Spectrum Analysis - Basic Notation and Principles

  • Spectrum Analysis Basic Notation and Principles - Frequency and Period
    The "wave length" of a sine or cosine function is typically expressed in terms of the number of cycles per unit time (Frequency), often denoted by the Greek letter nu (ν; some textbooks also use f). For example, the number of letters handled in a post office may show 12 cycles per year: On the first of every month a large amount of mail is sent (many bills come due then), then the amount of mail decreases in the middle of the month, then it increases again towards the end of the month. Therefore, every month the fluctuation in the amount of mail handled by the post office will go through a full cycle. Thus, if the unit of analysis is one year, then n would be equal to 12, as there would be 12 cycles per year. Of course, there will likely be other cycles with different frequencies. For example, there might be annual cycles (ν=1), and perhaps weekly cycles (ν=52 weeks per year).
  • Spectrum Analysis Basic Notation and Principles - The General Structural Model
  • Spectrum Analysis Basic Notation and Principles - A Simple Example
    Shumway (1988) presents a simple example to clarify the underlying "mechanics" of spectrum analysis. Let us create a series with 16 cases following the equation shown above, and then see how we may "extract" the information that was put in it. First, create a variable and define it as:
  • Spectrum Analysis Basic Notation and Principles - Periodogram
    The sine and cosine functions are mutually independent (or orthogonal); thus we may sum the squared coefficients for each frequency to obtain the periodogram. Specifically, the periodogram values above are computed as:
  • Spectrum Analysis Basic Notation and Principles - The Problem of Leakage
    In the example above, a sine function with a frequency of 0.2 was "inserted" into the series. However, because of the length of the series (16), none of the frequencies reported in the spreadsheet exactly "hits" on that frequency. In practice, what often happens in those cases is that the respective frequency will "leak" into adjacent frequencies. For example, you can find large periodogram values for two adjacent frequencies, when, in fact, there is only one strong underlying sine or cosine function at a frequency that falls in-between those implied by the length of the series. There are three ways in which one can approach the problem of leakage:
  • Spectrum Analysis Basic Notation and Principles - Padding the Time Series
    Because the frequency values are computed as N/t (the number of units of times) you can simply pad the series with a constant (e.g., zeros) and thereby introduce smaller increments in the frequency values. In a sense, padding allows one to apply a finer roster to the data. In fact, if we padded the example data file described in the example above with ten zeros, the results would not change, that is, the largest periodogram peaks would still occur at the frequency values closest to .0625 and .2. (Padding is also often desirable for computational efficiency reasons.)
  • Spectrum Analysis Basic Notation and Principles - Data Windows and Spectral Density Estimates
    In practice, when analyzing actual data, it is not usually of crucial importance to identify exactly the frequencies for particular underlying sine or cosine functions. Rather, because the periodogram values are subject to substantial random fluctuation, one is faced with the problem of very many "chaotic" periodogram spikes. In that case, one would like to find the frequencies with the greatest spectral densities, that is, the frequency regions, consisting of many adjacent frequencies, that contribute most to the overall periodic behavior of the series. This can be accomplished by smoothing the periodogram values via a weighted moving average transformation. Suppose the moving average window is of width m (which must be an odd number); the following are the most commonly used smoothers (note: p = (m-1)/2).
  • Spectrum Analysis Basic Notation and Principles - Tapering
    The so-called process of split-cosine-bell tapering is a recommended transformation of the series prior to the spectrum analysis. It usually leads to a reduction of leakage in the periodogram. The rationale for this transformation is explained in detail in Bloomfield (1976, p. 80-94). In essence, a proportion (p) of the data at the beginning and at the end of the series is transformed via multiplication by the weights:
  • Spectrum Analysis Basic Notation and Principles - Preparing the Data for Analysis
  • Spectrum Analysis Basic Notation and Principles - Results when no Periodicity in the Series Exists