Superposition of sinusoids

Fourier analysis is built from the complex exponential

$$e^{-i \omega t} \eq \cos\omega t - i\,\sin \omega t$$ (13)

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

$$B \eq {\rm Re\;}B + i {\rm Im\;}B$$ (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 B_j, namely,  

$$b(t) \eq \sum_j B_j \ e^{-i\omega_j t}$$ (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:  

$${\rm Re\;}\,b(t) \eq {1 \over 2} \sum_j \ ( B_j \ e^{-i\omega_j t} + \bar B_j \ e^{ i\omega_j t} )$$ (16)

In other words, for each positive $\omega_j$ with amplitude B_j, we add a negative $-\omega_j$ with amplitude $\bar B_j$(likewise, for every negative $\omega_j$ ...). The B_j are called the ``frequency function,'' or the ``Fourier transform.'' Loosely, the B_j 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

$$b(t) \eq \sum_j \, B_j \, e^{-i(j \, \Delta\omega) t}$$ (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
Figure 5 Superposition of two sinusoids.