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Fourier analysis is built from the complex exponential

| |
(13) |

A Fourier component of a time signal is a complex number,
a sum of real and imaginary parts,
say
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(14) |

which is attached to some frequency.
Let *j* be an integer
and be a set of frequencies.
A signal *b*(*t*) can be manufactured
by adding a collection of complex exponential signals,
each complex exponential being scaled by a complex coefficient *B*_{j},
namely,
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(15) |

This manufactures a **complex-valued signal**.
How do we arrange for *b*(*t*) to be real?
We can throw away the imaginary part, which is like
adding *b*(*t*) to its complex conjugate ,and then dividing by two:
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(16) |

In other words, for each positive with amplitude *B*_{j},
we add a negative with amplitude (likewise, for every negative ...).
The *B*_{j} are called the
``frequency function,''
or the ``Fourier transform.''
Loosely, the *B*_{j} are called the ``**spectrum**,'' though technically,
and in this book,
the word
``spectrum'' should be reserved for the product
.The words ``**amplitude spectrum**'' universally mean
.

In practice, the collection of frequencies is almost always evenly spaced.
Let *j* be an integer so that

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(17) |

Representing a signal by a sum of sinusoids
is technically known as
``inverse Fourier transformation.''
An example of this is shown in Figure 6.
**cosines
**

Figure 6
Superposition of two sinusoids.
(Press button to activate program `ed1D`.
See appendix for details.)

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Stanford Exploration Project

10/21/1998