In previous chapters we have considered Fourier series models only for discrete functions defined over a finite interval L. Such functions are common currency in experimental science since usually the continuous variables under study are quantified by sampling at discrete intervals. We found that if such a functions are defined for D sample points, then a Fourier series model with D Fourier coefficients will fit the data exactly. Consequently, the frequency spectra of such functions are also discrete and they exist over a finite bandwidth W=D/2L. In short, discrete functions on a finite interval have discrete spectra with finite bandwidth, and D data points produce D Fourier coefficients. This is Case 1 in Fig. 5.1.
We noted in Chapter 4 that if we increase the number of samples by lengthening the observation interval without changing the sampling rate, the result is an increase in the frequency resolution of the spectrum over the same bandwidth. The longer we take samples, the finer the frequency resolution of the spectrum. Pressing this argument further, we can imagine that if the observation interval grows infinitely long, then the resolution of the spectrum grows infinitesimally small and, in the limit, the spectrum becomes a continuous function. Thus a discrete function over an infinite interval has a continuous spectrum with finite bandwidth. This is Case 2 in Fig. 5.1.
Conversely, we noted that if we increase the sampling rate over a fixed, finite interval, then the bandwidth of the spectrum increases without changing the resolution of the spectrum. If the sampling rate grows infinitely high, then the resolution in the space/time domain grows infinitesimally small and, in the limit, the function becomes continuous. At the same time the bandwidth grows infinitely large. Thus, a continuous function over a finite interval has a discrete spectrum with an infinite bandwidth. This is Case 3 in Fig. 5.1.
Finally, if the space/time function is both continuous and defined over an infinite interval, then the spectrum is both continuous and defined over an infinite bandwidth. This is Case 4 in Fig. 5.1 and this is the province of the Fourier Transform proper. The student may appreciate that Case 4 is very general and it can be made to encompass the other three cases merely by throwing out points in the space or frequency domains to yield discrete functions as required. It is this generality which makes the Fourier Transform the primary focus in many engineering texts. We, on the other hand, will take a more pedestrian approach and march resolutely ahead, one step at a time.
Fig. 5.1 Schematic view of the 4 cases of Fourier analysis. Only the magnitude portion of spectrum is illustrated. In cases 2 and 3, continuous functions are depicted as the limiting case when resolution approaches zero.
The primary tool for calculating Fourier coefficients for discrete functions is the inner product. This suggests that it would be worthwhile trying to extend the notion of an inner product to include continuous functions. Consider the scenario in Case 3, for example, where the sampling rate grows large without bound, resulting in an infinitesimally small resolution Δx in the space/time domain. If we treat the sequence of samples as a vector, then the dimensionality of the vector will grow infinitely large as the discrete function approaches a continuous state. The inner product of two such vectors would be
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[5.1] |
which is likely to grow without bound as D grows infinitely large. In order to keep the sum finite, consider normalizing the sum by multiplying by Δx=L / D. That way, as D grows larger, Δx compensates by growing smaller thus keeping the inner product stable. This insight suggests that we may extend our notion of inner product of discrete functions to include continuous functions defined over the finite interval (a,b) by defining the inner product operation as
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[5.2] |
In words, this equation says that the inner product of two continuous functions equals the area under the curve (between the limits of a and b) formed by the product of the two functions. Similarly, we may extend our notion of orthogonality by declaring that if the inner product of two continuous functions is zero according to eqn. [5.2] then the functions are orthogonal over the interval specified.
In the study of discrete functions we found that the harmonic set of sampled trigonometric functions Ck and Sk were a convenient basis for a Fourier series model because they are mutually orthogonal. This suggests that we investigate the possible orthogonality of the continuous trigonometric functions. For example, consider cos(x) and sin(x) over an interval covering one period:
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[5.3] |
The easiest way to see why the integral is zero is by appeal to the symmetry of the sine function, which causes the area under the function for one full period to be zero. Note that since the two given functions are orthogonal over one period, they will be orthogonal over any integer number of periods.
To do another example, consider the inner product of cos(x) and cos(2x):
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[5.4] |
The last step was based on the fact that since both of these integrals have odd symmetry, the area under each curve separately is zero and so the total area is zero.
Based on the successful outcome of these examples, we will assert without proof that the harmonic family of sines and cosines are orthogonal over any interval of length equal to the period of the fundamental harmonic.
The inner product of a vector with itself yields the squared length of the vector according to the Pythagorean theorem for D-dimensional space. In the case of the sampled trigonometric functions, the squared length equals the dimensionality parameter D. To see the corresponding result for the case of continuous functions, consider the inner product of cos(x) with itself:
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[5.5] |
In a similar fashion it can be shown that the inner product of any harmonic cos(kx) with itself over the interval (-π, π) has area π.
The analogous result for a cosine function with period L is
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[5.6] |
where simplification was achieved by a substitution of variables y=2πx / L. Thus in general
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[5.7] |
The inner product of a continuous function with itself has an important interpretation in many physical situations. For example, Ohm's law of electrical circuits states that the power dissipated by a resister is given by the squared voltage divided by resistance. If v(t) describes the time course of voltage across a 1 ohm resistor, then the time-course of power consumption is v2(t) and the total amount of energy consumed over the interval from 0 to T seconds is
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[5.8] |
The average power consumption over the interval is given by dividing the total energy consumed by the length of the interval
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[5.9] |
For example, the average power consumption by a 1 Ohm resistor for a voltage waveform v(t) = A cos(x) equals A2/2.
Similarly, if v(t) is the instantaneous velocity of an object with unit mass then the integral in eqn. [5.8] equals the total amount of kinetic energy stored by the object. By analogy, even in contexts quite removed from similar physical situations, the inner product of a function with itself is often described as being equal to the amount of energy in the function.
The computation of Fourier coefficients for discrete functions involved forming the inner product of the data vector with the sampled trigonometric functions, so the student should not be too surprised when we find that the inner product also appears when computing Fourier coefficients for continuous functions. Since the inner product of continuous functions requires the evaluation of an integral, any shortcuts that ease this burden will be most useful. One such shortcut is based on symmetry and for this reason we make a short digression here on some general aspects of symmetrical functions.
Two types of symmetry are possible for ordinary, real-valued functions of 1 variable (e.g. y(x)=2x+5)). If y(x)=y(-x) then the function y(x) is said to have even symmetry, whereas if y(x)=-y(-x) then the function y(x) is said to have odd symmetry. The student may be surprised to learn that any particular y(x) can always be represented as the sum of some even function and an odd function. To demonstrate this fact, let E(x) be an even function and O(x) be an odd function defined by the equations
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[5.10] |
To verify that E(x) is even we replace the variable x by -x everywhere and observe that this substitution has no effect. In other words, E(x)=E(-x). To verify that O(x) is odd we replace the variable x by -x everywhere and observe that this substitution introduces a change in sign. In other words, O(x)=O(-x). Finally, we combine this pair of equations and observe that E(x)+O(x)=y(x).
The significance of the foregoing result is that often an integral involving y(x) can be simplified by representing y(x) as the sum of an even and an odd function and then using symmetry arguments to evaluate the result. The symmetry arguments being referred to are the following.
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[5.11] |
A third kind of symmetry emerges when considering complex valued functions of 1 variable (e.g. y(x)=2x + i 5x)). If some function y(x) is complex-valued, then it may be represented as the sum of a purely real function yR(x) and a purely imaginary function yI(x). If the function y(x) has even symmetry, then
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[5.12] |
Equating the real and imaginary parts of this equation separately we see that
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[5.13] |
In other words, if y(x) is even then both the real and imaginary parts of y(x) are even. A similar exercise will demonstrate to the student that if y(x) is odd, then both the real and imaginary components of y(x) are odd.
A different kind of symmetry mentioned previously is when the real part of y(x) is even but the imaginary part of y(x) is odd. In this case,
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[5.14] |
Generalizing the notion of complex conjugate developed earlier for complex numbers, we could say that the function y(x) has conjugate symmetry, or Hermitian symmetry.
Conjugate symmetry plays a large part in Fourier analysis of continuous functions. For example, the basis functions of the form eix = cosx + isinx are Hermitian. Furthermore, in Chapter 4 it was observed that the complex Fourier coefficients for real-valued data vectors have conjugate symmetry: ck = c*-k. When the spectrum becomes continuous, as in cases 2 and 4 in Fig. 5.1, then the spectrum is a complex-valued function. In the next chapter we will show that such a spectrum possesses conjugate symmetry. Some symmetry relations listed in Bracewell's textbook The Fourier Transform and Its Applications (p.14) are given in Table 12.2 in Chapter 12.