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
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.
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:
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
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  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.