Toeplitz and Circulant Matrices: A Review

Gray, Robert M.

doi:10.1561/9781933019680

Cited by 518 publications

(832 citation statements)

References 17 publications

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“…To analyze the asymptotic behavior of Toeplitz matrices, we define an N × N Toeplitz matrix T N whose elements t k satisfy ∞ k=−∞ |t k | < ∞. According to [32], T N is equivalent to a circulant matrix as N → ∞, and can be expressed as T N (t(λ)) where t(λ) = ∞ k=−∞ t k e jλk . Now we show that H θ is asymptotically equivalent to T N (t(λ)).…”

Section: B Training Sequence Design Via the Crbmentioning

confidence: 99%

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Optimal Training for Residual Self-Interference for Full-Duplex One-Way Relays

Tepedelenlioğlu

Şenol

2018

IEEE Trans. Commun.

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Full-duplex relays, residual self-interference, maximum likelihood estimation, optimal training sequence, Toeplitz matrix, frequency-selective channels, multiple relays AbstractChannel estimation and optimal training sequence design for full-duplex one-way relays are investigated.We propose a training scheme to estimate the residual self-interference (RSI) channel and the channels between nodes simultaneously. A maximum likelihood estimator is implemented with Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. In the presence of RSI, the overall source-to-destination channel becomes an intersymbol-interference (ISI) channel. With the help of estimates of the RSI channel, the destination is able to cancel the ISI through equalization. We derive and analyze the Cramer-Rao bound (CRB) in closed-form by using the asymptotic properties of Toeplitz matrices. The optimal training sequence is obtained by minimizing the CRB. Extensions for the fundamental one-way relay model to the frequency-selective fading channels and the multiple relays case are also considered. For the former, we propose a training scheme to estimate the overall channel, and for the latter the CRB and the optimal number of relays are derived when the distance between the source and the destination is fixed. Simulations using LTE parameters corroborate our theoretical results.Xiaofeng Li and Cihan Tepedelenlioglu are with the

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Section: B Training Sequence Design Via the Crbmentioning

confidence: 99%

“…Now we show that H θ is asymptotically equivalent to T N (t(λ)). First, since both H θ and T N (t(λ)) are banded Toeplitz matrices [32,Sec. 4.3], their strong norms (operator norms) are bounded.…”

Section: B Training Sequence Design Via the Crbmentioning

confidence: 99%

Optimal Training for Residual Self-Interference for Full-Duplex One-Way Relays

Tepedelenlioğlu

Şenol

2018

IEEE Trans. Commun.

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“…A well-known property of circulant matrices is the following formula for the determinant (see [16] for a proof of this formula and other properties of circulant matrices):…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…We consider a permutation σ ∈ T 24;3,16 (16,6). We have σ = C 1 C 2 , where (3.5) C 1 =(20, 23, 2, 18, 21, 24, 3, 19, 22, 1, 4) C 2 =(7, 10, 13,5,8,11,14,6,9,12,15), and the two fixed points of σ are 16 and 17.…”

Section: Examplementioning

confidence: 99%

Invariant CR mappings between hyperquadrics

Grundmeier¹,

Linsuain²,

Whitaker³

2018

Illinois J. Math.

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“…For quantum states with only nearest-neighbor entanglement the correlations appearing in c ( ) N can all be treated classically. The minimum of c ( ) N can then be derived with the help of circulant matrices[23] (see appendix B). As a consequence, all states with entanglement width  w 2 satisfy the inequality…”

mentioning

confidence: 99%

Characterizing the width of entanglement

Wölk

Gühne

2016

New J. Phys.

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The size of controllable quantum systems has grown in recent times. Therefore, the spatial degree of freedom becomes more and more important in experimental quantum systems. However, the investigation of entanglement in many-body systems mainly concentrated on the number of entangled particles and ignored the spatial degree of freedom so far. As a consequence, a general concept together with experimentally realizable criteria has been missing to describe the spatial distribution of entanglement. We close this gap by introducing the concept of entanglement width as measure of the spatial distribution of entanglement in many-body systems. We develop criteria to detect the width of entanglement based solely on global observables. As a result, our entanglement criteria can be applied easily to many-body systems since single-particle addressing is not necessary.

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Toeplitz and Circulant Matrices: A Review

Cited by 518 publications

References 17 publications

Optimal Training for Residual Self-Interference for Full-Duplex One-Way Relays

Optimal Training for Residual Self-Interference for Full-Duplex One-Way Relays

Invariant CR mappings between hyperquadrics

Characterizing the width of entanglement

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