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Superposition of sinusoids

Fourier analysis is built from the complex exponential
\begin{displaymath}
e^{-i \omega t} \eq \cos\omega t - i\,\sin \omega t\end{displaymath} (13)
A Fourier component of a time signal is a complex number, a sum of real and imaginary parts, say
\begin{displaymath}
B \eq {\rm Re\;}B + i {\rm Im\;}B\end{displaymath} (14)
which is attached to some frequency. Let j be an integer and $\omega_j$ 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 Bj, namely,  
 \begin{displaymath}
b(t) \eq \sum_j B_j \ e^{-i\omega_j t}\end{displaymath} (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 $\overline{b}(t)$,and then dividing by two:  
 \begin{displaymath}
{\rm Re\;}\,b(t) \eq {1 \over 2}
\sum_j \ 
( B_j \ e^{-i\omega_j t} +
 \bar B_j \ e^{ i\omega_j t}
 )\end{displaymath} (16)
In other words, for each positive $\omega_j$ with amplitude Bj, we add a negative $-\omega_j$ with amplitude $\bar B_j$(likewise, for every negative $\omega_j$ ...). The Bj are called the ``frequency function,'' or the ``Fourier transform.'' Loosely, the Bj are called the ``spectrum,'' though in formal mathematics, the word ``spectrum'' is reserved for the product $\bar B_j B_j$.The words ``amplitude spectrum'' universally mean

$\sqrt{ \bar B_j B_j}$.

In practice, the collection of frequencies is almost always evenly spaced. Let j be an integer $\omega = j \ \Delta \omega$ so that
\begin{displaymath}
b(t) \eq \sum_j \, B_j \, e^{-i(j \, \Delta\omega) t}\end{displaymath} (17)
Representing a signal by a sum of sinusoids is technically known as ``inverse Fourier transformation.'' An example of this is shown in Figure 5.

 
cosines
cosines
Figure 5
Superposition of two sinusoids.


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next up previous print clean
Next: Sampled time and Nyquist Up: FOURIER SUMS Previous: FOURIER SUMS
Stanford Exploration Project
3/1/2001