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Digital Filters Design for Signal and Image Processing by Mohamed Najim

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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 images of two functions imagesi(t) and imagesk(t) of finite energy, represented as follows:

images

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

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

images

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

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