Frequency hopping sequences (FHSs) play a significant role in modern frequency hopping spread spectrum communication and radar systems. In terms of application, the aperiodic Hamming correlation (HC) holds greater significance compared to the periodic HC as it directly impacts the communication performance. In addition, it is crucial for each user’s FHS to have a substantial wide-gap (WG) in order to prevent the received signals from interfering with each other. In this letter, we obtain a new bound by extending the aperiodic bound proposed by Peng-Fan and the WG FHS bound introduced by Li-Fan-Yang-Wang. The proposed bound is strict since they can be verified using specific parameters of aperiodic WG FHSs.

In 2004, Ryoh Fuji-Hara et al. (IEEE Trans. Inf. Theory. 50(10):2408-2420, 2004) proposed an open problem of finding a maximum multiplicative subgroup G in ℤn satisfying two conditions: (1) the sum of any two distinct elements in G is nonzero; (2) any difference from G is still a unit in ℤn. The subgroups satisfying Condition (2) is called difference unit group. Difference unit group is related to difference packing, zero-difference balanced function and partitioned difference family, and thus have many applications in coding and communication. Suppose the canonical factorization of n is ∏ki=1 peii. In this letter, we mainly answer the open problem with the result that the maximum cardinality of such a subgroup G is $\frac{d}{2^m}$, where d = gcd(p1 － 1, p2 － 1, ・・・, pk － 1) and m = ν2(d). Also an explicit construction of such a subgroup is introduced.

A novel temporal convolution network-gated recurrent unit (NTCN-GRU) algorithm is proposed for the greatest of constant false alarm rate (GO-CFAR) frequency hopping (FH) prediction, integrating GRU and Bayesian optimization (BO). GRU efficiently captures the semantic associations among long FH sequences, and mitigates the phenomenon of gradient vanishing or explosion. BO improves extracting data features by optimizing hyperparameters besides. Simulations demonstrate that the proposed algorithm effectively reduces the loss in the training process, greatly improves the FH prediction effect, and outperforms the existing FH sequence prediction model. The model runtime is also reduced by three-quarters compared with others FH sequence prediction models.

The low-hit-zone (LHZ) frequency hopping sequence (FHS) sets are widely applicable in quasi-synchronous frequency hopping multiple-access (QS-FHMA) systems. In order to reduce mutual interference (MI) in the zone around the signal origin between different users, we recommend the LHZ FHS set instead of the conventional FHS set. In this letter, we propose a design of LHZ FHS sets via interleaving techniques. The obtained sequences can be confirmed that they are near-optimal in relation to the Peng-Fan-Lee bound.

In frequency hopping communication, time delay and Doppler shift incur interference. With the escalating upgrading of complicated interference, in this paper, the time-frequency two-dimensional (TFTD) partial Hamming correlation (PHC) properties of wide-gap frequency-hopping sequences (WGFHSs) with frequency shift are discussed. A bound on the maximum TFTD partial Hamming auto-correlation (PHAC) and two bounds on the maximum TFTD PHC of WGFHSs are got. Li-Fan-Yang bounds are the particular cases of new bounds for frequency shift is zero.

In quasi-synchronous FH multiple-access (QS-FHMA) systems, no-hit-zone frequency-hopping sequences (NHZ-FHSs) can offer interference-free FHMA performance. But, outside the no-hit-zone (NHZ), the Hamming correlation of traditional NHZ-FHZs maybe so large that the performance becomes not good. And in high-speed mobile environment, Doppler shift phenomenon will appear. In order to ensure the performance of FHMA, it is necessary to study the NHZ-FHSs in the presence of transmission delay and frequency offset. In this paper, We derive a lower bound on the maximum time-frequency two-dimensional Hamming correlation outside of the NHZ of NHZ-FHSs. The Zeng-Zhou-Liu-Liu bound is a particular situation of the new bound for frequency shift is zero.

The construction of frequency hopping sequences with good Hamming correlation is the foundation of research in frequency hopping communication. In this letter, classes of optimal low hit zone frequency hopping sequence set are constructed based on the interleaving technology. The results of the study show that the sequence set with large family size is optimal for the Peng-Fan-Lee bound. And all the sequences in the set are inequivalent.

In quasi-synchronous frequency-hopping multiple access (QS-FHMA) systems, relative delays are allowed to vary in a domain around the origin. Under such condition, the low hit zone (LHZ) frequency-hopping sequence (FHS) set is more propitious than the conventional FHS set to be applied by the systems. In this paper, a construction based on the interleaving techniques of FHS set with LHZ is proposed. Besides the requirement for this constructed LHZ FHS set to get the optimality or the near optimality with respect to the Peng-Fan-Lee bound is also given. It turns out that the constructed LHZ FHS set has new parameters not covered in the literature, thus it does have great significance in practice.

New classes of zero-difference balanced (ZDB) functions derived from Fermat quotients are proposed in this letter. Based on the new ZDB functions, some applications, such as the construction of optimal frequency hopping sequences set and perfect difference systems of sets, are introduced.

In this paper, with a modification of our earlier construction in [12], new classes of optimal LHZ FHS sets with new parameters are obtained which are optimal in the sense that their parameters meet the Peng-Fan-Lee bound. It is shown that all the sequences in the proposed FHS sets are shift distinct. The proposed FHS sets are suitable for quasi-synchronous time/frequency hopping code division multiple access systems to eliminate multiple-access interference.

For frequency hopping spread spectrum communication systems, the average Hamming correlation (AHC) among frequency hopping sequences (FHSs) is an important performance indicator. In this letter, a sufficient and necessary condition for a set of FHSs with optimal AHC is given. Based on interleaved technique, a new construction for optimal AHC FHS sets is also proposed, which generalizes the construction of Chung and Yang. Several optimal AHC FHS sets with more flexible parameters not covered in the literature are obtained by the new construction, which are summarized in Table 1.

In order to evaluate the goodness of frequency hopping (FH) sequence design, the periodic Hamming correlation function is used as an important measure. But aperiodic Hamming correlation of FH sequences matters in real applications, while it received little attraction in the literature compared with periodic Hamming correlation. In this paper, the new aperiodic Hamming correlation lower bounds for FH sequences, with respect to the size of the frequency slot set, the sequence length, the family size, the maximum aperiodic Hamming autocorrelation and the maximum aperiodic Hamming crosscorrelation are established. The new aperiodic bounds are tighter than the Peng-Fan bounds. In addition, the new bounds include the second powers of the maximum aperiodic Hamming autocorrelation and the maximum aperiodic Hamming crosscorrelation but the Peng-Fan bounds do not include them. For the given sequence length, the family size and the frequency slot set size, the values of the maximum aperiodic Hamming autocorrelation and the maximum aperiodic Hamming crosscorrelation are inside of an ellipse which is given by the new aperiodic bounds.

This paper proposes applying intra-subframe frequency hopping (FH) to closed-loop (CL) type transmit diversity using codebook based precoding for a shared channel carrying user traffic data in discrete Fourier transform (DFT)-precoded Orthogonal Frequency Division Multiple Access (OFDMA). In the paper, we present two types of precoding schemes associated with intra-subframe FH: individual precoding vector selection between 2 slots where a 1-ms subframe comprises 2 slots among the reduced precoding codebooks, and common precoding vector selection between 2 slots. We investigate the effect of intra-subframe FH on the codebook based transmit diversity in terms of the average block error rate (BLER) performance while maintaining the same number of feedback bits required for notification of the selected precoding vector as that for the conventional CL transmit diversity without FH. Computer simulation results show that the codebook based transmit diversity with intra-subframe FH is very effective in decreasing the required average received signal-to-noise power ratio (SNR) when the fading maximum Doppler frequency, fD, is higher than approximately 50 Hz both for 2- and 4-antenna transmission in the DFT-precoded OFDMA.

In this paper, new design of optimal frequency hopping sequences (FHSs) with low hit zone (LHZ) with respect to the Peng-Fan-Lee bound is presented based on interleaving techniques. By the new design, new classes of optimal LHZ FHS sets with large family size are obtained. It is shown that all the sequences in the proposed FHS sets are shift distinct. The proposed FHS sets are suitable for quasi-synchronous time/frequency hopping code division multiple access systems to eliminate multiple-access interference.

In this letter, we determine the linear complexity and minimum polynomial of the frequency hopping sequences over GF(q) introduced by Chung and Yang, where q is an odd prime. The results of this letter show that these sequences are quite good from the linear complexity viewpoint. By modifying these sequences, another class of frequency hopping sequences are obtained. The modified sequences also have low Hamming autocorrelation and large linear complexity.

Hop-timing detection is of extreme importance for the reception of frequency hopping (FH) signals. Any error in the hop-timing detection has a deleterious effect on the performance of the receiver in frequency hopping (FH) communication systems. However, it is not easy to detect the hop-timing under low signal to noise power ratio (SNR) environments. Adaptive array antennas (AAA) have been expected to improve the performance of FH communication systems by beamforming for the direction of arrival of the desired signal. Since the conventional AAA exploits at least the coarse synchronization for dehopping of FH signals before achieving the beamforming, any fault in the hop-timing detection causes the deterioration of the performance of AAA. Using AAA based on the constant modulus algorithm (CMA), this paper proposes a new method for blind beamforming and hop-timing detection for FH signals. The proposed method exploits both the spatial and temporal characteristics of the received signal to accomplish the beamforming and detect the hop-timing without knowing any a priori information such as fine/coarse time synchronization and training signal. The performance verifications of the proposed method based on pertinent simulations are presented.

In order to evaluate the goodness of frequency hopping sequence design, the periodic Hamming correlation function is used as an important measure. Usually, the length of correlation window is shorter than the period of the chosen frequency hopping sequence, so the study of the partial Hamming correlation of frequency hopping sequence is particularly important. In this paper, the maximum partial Hamming correlation lower bounds of frequency hopping sequences with low hit zone, with respect to the size of the frequency slot set, the length of correlation window, the family size, the low hit zone, the maximum partial Hamming autocorrelation and the maximum partial Hamming crosscorrelation are established. It is shown that the new bounds include the known Lempel-Greenberger bound, Peng-Fan bounds, Eun-Jin-Hong-Song bound and Peng-Fan-Lee bounds as special cases.

In order to evaluate the goodness of frequency hopping sequence design, the aperiodic Hamming correlation function is used as an important measure. In this letter, the aperiodic Hamming correlation lower bounds for frequency hopping sequences with low hit zone which have not yet been reported previously are established.

In this paper, we propose a novel frequency-hopping scheme in order to improve the BER (Bit Error Rate) performance of the Partial Block MC-CDMA (PB/MC-CDMA) system. The joint carrier distribution and frequency hopping (JDFH) scheme achieves the optimal frequency diversity gain while avoiding interference. By contrast, the conventional FH scheme only avoids interference, and the frequency interleaving scheme achieves only frequency diversity. The JDFH scheme thus performs better than conventional schemes, such as carrier FH, block FH, or frequency interleaving. Through computer simulations, we confirmed the superior performance of the PB/MC-CDMA system when using the JDFH scheme.

In this paper, we propose a new modulation named parallel combinatory/high compaction multi-carrier modulation (PC/HC-MCM) using the techniques of parallel combinatory orthogonal frequency division multiplexing (PC-OFDM) and high compaction multi-carrier modulation (HC-MCM). Two types of PC/HC-MCM systems, which are named as modulated PC/HC-MCM system and (unmodulated) PC/HC-MCM system, can be designed. The modulated PC/HC-MCM system achieves better bit-error rate (BER) performance than that of HC-MCM system with equal bandwidth efficiency (BWE). The PC/HC-MCM system can obtain the better peak-to-average power ratio (PAPR) characteristics by selecting appropriate constellation for each subcarrier. On the other hand, since PC/HC-MCM can divide the PC-OFDM symbol duration into multiple time-slots, the advantages of frequency hopping (FH) can be applied in the PC/HC-MCM system. Therefore, we also combine the PC/HC-MCM and frequency hopping multiple access (FHMA) to propose a novel multiple access (MA) system. It can simultaneously transmit multiple users' data within one symbol duration of PC-OFDM.