Abstract
In Sect. 1.5 we defined filters as operations in continuous time which preserved frequencies. Such operations are important since they can change the frequency content in many ways. They are difficult to use computationally, however, since they are defined for all instances in time. This will now be addressed by changing focus to discrete-time. Filters will now be required to operate on a possibly infinite vector \(\mathbf {x}=(x_n)_{n=-\infty }^{\infty }\) of values, corresponding to concrete instances in time. Such filters will be called discrete time filters, and they too will be required to be frequency preserving. We will see that discrete time filters make analog filters computable, similarly to how the DFT made Fourier series computable in Chap. 2. The DFT will be a central ingredient also now.
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Recall that the orthogonal diagonalization of S takes the form S = PDP T, where P contains as columns an orthonormal set of eigenvectors, and D is diagonal with the eigenvectors listed on the diagonal (see Section 7.1 in [32]).
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Ryan, Ø. (2019). Discrete Time Filters. In: Linear Algebra, Signal Processing, and Wavelets - A Unified Approach. Springer Undergraduate Texts in Mathematics and Technology. Springer, Cham. https://doi.org/10.1007/978-3-030-02940-1_3
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