crossorthogonality

Crossorthogonality is a concept that arises in various fields, particularly in mathematics and signal processing. It describes a specific relationship between two vectors or functions. When two vectors are crossorthogonal, their inner product is zero. The inner product is a mathematical operation that takes two vectors and returns a scalar value, and in Euclidean space, it's often calculated as the sum of the products of their corresponding components. If this sum equals zero, the vectors are said to be orthogonal.

The term "crossorthogonality" is often used to distinguish this property from a vector being orthogonal to itself, which is only true for the zero vector. In the context of signal processing, crossorthogonality is crucial for techniques like orthogonal frequency-division multiplexing (OFDM). In OFDM, different data streams are transmitted on closely spaced subcarriers. These subcarriers are designed to be orthogonal to each other over the symbol duration. This crossorthogonality allows the receiver to easily separate and decode the individual data streams without significant interference, even though the subcarriers overlap in frequency.

This property also appears in linear algebra, where it can refer to the orthogonality of vectors from different vector spaces or subspaces. For instance, in the context of projection, a vector might be projected onto a subspace, and the original vector will be crossorthogonal to the error vector, which represents the difference between the original vector and its projection. The concept is fundamental to understanding and manipulating vector spaces and their relationships.