Golden Modulation: a New and Effective Waveform for Massive IoT

This paper considers massive Internet of Things systems, especially for LoW Power Wide Area Networks, that aim at connecting billions of low-cost devices with multi-year battery life requirements. Current systems for massive Internet of Things exhibit severe problems when trying to pursue the target of serving a very large number of users. In this paper, a novel asynchronous spread spectrum modulation, called Golden Modulation, is introduced. This modulation provides a vast family of equivalent waveforms with very low cross-interference even in asynchronous conditions, hence enabling natural multiuser operation without the need for inter-user synchronization or for interference cancellation receivers. Thanks to minimal interference between waveforms, coupled with the absence of coordination requirements, this modulation can accommodate very high system capacity. The basic modulation principles, relying on spectrum spreading via direct Zadoff-Chu sequences modulation, are presented and the corresponding theoretical bit error rate performance in an additive white Gaussian noise channel is derived and compared by simulation with realistic Golden Modulation receiver performance. The demodulation of the Golden Modulation is also described, and its performance in the presence of uncoordinated multiple users is characterized.


I. INTRODUCTION
M Assive access for the Internet of Things (IoT) has been investigated for quite some time (see, e.g., [1]) and is a hot research topic for current and future systems, such as 6G (see, e.g., [2]).Solutions avoiding the need for device coordination (which typically requires significant energy and communications resources) are considered as particularly promising (see, e.g., [3]).For IoT systems operating in the licensed spectrum, of which NB-IoT [4] is currently the most popular solution, the research focuses on access schemes, and the waveform design is based on variations of multi-carrier modulations (see [5] in [6]).Instead, the waveforms used for IoT systems operating in the unlicensed spectrum exhibit a wide variety.Traditional solutions range from the usual Cyclic Prefix-OFDM multi-carrier modulation for the DECT-2020 New Radio [7] to ultranarrowband D-BPSK (Differential Binary Phase-Shift Keying) for the SigFox systems [8].A new spread spectrum waveform emerged with the LoRa/LoRaWAN ® system [9], [10], which gained strong momentum and, at the time of writing, is worldwide the most A short (4-page) version of this manuscript has been submitted to IEEE MedComNet 2023.
popular system for massive IoT in unlicensed bands.The wide deployment of LoRaWAN networks highlighted some critical aspects related specifically to situations in which the density of End Nodes (ED) is very high [11] [12].The ability of LoRa/LoRaWAN to multiplex concurrent data transmissions from the EDs relies on the availability of different "orthogonal" waveforms, also referred to as logical channels.When the density of EDs becomes high, the number of logical channels becomes a bottleneck, making the network congested.The fact that the "orthogonal" waveforms are actually not fully orthogonal (see [13]) makes the problem even worse.
From the above discussion, it emerges that a major open problem in massive access for the Internet of Things (IoT) lies fundamentally in the design of the waveform, which must be such that a very large number of distinguishable (ideally orthogonal) channels are available.In the LoRa systems, in each frequency channel separate logical channels are obtained by using different Spreading Factors (defined as the number of transmitted samples per information symbol [9], [10]), whose number is limited to 6 since they can take a value in the set {7, 8, . . ., 12} [14].It must be pointed out that the research for good linear frequency modulated signals, such as those used in LoRa, is still active [15].
The main contribution of this paper is the design and evaluation of a novel set of waveforms, called Golden Modulation (GM), which departs conceptually from linear frequency modulated signals, a characteristic of the LoRa Modulation, and allows a huge increase in the number of parallel channels, compared to the state of the art mentioned above.GM provides both leading edge sensitivity and very high multi-user capacity.This modulation supports flexible bandwidth and spreading factor as well as very good uncoded sensitivity.In addition, it provides a vast family of equivalent waveforms with very low interference between them even in asynchronous conditions.More precisely, it achieves very low interference between noncoordinated users, hence enabling natural multiuser operation without the need for inter-user synchronization or for interference cancellation receivers.Thanks to minimal interference between waveforms, coupled with the absence of coordination requirements, this modulation can accommodate very high system capacity simply relying on the multiplexing between autonomously operating users, separated by their waveforms, freely transmitting whenever they need to.As a comparison, LoRa modulation provides one waveform per spreading factor while hundreds or even thousands of available waveforms (see Table II) are possible using Golden Modulation.
GM is based on a novel and generalized perspective on the use of the well known Zadoff-Chu sequences.This feature is enabled by using the Zadoff-Chu sequences to carry information rather than (as it is usually done) for synchronization and/or channel estimation in the headers of a packet in a digital communication link (see, e.g., [16]).In particular, although the use of Zadoff-Chu sequences was recently discussed for signaling in IoT scenarios [17], [18], we are the first to recognize their potential to provide a large number of distinct logical channels, thereby providing an effective solution for true IoT massive access.The efficient implementation of GM is also described in this paper at both the transmitter and the receiver sides.
The paper is organized as follows.In Section II the system requirements for Massive IoT systems are reviewed, with emphasis on the effect of network synchronization on the possibility to fulfill the requirements.In Section III, the notation used in the paper is introduced.In Section IV, a short review and description are provided for the frequency shifted Zadoff-Chu Sequences, which are employed in GM.In Section V, the use of GM in a single user system is presented.Section VI describes the GM receiver.In Section VII, the use of GM in the general setting of multiple users with multiple spreading factors is presented.In Section VIII, GM is framed in the general perspective of a system including higher layers on top of the physical layer.Finally, conclusions are drawn in Section IX.

II. MASSIVE IOT SYSTEM REQUIREMENTS
Massive IoT aims at connecting billions of low-cost devices with multi-year battery life.To achieve such stringent requirements, some key bottlenecks must be removed: in particular, the system capacity shortage faced in current LPWAN networks, both in unlicensed and licensed spectrum, must be solved.An overall capacity several orders of magnitude higher than what is available in current state-of-the-art LPWANs is required in order to enable the expected number of connections per IoT gateway.In addition, the device's power consumption must be minimized to enable multi-year battery life.The device power consumption is driven on one side by its required transmit power and on the other side by the energy required for synchronization onto the network.The required degree of network synchronization varies drastically depending on the physical layer and associated link layer of the system: OFDM-based systems such as NB-IoT, LTE cat M or the recently introduced DECT 2020 NR require stringent time and frequency synchronization among all devices, coupled with strict resource request and assignment mechanisms in order to maintain orthogonality of the PHY signals and to avoid packet collisions.This has a significant impact not only in terms of energy consumption but also in terms of protocol overhead, since specific channels for downlink synchronization, resource request and resource grant are then required, and consume radio resources.On the contrary, non-cellular LPWAN systems such as LoRa or SigFox can operate nearly without network synchronization.It is interesting to note that, due to the characteristics of massive IoT traffic, where each device transmits infrequently small amounts of data, the energy needed for network synchronization may become dominant over the energy required to effectively transmit data packets.This is especially true for cellular-based systems such as NB-IoT (also known as 5G IoT) or LTE cat M, where the device is a slave to the network which fully controls the access to the medium, implying high power consumption and high control overhead, hence reducing further the effective capacity of the system.A comparison between LoRaWAN and NB-IoT has been performed in [19], showing that LoRaWAN and NB-IoT have similar performance in terms of network capacity, with NB-IoT slightly outperforming LoRaWAN for small cells, at the cost of a significantly higher energy consumption even without considering the energy cost of network synchronization.The impact of NB-IoT synchronization on the energy budget for each NB-IoT Uplink data packet transmission in real conditions has been analyzed in [20].Overall, it seems reasonable to state that a scalable massive IoT system must avoid device network synchronization to make the device operation as autonomous as possible.It is hence of crucial importance to have a PHY layer able to operate with minimum or no device synchronization requirements, while ensuring in addition high sensitivity and high capacity.

III. NOTATION
For the sake of readability, in this section we provide a summary of the notation used in this paper.
A generic signal in the continuous time domain is denoted as x(t), t ∈ R, where R is the set of real numbers and x(t) ∈ R or x(t) ∈ C, where C is the set of complex numbers.
A generic discrete time signal x d (n) can be denoted as x(nT c ) where n ∈ Z and T c is the temporal interval separating each sample of the signal.While we are well aware that discrete time signals are often denoted (with a slight abuse of notation) simply as x(n), our notation is useful in this paper since we will be dealing with discrete signals with different temporal intervals.Dropping T c would possibly create confusion, while adding it improves readability at a minimal cost in terms of complexity of the notation.
Our notation is useful also when dealing with signals characterized by "temporal blocks."We refer to the situation where a signal is made by appending sets of samples of cardinality N .For example, each frame can result from the modulation of a signal s( T ) carrying a symbol from the set S (e.g., QAM symbols) in the th symbol time T : the temporal quantum between different signal samples of the modulated signal is T c = T N where N is the interpolating factor.So we can write the modulated signal as x(nT c ).Our notation is particularly useful when we want to refer to a certain part of the signal, such as the part of the signal modulated by the th symbol: we On the other hand, when no temporal information is involved, we use the common notation where a sequence (such as a Zadoff-Chu sequence) is simply denoted by x d (n).
We refer the reader to [21] for a general treatment of signal theory according to our notation, and to [22] for a paper on LoRaWAN using the proposed notation.Table I provides a reference summary of our notation, for the convenience of the reader.
Finally, in the rest of the paper, we denote as link the communication between a single transmitter and a (single) receiver.

IV. THE FREQUENCY SHIFTED ZADOFF-CHU SEQUENCES
According to [23], we can express the frequency shifted Zadoff-Chu Sequences as follows where q ∈ Z , N ∈ N, N = 0, r ∈ N, gcd(r, N ) = 1, and k = 0, . . ., N −1.Usually, r is called the "root" of the sequence and N its length.It must be noted that, although in general q ∈ Z, meaningful values for q are in the set {0, 1, . . ., N − 1}.For example, when q ≥ N , Equation (1) becomes From Equations ( 9) and (10) in [23], we have that the cyclic autocorrelation of the sequence Z r,q N (•) is non-zero only for a shift equal to 0. From [24], we have that by selecting N odd and any pair of roots r 1 , r 2 such that gcd(r 1 − r 2 , N ) = 1, the cyclic cross-correlation between the two Zadoff-Chu sequences Z r1,q N (•) and Z r2,q N (•) is equal to1 √ N .In particular, if N is a prime number, (r 1 − r 2 ) is always relatively prime to N , whichever the selected pair of roots (r 1 ,r 2 ), 1 and the whole set of possible Zadoff-Chu roots can be used with minimal cross-correlations between the corresponding sequences.

A. Discrete Time Signal Analysis
We suppose to have a source of M -ary symbols M < N (see Section IV and Equation ( 2)), emitting symbols at rate R. We indicate the generic symbol in the symbol period T as q( T ) = q ∈ {0, 1, . . ., M − 1}, T = 1/R, ∈ N. We denote the "chip" period as T c = T /N .Then, the discretetime modulated GM signal in the symbol period T is In order to obtain an expression of the modulated signal highlighting its dependence on the set {q , N, r} defining the parameters and the symbol, we rewrite Eq. ( 3), providing an alternative expression for the modulated signal, as For a generic time nT c , defining (n = n mod N , we can write the discrete time modulated signal as These alternative expressions for the modulated signals will be needed in the following, depending on the context, for example to assess the orthogonality or in Section VII.
As in [10], we set T c = 1 B , where B is the minimum bandwidth occupied by a GM link.As a consequence, the symbol duration T is T = N B , so that the symbol rate is R = B N and the bit rate We should note that by selecting N as a power of 2, we can define by analogy the Spreading Factor for GM as If N is not a power of 2, SF in Equation ( 6) results to be a non integer number; in that case we define where the operator • performs the rounding to the closest integer.
GM demodulation is based on the orthogonality of Zadoff-Chu sequences with the same root and offsets q, q .We have that (see Section IV) so the discrete-time signal space for the GM modulation admits as an orthonormal basis the set of sequences 1 √ N x(kT c ; q, N ; r) indexed by the possible transmitted symbol q ∈ {0, 1, . . ., M − 1} and for the running time index kT c , k ∈ {0, 1, . . ., N − 1}.We then observe that we are in the same conditions as in [25]; by re-using the same arguments and the proof as in [25], Sections II and III-A, in our case (see the Appendix for details), the BER performance in an Additive White Gaussian Channel (AWGN) can be accurately approximated by where Γ is the Signal-to-Noise Ratio (SNR) and is the complementary Gaussian distribution function.

B. Continuous Time Signal Analysis
To be transmitted, the discrete-time GM signals must be converted in the continuous time domain through a pulse shaping filter g(t).The complete chain to produce the analog continuous time GM modulated signal is shown in Fig. 1.A practical choice for the pulse shaping filter g(t) typically results in a signal bandwidth B ch > 1 Tc .For example, for the widely used root raised cosine waveform, we have B ch = (1 + β)B, where β is the roll-off factor and determines the excess bandwidth.The performance of the Golden modulation in an AWGN channel is provided in Fig. 2 for the case of ideal Nyquist filtering, where the uncoded Bit Error Rate (BER) obtained via computer simulations is compared to that obtained via the analytical approximation of Equation (9).Simulation results performed for Golden Modulation with SF 8, 12 and 16 show a very good match with the analytical curves.
To conclude the characterization of the Golden Modulation, in Fig. 3 the simulation result of the Power Spectral Density for a bandwidth of 100 kHz and SF12 is shown.

VI. THE RECEIVER
Ignoring for the sake of simplicity the effect of the transmission channel and assuming N odd, to demodulate the symbol transmitted in the period T we consider the received signal where w(•) is a Complex Additive White Gaussian Noise (CAWGN), with variance σ 2 w .Since the receiver is assumed to know the parameters r and N , it can compute the signal where The value of q can be obtained by forming the N dimensional vector computing Z( T ) = DFT(Y), and taking the index of the entry with largest magnitude.Notably, if N is prime the DFT can still be made at a complexity similar to the usual radix-2 one via the FFTW algorithm [26].This complexity is the same as in LoRa systems.

VII. GOLDEN MODULATION: MULTIPLE ACCESS AND MULTIPLE RATE
From Section V, one can see that for any link employing the Golden Modulation we have three degrees of freedom (by considering fixed both the bandwidth B and the pulse shaping filter), namely: • the root r; • the length N ; and • the cardinality of the modulation M = 2 b < N .
We suppose, as a first step, to fix the length N as odd and the cardinality M = 2 b < N .This is equivalent to fixing the rate of the information source whose output is carried in the link.We can see from Eq. ( 4) that for the transmission of the symbols we still have one degree of freedom, i.e., the root r, which can be used to separate transmissions issued by different end devices, as discussed in the following.

A. Multiple users with the same Spreading Factor
In this subsection we show how we can multiplex in a single channel of band B ch multiple users with the same rate.
We know from Section IV that, by selecting N odd and any pair of distinct roots r 1 , r 2 such that gcd(r 1 − r 2 , N ) = 1, the signals in the time interval { T, ( + 1)T c , . . .+ ( + N − 1)T c )}, namely x( T + kT c ; q (1) , N ; r 1 ) and x( T +kT c ; q (2) ; N ; r 2 ), have their cyclic cross-correlation constant and equal to 1 √ N .Given N prime, a set of K ≤ N − 1 distinct roots R N,K = {r 1 , r 2 , . . .r K } can be selected such that gcd(r i − r j , N ) = 1 for r i , r j ∈ R N,K , i = j.Any link such as the one described in Section V, using the root r i ∈ R N,K , is then "almost" orthogonal to any other link of the same type using a different root r j ∈ R N,K , i = j.More precisely, the signals x(nT c ; q (i) , N ; r i ) and x(nT c ; q (j) , N ; r j ) have a low (equal ) cyclic cross-correlation and the symbols q (i) and q (j) transmitted by the two links can be recovered by a suitable receiver (described in Section VI) for SNR in the region of operation of the system.This property makes it possible to define up to N − 1 different links, thereby enabling massive access in IoT systems.
In a given frequency channel, a user is characterized by its root r and its spreading factor SF , and the level of interference between users directly depends on the circular crosscorrelation between their respective signals x(nT c ; q (i) , N ; r i ) and x(nT c ; q (j) , N ; r j ).Golden Modulation is designed to exploit the conditions for perfect auto-correlation and minimal cross-correlation properties of Zadoff-Chu sequences equal respectively to 1 and 1/ √ N .
The above performance applies identically to the full set of Zadoff-Chu sequences of length N as illustrated by Fig. 4, which shows the cross-correlation results between any two sequences from a set of 100 Zadoff-Chu sequences of length 2053.The set of "quasi orthogonal" Golden Modulation signals to be used by devices is hence very large (up to N −1) and increases with the spreading factor.Similarly, the orthogonality level between users increases with the spreading factor as shown in Table II, where "interference rejection" is the highest value of the cross-correlation between a symbol with a certain SF and any other symbol with the same SF. 2 remark on Figure 4 is in order.As expected, when the two sequences are the same, i.e., i = j, we observe a large peak equal to N .In all other cases (i.e., i = j) the maximum peak is much smaller (with a value of √ N ) and independent of i and j (i.e., it is flat, as can be observed from Figure 4).
It must be noted that comparable performance can be obtained when considering two "uncoordinated" links, i.e., links that are not aligned in time.In this case, since signals issued by different users reach the receiver with independent timing, the underlying Zadoff-Chu sequences of the different links overlap asynchronously and interference will appear at the correlator output, depending on their aperiodic cross-correlation.Correlation properties of polyphase sequences have been extensively studied in the literature, e.g., see [27], [28], [29] and [30], and [31] eventually derived the aperiodic cross-correlation bounds for Zadoff-Chu sequences.These bounds show the very good aperiodic correlation properties of Zadoff-Chu sequences, as illustrated in the example of Fig. 5 showing the maximum interference level experienced by a target link with SF = 9 due to all possible other uncoordinated interfering signals with SF = 9.
Fig. 6 shows that the maximum interference level between different GM signals with the same SF is almost independent of the fractional T c delay.In particular, to study the effect of fractional misalignment between the two sequences, we plotted the cross-correlation values for all time-shifts that are multiples of T c /4.While for time-shifts multiples of T c the normalized interference level is 1/ √ N ( 0.438 for N = 521 in this case, represented by the horizontal dashed line), for fractional time-shifts the possible values of the interference level vary within the range from 0 to about 0.65, showing that fractional misalignment has a modest price in terms of a (uniformly bounded) interference increase.
Golden Modulation leverages these low cross-correlation properties of the filtered Zadoff-Chu sequences to provide ultra-high capacity without compromising range or power consumption.This is a major advantage from a system point of view since multiple devices can transmit simultaneously using different Zadoff-Chu roots without the need for any synchronization between the different transmissions, hence avoiding the need for complex and energy consuming synchronization mechanisms.This permits to increase by several orders of magnitude the system capacity of IoT networks without paying any penalty in terms of single link performance, power  The performance of GM in a multi user configuration is presented for SF 12 in Fig. 7, where packets from the different users are received with equal power and random delays.It can be seen that the demodulation performance of Golden Modulation remains very good in the presence of several tens of simultaneous interfering devices while LoRa is limited to the reception of a single packet per SF (provided that it is free Last but not least, this major capacity boost is obtained without the need for advanced receivers thanks to the "quasi orthogonality" of the frames received from the different users.In particular, the received signals can be demodulated by single user receivers without the need for complex interference cancellation techniques.More precisely, we can state that, considering two links with the same rate x(nT c ; q (i) (n) , N ; r i ) and x((n + n 0 )T c ; q (j) (n) , N ; r j ) with 0 ≤ n 0 < N and r i = r j , the respective sequences of symbols q (i) (n) and q transmitted by the two links can be recovered by a suitable receiver without the need for costly interference cancellation algorithms nor complex channel coding schemes.

B. Multiple users with different Spreading Factors
In this subsection we consider multiple users with different Spreading Factors, to explain the case when two users may use two generic lengths N 1 and N 2 , not necessarily equal, using roots r i ∈ R N1,K1 and r j ∈ R N2,K2 .The generic two users may also be uncoordinated, i.e., not aligned in time.We note that this case corresponds to the actual situation where End-Nodes (ED) in a Low Power Wide Area Network (LPWAN) autonomously trigger their transmissions at the rate suitable for their respective link budget and the time suitable for their respective application.
Golden Modulation, relying on the properties of its underlying Zadoff-Chu sequences, exhibits a "quasi-orthogonality" between signals with different Spreading Factors as well, whichever their roots.As an example, the interference generated by a signal with different Spreading Factor (i.e., different N ) is shown in Fig. 8, which plots the output of the receiver that results from the superposition of an intended signal and an uncoordinated interferer operating with SF = 11 and SF = 10, respectively.The intended signal produces the expected peak of value N = 2053 at the output of the correlator according to its time-shift (around 1000 samples in this case) and zero autocorrelation otherwise, whereas the interferer produces low cross-correlation values for all possible time-shifts, confirming the good cross-correlation properties of these waveforms even for different spreading factors.The Golden Modulation frame is made of a series of concatenated Zadoff-Chu sequences covering a preamble and a payload section (see the Golden Modulation frame format in Fig. 9).Frames issued by different devices can advantageously use different roots or sets of roots; this allows easy multiuser reception without the need for device coordination.It must be highlighted that there are N − 1 possible roots for Zadoff-Chu sequences of length N , hence providing the potential for the generation of thousands of "quasi orthogonal" frames with the same or different SF , totally unlocking the IoT networks capacity.In addition, the ability to support the highly desirable uncoordinated mode of operation guarantees robustness and extremely low protocol overhead.
It must be noticed that, despite its baseline design using oddlength Zadoff-Chu sequences, Golden Modulation can also be implemented based on even-length sequences.For example, an implementation based on truncated odd length Zadoff-Chu sequences leading to an effective sequence length equal to a power of 2 is also possible.In the latter case, the degradation of GM's orthogonality is marginal, as shown in Fig. 10, and has no noticeable effect on its performance.Again, looking at Figure 10, we observe that for i = j the maximum peak is small and almost independent of the indices of the two sequences.
An implementation based on truncated odd-length Zadoff-Chu sequences leading to an effective sequence length equal to a power of 2 is interesting for cases where adaptation to processing units or to size-constrained lower layer protocols is required.The latter case is actually very much of practical use.For example, it is straightforward to implement the LoRaWAN protocols on top of GM with Zadoff-Chu truncated sequences to achieve an effective sequence length equal to a power of 2. As a matter of fact, once the value of the SF for the LoRaWAN is selected, the same SF can be selected for GM.Moreover, once a certain bandwidth B is selected for the LoRaWAN protocol, the same bandwidth B can be selected for GM.All of the specifications (for example [14]) refer to bytes for the fields of any message and the messages can then be immediately translated in bytes carried by GM.As one can notice from Figure 9, the structure (preamble followed by data) is the same.All the "regional parameters" as per [32] are automatically reusable (for the same bandwidth B), ensuring compliance with the regulations worldwide.In general, we can safely state that the complete LoRaWAN system of specifications (including the backend, the Firmware Update Over The Air (FUOTA), etc.) becomes completely re-usable.For the commercial viability of GM, this represents a decisive strength.Of course, here LoRaWAN is just an example, although important, and the byte oriented transmission made possible by the use of an effective sequence length equal to a power of 2 opens the possibility to use GM in many other systems.

IX. CONCLUSION
In this paper, we introduced Golden Modulation, a novel modulation with ultra-long range capability and massive capacity.The basic modulation principles relying on spectrum spreading via direct Zadoff-Chu sequences modulation have been presented and the corresponding theoretical BER performance in AWGN channel has been derived and compared by simulation with realistic Golden Modulation receiver performance.A basic low complexity DFT-based receiver has been described.The fundamentals of multi-user operation have been described and the corresponding multi-user interference evaluated from both a theoretical and a simulation point view.The very low multi-user interference level, even in uncoordinated conditions, coupled with the numerous available waveforms, unlocks the well-known capacity bottleneck of IoT networks.Overall, low complexity receiver, flexible frame structure and minimal protocol constraints pave the way for the deployment of LPWAN networks with truly massive capacity, or for the upgrade of existing ones.Furthermore, the possibility to use an effective sequence length equal to a power of 2 makes GM suitable as a physical layer for many existing protocol suites.

APPENDIX CLOSED FORM EXPRESSION OF GOLDEN MODULATION BER
According to Equation (8), the discrete-time signal space for the Golden Modulation admits as an orthonormal basis the set of sequences 1 √ N x(kT c ; q, N ; r) indexed by the possible transmitted symbol q ∈ {0, 1, . . ., M − 1} and for the running time index kT c , k ∈ {0, 1, . . ., N −1}.One can easily see that the detection problem is the same (apart from the notation) as in [25] As demonstrated in [25], then the bit error probability can be expressed as where σ 2 = N0 2 , with N 0 the single-sided noise power spectral density, and f β (β) is the probability density function for the Rician distributed β with shape parameter κ β = E S N0 = SF N0 .Further noticing the very high spreading gains used and applying Gaussian approximations, [25] derives an approximation of the corresponding bit error probability valid for high spreading gains where Γ is the SNR.This expression is also valid for GM.Corroborating this statement, the simulations performed with SF = 8, 12, 16 show a very good match with the curves predicted by Equation ( 14), as shown in Fig. 2.

ACKNOWLEDGMENTS
The authors would like to thank Prof. Fabien Ferrero from University of Nice (UCA-LEAT) for the constructive discussions and for his support for the experimental validations of the Golden Modulation performed in his lab premises.This

Fig. 3 .
Fig. 3. Power spectral density of the Golden Modulation for a bandwidth of 100 kHz and SF=12

Fig. 4 .
Fig. 4. Cyclic cross-correlation maximum peaks for N = 2053 Zadoff-Chu sequences Z r i ,0 N (k) and Z r j ,0 N (k) (subset of 100 roots); indices i and j are plotted on the two horizontal axes.

Fig. 5 .
Fig. 5. Crosscorrelation maxima between symbols of an SF = 9 link and any uncoordinated (not time aligned) interfering link with SF = 9 and a different root

Fig. 8 .
Fig. 8. Correlator output for an SF = 11 link in the presence of an uncoordinated interfering link with SF = 10

Fig. 10 .
Fig. 10.Cyclic cross-correlation maximum peaks for the N = 2053 Zadoff-Chu sequences Z r i ,0 N (k) and Z r j ,0 N (k) truncated to N = 2048 (subset of 100 roots); indices i and j are plotted on the two horizontal axes.

TABLE I A
SUMMARY OF THE NOTATION

TABLE II NUMBER
OF GOLDEN MODULATION WAVEFORMS AND INTERFERENCE LEVELS PER SF.