## 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:*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

*Euler's Identity*:

*sampled complex sinusoid*, and we'll talk about sinusoids in some detail, particularly from an audio perspective. Finally, we need to understand what the summation over is doing in the definition of the DFT. We'll learn that it should be seen as the computation of the

*inner product*of the signals and defined above, so that we may write the DFT, using inner-product notation, as

*sum of projections*of onto ;

*i.e.*, we'll show

*coefficient of projection*of onto . Using the notation to mean the whole signal for all , the IDFT can be written more simply as

*basis sinusoids*and their coefficients of projection are

*complex valued*in general. Having completely understood the DFT and its inverse mathematically, we go on to proving various

*Fourier Theorems*, such as the ``shift theorem,'' the ``convolution theorem,'' and ``Parseval's theorem.'' The Fourier theorems provide a basic thinking vocabulary for working with signals in the time and frequency domains. They can be used to answer questions such as

``What happens in the frequency domain if I do [operation x] in the time domain?''Usually a frequency-domain understanding comes closest to a

*perceptual*understanding of audio processing. Finally, we will study a variety of practical spectrum analysis examples, using primarily the

`matlab`programming language [67] to analyze and display signals and their spectra.

**Next Section:**

DFT Math Outline

**Previous Section:**

Inverse DFT