# 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

The sampling interval is also called the sampling *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:

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

With , , and imaginary exponents understood, we can go on to prove
*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

After the foregoing, the inverse DFT can be understood as the
*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

We will additionally discuss various practical aspects of working with signals and spectra.

**Next Section:**

Complex Numbers

**Previous Section:**

Preface