8.4 The variance spectrum  8. Spectral Analysis  8.2 The Fourier series

8.3 The Fourier integral

As the period of the Fourier series becomes infinite, the number of coefficients in the series also becomes infinite, and each individual coefficient approaches zero. Such a series might be written as

$$f(t) = \sum_{k=-\infty}^\infty c_k e^{i\omega_kt} = {{1}\over{T}}\sum_{k=-\infty}^\infty (Tc_k) e^{i2\pi kt/T} .$$ (8.12)

 

The individual frequencies in this series are $\nu_k=k/T$, and the interval between consecutive frequencies is $\Delta\nu=1/T$. As the period T becomes infinitely large, the frequency interval becomes infinitesimally small, and Tc_k approaches a continuous function $\tilde f(\nu)$. The limit as T becomes infinite is thus

$$f(t) = \int_{-\infty}^\infty \tilde f(\nu)e^{-i2\pi\nu t}d\nu$$ (8.13)

 

where

$$\tilde f(\nu) = \int_{-\infty}^\infty f(t) e^{+i2\pi\nu t} dt.$$ (8.14)

 

This differs from the Fourier series representation in that $\tilde f(\nu)$ is a continuous function of frequency, while the coefficients c_j represent discrete frequencies. Often the physical process is best represented as a continuous function having a continuous spectrum, even though the series is measured at discrete times. In those cases, c_j is used as an estimate of the continuous density $\tilde f(\nu)$ over an appropriate frequency interval, as described in the next section.

   
 8.4 The variance spectrum  8. Spectral Analysis  8.2 The Fourier series