Publication

Tree structures for orthogonal transform and application to the Hadamard transform

Martin Vetterli
1983
Journal Articles

Abstract

Tree structures for computing orthogonal transforms are introduced. Two cases, delay trees and decimation trees, are investigated. A simple condition, namely the orthogonality of branches with a common root, is shown to be necessary and sufficient for the overall transform to be orthogonal. Main advantages are structural simplicity and a number of operations proportional to N Log2N. Application of the tree structures to the Walsh-Hadamard Transform (in natural, sequency and dyadic order) is presented. A single module can be multiplexed or used in parallel in order to perform all operations. Such a system is shown to be well suited for hardware implementation.

Official source

https://infoscience.epfl.ch/entities/publication/28175b96-5fb7-4adf-833a-7ebbb8fc64a7

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

Martin Vetterli
1983
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/28175b96-5fb7-4adf-833a-7ebbb8fc64a7

About this result

Related concepts (32)

Hadamard transform

The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal, symmetric, involutive, linear operation on 2m real numbers (or complex, or hypercomplex numbers, although the Hadamard matrices themselves are purely real). The Hadamard transform can be regarded as being built out of size-2 discrete Fourier transforms (DFTs), and is in fact equivalent to a multidimensional DFT of size 2 × 2 × ⋯ × 2 × 2.

Orthogonal matrix

In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is where QT is the transpose of Q and I is the identity matrix. This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse: where Q−1 is the inverse of Q. An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT), unitary (Q−1 = Q∗), where Q∗ is the Hermitian adjoint (conjugate transpose) of Q, and therefore normal (Q∗Q = QQ∗) over the real numbers.

Quantum Fourier transform

In quantum computing, the quantum Fourier transform (QFT) is a linear transformation on quantum bits, and is the quantum analogue of the discrete Fourier transform. The quantum Fourier transform is a part of many quantum algorithms, notably Shor's algorithm for factoring and computing the discrete logarithm, the quantum phase estimation algorithm for estimating the eigenvalues of a unitary operator, and algorithms for the hidden subgroup problem. The quantum Fourier transform was discovered by Don Coppersmith.

Related publications (19)

Tree structures for orthogonal transform and application to the Hadamard transform

Chat with Graph Search

On Attestation Dilution

Horizontally Scaling Heterogeneous Storage for Combined Capabilities with PolyStore

A tale of two shuffle algebras