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
![$ n\in [0,N-1]$](http://www.dsprelated.com/josimages_new/mdft/img39.png)
, 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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