DFT as an Orthogonal Basis Change

1. Is DFT orthogonal?
2. How do you change an orthogonal basis to an orthonormal basis?
3. Is a change of basis matrix orthogonal?
4. Is Fourier series an orthogonal basis?

Is DFT orthogonal?

The DFT belongs to a class of transforms called orthogonal transforms, and it is not the only member of this calss used in DSP applications. Some of the more popular are the Walsh, slant, and COSINE transforms.

How do you change an orthogonal basis to an orthonormal basis?

Since a basis cannot contain the zero vector, there is an easy way to convert an orthogonal basis to an orthonormal basis. Namely, we replace each basis vector with a unit vector pointing in the same direction. normalized vectors ui = vi/ vi , i = 1,...,n, form an orthonormal basis.

Is a change of basis matrix orthogonal?

A matrix P is orthogonal if P−1=PT. A change of basis matrix P relating two orthonormal bases is an orthogonal matrix. i.e., P−1=PT.

Is Fourier series an orthogonal basis?

The Fourier series will provide an orthonormal basis for images. 2.1 Image Representations: To simplify, I'll do everything in terms of a 1D function f(t), but all this extends to 2D images. We'll start by considering periodic functions that go from 0 to 2π, which turn out to be easier.