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Register a new userOn Complex Conjugate Pair Sums and Complex Conjugate Subspaces
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Added by
Arikatla Satyanarayana Reddy
Publication date
2021
fields
Electronic Engineering
and research's language is
English
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In this letter, we study a few properties of Complex Conjugate Pair Sums (CCPSs) and Complex Conjugate Subspaces (CCSs). Initially, we consider an LTI system whose impulse response is one period data of CCPS. For a given input x(n), we prove that the output of this system is equivalent to computing the first order derivative of x(n). Further, with some constraints on the impulse response, the system output is also equivalent to the second order derivative. With this, we show that a fine edge detection in an image can be achieved using CCPSs as impulse response over Ramanujan Sums (RSs). Later computation of projection for CCS is studied. Here the projection matrix has a circulant structure, which makes the computation of projections easier. Finally, we prove that CCS is shift-invariant and closed under the operation of circular cross-correlation.
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This letter introduces a real valued summation known as Complex Conjugate Pair Sum (CCPS). The space spanned by CCPS and its one circular downshift is called {em Complex Conjugate Subspace (CCS)}. For a given positive integer $Ngeq3$, there exists $frac{varphi(N)}{2}$ CCPSs forming $frac{varphi(N)}{2}$ CCSs, where $varphi(N)$ is the Eulers totient function. We prove that these CCSs are mutually orthogonal and their direct sum form a $varphi(N)$ dimensional subspace $s_N$ of $mathbb{C}^N$. We propose that any signal of finite length $N$ is represented as a linear combination of elements from a special basis of $s_d$, for each divisor $d$ of $N$. This defines a new transform named as Complex Conjugate Periodic Transform (CCPT). Later, we compared CCPT with DFT (Discrete Fourier Transform) and RPT (Ramanujan Periodic Transform). It is shown that, using CCPT we can estimate the period, hidden periods and frequency information of a signal. Whereas, RPT does not provide the frequency information. For a complex valued input signal, CCPT offers computational benefit over DFT. A CCPT dictionary based method is proposed to extract non-divisor period information.
Nebe, Rains and Sloane studied the polynomial invariants for real and complex Clifford groups and they relate the invariants to the space of complete weight enumerators of certain self-dual codes. The purpose of this paper is to show that very similar results can be obtained for the invariants of the complex Clifford group $mathcal{X}_m$ acting on the space of conjugate polynomials in $2^m$ variables of degree $N_1$ in $x_f$ and of degree $N_2$ in their complex conjugates $overline{x_f}$. In particular, we show that the dimension of this space is $2$, for $(N_1,N_2)=(5,5)$. This solves the Conjecture 2 given in Zhu, Kueng, Grassl and Gross affirmatively. In other words if an orbit of the complex Clifford group is a projective $4$-design, then it is automatically a projective $5$-design.
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Correlating the signals from simultaneous atom interferometers has enabled some of the most precise determinations of fundamental constants. Here, we show that multiple interferometers with strategically chosen initial conditions (offset simultaneous conjugate interferometers or OSCIs) can provide multi-channel readouts that amplify or suppress specific effects. This allows us to measure the photon recoil, and thus the fine structure constant, while being insensitive to gravity gradients, general acceleration gradients, and unwanted diffraction phases - these effects can be simultaneously monitored in other channels. An expected 4-fold reduction of sensitivity to spatial variations of gravity (due to higher-order gradients) and a 6-fold suppression of diffraction phases paves the way to measurements of the fine structure constant below the 0.1-ppb level, or to simultaneous sensing of gravity, the gravity gradient, and rotations.
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