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.
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