8.3 The Fourier integral  8. Spectral Analysis  8.1 Introduction

8.2 The Fourier series

The basic tool used in most spectral analysis is the Fourier series. This series is only one of many possible choices, but is one particularly suited to periodic functions. Consider a function that repeats with period T, such that

$$f(t) = f(t+T) = f(t+2T) = \cdots.$$ (8.1)

 

Such a function can be represented by f^*(t), a superposition of sine and cosine waves with various frequencies having amplitudes $\{a_k\}$ and $\{b_k\}$:

$$f^*(t) = a_0 + \sum_{k=1}^\infty \bigl(a_k\cos(\omega_kt) +b_k\sin(\omega_kt)\bigr)$$ (8.2)

 

or

$$f^*(t) = \sum_{k=-\infty}^\infty c_ke^{i\omega_kt}$$ (8.3)

 

where

$$\omega_k=2\pi\nu_k = {{2\pi k}\over{T}} = {{2\pikV}\over{\lambda}}$$ (8.4)

 

relates the angular frequency $\omega_k$ to the frequency $\nu_k$, the period T, and (for measurements made during motion through the field with velocity V) the wavelength $\lambda$, and where c_k is a complex number. The lowest angular frequency represented in such an expansion (other than the constant term) is $\omega_1$, corresponding to one sine-wave oscillation covering the period T; other frequencies are integer multiples of this fundamental frequency.

The coefficients c_k can be determined from

$$c_k = {{1}\over{T}} \int_{-T/2}^{T/2} f(t) e^{-i\omega_kt}dt$$ (8.5)

 

as can be demonstrated by substituting (8.3) into (8.5) and using the orthogonality relationships for the exponential functions (or, equivalently, for sine and cosine functions). Using this formula, any periodic time series can be represented by Fourier components. Furthermore, any series observed only from -T/2 to T/2 can be assumed to be periodic for values outside those limits if the values at -T/2 and at T/2 are equal. Although any finite set of coefficients will be an inexact representation of a continuous series, N coefficients can provide an exact representation of a set of N points sampled from that series.

The variance in the time series is

$${{1}\over{T}} \int_{-T/2}^{T/2} \bigl(f(t)-\overline{f(t)}\b......int_{-T/2}^{T/2} \bigl(f^2(t) -(\overline{f(t)})^2\bigr) dt .$$ (8.6)

 

In the complex representation of the time series,

$${rl}{{1}\over{T}} \int_{-T/2}^{T/2}{f^*}^2(t)dt ={{1}\over{T}}\int_{-T/2}^{T/2}(\sum_j c_je^{i\omega_jt})(\sum_m c_m^*e^{-i\omega_mt})dt$$

$$= \sum_j\sum_m {{c_jc_m^*}\over{T}} \int_{-T/2}^{T/2}e^{i(\omega_j-\omega_m)t} dt$$ (8.7)

$$= \sum_j c_jc_j^* = \sum_j \vert c_j\vert^2$$

 

because the integrals for terms with $j\ne m$ are zero.

Since

$$\overline{f^*(t)} = {{1}\over{T}} \int_{-T/2}^{T/2} f^*(t) dt = c_0 ,$$ (8.8)

 

the variance in f^*(t) is

$$V^*_{ff} = \sum_{j\ne 0} \vert c_j\vert^2 .$$ (8.9)

 

The term |c_j|² is the contribution to the variance of the Fourier series made by the component with angular frequency $\omega_j$. A plot of |c_j|² vs $\omega_j$, often called a periodogram, can be considered a representation of the spectral density of the variance, or the "variance spectrum," of that time series. Because of the history of this analysis in electrical engineering applications, the variance spectrum is often called the "power spectrum." Note that the periodogram is a discrete spectrum, not a continuous spectrum, and it represents the Fourier series but not necessarily the underlying function f(t).

If a measurement is made at discrete times spaced $\Delta T$ apart and spanning an interval T, then there will be $N=T/\Delta T$ samples and a choice of NFourier components can give an exact representation of the data. This does not require that such a Fourier description be the true time series, only that it match the measured values. Other choices could also match the measurements, for example by using higher-frequency components, but the choice would not be unique. If the angular frequencies are selected to be multiples of the lowest frequency $\omega_1=2\pi/T$, the problem is reduced to solving the N simultaneous equations relating the observations to the Fourier coefficients:

$$f_k = f(k\Delta T) = \sum_j c_je^{i\omega_jk\Delta T}= \sum_j c_j e^{ijk\omega_1\Delta T}$$ (8.10)

 

where the N observations lead to N equations of the form (8.10) at the times $k\Delta T$. For N even, j can be selected to vary from -N/2 to (N/2)-1.

Series relationships equivalent to the integral orthogonality relationships then lead to the particularly simple result that

$$c_j = {{1}\over{N}} \sum_k f_ke^{-ikj\omega_1\Delta T} \ .$$ (8.11)

 

Although N parameters can be determined from the N observations, the coefficients in (8.11) are complex, so only N/2 coefficients can be determined from the data. The two variables per coefficient can be considered the amplitude and the phase, or alternately the sine-wave and cosine-wave contributions.

   
 8.3 The Fourier integral  8. Spectral Analysis  8.1 Introduction 

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