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Sec. 2–2 Fourier Transform and Spectra 49
This can easily be demonstrated by writing the spectrum in polar form:
Then
and
W(f) = |W(f)|e ju(f)
W(-f) = |W(-f)|e ju(-f)
W * (f) = |W(f)|e -ju(f)
Using Eq. (2–35), we see that Eqs. (2–38) and (2–39) are true.
In summary, from the previous discussion, the following are some properties of the
Fourier transform:
• f, called frequency and having units of hertz, is just a parameter of the FT that specifies
what frequency we are interested in looking for in the waveform w(t).
• The FT looks for the frequency f in the w(t) over all time, that is, over
-q 6 t 6q
• W(f) can be complex, even though w(t) is real.
• If w(t) is real, then W(-f) = W * (f).
Parseval’s Theorem and Energy Spectral Density
Parseval’s Theorem.
L
If w 1 (t) = w 2 (t) = w(t), then the theorem reduces to
q
-q
q
w 1 (t)w * 2 (t)dt = W 1 (f)W * 2 (f)df
L
-q
(2–40)
Rayleigh’s energy theorem, which is
q
q
E = |w(t)| 2 dt = |W(f)| 2 df
L L
(2–41)
PROOF. Working with the left side of Eq. (2–40) and using Eq. (2–30) to replace
w 1 (t) yields.
L
q
-q
-q
q q
w 1 (t)w * 2 (t) dt = c W 1 (f)e j2pft df dw * 2 (t) dt
L L
-q
=
L
q
-q
q
-q L-q
-q
W 1 (f)w 2 * (t)e j2pft df dt