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
Thus, right off the bat, we need calculus
. The DFT, on the
other hand, replaces the infinite integral with a finite sum:
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
is finite, which is
always true in practice). Moreover, in the field of digital signal
, signals and spectra
are processed only in sampled
form, so that the DFT is what we really need anyway (implemented using
when possible). In summary, the DFT is simpler
, and more relevant computationally
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 Discrete Fourier Transform (DFT) of a signal
may be defined by
' means ``is defined as'' or ``equals by definition'', and
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
means is any integer.
The inverse DFT (the IDFT) is given by
The inverse DFT is written using `
' instead of `
the result follows from the definition of the DFT, as we will show
in Chapter 6
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
There are two remaining symbols in the DFT we have not yet defined:
, 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).
, 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
We will systematically develop what we mean by imaginary exponents
that such mathematical expressions are well defined.
With , , and imaginary exponents understood, we can go on to prove
Euler's Identity is the key to understanding the meaning of expressions like
We'll see that such an expression defines a sampled complex
, 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
is the sampled complex sinusoid
(normalized) radian frequency
, and the inner product
is defined by
We will show that the inner product of
is a measure of ``how much'' of
is present in
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 onto
is the coefficient of projection
Using the notation
to mean the whole signal
, the IDFT can be written more simply as
Note that both the basis sinusoids
and their coefficients of
are complex valued
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
 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.
Next Section: Complex NumbersPrevious Section: Preface