Page 277 - Analog and Digital Filter Design

274 Analog and Digital Filter Design

The two quadrature signals are then used to modulate a carrier in separate
mixers. The carrier signal to be modulated is applied directly to one mixer, but
through a 90" phase-shift network for the other mixer (alternatively the carrier
phase is shifted by +45" at one mixer and by 45" at the other). In one mixer
the output is:

Sin(m1.t) .sin(m2t) = 1/2 .cos([ml - w2] .t) - 1/2 .cos([ml+ m2] .t).

At the other mixer the output is:

The outputs from the two mixers can now be added or subtracted to give the
required sideband. Adding gives cos([wl - m2].t), which is the lower sideband.
Subtracting gives cos([ml + m2].t), which is the upper sideband. Notice that the
amplitude is unity, rather than the half of each sideband produced by simply
ftltering out the unwanted sideband.

S. D. Bedrosian has studied the problem of producing quadrature phase-shift
circuits. He has written a paper' that gives pole position formulae for quadra-
ture networks. These formulae can be used to produce active or passive quad-
rature circuits. The quadrature circuit comprises two delay networks, known as
the P net and the N net because calculations give positive (P) and negative (N)
pole locations on the real axis. The P net and the N net have a common input
and separate outputs. Each network produces a phase shift across the frequency
band of interest, but the phase shift of one network is 90" more than the other.
Only the relative phase difference is important; the absolute phase shift is
irrelevant for our purpose.

Active or passive first-order equalizer sections, described earlier in Figures 9.3
and 9.10, respectively, can be used in cascade to form the P and N networks.
The number of first-order equalizer sections in each P or N network is numer-
ically half the order of the quadrature network. For example, a fourth-order
quadrature circuit has two iirst-order equalizer sections in each network. Tables
9.14 to 9.17 give the normalized pole locations for equalizers with ratios of
upper to lower passband frequency of 11.35,20, 50, and 100. The ratio of 11.35
was chosen for the popular 300Hz to 3.4kHz band.