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Bandpass Signaling Principles and Circuits Chap. 4
where s(t) is a bipolar switching waveform, as shown in Fig. 4–11d. Since the switching waveform
arises from the LO signal, its period is T 0 = 1f 0 . The switching waveform is described by
s(t) = 4 a
q
n=1
sin(np/2)
np
cos nv 0 t
(4–70)
so that the mixer output is
v 1 (t) = [v in (t)]c4K a
q
n=1
sin(np/2)
np
cos nv 0 td
(4–71)
This equation shows that if the input is a bandpass signal with nonzero spectrum in the vicinity
of f c , the spectrum of the input will be translated to the frequencies |f c ± nf 0 |, where n = 1,
3, 5, . . . . In practice, the value K is such that the conversion gain (which is defined as the
desired output level divided by the input level) at the frequency |f c ± f 0 | is about -6 dB.
Of course, an output filter may be used to select the up-converted or down-converted
frequency band.
In addition to up- or down-conversion applications, mixers (i.e., multipliers) may be
used for amplitude modulators to translate a baseband signal to an RF frequency band, and
mixers may be used as product detectors to translate RF signals to baseband. These applications
will be discussed in later sections that deal with transmitters and receivers.
4–12 FREQUENCY MULTIPLIERS
Frequency multipliers consist of a nonlinear circuit followed by a tuned circuit, as illustrated
in Fig. 4–12. If a bandpass signal is fed into a frequency multiplier, the output will appear in
a frequency band at the nth harmonic of the input carrier frequency. Because the device is
nonlinear, the bandwidth of the nth harmonic output is larger than that of the input signal. In
general, the bandpass input signal is represented by
v in (t) = R(t) cos[v c t + u(t)]
(4–72)
The transfer function of the nonlinear device may be expanded in a Taylor’s series, so that the
nth-order output term is
or †
v 1 (t) = K n v n in (t) = K n R n (t) cos n [v c t + u(t)]
v 1 (t) = CR n (t) cos[nv c t + nu(t)] + other terms
† mth-order output terms, where m 7 n, may also contribute to the nth harmonic output, provided that K m
is sufficiently large with respect to K n . This condition is illustrated by the trigonometric identity 8cos 4 x = 3 + 4
cos 2x + cos 4x, in which m = 4 and n = 2.