Frequency hopping (FH) is a spread spectrum technique used to protect against detection, interception, location, and jamming where the transmission frequency is changed in a seemingly random manner, only occupying a given frequency band for a very short amount of time. FH systems provide low probability of intercept (LPI) capabilities mainly by using large hop bandwidths. Using large portions of the electromagnetic spectrum is beneficial because it makes it potentially more difficult for a third party to monitor the entire bandwidth at once. A popular way to implement FH is by using specially designed pseudorandom sequences known only to the intended users. The pseudorandom sequences must be designed according to certain mathematical properties in order to guarantee that an attacker cannot easily learn the hopping sequence and defeat the protection. Prior research has revealed an interesting equivalence relation between mathematically optimal FH sequences and partitioned difference families in cyclic groups. Using this relationship, we provide a method that yields new families of optimal FH sequences, inequivalent to known ones. The resulting FH sequence families contain several members whose underlying pseudorandom sequences possess high linear span, thereby making them desirable for secure communications. Our research shows exponential growth of the linear span, which is a significant increase in security over the state of the art (SOA).

Constant Amplitude (CA), Zero Auto Correlation (ZAC) sequences (or CAZAC sequences, aka perfect sequences) have numerous applications. We generalize the CAZAC notion to what we term as CASAC by permitting small autocorrelations (SAC). We extend Björck’s classification result of two-valued CAZAC sequences by providing a complete classification of all almost 2-valued (i.e., two-valued except for the first position which uses a third value) CASAC sequences. While Björck’s original work dealt only with primes p, we extend his ideas to any abelian group of order $v\equiv 1\pmod {4}$ , as opposed to restricting just to the prime fields GF(p). Björck sequences have better ambiguity function than Zadoff-Chu sequences, making them suitable for radar and communications applications in the presence of high Doppler shifts. In fact, the discrete narrow band ambiguity function has an optimal bound in case of Björck sequences (as opposed to Gauss sequences). A one-parameter infinite family of CASAC we construct would have applications in Multiple-Input Multiple-Output (MIMO) areas. Toward MIMO applications, we introduce a performance measure we term as cross merit factor to study cross correlation behavior, generalizing the well-known notion of Golay Merit Factor (GMF).

Recently deep learning has demonstrated much success within the fields of image and natural language processing, facial recognition, and computer vision. The success is attributed to large, accessible databases and deep learning's ability to learn highly accurate models. Thus, deep learning is being investigated as a viable end-to-end approach to digital communications design. This work investigates the use of adversarial deep learning to ensure that a radio can communicate covertly, via Direct Sequence Spread Spectrum (DSSS), with another while a third (the adversary) is actively attempting to detect, intercept and exploit their communications. The adversary's ability to detect and exploit the DSSS signals is hindered by: (i) generating a set of spreading codes that are balanced and result in low side lobes as well as (ii) actively adapting the encoding scheme. Lastly, DSSS communications performance is assessed using energy constrained devices to accurately portray IoT and IoBT device limitations.

A weighing matrix W = (wi,j) is a square matrix of order n and entries wi,j in {0,± 1} such that WWT = kIn. In his thesis, Strassler gave a table of existence results for circulant weighing matrices with n ≤ 200 and k ≤ 100. In the latest version of Strassler’s table given by Tan, there are 34 open cases remaining. In this paper we give nonexistence proofs for 12 of these cases, report on preliminary searches outside Strassler’s table, and characterize the known proper circulant weighing matrices.

We investigate how Legendre $G$-array pairs are related to several different perfect binary $G$-array families. In particular we study the relations between Legendre $G$-array pairs, Sidelnikov-Lempel-Cohn-Eastman $\mathbb{Z}_{q-1}$-arrays, Yamada-Pott $G$-array pairs, Ding-Helleseth-Martinsen $\mathbb{Z}_{2}\times \mathbb{Z}_p^{m}$-arrays, Yamada $\mathbb{Z}_{(q-1)/2}$-arrays, Szekeres $\mathbb{Z}^m_{p}$-array pairs, Paley $\mathbb{Z}^m_{p}$-array pairs, and Baumert $\mathbb{Z}^{m_1}_{p_1}\times \mathbb{Z}^{m_2}_{p_2}$-array pairs. Our work also solves one of the two open problems posed in Ding~[J. Combin. Des. 16 (2008), 164-171]. Moreover, we provide several computer search based existence and non-existence results regarding Legendre $\mathbb{Z}_n$-array pairs. Finally, by using cyclotomic cosets, we provide a previously unknown Legendre $\mathbb{Z}_{57}$-array pair.

Unlike conventional processing, we demonstrate that separation of waveforms at the receivers is indeed possible in the MIMO and multistatic radar setting in a fractional Fourier domain. We advocate for the simultaneous transmission of linear frequency modulated (LFM) waveforms. i.e. LFM chirps. In MIMO radar literature, a frequently employed assumption (either explicit or implicit) is that: orthogonality of waveforms is maintained for all delay and for all Doppler shifts. Our mathematical proof, via several theorems and corollaries, demonstrates that two waveforms may not have zero cross-correlations for all delays and Dopplers, even when they are strictly orthogonal.

We propose using periodic binary sequences with optimal correlation energy (CE) to generate near E(s(2))-optimal supersaturated designs (SSDs) and near D-optimal 2-symbol fractional factorial designs for the all main effects and the intercept model. We derive a lower bound for the CE of odd length periodic sequences and provide previously unknown odd length periodic sequences with optimal CE up to length 43.

In this article we formulate an optimization problem of minimizing the distance from the uniform van der Waerden matrices to orthostochastic matrices of different orders. We find a lower bound for the number of stationary points of the minimization problem, which is connected to the number of possible partitions of a natural number. The existence of Hadamard matrices ensures the existence of global minimum orthostochastic matrices for such problems. The local minimum orthostochastic matrices have been obtained for all other orders except for 11 and 19. We explore the properties of Hadamard, conference and weighing matrices to obtain such minimizing orthostochastic matrices.

Cheng and Tang [Biometrika, 88 (2001), pp. 1169-1174] derived an upper bound on the maximum number of columns B (n, t) that can be accommodated in a twosymbol supersaturated design (SSD) for a given number of rows (n) and a maximum in absolute value correlation between any two columns (t/n). In particular, they proved that B (n, 2) = n + 2 for n = 2 (mod 4) and n > 6. However, the only known SSD satisfying this upper bound is when n = 10. By utilizing a computer search, we prove that B (n, 2) = n + 1 for n = 18, 22, 30, and B (14, 2) = 15. These results are obtained by proving the nonexistence of certain resolvable incomplete blocks designs. The combinatorial properties of the RIBDs are used to reduce the search space. Our results improve the E (s2) lower bound for SSDs with n rows and n + 2 columns, for n = 14, 18, 22, and 30. Finally, we show that a skew-type Hadamard matrix of order n can be used to construct an SSD with n -2 rows and n -1 columns that proves B (n -2, 2) = n -1. Hence, we establish B (n, 2) = n + 1 for n = 14, 18, 22, 30 and B (n, 2) = n + 1 for all n = 2 (mod 4) such that n = 270. Our result also implies that B (n, 2) = n + 1 when n + 1 is a prime power and n + 1 = 3 (mod 4). We conjecture that n + 1 = B (n, 2) < B' (n, 2) = n + 2 for all n > 10 and n = 2 (mod 4), where B' (n, 2) is the maximum number of equiangular lines in. n-1 with pairwise angle arccos(2/n).

The optimization problems involving orthogonal matrices have been formulated in this work. A lower bound for the number of stationary points of such optimization problems is found and its connection to the number of possible partitions of natural numbers is also established. We obtained local and global optima of such problems for different orders and showed their connection with the Hadamard, conference and weighing matrices. The application of general theory to some concrete examples including maximization of Shannon, Reny, Tsallis and Sharma-Mittal entropies for orthogonal matrices, minimum distance orthostochastic matrices to uniform van der Waerden matrices, Cressie-Read and K-divergence functions for orthogonal matrices, etc are also discussed. Global optima for all orders has been found for the optimization problems involving unitary matrix constraints.

In the context of radar waveforms, there are many references to “Doppler Tolerance” in the literature, but a formal, complete, precise, and reasonable definition has not been forthcoming. We attempt to fill this void in this paper. We revisit existing definitions and demonstrate, that they are either too restrictive for any practical use, incomplete, or imprecise. Our definition uses the ambiguity function as its main ingredient. We emphasize that the Doppler tolerance is a 3D function. The first parameter is a spatial variable which relates to the measures of connectedness of possible disjoint ambiguity function peaks. The second parameter is the time delay at which the Doppler tolerance is itself specified, and the third parameter is similar to a threshold and is related to height of the ambiguity function used in measuring the Doppler tolerance. As a byproduct of our definition, we analytically conclude that for small time bandwidth products the linear frequency modulated (LFM) waveform is only as Doppler tolerant as an unmodulated rectangular pulse. We therefore bust the well known myth that “(all) chirps are Doppler tolerant”.

Binary array pairs with optimal/ideal correlation values and their algebraic counterparts textquotedblleft difference set pairstextquotedblright;(DSPs) in abelian groups are studied‎. ‎In addition to generalizing known 1-dimensional (sequences) examples‎, ‎we provide four new recursive constructions‎, ‎unifying previously obtained ones‎. ‎Any further advancements in the construction of binary sequences/arrays with optimal/ideal correlation values (equivalently cyclic/abelian difference sets) would give rise to richer classes of DSPs (and hence binary perfect array pairs)‎. ‎Discrete signals arising from DSPs find applications in cryptography‎, ‎CDMA systems‎, ‎radar and wireless communications‎.

Binary array pairs with optimal/ideal correlation values and their algebraic counterparts textquotedblleft difference set pairstextquotedblright;(DSPs) in abelian groups are studied‎. ‎In addition to generalizing known 1-dimensional (sequences) examples‎, ‎we provide four new recursive constructions‎, ‎unifying previously obtained ones‎. ‎Any further advancements in the construction of binary sequences/arrays with optimal/ideal correlation values (equivalently cyclic/abelian difference sets) would give rise to richer classes of DSPs (and hence binary perfect array pairs)‎. ‎Discrete signals arising from DSPs find applications in cryptography‎, ‎CDMA systems‎, ‎radar and wireless communications‎.

Synchronization technology is an indispensable tool in wireless communication. The probability of missing synchronization and false synchronization is a good metric to evaluate the performance of synchronization. In this paper, we utilize some new classes of ideal punctured binary sequence pairs and compare their synchronization performance with known sequences including the well-known Barker codes. Our simulation results show that the new sequences do out-perform their previously known compeers in several instances.

In this paper, we prove the nonexistence of two weighing matrices of weight 81, namely CW(88,81) and CW(99,81). We will apply two very different methods to do so; for the case of CW(88,81), we will use almost purely counting methods, while for CW(99,81), we will use algebraic methods.

We give a new existence criterion for p-ary sequences which have ideal two-level autocorrelation; and we use it to obtain four general families of such sequences: one for p=2, one for general odd primes p and two special ones for p=3. The binary family turns out to be equivalent to that discovered by Dillon and Dobbertin and published in 2004. The general p-ary family is equivalent to that discovered by Gong and Helleseth, by Dillon and, when p=3, by Helleseth, Kumar, and Martinsen. All of these p-ary results were published in 2001 and 2002. The special ternary families are new and give as special cases the sequences conjectured by Alfred Lin in his 1998 Ph.D. thesis as well as most of those conjectured in 2001 by Ludkovski and Gong. Our sequences may also be used to construct (relative) difference sets, their corresponding block designs and generalized weighing matrices.

In this paper, we present some construction methods for punctured binary array/sequence pairs (PBAPs/PBSPs) with ideal/optimal correlation constant using their algebraic counterparts punctured difference set pairs in Abelian groups. In addition, we provide new construction techniques of PBAPs/PBSPs via geometry and also using the embeddable sequence pairs of smaller lengths to obtain larger ones. PBAPs/PBSPs find a plethora of applications in radar systems, cryptography, frame synchronization, mismatched filtering, and various other engineering fields.

We define a special type of weighing matrix called block weighing matrices . Motivated by questions arising in the context of optical quantum computing, we prove that infinite families of anticirculant block weighing matrices can be obtained from generic weighing matrices. The classification problem is left open.

We provide constructions of cyclic 2-class PBIBD's using cyclotomy in finite fields. Our results give theoretical explanations of the two sporadic examples given by Agrawal (1987).