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Sec. 4–5 Bandpass Filtering and Linear Distortion 249
K(f) with the help of Eqs. (4–11) and (4–12). Figure 4–3b shows a typical bandpass frequency
response characteristic |H(f)|.
THEOREM. The complex envelopes for the input, output, and impulse response of a
bandpass filter are related by
1
2 g 2 (t) = 1 2 g 1(t) * 1 2 k(t)
(4–22)
where g 1 (t) is the complex envelope of the input and k(t) is the complex envelope of the
impulse response. It also follows that
1
2 G 2(f) = 1 2 G 1(f) 1 2 K(f)
(4–23)
Proof. We know that the spectrum of the output is
V 2 (f) = V 1 (f)H(f)
(4–24)
Because v 1 (t), v 2 (t), and h(t) are all bandpass waveforms, the spectra of these waveforms are
related to the spectra of their complex envelopes by Eq. (4–15); thus, Eq. (4–24) becomes
1
2 [G 2(f - f c ) + G * 2(-f - f c )]
= 1 2 [G 1(f - f c ) + G * 1(- f - f c )] 1 2 [K(f - f c) + K * (- f - f c )]
(4–25)
= 1 4 [G 1(f - f c )K(f - f c ) + G 1 (f - f c )K * (- f - f c )
+ G 1 * (-f - f c )K(f - f c ) + G 1 * (-f - f c )K * (-f - f c )]
But G 1 (f - f c ) K * (-f - f c ) = 0, because the spectrum of G 1 (f - f c ) is zero in the region of
frequencies around -f c , where K * (-f - f c ) is nonzero. That is, there is no spectral overlap
of G 1 (f - f c ) and K * (-f - f c ), because G 1 (f) and K(f) have nonzero spectra around only
f = 0 (i.e., baseband, as illustrated in Fig. 4–3d). Similarly, G * 1(-f-f c ) K(f - f c ) = 0.
Consequently, Eq. (4–25) becomes
3 1 2 G 2(f-f c )4 + 3 1 2 G* 2(-f - f c )4
= 3 1 2 G 1(f-f c ) 1 2 K(f-f c)4 + 3 1 2 G* 1(-f-f c ) 1 2 K * ( -f-f c )4
(4–26)
1
Thus,
2 G 2(f) = 1 2 G 1(f) 1 2 K(f), which is identical to Eq. (4–23). Taking the inverse
Fourier transform of both sides of Eq. (4–23), Eq. (4–22) is obtained.
This theorem indicates that any bandpass filter system may be described and analyzed by
using an equivalent low-pass filter as shown in Fig. 4–3c. A typical equivalent low-pass frequency
response characteristic is shown in Fig. 4–3d. Equations for equivalent low-pass filters are usually
much less complicated than those for bandpass filters, so the equivalent low-pass filter system
model is very useful. Because the highest frequency is much smaller in the equivalent low-pass
filter, it is the basis for computer programs that use sampling to simulate bandpass communication
systems (discussed in Sec. 4–6). Also, as shown in Prob. 4–17 and Fig. P4–17, the equivalent
low-pass filter with complex impulse response may be realized by using four low-pass filters with
real impulse response; however, if the frequency response of the bandpass filter is Hermitian symmetric
about f = f c , only two low-pass filters with real impulse response are required.