Power Spectrum

One way to look at a signal is in the discrete time domain, which puts a series of values consecutively in time. In this way we can tell something about the behavior of the signal at every moment in time, and can also make some simple statements about its long-term behavior. However, it is rather difficult to say anything about how the long-term behavior is related to the short-term development of the signal. Another way to look at a signal is to view its spectral density (i.e., the Fourier transform of the signal). The Fourier transform views the signal as a whole. It swaps the dimension of time with the dimension of frequency. One can think of the Fourier transform as a combination of slow and fast oscillations with different amplitude. A very strong and slow component in the frequency domain implies that there is a high correlation between the large-scale pieces of the signal in time (macro-structures), while a very strong and fast oscillation implies correlation in the micro-structures. Therefore, if our signal $f(t)$ represents values in every single moment of time, its Fourier transform $F(\omega)$ represents the strength of every oscillation in a holistic way in that chunk of time. These two signals are related to each other by the following formula[48]:

$$F(\omega) = \int_{-\infty}^{\infty} f(t){e}^{-j\omega t} dt.$$ (4.5)

One can think of the time domain function as how one listens to a melody and the frequency domain function as how one listens to a chord. Even though the situation in musical communication is not as simple as that (i.e. the time scales in which we listen to melodies and chords are different), this metaphor can give us a starting point in understanding this analysis.

In the Fourier transform, oscillations are characterized with sinusoid functions. Auditorily speaking, these functions are the purest sounds one can create (i.e. they are ``clean as a whistle''). The average value of any smooth oscillation, fast or slow, strong or weak, is zero. If we use the square of the values in time we can study the power of these oscillations in the same way we studied the original signal (i.e. take its Fourier transform). Parseval's theorem for energy signals states that:

$$\int_{-\infty}^{\infty} {\mid f(t) \mid}^{2} dt = \frac{1}{2\pi}\int_{-\infty}^{\infty}{\mid \, F(\omega)\mid}^{2} d\omega .$$ (4.6)

The Fourier transform analysis assumes the life of a signal from $-\infty$ to $\infty$. For that reason when an analysis is carried out for a finite amount of time, it is either assumed that the signal is periodic or that it has a finite amount of energy. A true power spectrum of a signal has to consider the signal from $-\infty$ to $\infty$. However, we are not always able to observe a signal that way or derive precise functions for it. We can define $F_{T}(\omega)$ which is the fourier transform of the signal in period T, and define the power spectrum as the following:

$$S_{f}(\omega) = \lim_{T \rightarrow \infty} \frac{1}{T}{\mid F_{T}(\omega)\mid}^2.$$ (4.7)

The power spectrum itself is the Fourier transform of the auto-correlation function. Auto-correlation function represents the relationship of long and short-term correlation within the signal itself.

$$<f(t)f(t+\tau)> = \frac{1}{2\pi}\int_{0}^{\infty} S_{f}(\omega){e}^{j\omega t}d\omega.$$ (4.8)

In this experiment, it is this last relationship which is of immediate interest to us. The power spectrum is a function in the frequency domain, which means that we can examine the long-term behavior of fast and slow oscillations. We will be looking at power spectrums approximately in the range of 0.001 to 5 Hz, which corresponds to oscillations which happen from 0.2 to 1000 seconds. Thus, a high value in the low spectral region, close to 0.001 Hz, means a high correlation in a very long time scale (i.e. in macro-structures) and a high value in the high region of the spectrum close to 5 Hz implies high correlations in the micro-structures^4.2. A relationship between the different sections of the power spectrum implies a relationship between the auto-correlation of the signal in the time domain to which those frequency sections are referring. In the following section we will examine the effects of changing some parameters of a random signal on its power spectrum.