# Introduction to the DFT

This chapter introduces the Discrete Fourier Transform (DFT) and points out the mathematical elements that will be explicated in this book. To find motivation for a detailed study of the DFT, the reader might first peruse Chapter 8 to get a feeling for some of the many practical applications of the DFT. (See also the preface on page .)Before we get started on the DFT, let's look for a moment at the

*Fourier transform*(FT) and explain why we are not talking about it instead. The Fourier transform of a continuous-time signal may be defined as

*calculus*. The DFT, on the other hand, replaces the infinite integral with a finite sum:

*sampled*form, so that the DFT is what we really need anyway (implemented using an FFT when possible). In summary, the DFT is

*simpler mathematically*, and

*more relevant computationally*than the Fourier transform. At the same time, the basic concepts are the same. Therefore, we begin with the DFT, and address FT-specific results in the appendices.

## DFT Definition

The*Discrete Fourier Transform (DFT)*of a signal may be defined by

*period*. For a tutorial on sampling continuous-time signals to obtain non-aliased discrete-time signals, see Appendix D. When all signal samples are real, we say . If they may be complex, we write . Finally, means is any integer.

## Inverse DFT

The*inverse*DFT (the IDFT) is given by

## 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.

## DFT Math Outline

In summary, understanding the DFT takes us through the following topics:- Complex numbers
- Complex exponents
- Why ?
- Euler's identity
- Projecting signals onto signals via the inner product
- The DFT as the coefficient of projection of a signal onto a sinusoid
- The IDFT as a sum of projections onto sinusoids
- Various Fourier theorems
- Elementary time-frequency pairs
- Practical spectrum analysis in matlab

**Next Section:**

Complex Numbers

**Previous Section:**

Preface