crossorthogonal

Crossorthogonal (often written cross-orthogonal) is a term used in mathematics, signal processing, and related fields to describe a relationship in which cross interactions between two sets of entities are zero or minimized. It is not a single formal concept with one universally accepted definition, but rather a descriptive label applied in contexts where two groups are required to be orthogonal to each other in a cross-wise sense.

In linear algebra, cross-orthogonality can refer to two sets of vectors whose cross-interactions vanish. For example, if A and B are matrices with column vectors {a_i} and {b_j}, one may require a_i^T b_j = 0 for chosen indices i and j, indicating that the column spaces are orthogonal in the cross sense. This is often used in decoupling problems or in constructing bases with mutually orthogonal cross-terms.

In signal processing, two signals x(t) and y(t) are cross-orthogonal if their cross-correlation is zero for all time shifts: R_xy(τ) = ∫ x(t) y(t+τ) dt = 0 for all τ. In discrete time, the sum over n of x[n] y[n+k] vanishes for all shifts k. Such a property implies no cross-talk or interference between the signals in the relevant timing window.

In practice, cross-orthogonality is often enforced via constraints or penalties in optimization, or achieved through specially designed code sequences (for example, codes with low or zero cross-correlation).

See also: orthogonality, cross-correlation, Gram-Schmidt, Walsh codes.