In these notes, we will adopt the engineering convention of using 
as the imaginary unit. We will in general denote the Fourier transform of
a function f as 
.
Consider a function 
defined on the entire real line 
. The Fourier transform is defined as:
and the inverse Fourier transform as:
Some comments on conventions are necessary. First, the transform involves a
negative sign in the exponent, and the inverse transform involves a positive
sign. In certain contexts, and in particular depending upon whether a
mathematical, physical or engineering perspective is used, this situation
may be reversed. As we shall see later in the case of plane waves, this
convention is sometimes mixed within the same context! The other remark
involves the 
factor present in the inverse transform. Again,
sometimes this factor is present in the forward transform, or sometimes both
(1.1) and (1.2) contain the factor 
. In other
contexts, the formulas are written in terms of 
; for
example, if u=t, a time variable with units of 
, then 
, called radian frequency, and having units of 
, while 
has units of Hz (Hertz). Making this change of variables eliminates the 
factor all together.
The formulas (1.1) and (1.2) are the basis for Fourier analysis
of functions defined on the unbounded 1-dimensional continuum from 
to 
. The inverse transform (1.2) has the additional
interpretation of representing any such function as the superposition of a
collection of sinusoidal functions:
The parameter 
must range over the unbounded continuum 
in order for all functions to be so represented. An important
parameter in Fourier analysis is the total energy in a function f,
defined as:
The total energy in the frequency domain is defined as:
and an important result is Parseval's theorem:
Alternatively, consider a function defined on finite
interval 
. Then the Fourier transform and inverse transform
relations are, respectively:
and:
where:
Actually, (1.8) is sometimes called the Fourier series, and
the values 
are called the Fourier coefficients of f. Note
that (1.8) represents the function f as the superposition of a
discrete collection of sinusoids at frequencies 
, where the
fundamental frequency 
is related to the length of the domain
of definition T via (1.9). This form of Fourier analysis has a
different interpretation. If we extend the definition of f to beyond the
interval , then the extended function is periodic with
period T. Indeed, the basis sinusoidal functions:
are precisely the sinusoids which have period T.
Thus, whereas (1.1) and (1.2) define Fourier analysis for
aperiodic functions, (1.7) and (1.8) define Fourier
analysis for periodic functions. Whereas the frequency domain for aperiodic
functions is continuous, the frequency domain for periodic functions is discrete, and indeed the representation of a periodic function with a sequence of values at the frequencies
is called the line spectrum of f. Indeed, if we attempt to compute
the aperiodic Fourier transform (1.1) of a periodic function f, we
get an impulse train:
where 
denotes the Dirac delta function.
Now, if we consider a function f specified on the interval
which is not extended periodically but is instead zero outside
this interval, the two Fourier analyses do not coincide. To clarify this,
let be periodic with period T and Fourier coefficients
, and let 
be the window function
defined as:
and define the truncated function 
as:
Then the Fourier transform (1.1) 
of is:
where:
A comparison of (1.14) and (1.11) reveals that
truncated a periodic function to one period causes the line spectrum to
become continuous, but ``approximately'' discrete to the extent which the 
functions in (1.14) have nonzero width and rippling tails.
Let us generalize to the case where an otherwise periodic function is
truncated to several periods. Thus, let f have period T and Fourier
coefficients corresponding to a discrete line spectrum. We
truncate f with the window function 
to N complete periods, thus
obtaining:
Then the Fourier transform of 
is:
The continuous spectrum 
is an approximation of the discrete
spectrum in the sense that narrow pulses ( functions) of
width (measured from first null to first null) 
. These pulses,
corresponding to spectral resolution, get narrower as N increases.
As we include more periods of f in our ``window'', the spectrum becomes
more and more discrete.
Consider, for example, a wave function present in a one-dimensional
crystalline structure (i.e., a periodic potential structure). If the
periodic structure were extended to infinity, without bound, then the wave
function could be represented by a line spectrum, and the ``allowable''
frequencies would be discrete, namely . However, in any real
device or material, the crystal has boundaries. The impact of this upon the
wave function depends on the nature of the boundary. The case presented
above assumes the function is truncated to identically 0 outside the
material region, but regardless of the precise nature of the boundary
conditions, we are left with a quasi-periodic function with a continuous spectrum (that is, a continuum of allowed frequecies); the
continuous spectrum is ``approximately'' discrete, with the approximate
model becoming more appropriate as the dimensions of the material encompass
a sufficiently large number of fundamental periods of the wave function.
