The methods of Fourier analysis described in previous chapters have as their domain three classes of functions: discrete data vectors with finite number of values, discrete vectors with infinite number of values, and continuous functions which are confined to a finite interval. These are cases 1, 2, and 3, respectively, illustrated in Fig. 5.1. The fourth case admits continuous functions defined over an infinite interval. This is the province of the Fourier Transform.
The student may be wondering why it is important to be able to do Fourier analysis on functions defined over an infinite extent when any "real world" function can only be observed over a finite distance, or for a finite length of time. Several reasons can be offered. First, case 4 is so general that it encompasses the other 3 cases. Consequently, the Fourier transform provides a unified approach to a variety of problems, many of which are just special cases of a more general formulation. Second, physical systems are often modeled by continuous functions with infinite extent. For example, the optical image of a point source of light may be described theoretically by a Gaussian function of the form exp(-x2), which exists for all x. Another example is the rate of flow of water out of a tap at the bottom of a bucket of water, which can be modeled as exp(-kt). Fresh insight into the behavior of such systems may be gained by spectral analysis, providing we have the capability to deal with functions defined over all time or space. Third, by removing the restriction in case 3 that continuous functions exist over only a finite interval, the methods of Fourier analysis have developed into a very powerful analytical tool which has proved useful throughout many branches of science and mathematics.
Our approach to the Fourier transform (case 4) will be by generalizing our earlier results for the analysis of continuous functions over a finite interval (case 3). In Chapter 6 we found that an arbitrary, real-valued function y(x), can be represented exactly by a Fourier series with an infinite number of terms. If y(x) is defined over the interval (-L/2, L/2) then the Fourier model is
|
[11.1] |
As defined in eqn. [4.2], the fundamental
frequency of this model is Δf = 1 / L and
all of the higher harmonic frequencies
fkare
integer multiples of this fundamental
frequency. That is, fk = k / L = kΔf.
Now we are faced with the prospect of letting L 
∞ which
implies that Δf 0 and consequently
the concept of harmonic frequency will cease
to
be useful.
In order to rescue the situation, we need to disassociate the concepts of physical frequency and harmonic number. To do this, we first change notation so that we may treat the Fourier coefficients as functions of the frequency variable fk which takes on discrete values that are multiples of Δf. Thus eqn. [11.1] becomes
 |
[11.2] |
Next, to get around the difficulty of a vanishingly small Δf we multiply every term on the right hand side of eqn. [11.2] by the quantity Δf / Δf to get
 |
[11.3] |
This maneuver places the Fourier series is subtly different form illustrated in Fig. 11.1. Previously, the frequency spectrum was conceived as a discrete "stem" plot with the height of each stem representing the Fourier coefficients. Now, the spectrum is a "stairs" plot in which the area of each stair step represents the Fourier coefficients. One such step is highlighted in Fig. 11.1. The width of each stair step is Δf and height of each step is the amplitude of the Fourier coefficient divided by Δf, the frequency resolution of the spectrum. Thus, it would be appropriate to consider this form of the Fourier spectrum as an amplitude density graph. This is a critical change in viewpoint because now it is the areaof the cross-hatched rectangle (rather than the ordinate value) which represents the amplitude ak of this particular trigonometric component of the model.
From this new viewpoint the two summations in eqn. [11.3] represent
areas under discrete curves. This is most easily seen by considering the example
of x=0,
in which case cos(2πfx)=1 and so
the middle term in eqn. [11.3] represents
the
combined area of all the rectangles in the spectrum in Fig.
11.1. This interpretation
remains true for other values of x as well, the only difference being
that the
heights of the rectangles are modulated by cos(2πfx)
before computing their areas. Consequently, it is apparent that, in the limit
as Δf 0,
the stairstep graph becomes a smooth curve and the summation terms in eqn. [11.3] become
integrals which represent the areas under smooth spectral density functions.
That is, we implicitly
define two new functions, C(f) and S(f) based on the following
equations
 |
[11.4] |
Thus the functions C(f) and S(f) represent the limiting case
of amplitude density functions (for the cosine and sine portions of the Fourier
spectrum, respectively)
when Δf approaches zero. Given these definitions,
we may conclude that in the
limit as Δf 0 eqn. [11.3] becomes
 |
[11.5] |
This result is the inverse Fourier transform equation in trigonometrical form. It defines how to reconstruct the continuous function y(x) from the two spectral density functions C(f) and S(f). Methods for determining these spectral density functions in the first place are given next.
In order to specify the forward Fourier transform, we examine more closely the behavior of the Fourier coefficients ak and bk in the limit. To do this, recall the definition of ak given by eqn. [6.5]
 |
[11.6] |
which holds for all values of k, including k=0. Substituting Δf = 1 / L and fk = k / L we have
 |
[11.7] |
As L ∞ the
discrete harmonic frequencies 1/L, 2/L , etc. become a continuum and
the ratio
ak/Δf becomes a continuous
function of frequency denoted by C(f). Similarly,
the ratio bk/Δf becomes
a continuous function of frequency denoted by S(f).
That is,
 |
[11.8] |
The functions C(f) and S(f) are known as the cosine Fourier transform of y(x)and the sine Fourier transform of y(x), respectively. Notice that both equations are valid for the specific case of f=0, which accounts for the lack of an explicit constant term in eqn. [11.5]. Notice also that C(-f) = C(f) and S(-f) = -S(f), which means that C(f) has even symmetry and S(f) has odd symmetry.
To help appreciate the transition from Fourier series to the Fourier transform, consider the function defined in Fig. 11.2. This pulse is defined over an interval of length equal to one second. The Fourier series for a pulse may be found in a reference book or is easily computed by hand to be
 |
[11.9] |
Now if the observation interval is doubled without changing the duration of the pulse as illustrated in Fig. 11.3, we have the new Fourier series
 |
[11.10] |
Notice that, as expected, there are twice as many harmonics in the same physical bandwidth in Fig. 11.3 and the amplitudes of corresponding components are half as large in Fig. 11.3 as compared to Fig. 11.2. Thus, if we were to increase the observation interval even more, the Fourier coefficients would continue to decline and would become vanishingly small as the observation interval grows arbitrarily large. To rescue the concept of a spectrum, we should plot instead the spectral density function a(f)/Δf. By dividing by Δf, we effectively compensate for the lengthening interval. As this example demonstrates, the shape of the spectral density function for a finite-duration signal remains the same as the length of the observation interval increases, provided the extension of the interval is accomplished by "padding" with zeros (otherwise the extension will affect a(0)).
Pursuing this example, we compute the formal Fourier transform of a pulse of unit height and width w defined over an infinite interval. Applying eqn. [11.8]we find
 |
[11.11] |
This form of result occurs frequently in Fourier analysis, which has lead to the definition of the special function sinc(x) = sin(πx)/πx. Thus the final result is
| C(f) = 2wsinc(wf) | [11.12] |
for which w=0.5 in the present example.
The normalized Fourier series a(f)/Δf of a pulse seen in a 2 sec. window is compared in Fig. 11.4 with the cosine Fourier transform of the same pulse seen in an infinite window. Note that the continuous Fourier transform interpolates the discrete spectrum. This is an example of the notion that the Fourier transform is a more general tool which encompasses the Fourier series as a special case.
Although the preceding development is useful for developing an intuitive understanding of the transition from Fourier series to the Fourier transform, the actual results are largely of historical interest since modern authors invariably choose to represent the Fourier transform in complex form. Starting with the basic formulation of the complex Fourier series given in eqn. [6.6]
 |
[6.6] |
we would follow the same approach taken above. Introducing the necessary change
of variables and multiplication by Δf/Δf,
in the limit as Δf 0 this
equation becomes
the inverse Fourier transform
 |
[11.13] |
where Y(f) is the complex frequency spectrum of y(x).
To obtain the forward transform, we begin with the definition of the complex Fourier coefficients for finite continuous functions
 |
[6.7] |
As L ∞ the
discrete harmonic frequencies 1/L, 2/L , etc. become a continuum and the ratio
ck/Δf becomes a continuous function of frequency called Y(f). Thus,
the
forward Fourier transform is defined by the equation
 |
[11.14] |
By observing the striking similarity between equations [11.13] and [11.14] the student may begin to appreciate the reason for the popularity of the complex form of the Fourier transform operations. The only difference between the forward and reverse transforms is the sign of the exponent in the complex exponential. This exponential term is called the kernel of the transform and thus we observe that the kernels of the forward and reverse transforms are complex conjugates of each other. Another advantage of the complex form over the trigonometric form is that there is only one integral to evaluate rather than two. Lastly, the complex form of the transform will admit complex-valued functions for y(x).
Fourier's theorem is simply a restatement of the preceding results:
| if | |
[11.14] |
| then | |
[11.13] |
If we substitute trigonometric forms for the kernel in eqn. [11.14] using Euler's theorem eiθ = cosθ + isinθ we obtain
 |
[11.15] |
These integrals are recognized from eqn. [11.8] as the sine and cosine Fourier transforms. Thus we conclude that
 |
[11.16] |
which is the analogous result to that for Fourier coefficients described in eqn. [6.7]. This result also indicates that the Fourier transform Y(f) has Hermitian (conjugate) symmetry.
