## 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

We will systematically develop what we mean by imaginary exponents in order that such mathematical expressions are well defined.

With , , and imaginary exponents understood, we can go on to prove Euler's Identity:

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

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 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

where is the sampled complex sinusoid at (normalized) radian frequency , and the inner product operation is defined by

We will show that the inner product of with the th basis sinusoid'' 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 ; i.e., we'll show

where

is the coefficient of projection of onto . Using the notation to mean the whole signal for all , the IDFT can be written more simply as

Note that both the 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.

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DFT Math Outline
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Inverse DFT