3. Properties of the DFT

3.1. Alternate Forms for the DFT The DFT arises in so many different settings and is used by practitioners in so many different fields that, not surprisingly, it appears in many different disguises. In defining the DFT in the previous chapter, we issued the proviso that the definition used primarily in this book, namely Fk = 1 N ∑ n=− N 2 +1 N 2 ƒn ω N −nk , for , is only one of many that appear in the literature. To underscore this point we will occasionally use different forms of the DFT even in this book. Given this state of affairs, it seems reasonable to devote just a few moments to other forms of the DFT that might be encountered in practice. Knowing the protracted deliberations that led to our choice of a DFT definition, it would be foolish to suggest that one form is superior among all others. There is no single DFT that has a clear advantage over all others. The best attitude is to accept the DFT's multiple personalities and to deal with them however they appear. This approach is particularly valuable in working with DFT (or FFT) software, an issue we will also discuss briefly.