(12) |

(13) |

(14) |

Consider a simple, real, even signal
such as (*b _{-1}*,

Consider the real, odd signal (*b _{-1}*,

Likewise, the transform of the imaginary even function (*i*, 0, *i*)
is the imaginary even function .Finally, the transform of the imaginary odd function (-*i*, 0, *i*)
is real and odd.

Let *r* and *i* refer to real and imaginary,
*e* and *o* to even and odd,
and lower-case and upper-case letters to time and frequency functions.
A summary of the symmetries
of Fourier transform is shown in Figure 6.

reRE
Odd functions swap real and imaginary.
Even functions do not get mixed up with complex numbers.
Figure 5 |

More elaborate signals can be made
by adding together the three-point functions we have considered.
Since sums of even functions are even,
and so on,
the diagram in Figure 6
applies to all signals.
An arbitrary signal is made from these four parts only,
i.e., the function has the form
.On transformation of *b*_{t},
each of the four individual parts transforms according to the table.

Most ``industry standard'' methods of Fourier transform
set the zero frequency
as the first element in the vector array holding the transformed signal,
as implied by equation (3).
This is a little inconvenient, as we saw a few pages back.
The Nyquist frequency is then the first point past the middle
of the even-length array,
and the negative frequencies lie beyond.
Figure 7
shows an example of an **even function** as it is customarily stored.

Figure 6

10/21/1998