is the imaginary error function, defined by  where z is a real number. We show an example result in ‎Fig. 4 for cross correlation between two nonlinear user signals for delay of  using this method. In this particular signal set, the actual TF functions change as the total number of users N changes; hence we show  for two values of N. We can see as the number of segments per symbols used in the integral approximation increases, the approximation gets closer to the cross correlation value computed via direct numerical integration of (9).

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 mean correlations is shown in ‎Fig. 5.

This figure shows results for a set of N=25 linear chirps, and for two sets of N=25 nonlinear chirps, both sinusoidal and quartic. Here we found cross correlation between each pair of two users in a set with relative delay , and averaged over the ( ) for each delay. We have shown both analytical (equations (11) and ) and numerically computed results. ‎Fig. 6 (a) to (c) show average correlation values for these three chirp types for three different values of the number of signals N. Insets in the figure show these correlations at two smaller delay ranges, 0.05T and 0.01T, for illustration.

                                                                   (a)                                                                              (b)

Fig. 4.    Approximation of nonlinear chirp by linear segments: (a) TF traces for two signals in a set of N=15 users, (b) Cross correlation vs. number of segments ( ) for user two and five when in  sets of N=15 and N=30 users.

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 N, whereas the sinusoidal signals have approximately the same mean correlations as the linear case.

Fig. 5.    Average values of analytical and numerically evaluated cross correlation versus delay for three different chirp signal sets, each with N= 25 users.

Note that correlation plots are symmetric around 0.5T as ‎Fig. 5 depicts, therefore only delays up to this value are shown in ‎Fig. 6. Even for the largest value of N, the quartic signal set has lower mean correlations at delays above some very small value (~0.005T) up to a substantial delay value of approximately 0.1T; these results illustrate the quartic set’s suitability for QS operation.

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 Evaluation

For a multiuser M-ary orthogonal linear chirp spread spectrum system the k’th user’s transmitted baseband signal is,

where A_k is the signal amplitude, k  is user index, u is an equiprobable M-ary symbol , T is the symbol duration and function p(t) is the unit rectangular pulse equal to one for and zero otherwise. This equation expresses transmission of a block of J symbols.

‎Fig. 8 illustrates the system block diagram. In the transmitter, for each user’s data, a block of b bits is translated to one of M=2^b symbols. Each symbol is mapped to a specific one of M sub-bands, and within each sub-band, a set of N chirp waveforms is used to accommodate the N users.

(a)                                                              (b)                                                           (c)

Fig. 6.    Normalized average cross correlation versus delay for linear, sinusoidal, and quartic cases for (a) N=10, (b) N=20 and (c) N=50 signals.

(a)                                                              (b)                                                           (c)

Fig. 7.    Probability of occurrence of normalized cross-correlations for linear, sinusoidal and quartic chirps, for (a) N=10, (b) N=20 and (c) N=50 signals.

Each sub-band has bandwidth 2N/T, so the entire system bandwidth is 2NM/T and the spectral efficiency of a fully loaded system is log₂(M)/(2M) bps/Hz. In this paper we restrict our analysis to the binary CSS (BCSS) case. We also conduct the derivation beginning with linear chirps, but as shown in the Appendix, our actual result is applicable to any nonlinear chirp set as well, with the key requirement being that we have the cross correlation expressions to use within the BER formula.

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 k with N user signals present, as illustrated in in ‎Fig. 8 After coherent downconversion, the baseband signal including noise at user k’s receiver can be written as,

where  is complex noise  stationary and Gaussian with zero mean,  is the signal of user i at the receiver of user k, and =0. Note that in quasi-synchronous mode the delays ( ) can take any values; in general they are arbitrary and modeled as random, but we assume that they are constant for at least J symbols.

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  sends symbol “0”, the BCSS decision block inputs for the two correlator branches can be written as,

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  (the noise variance), and means dependent on the data symbols and cross-correlation values. Hence, we can find the bit error probability in terms of the well-known Q-function, the tail integral of the zero-mean, unit-variance Gaussian probability density function. For more details on the derivation, see the Appendix. For N users in a binary (M=2) CSS system, the resulting BER for user k can be expressed as,

Fig. 8.    Multiuser M-CSS complex baseband system block diagram

where  is a vector of size , with elements in the set {-1,1}, defined as,

with } the ith coefficient in the binary expansion for decimal number , i.e,

and  is the cross correlation vector of dimension , with superscript T denoting transpose. This vector is

i.e., vector  includes all cross-correlation values  except . For unequal received signal energies, the cross-correlation vector can be written as,

As an example for evaluating (20) and (21), the b vectors for N=4 are as follows,

Fig. 9.    BER vs. SNR for analysis and simulation for two users (N=2) in a linear BCSS system, for several values of delay and for both equal and unequal bit energies.

A comparison of simulation and analytical results for a binary linear chirp system with two users and different fixed delay values is presented in ‎Fig. 9, showing essentially perfect agreement. An example for unequal energies is also included. ‎Fig. 10 shows BER vs. SNR for a fixed delay value of 0.1T for different numbers of users. Here we assumed the desired user is synchronized and all other user signals have delay . We again include an example for unequal energies. In both ‎Fig. 9 and 10 we see excellent agreement between analytical and simulation results. Note that each user’s performance is related to their position within the set and the delays of other user signals and sampling rate should satisfy the minimum Nyquist sampling rate for different chirp bandwidths (sampling rate>2B).

Fig. 10.  BER vs. SNR comparing simulation versus theoretical results for different number of users with fixed timing offset and equal/unequal energy.

                                                                                                                                                                                  VI.        BER Performance Results

As previously noted, there are multiple ways to modulate chirps with data: mapping M-ary symbols to M of the N chirps in the set, using chirps of the opposite slope (e.g., “downchirps” as well as “upchirps”), on-off signaling, and even using different starting/stopping frequencies. This latter method is used in the LoRa technology ‎[6], where with a linear chirp frequency f_chirp in the range [f_min, f_max], two different symbols can be represented during a symbol interval by either (a) a sweep from f_min to f_max, or by (b) a sweep from f_m to f_max immediately followed by sweep from f_min to f_m, with the second symbol’s start frequency f_m in the range f_min < f_{m <} f_max.

As also noted, each user signal’s delay is assumed perfectly known at its receiver. Delay tracking using coherent delay-locked loops (DLLs), similar to previous efforts for CDMA systems ‎[27], can address this, but this is out of the scope of this paper. Our simulated performance results assume perfect delay estimation for each single-user receiver. Specifically, we model each delay as a zero-mean Gaussian random variable with standard deviation ( ). We define a system as partially loaded when fewer than N signals are being transmitted. In this case, the chirp signals are distributed with equal spacing in frequency to occupy the same area within the TF plane as in the fully-loaded case.

A.    Quasi-synchronous vs. Fully Synchronized

‎Fig. 11 (a) to (c) depict simulated bit error ratio performance versus bit energy to noise density ratio (E_b/N₀) for fully loaded linear and nonlinear designs for both synchronized and quasi-synchronous conditions for N = 10, 20, and 50. For these results, the zero-mean Gaussian random delays have standard deviation of 0.01T and 0.1T. ‎Fig. 11 (a) shows system performance in a perfectly synchronized system. The first thing to observe is that the nonlinear chirp signals are not orthogonal. Hence their performance degrades as the number of signals and MAI increase, particularly for the sinusoidal nonlinear case. However, as we can see in ‎Fig. 11 (b), a very small set of random delays with  significantly degrades the performance of the linear chirps, whereas the degradation of the quartic nonlinear case is moderate. For the largest value of  in ‎Fig. 11 (c), the nonlinear quartic case 2 is superior to the other sets of waveforms for any system loading. To show that our BER equation (19) can be used for any TF waveform shape, we also show analytical sinusoidal and quartic chirp performance in ‎Fig. 11 (c).

Note that for actual random delays, we model the ’s as (Gaussian) random variables. To analytically assess performance in this case we would consider our cross correlations ((11) or ) to be conditioned on delay . Average BER would then be expressed as the integral of (19) multiplied by the Gaussian probability density function for . In general, the resulting complicated expression is not integrable in closed form; we leave exploration of this for future work.

Since the sinusoidal chirps do not extend the TF plane area coverage by much over the linear chirps, the sinusoidal chirps only slightly outperforms the linear case in ‎Fig. 11 (b) and (c) in QS conditions. Synchronization on the order of  is very close to perfect, but the 0.1T value is more practical, particularly for mobile platforms.

Also worth study is performance of partially-loaded systems. As ‎Fig. 12 depicts, the quartic CSS performance gain over the linear chirps in QS conditions will increase even more with the use of fewer signals (K < N=40) when the K signals are selected to be maximally and equally spaced in the TF plane. This behavior has been observed for any arbitrary value of N.

B.    Proposed Nonlinear Chirps Versus Chirps from Literature

To further illustrate performance gains of our nonlinear chirp designs, we compare the performance of our nonlinear chirps with other chirp waveforms in the literature ‎[7], ‎[9] and ‎[10] in quasi-synchronous conditions. This includes the amplitude-varying linear chirp, the quadratic, exponential, and hyperbolic sinusoidal. Fig. 13 shows BER vs. E_b/N₀ for  and .

Our quartic nonlinear set outperforms the other chirp waveforms for the practical case of . For the smaller value of , performance of all sets is very close except for the poorest-performing exponential case of ‎[9]. Note that this plot is for a fully loaded system of 10 users. Waveforms from ‎[7], ‎[9] and ‎[10] have different bandwidths for each user signal but the same total bandwidth was set to be identical for all selected waveforms. Moreover, all the other chirp signals in these references have amplitude variation, yielding a larger peak-to-average power ratio, whereas our waveforms have a constant envelope.

(a)                                                                        (b)                                                                    (c)

Fig. 11.  Simulated multiuser binary chirp spread spectrum BER vs. E_b/N₀ for fully loaded systems for, (a) synchronized CSS system for N=10, N=20 and N=50, (b) quasi-synchronized CSS system with  for N=10, N=20 and N=50, and (c) quasi-synchronized CSS system with  for N=10, N=20 and N=50, including two analytical results.

Fig. 12.  Half- and quarter-loaded system simulated BER vs. E_b/N₀ for linear and quartic chirp signals in fully synchronized and QS conditions with =0.1T, and N=40.

C.    Performance in an Empirical Air-Ground Channel

To finish description of our performance results, we simulated CSS performance over a dispersive air-ground channel. The channel models are based on empirical air to ground measurement results sponsored by NASA, reported in ‎[28] – ‎[31]. Table I lists channel parameters for two locations: suburban Palmdale, CA, and the near urban setting for Cleveland, OH.

Table I. Air Ground Channel Parameters ‎[30].

As can be seen, RMS delay spreads are larger for suburban Palmdale than for the near urban Cleveland channel. Since we employ no equalization or multipath mitigation in these initial results, we expect poorer performance in the suburban case.

Fig. 13.  Simulated multiuser binary chirp spread spectrum BER vs. E_b/N₀ for fully loaded systems for, (a) synchronized CSS system for N=10, N=20 and N=50, (b) quasi-synchronized CSS system with  for N=10, N=20 and N=50, and (c) quasi-synchronized CSS system with  for N=10, N=20 and N=50, including two analytical results.

The links for the example AG channels are a set of air to ground links emulating a multipoint to point air to ground system with total data rate of 100 kbits/s and total bandwidth of 400 kHz. There are N=10 users, each transmitting at 10 kbps over this bandwidth, the value of which is comparable to that proposed for other AG systems ‎[32]. Transmissions from aircraft are received at the ground station quasi-synchronously, with zero-mean Gaussian distributed timing offsets with =0.1T, with each AG signal encountering its own unique channel.

Fig. 14 shows BER performance of the CSS signals over these realistic AG channels. For this relatively small bandwidth, the channel fading is essentially flat, except for the largest values of delay spread, which occur with low probability.

As expected based on delay spreads, results are better for near urban Cleveland than for suburban Palmdale. Performance of the quarter-loaded quartic system again illustrates the substantial effect of MAI in this system, but once again, our non-linear designs outperform the traditional linear chirps.

Fig. 14.  Simulated BER vs. E_b/N₀ for CSS signals over simulated air-ground channels based on models in ‎[30].

                                                                                                                                                                                                       VII.       Conclusion

In this paper, we investigated multi user chirp spread spectrum system performance in quasi-synchronous conditions, for the classic linear chirp, several existing nonlinear chirp designs in the literature, and two new nonlinear chirps. We derived a closed-form expression for the cross correlation for the linear chirps and a closed-form approximation for nonlinear chirp cross correlations. From these we developed an expression for the error probability for any TF shape waveform for binary multi-user chirp spread spectrum in quasi-synchronous conditions. We validated our analysis via numerical and simulation results, and provided example correlation statistics for the linear and our new nonlinear chirps. The linear chirps are generally best in perfectly synchronized cases, but we showed that since our nonlinear cases use more “time-frequency space,” they can outperform nearly all other chirp designs we have evaluated, for a range of assumed Gaussian-distributed timing offsets. The performance of our new designs is particularly superior in non-fully-loaded systems. Our new quartic nonlinear chirp design performs best. We also illustrated performance improvements of our new designs over a realistic dispersive air-ground channel. For future research, we will investigate the effects of Doppler shifts, and non-coherent detection for even more practical conditions.

Appendix

Here we derive the BER expression for any user in the BCSS system, in the presence of up to N-1 asynchronous other-user signals. The block diagram for the M-ary CSS system was presented in ‎Fig. 8 and the transmitted baseband waveform expression for user k’s signal is (16).

Without loss of generality, we derive the BER for user k=0, in the presence of a single interfering (asynchronous) user signal, user m=1. We then show that it is straightforward to generalize to an arbitrary number of interfering users, and that the expression holds for any selected user. We also analyze assuming that user 0 transmits a symbol u=0; with equally-likely data symbols, the derivation and results are identical for transmission of the symbol u=1. For our AWGN channel we can analyze detection of a single symbol to determine BER, and we assume transmission of the first bit, from time t=0 to T.

In our system, the 0 symbols all lie within the same sub-band, whereas all the 1 symbols lie in the adjacent sub-band; for the AWGN channel there is no “inter-sub-band interference.” We assume that user 0’s receiver is synchronized to its transmission, and the receiver correlates the received signal with the complex conjugate of the user 0 transmitted signal for both possible symbols 0 and 1. The bit decision is made by selecting the largest correlator output. Interfering user m=1’s transmission is equally likely to be either a 0 or a 1, so we account for both possibilities. The interfering user m=1 signal is delayed by  relative to user 0. The cross correlation between these two signals in the same sub-band (“0” sent) is given in (11) and . User k’s (=0) correlator outputs are given by,

where r_kv denotes the user k correlator output for symbol u, s_kv(t) denotes user k’s signal for symbol u, and w(t) is the AWGN. By expanding (A1), and using the definition of cross correlation we can write,

Variable  consists of the desired signal, MAI, and noise components when user 1 sends a 0, and desired signal and noise terms when user 1 sends a 1, whereas  consists of noise only when user 1 sends a 0, and noise plus an MAI term when user 1 sends 1. The integrals involving the noise terms are zero-mean Gaussian variables with variance N₀E_s,0.

In the binary case the decision can be cast as comparing the Gaussian variable r_0,0-r_0,1 with a threshold, and using the well-known tail integral of the zero-mean, unit variance Gaussian density function—the Q-function—we can express the error probability as,

As required, when synchronized ( ), the result reduces to the well-known result for coherent binary FSK.

For the performance with three users we have two additional Q-function terms, with each of the four terms multiplied by ¼ to account for the four equiprobable possibilities for the two interfering user symbol values

By continuing this process of including additional asynchronous users, by induction we arrive at the final expression (19). In addition, since the BER equations account for the type of chirps only through the cross correlations, we also deduce that they pertain to any chirp type—linear or nonlinear—as long as we have the expressions for cross correlation to use within these BER equations. This claim is validated via our results in Section V.

Acknowledgment

The authors thank Dr. H. Jamal of the University of South Carolina for development of the AG channel routines.

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