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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.
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 Spring '12
 MUNK
 Signal Processing, DFT, inner product, inverse DFT, Julius O. Smith

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