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Sec. 4–11 Mixers, Up Converters, and Down Converters 267
up-conversion frequency band, where f u = f c + f 0 , and one at the down-conversion band,
where f d = f c - f 0 . A filter, as illustrated in Fig. 4–8, may be used to select either the upconversion
component or the down-conversion component. This combination of a mixer
plus a filter to remove one of the mixer output components is often called a single-sideband
mixer. A bandpass filter is used to select the up-conversion component, but the
down-conversion component is selected by either a baseband filter or a bandpass filter,
depending on the location of f c - f 0 . For example, if f c - f 0 = 0, a low-pass filter would be
needed, and the resulting output spectrum would be a baseband spectrum. If f c - f 0 7 0,
where f c - f 0 was larger than the bandwidth of g in (t), a bandpass filter would be used, and
the filter output would be
v 2 (t) = Re{g 2 (t)e j(v c-v 0 )t } = A 0
2 Re{g in(t)e j(v c-v 0 )t }
(4–58)
For this case of f c 7 f 0 , it is seen that the modulation on the mixer input signal v in (t) is preserved
on the mixer up- or down-converted signals.
If f c 6 f 0 , we rewrite Eq. (4–57), obtaining
v 1 (t) = A 0
(4–59)
2 Re{g in(t)e j(v c+v 0 )t } + A 0
2 Re{g in *(t)ej(v 0-v c )t }
because the frequency in the exponent of the bandpass signal representation needs to be
positive for easy physical interpretation of the location of spectral components. For this case
of f c 6 f 0 , the complex envelope of the down-converted signal has been conjugated compared
to the complex envelope of the input signal. This is equivalent to saying that the sidebands
have been exchanged; that is, the upper sideband of the input signal spectrum becomes the
lower sideband of the down-converted output signal, and so on. This is demonstrated mathematically
by looking at the spectrum of g*(t), which is
q
q
[g * in (t)] = g * in (t)e -jvt dt = c g in (t)e -j(-v)t *
dt d
L L
-q
= G in * (-f)
(4–60)
The -f indicates that the upper and lower sidebands have been exchanged, and the conjugate
indicates that the phase spectrum has been inverted.
In summary, the complex envelope for the signal out of an up converter is
g 2 (t) = A 0
(4–61a)
2 g in(t)
where f u = f c + f 0 7 0. Thus, the same modulation is on the output signal as was on the input
signal, but the amplitude has been changed by the A 0 2 scale factor.
For the case of down conversion, there are two possibilities. For f d = f c - f 0 7 0, where
f 0 6 f c ,
-q
g 2 (t) = A 0
2 g in(t)
(4–61b)