567
CHAPTER
31
Re X
[
k
]
’
2
N
j
N
1
n
’
0
x
[
n
] cos(2
B
kn
/
N
)
Im X
[
k
]
’
2
N
j
N
1
n
’
0
x
[
n
] sin(2
B
kn
/
N
)
EQUATION 311
The real DFT.
This is the forward transform,
calculating the frequency domain from the
time domain. In spite of using the names:
real
part
and
imaginary part
, these equations
only involve ordinary numbers.
The
frequency index,
k
, runs from 0 to
N
/2. These
are the same equations given in Eq. 84,
except that the
2/
N
term has been included in
the forward transform.
The Complex Fourier Transform
Although complex numbers are fundamentally disconnected from our reality, they can be used to
solve science and engineering problems in two ways.
First, the parameters from a real world
problem can be substituted into a complex form, as presented in the last chapter.
The second
method is much more elegant and powerful, a way of making the complex numbers
mathematically equivalent to the physical problem.
This approach leads to the
complex Fourier
transform
, a more sophisticated version of the
real Fourier transform
discussed in Chapter 8.
The complex Fourier transform is important in itself, but also as a stepping stone to more
powerful complex techniques, such as the
Laplace
and
ztransforms
. These complex transforms
are the foundation of theoretical DSP.
The Real DFT
All four members of the Fourier transform family (DFT, DTFT, Fourier
complex numbers.
Since DSP is mainly concerned with the DFT, we will use
it as an example.
Before jumping into the complex math, let's review the real
DFT with a special emphasis on things that are awkward with the mathematics.
In Chapter 8 we defined the
real
version of the Discrete Fourier Transform
according to the equations:
In words, an
N
sample time domain signal,
, is decomposed into a set
x
[
n
]
of
cosine waves, and
sine waves, with frequencies given by the
N
/2
%
1
N
/2
%
1
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568
index,
k
.
The amplitudes of the cosine waves are contained in
, while
ReX
[
k
]
the amplitudes of the sine waves are contained in
. These equations
Im X
[
k
]
operate by
correlating
the respective cosine or sine wave with the time domain
signal.
In spite of using the names:
real part
and
imaginary part
, there are no
complex numbers in these equations.
There isn't a
j
anywhere in sight!
We
have also included the normalization factor,
in these equations.
2/
N
Remember, this can be placed in front of either the synthesis or analysis
equation, or be handled as a separate step (as described by Eq. 83). These
equations should be very familiar from previous chapters.
If they aren't, go
back and brush up on these concepts before continuing.
If you don't understand
the
real
DFT, you will never be able to understand the
complex
DFT.
Even though the real DFT uses only real numbers,
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 Spring '11
 Staff
 Digital Signal Processing, complex fourier transform

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