2.2.1 Introduction to the Fourier Transform

Signals such as the Gaussian pulse in Figure 2.1 can be represented as either a time waveform or as a complex spectrum that has both magnitude and phase. These forms are alternate but completely equivalent ways of describing the same pulse. Some problems are more easily solved in the frequency domain, while others are better done in the time domain. Consequently, it will be necessary to use a method to switch from one domain to another. Joseph Fourier, a nineteenth century French mathematician, had an important insight that a waveform repeating in time could be synthesized from a sum of simple sines and cosines of different frequencies and phases (Bracewell, 2000). These frequencies are harmonically related by integers: a fundamental frequency (f0) and its harmonics, which are integral multiples (2f0, 3f0, etc.). This sum forms the famous Fourier series.

While the Fourier series is interesting from a historical point of view and its applicability to certain types of problems, there is a much more convenient way of doing Fourier analysis. A continuous spectrum can be obtained from a time waveform through a single mathematical operation called the “Fourier transform.” The minus i Fourier transform, also known as the Fourier integral, is defined as:

(2.1)

in which H(f) (with an upper-case letter convention for the transform) is the minus i Fourier transform of h(t) (lower-case letter for the function), “i” is , and symbolizes the minus i Fourier transform operator. Note that, in general, both h(t) and H(f) may be complex, with both real and imaginary parts. Another operation, the minus i inverse Fourier transform, can be used to recover h(t) from H(f) as follows:

(2.2)

In this equation, is the symbol for the inverse minus i Fourier transform. A sufficient but not necessary condition for a Fourier transform is the existence of the absolute value of the function over the same infinite limits; another condition is a finite number of discontinuities in the function to be transformed. If a function is physically realizable, it most likely will have a transform. Certain generalized functions that exist in a limiting sense and that may represent measurement extremes (such as an impulse in time or a pure tone) are convenient and useful abstractions. The Fourier transform also provides an elegant and powerful way of calculating a sequence of operations represented by a series of building blocks, as shown shortly.

For applications involving a sequence of numbers or data, a more appropriate form of the Fourier transform, the discrete Fourier transform (DFT), has been devised. The DFT consists of a discrete sum of N-weighted complex exponents, exp(−i2π mn/N), in which m and n are integers. J. W. Cooley and J. W. Tukey (1965) introduced an efficient way of calculating the DFT called the fast Fourier transform (FFT). The DFT and its inverse are now routine mathematical algorithms and have been implemented directly into signal processing chips.

2.2.2 Fourier Transform Relationships

The most important relationships for the Fourier transform, the DFT, and their application are reviewed in Appendix A. This section emphasizes only key features of the Fourier transform, but additional references are provided for more background and details.

A key Fourier transform relationship is that time lengths and frequency lengths are related reciprocally. A short time pulse has a wide extent in frequency, or a broad bandwidth. Similarly, a long pulse, such as a tone burst of n cycles, has a narrow band spectrum. These pulses are illustrated in Figure 2.2 and Figure 2.3. If, for example, a tone burst of 10 cycles in Figure 2.2 is halved to 5 cycles in Figure 2.3, its spectrum is doubled in width. All of these effects can be explained mathematically by the Fourier transform scaling theorem:

(2.3)

For this example, if g(t) is shown in Figure 2.2, then for the shorter length signal in Figure 2.3, if a=2.0, then the spectrum is halved in amplitude and its width is stretched by a factor of two in its frequency extent. Many other Fourier transform theorems are listed in Table A.1 of Appendix A.

Consider the Fourier transform pair from this table for a Gaussian function:

(2.4)

To find the minus i Fourier transform of a following given time domain Gaussian analytically, for example,

(2.5A)

first put it into a form appropriate for the scaling theorem, Eqn 2.3, and the Gaussian, Eqn 2.4,

(2.5B)

so that Then by the scaling theorem, the transform is:

(2.6)

The Gaussian is well behaved and has smooth time and frequency transitions. Fast time transitions have a wide spectral extent. An extreme example of this characteristic is the impulse in Figure 2.4. This pulse is so short in time that, in practical terms, it appears as a spike or as a signal amplitude occurring only at the smallest measurable time increment. The ideal impulse would have a flat spectrum (or an extremely wide one in realistic terms). The converse of the impulse in time is a tone burst so long that it would mimic a sine wave as in Figure 2.5. The spectrum of this nearly pure tone would appear on a spectrum analyzer (an instrument for measuring the spectra of signals) as either an amplitude at a single frequency in the smallest resolvable frequency resolution cell or as a spectral impulse. Note that instead of a pair of spectral lines representing impulse functions in Figure 2.5, finite width spectra are shown as a consequence of the finite length time waveform used for this calculation by a digital Fourier transform. All of these effects can be demonstrated beautifully by the Fourier transform. The Fourier transform operations for Figures 2.1–2.5 were implemented by MATLAB program chap2figs.m.

2.3.1 Time and Frequency Building Blocks

One of the motivations for using the Fourier transform is that it can describe how a signal changes its form as it propagates or when it is sent through a device or filter. Both of these changes can be represented by a building block. Assume there is a filter that has a time response, q(t), and a frequency response, Q(f). Each of these responses can be represented by a building...