Digital Filters Design for Signal and Image Processing

3.2. The Fourier transform of continuous signals

3.2.1. Summary of the Fourier series decomposition of continuous signals

3.2.1.1. Decomposition of finite energy signals using an orthonormal base

Let x(t) be a finite energy signal. We consider the scalar product of two functions _i(t) and _k(t) of finite energy, represented as follows:

where _k*(t) denotes the complex conjugate of _k(t).

A family {_k(t)} of finite energy functions is called orthonormal if it verifies the following relations:

A family {_k(t)} is complete if any vector of the space can be approximated as closely as possible by a linear ...