Search Results (5)

For an undirected graph G, a zero-sum flow is an assignment of nonzero integer weights to the edges such that each vertex has a zero-sum, namely the sum of all incident edge weights with each vertex is zero. This concept is an undirected analog of nowhere-zero flows for directed graphs. We study a more general one, namely constant-sum $\mathbb{A}$-flows, which gives edge weights using nonzero elements of an additive Abelian group $\mathbb{A}$ and requires each vertex to have a constant-sum instead. In particular, we focus on two special cases: $\mathbb{A}={\mathbb{Z}}_k$, the finite cyclic group of integer congruence modulo k, and $\mathbb{A}=\mathbb{Z}$, the infinite cyclic group of integers. The constant sum under a constant-sum $\mathbb{A}$-flow is called an index of G for short, and the set of all possible constant sums (indices) of G is called the constant sum spectrum. It is denoted by $I_k\left(G\right)$ and $I\left(G\right)$ for $\mathbb{A}={\mathbb{Z}}_k$ and $\mathbb{A}=\mathbb{Z}$, respectively. The zero-sum flows and constant-sum group flows for regular graphs regarding cases $\mathbb{Z}$ and ${\mathbb{Z}}_k$ have been studied extensively in the literature over the years. In this article, we study the constant sum spectrum of nearly regular graphs such as wheel graphs $W_n$ and fan graphs $F_n$ in particular. We completely determine the constant-sum spectrum of fan graphs and wheel graphs concerning ${\mathbb{Z}}_k$ and $\mathbb{Z}$, respectively. Some open problems will be mentioned in the concluding remarks. Full article

Accurate direction of arrival (DoA) estimation is paramount in various fields, from surveillance and security to spatial audio processing. This work introduces an innovative approach that refines the DoA estimation process and demonstrates its applicability in diverse and critical domains. We propose a two-stage method that capitalizes on the often-overlooked secondary peaks of the cross-correlation function by introducing a reduced complexity DoA estimation method. In the first stage, a low complexity cost function based on the zero cyclic sum (ZCS) condition is used to allow for an exhaustive search of all combinations of time delays between pairs of microphones, including primary peak and secondary peaks of each cross-correlation. For the second stage, only a subset of the time delay combinations with the lowest ZCS cost function need to be tested using a least-squares (LS) solution, which requires more computational effort. To showcase the versatility and effectiveness of our method, we apply it to the challenging acoustic-based drone DoA estimation scenario using an array of four microphones. Through rigorous experimentation with simulated and actual data, our research underscores the potential of our proposed DoA estimation method as an alternative for handling complex acoustic scenarios. The ZCS method demonstrates an accuracy of $89.4\%\pm 2.7\%$, whereas the ZCS with the LS method exhibits a notably higher accuracy of $94.0\%\pm 3.1\%$, showcasing the superior performance of the latter. Full article

With the advancement of information technology construction, the integration of radar and communication represents a crucial technological evolution. Driven by the research boom of integrated sensing and communications (ISACs), some scholars have proposed utilizing orthogonal frequency-division multiplexing (OFDM) to separately modulate radar and communication signals. However, the OFDM symbols in this paper incorporate a cyclic prefix (CP) and a virtual carrier (VC) instead of zero padding (ZP). This approach mitigates out-of-band power caused by ZP, in addition to reducing adjacent channel interference (ACI). In addition, we introduce low-density parity-check (LDPC) and use an improved normalized min-sum algorithm (NMSA) in decoding. The enhanced decoding efficiency and minimized system errors render the proposed waveform more suitable for complex environments. In terms of signal processing methods, this paper continues to use radar signals as a priori information to participate in channel estimation. Further, we consider the symbol timing offset (STO) and carrier frequency offset (CFO) issues. In order to obtain more reliable data, we use the minimum mean-square error (MMSE) estimation based on the discrete Fourier transform (DFT) to evaluate the channel. Simulation experiments verify that the system we propose not only realizes the transmission and detection functions but also improves the performance index of the integrated signal, such as the bit error rate (BER) of 7 × 10⁻⁵, the peak side lobe ratio (PSLR) of −13.81 dB, and the integrated side lobe ratio (ISLR) of −8.98 dB at a signal-to-noise ratio (SNR) of 10 dB. Full article

As an analogous concept of a nowhere-zero flow for directed graphs, zero-sum flows and constant-sum flows are defined and studied in the literature. For an undirected graph, a zero-sum flow (constant-sum flow resp.) is an assignment of nonzero integers to the edges such that the sum of the values of all edges incident with each vertex is zero (constant h resp.), and we call it a zero-sum k-flow (h-sum k-flow resp.) if the values of the edges are less than k. We extend these concepts to general constant-sum A-flow, where A is an Abelian group, and consider the case $A={\mathbb{Z}}_k$ the additive Abelian cyclic group of integer congruences modulo k with identity 0. In the literature, a graph is alternatively called ${\mathbb{Z}}_k$-magic if it admits a constant-sum ${\mathbb{Z}}_k$-flow, where the constant sum is called a magic sum or an index for short. We define the set of all possible magic sums such that G admits a constant-sum ${\mathbb{Z}}_k$-flow to be $I_k\left(G\right)$ and call it the magic sum spectrum, or for short, the index set of G with respect to ${\mathbb{Z}}_k$. In this article, we study the general properties of the magic sum spectrum of graphs. We determine the magic sum spectrum of complete bipartite graphs $K_{m,n}$ for $m\ge n\ge 2$ as the additive cyclic subgroups of ${\mathbb{Z}}_k$ generated by $\frac{k}{d}$, where $d=gcd(m-n,k)$. Also, we show that every regular graph G with a perfect matching has a full magic sum spectrum, namely, $I_k\left(G\right)={\mathbb{Z}}_k$ for all $k\ge 3$. We characterize a 3-regular graph so that it admits a perfect matching if and only if it has a full magic sum spectrum, while an example is given for a 3-regular graph without a perfect matching which has no full magic sum spectrum. Another example is given for a 5-regular graph without a perfect matching, which, however, has a full magic sum spectrum. In particular, we completely determine the magic sum spectra for all regular graphs of even degree. As a byproduct, we verify a conjecture raised by Akbari et al., which claims that every connected $4k$-regular graph of even order admits a 1-sum 4-flow. More open problems are included. Full article

Varying compliance (VC) is an unavoidable form of parametric excitation in rolling bearings and can affect the stability and safety of the bearing and its supporting rotor system. To date, we have investigated VC primary resonance in ball bearings, and in this paper other parametric VC resonance types are addressed. For a classical ball bearing model with Hertzian contact and clearance nonlinearities between the rolling elements and raceway, the harmonic balance and alternating frequency/time domain (HB–AFT) method and Floquet theory are adopted to analyze the VC parametric resonances and their stabilities. It is found that the 1/2-order subharmonic resonances, 2-order superharmonic resonances, and various VC combination resonances, such as the 1-order and 2-order summed types, can be excited, thus resulting in period-1, period-2, period-4, period-8, period-35, quasi-period, and even chaotic VC motions in the system. Furthermore, the bifurcation and hysteresis characteristics of complex VC resonant responses are discussed, in which cyclic fold, period doubling, and the second Hopf bifurcation can occur. Finally, the global involution of VC resonances around bearing clearance-free operations (i.e., adjusting the bearing clearance to zero or one with low interference) are provided. The overall results extend the investigation of VC parametric resonance cases in rolling bearings. Full article