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46
Signals and Spectra Chap. 2
These evaluation techniques are developed throughout the remainder of this chapter.
From Eq. (2–26), since is complex, W(f) is a complex function of frequency.
W(f) may be decomposed into two real functions X(f) and Y(f) such that
e -j2pft
W(f) = X(f) + jY(f)
(2–27)
which is identical to writing a complex number in terms of pairs of real numbers that can be
plotted in a two-dimensional Cartesian coordinate system. For this reason, Eq. (2–27) is
sometimes called the quadrature form, or Cartesian form. Similarly, Eq. (2–26) can be
written equivalently in terms of a polar coordinate system, where the pair of real functions
denotes the magnitude and phase:
That is,
W(f) = |W(f)|e ju(f)
|W(f)| = 3X 2 (f) + Y 2 (f) and u(f) = tan -1 a Y(f)
X(f) b
(2–28)
(2–29)
This is called the magnitude–phase form, or polar form. To determine whether certain
frequency components are present, one would examine the magnitude spectrum ƒW(f)ƒ,
which engineers sometimes loosely call the spectrum.
The time waveform may be calculated from the spectrum by using the inverse Fourier
transform
q
w(t) = W(f)e j2pft df
L
-q
(2–30)
The functions w(t) and W(f) are said to constitute a Fourier transform pair, where w(t)
is the time domain description and W(f) is the frequency domain description. In this book, the
time domain function is usually denoted by a lowercase letter and the frequency domain
function by an uppercase letter. Shorthand notation for the pairing between the two domains
will be denoted by a double arrow: w(t) 4 W(f).
The waveform w(t) is Fourier transformable (i.e., sufficient conditions) if it satisfies
both Dirichlet conditions:
• Over any time interval of finite width, the function w(t) is single valued with a finite
number of maxima and minima, and the number of discontinuities (if any) is finite.
• w(t) is absolutely integrable. That is,
q
|w(t)| dt 6q
(2–31)
L-q
Although these conditions are sufficient, they are not necessary. In fact, some of the examples
given subsequently do not satisfy the Dirichlet conditions, and yet the Fourier transform can
be found.