Hadamard transform

About: Hadamard transform is a research topic. Over the lifetime, 7262 publications have been published within this topic receiving 94328 citations.

On Universal and Fault-Tolerant Quantum Computing

Abstract: A novel universal and fault-tolerant basis (set of gates) for quantum computation is described. Such a set is necessary to perform quantum computation in a realistic noisy environment. The new basis consists of two single-qubit gates (Hadamard and ${\sigma_z}^{1/4}$), and one double-qubit gate (Controlled-NOT). Since the set consisting of Controlled-NOT and Hadamard gates is not universal, the new basis achieves universality by including only one additional elementary (in the sense that it does not include angles that are irrational multiples of $\pi$) single-qubit gate, and hence, is potentially the simplest universal basis that one can construct. We also provide an alternative proof of universality for the only other known class of universal and fault-tolerant basis proposed by Shor and by Kitaev.

Multiple block-size transform video coding using an asymmetric sub-band structure

Abstract: A sub-band coder for video signals is disclosed, which is easily realizable in VLSI due to its simple structure. The coder uses sets of two-tap finite impulse response (FIR) filters having equal-magnitude coefficients to decompose the video signal at multiple stages. In the preferred embodiment the sub-band coder has an asymmetrical structure. With the arrangement, the sub-band decomposition effects what is equivalent to a multiple block-size Hadamard transform of the video signal in which coefficients are transmitted from various block-sized transforms of larger blocks of pixel data.

Binary Discrete Cosine and Hartley Transforms

Abstract: In this paper, a systematic method for developing a binary version of a given transform by using the Walsh-Hadamard transform (WHT) is proposed. The resulting transform approximates the underlying transform very well, while maintaining all the advantages and properties of WHT. The method is successfully applied for developing a binary discrete cosine transform (BDCT) and a binary discrete Hartley transform (BDHT). It is shown that the resulting BDCT corresponds to the well-known sequency-ordered WHT, whereas the BDHT can be considered as a new Hartley-ordered WHT. Specifically, the properties of the proposed Hartley-ordering are discussed and a shift-copy scheme is proposed for a simple and direct generation of the Hartley-ordering functions. For software and hardware implementation purposes, a unified structure for the computation of the WHT, BDCT, and BDHT is proposed by establishing an elegant relationship between the three transform matrices. In addition, a spiral-ordering is proposed to graphically obtain the BDHT from the BDCT and vice versa. The application of these binary transforms in image compression, encryption and spectral analysis clearly shows the ability of the BDCT (BDHT) in approximating the DCT (DHT) very well.

On some applications of Hadamard matrices

Abstract: Modern communications systems are heavily reliant on statistical techniques to recover information in the presence of noise and interference. One of the mathematical structures used to achieve this goal is Hadamard matrices. They are used in many different ways and some examples are given. This paper concentrates on code division multiple access systems where Hadamard matrices are used for user separation. Two older techniques from design and analysis of experiments which rely on similar processes are also included. We give a short bibliography (from the thousands produced by a google search) of applications of Hadamard matrices appearing since the paper of Hedayat and Wallis in 1978 and some applications in telecommunications.

Conformable Fractional Integrals Versions of Hermite-Hadamard Inequalities and Their Generalizations

Abstract: We prove new Hermite-Hadamard inequalities for conformable fractional integrals by using convex function, -convex, and coordinate convex functions. We prove new Montgomery identity and by using this identity we obtain generalized Hermite-Hadamard type inequalities.