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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 $ x(t)$ may be defined as

$\displaystyle X(\omega) = \int_{-\infty}^\infty x(t)e^{-j\omega t} dt, \qquad \omega\in(-\infty,\infty). $

Thus, right off the bat, we need calculus. The DFT, on the other hand, replaces the infinite integral with a finite sum:

$\displaystyle X(\omega_k ) \isdef \sum_{n=0}^{N-1}x(t_n)e^{-j\omega_k t_n}, \qquad k=0,1,2,\ldots,N-1, $

where the various quantities in this formula are defined on the next page. Calculus is not needed to define the DFT (or its inverse, as we will see), and with finite summation limits, we cannot encounter difficulties with infinities (provided $ x(t_n)$ is finite, which is always true in practice). Moreover, in the field of digital signal processing, signals and spectra are processed only in 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 $ x$ may be defined by

$\displaystyle X(\omega_k ) \isdef \sum_{n=0}^{N-1}x(t_n)e^{-j\omega_k t_n}, \qquad k=0,1,2,\ldots,N-1, $

where ` $ \isdeftext $' means ``is defined as'' or ``equals by definition'', and
\begin{eqnarray*} \sum_{n=0}^{N-1} f(n) &\isdef & f(0) + f(1) + \dots + f(N-1)\\... ...mbox{number of time samples = no.\ frequency samples (integer).} \end{eqnarray*}
The sampling interval $ T$ 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 $ N$ signal samples $ x(t_n)$ are real, we say $ x\in{\bf R}^N$. If they may be complex, we write $ x\in{\bf C}^N$. Finally, $ n\in{\bf Z}$ means $ n$ is any integer.

Inverse DFT

The inverse DFT (the IDFT) is given by

$\displaystyle x(t_n) = \frac{1}{N}\sum_{k=0}^{N-1}X(\omega_k )e^{j\omega_k t_n}, \qquad n=0,1,2,\ldots,N-1. $

The inverse DFT is written using `$ =$' instead of ` $ \isdeftext $' because the result follows from the definition of the DFT, as we will show in Chapter 6.

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 $ T=1$ in the previous definition:
\begin{eqnarray*} X(k) &\isdef & \sum_{n=0}^{N-1}x(n)e^{-j{2\pi n k/N}}, \qquad ... ...\sum_{k=0}^{N-1}X(k)e^{j{2\pi n k/N}}, \qquad n=0,1,2,\ldots,N-1 \end{eqnarray*}
where $ x(n)$ denotes the input signal at time (sample) $ n$, and $ X(k)$ denotes the $ k$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:
\begin{eqnarray*} j &\isdef & \sqrt{-1} \\ e &\isdef & \lim_{n\to\infty} \left(1+\frac{1}{n}\right)^{n} = 2.71828182845905\ldots \end{eqnarray*}
The first, $ j=\sqrt{-1}$, 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, $ e = 2.718\ldots\,$, is a (transcendental) real number defined by the above limit. We will derive $ e$ 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

$\displaystyle s_k(n) \isdef e^{j{2\pi n k/N}}. $

We will systematically develop what we mean by imaginary exponents in order that such mathematical expressions are well defined. With $ e$, $ j$, and imaginary exponents understood, we can go on to prove Euler's Identity:

$\displaystyle e^{j\theta} = \cos(\theta) + j\sin(\theta) $

Euler's Identity is the key to understanding the meaning of expressions like

$\displaystyle s_k(t_n) \isdef e^{j\omega_k t_n}= \cos(\omega_k t_n) + j\sin(\omega_k t_n). $

We'll see that such an expression defines a 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 $ n$ 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 $ x$ and $ s_k$ defined above, so that we may write the DFT, using inner-product notation, as

$\displaystyle X(k) \isdef \left<x,s_k\right> $

where $ s_k(n) \isdef e^{j{2\pi n k/N}}$ is the sampled complex sinusoid at (normalized) radian frequency $ \omega_k T=2\pi k/N$, and the inner product operation $ \left<\,\cdot\,,\,\cdot\,\right>$ is defined by

$\displaystyle \left<x,y\right> \isdef \sum_{n=0}^{N-1}x(n) \overline{y(n)}. $

We will show that the inner product of $ x$ with the $ k$th ``basis sinusoid'' $ s_k$ is a measure of ``how much'' of $ s_k$ is present in $ x$ and at ``what phase'' (since it is a complex number). After the foregoing, the inverse DFT can be understood as the sum of projections of $ x$ onto $ \{s_k\}_{k=0}^{N-1}$; i.e., we'll show

$\displaystyle x(n) = \sum_{k=0}^{N-1}\tilde{X}_k s_k(n), \qquad n=0,1,2,\ldots,N-1 $

where

$\displaystyle \tilde{X}_k \isdef \frac{X(k)}{N} $

is the coefficient of projection of $ x$ onto $ s_k$. Using the notation $ x\isdef x(\cdot)$ to mean the whole signal $ x(n)$ for all $ n\in [0,N-1]$, the IDFT can be written more simply as

$\displaystyle x = \sum_k \tilde{X}_k s_k. $

Note that both the basis sinusoids $ s_k$ and their coefficients of projection $ \tilde{X}_k$ 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: We will additionally discuss various practical aspects of working with signals and spectra.
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Complex Numbers
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Preface