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
We will show that the inner product of
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
Note that both the basis sinusoids
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 [70] to analyze and display signals and their spectra.