Fourier tutorial: Preface

Outline

The ‘FFT’ is ubiquitous in the sciences, but what is it? Joseph Fourier (1768-1830) made the initial discovery that segments of signals — functions with any shape but with some finite duration — can be represented as the sum of sine (and cosine) functions. This is neat. To mathematicians it has led to generalizations into quite abstract realms (viz ‘Harmonic Analysis’), and it has given to applied scientists the ‘frequency domain’. Nowadays it is hard to imagine life without Fourier.

The value of the frequency domain is that allows a complementary view of the data, one that is complementary in a very well-defined sense. Moreover, the frequency domain is often preferable to the ‘real’ domain because some phenomena (like waves) are clearly oacillatory, so appear simpler in the frequency domain than in the real domain.

A few terms

Later pages will demonstrate the complementarity and the precise relationship between between the two domains. The main point of this introductory page is to clarify the terminology. You might come across the terms Fourier Transform (FT), Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT), so what is the difference, if any?

The three terms are commonly used interchangeably; but there are differences, which should be respected in formal situations such as papers. In brief, there is a progression of increasingly special cases:

• FT is a general mathematical formulation using integrals and allowing for signals that are continuous (not discretized in time) and of infinite length. It is the most compact and insightful representation of the core idea of Fourier.
• DFT is a special case of the FT for discretized data of finite duration, i.e. for just about everything in science these days. This leads to some computational advantages, since the equations defining the transform no longer involve integrals but summations. The DFT is much more computer-friendly. But it also leads to some subtle difficulties with interpreting the results, since the algorithm operates only on a finite segment of data, so must then make some assumption about what lies beyond. Exactly what that assumption is will be explained later.
• FFT is a special case of the DFT. It was noticed in 1969 that the DFT can be calculated very quickly if the number of discretized values is some power of 2: i.e. 2, 4, 8, 16, etc. For typical cases the speedup can be several orders of magnitude. The power of 2 requirement is usually easy to meet, and conseqently the Fast Fourier Transform helped to make frequency domain analysis very popular. The only problem with the FFT is when the power of 2 requirement can be met only by arbitrary truncation or extrapolation of the available data. Then there are some subtle effects that have to be recognized, as will be explained later.

Much of the time you will use the FFT to transform time series data into the frequency domain, whereupon you can calculate power spectra, or coherence. However you can also modify the frequency domain version of the data and reverse the effect of the FFT, and reconstitute a (modified) time series. This is called Fourier filtering, and makes use of both a forward and inverse FFT transform. The latter is sometimes referred to as an IFFT. The fact that the FFT is invertible is notable, and underlines the fact that performing either transformation loses no information: the superficial appearance of the data is quite different in the real and frequency domains, but the two representations are interchangeable. This is the hallmark of a ‘transform’, in its technical sense.

You may have noticed in the preceding paragraph that estimation of the power spectrum was described as involving a forward FFT followed by some additional step. This is a significant point: the FFT is not to be equated with a power spectrum, as will be explained later. The FFT produces a ‘complex spectrum’ (which is invertible), and it is only with another step that it becomes a ‘power spectrum’ (which is not invertible). This clear distinction will be reinforced in later pages.

Other important terms are 'spectrum' and 'spectra'. They are the singular and plural forms, respectively, of the word. The complete story is that 'spectrum' is a second declension neuter noun in Latin, and its full declension is

Thus, for consistency, one should talk about 'the resolution of a spectri'. Fortunately common usage trumps consistency.

A preliminary look at DFTs

The following applet performs a forward DFT followed by an inverse DFT.
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If you don't see any applet immediately above, then read the notes in the first page of this tutorial concerning applets.

Play with the above applet, and note:

• For any input function, a forward transform followed by an inverse transform has no net effect.
• Calculation of the power spectrum is an additional step.
• Sine functions have 'clean' transforms, in the sense of producing a single peak; many other functions show harmonics (a series of peaks).
• Conversely, a spike in the time domain has a quite featureless representation in the frequency domain. Curious.
• Log transformation of the power spectrum is useful whenever you want to see both large and small spectral features.
• The imaginary part of the forward transform is always small when there is no noise. Why?
• The forward transform has obvious symmetry. Why?

All will be explained in later pages. We will also look into the many uses for discrete Fourier transforms, and will point out the many pitfalls for the unwary.