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Mathematics_DFT

# Mathematics_DFT - Mathematics of the DFT Page 1 of 3 Next |...

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Next | Prev | Up | Top | Index | JOS Index | JOS Pubs | JOS Home | Search Mathematics of the DFT In the signal processing literature, it is common to write the DFT and its inverse in the more pure form below, obtained by setting in the previous definition: where denotes the input signal at time (sample) , and denotes the th spectral sample. This form is the simplest mathematically, while the previous form is easier to interpret physically. There are two remaining symbols in the DFT we have not yet defined: The first, , is the basis for complex numbers . 1.1 As a result, complex numbers will be the first topic we cover in this book (but only to the extent needed to understand the DFT). The second, , is a ( transcendental ) real number defined by the above limit. We will derive and talk about why it comes up in Chapter 3 . Note that not only do we have complex numbers to contend with, but we have them appearing in exponents, as in We will systematically develop what we mean by imaginary exponents in order that such mathematical expressions are well defined.

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