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48
Signals and Spectra Chap. 2
The magnitude–phase form is
See Example2_03B.m for plots.
|W(f)| = C
1
1 + (2pf) 2 and u(f) = -tan -1 (2pf)
More examples will be given after some helpful theorems are developed.
Properties of Fourier Transforms
Many useful and interesting theorems follow from the definition of the spectrum as given by
Eq. (2–26). One of particular interest is a consequence of working with real waveforms. In
any physical circuit that can be built, the voltage (or current) waveforms are real functions (as
opposed to complex functions) of time.
THEOREM.
Spectral symmetry of real signals. lf w(t) is real, then
(The superscript asterisk denotes the conjugate operation.)
PROOF.
From Eq. (2–26), we get
W(-f) = W * (f)
q
W(-f) = w(t)e j2pft dt
L
-q
and taking the conjugate of Eq. (2–26) yields
q
W * (f) = w * (t)e j2pft dt
L
-q
(2–35)
(2–36)
(2–37)
But since w(t) is real, w*(t) = w(t), and Eq. (2–35) follows because the right sides of Eqs.
(2–36) and (2–37) are equal. It can also be shown that if w(t) is real and happens to be an even
function of t, W(f) is real. Similarly, if w(t) is real and is an odd function of t, W( f) is
imaginary.
Another useful corollary of Eq. (2–35) is that, for w(t) real, the magnitude spectrum is
even about the origin (i.e., f = 0), or
|W(-f)| = |W(f)|
and the phase spectrum is odd about the origin:
u(-f) = -u(f)
(2–38)
(2–39)