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Sec. 4–1 Complex Envelope Representation of Bandpass Waveforms 239
when s(t) K v(t), the noise when n(t) K v(t), the filtered signal plus noise at the channel output
when r(t) K v(t), or any other type of bandpass waveform. †
THEOREM.
Any physical bandpass waveform can be represented by
v(t) = Re{g(t)e jvct }
(4–1a)
Here, Re{·} denotes the real part of {·}, g(t) is called the complex envelope of v(t), and
f c is the associated carrier frequency (in hertz) where v c = 2p f c . Furthermore, two other
equivalent representations are
and
where
and
v(t) = R(t) cos[v c t + u(t)]
v(t) = x(t) cos v c t - y(t) sin v c t
g(t) = x(t) + jy(t) = ƒ g(t) ƒ e jlg(t) K R(t)e ju(t)
x(t) = Re{g(t)} K R(t) cos u(t)
y(t) = Im{g(t)} K R(t) sin u(t)
R(t) ! |g(t)| K 3x 2 (t) + y 2 (t)
u1t2 ! g1t2
= tan -1 a y(t)
x(t) b
(4–1b)
(4–1c)
(4–2)
(4–3a)
(4–3b)
(4–4a)
(4–4b)
Proof. Any physical waveform (it does not have to be periodic) may be represented
over all time, T 0 : q, by the complex Fourier series:
v(t) =
n=q
a
n=-q
c n e jnv 0t , v 0 = 2p>T 0
(4–5)
Furthermore, because the physical waveform is real, c-n = c and, using
+ 1 2 { #
n
*,
} * , we obtain
Re{# } =
1
2 { # }
v(t) = Ree c 0 + 2 a
q
n=1
c n e jnv 0t f
(4–6)
Furthermore, because v(t) is a bandpass waveform, the c n have negligible magnitudes for n in
the vicinity of 0 and, in particular, c 0 = 0. Thus, with the introduction of an arbitrary parameter
f c , Eq. (4–6) becomes ‡
† The symbol K denotes an equivalence, and the symbol ! denotes a definition.
‡ Because the frequencies involved in the argument of Re{·} are all positive, it can be shown that the complex
function 2g n=1 q c n e jnv 0 t
is analytic in the upper-half complex t plane. Many interesting properties result because
this function is an analytic function of a complex variable.