Page 211 - Fundamentals of Communications Systems

Amplitude Modulation 6.33

This problem tries to lead you to an explanation of why the filtering does not
significantly affect the envelope detector’s performance. Consider the simple
case of a sinusoidal message signal
m(t) = cos(2π f m t)

with f m > 0 and a VSB modulator that produces a complex envelope signal of
the form

A c (1 + acos(2π f m t) + jasin(2π f m t)) f m > f v
x z (t) = (6.32)
A c (1 + acos(2π f m t)) f m ≤ f v
(a) Sketch the modulator that would produce Eq. (6.32) and derive the transfer
function of the quadrature filter, H Q (f ), that is necessary to produce this
signal.
(b) Calculate the output envelope, |y z (t)|.
(c) Show that the envelope detector output is the desired signal (the message
signal plus a DC offset).
(d) Consider the case where modulator is exactly the same as above and the
message signal is the sum of two sinusoids
m(t) = A 1 cos(2π f 1 t) + A 2 cos(2π f 2 t) (6.33)

what would the form of x z (t) be?
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(e) Show if a is chosen such that a is small that the envelope detector output
is approximately the desired signal (the message signal plus a DC offset).
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(f) Compute the MCPR for f m < f v and f m > f v assuming a is small.
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(g) Choose a value of a for which a is small compared to a and the envelope
detector can be used with a small resulting distortion. What does this say
about the MCPR of typical TV broadcast?

Problem 6.12. RF engineers that design and build quadrature upconverters (see
Figure 6.37) need tests to estimate how close to ideal their circuits are perform-
ing. The standard test used is known as a single sideband rejection test. This
test uses an input of x I (t) = cos(2π f m t) and x Q (t) = sin(2π f m t) and measures
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the resulting bandpass power spectrum, |X c (f )| on a spectrum analyzer.
2
(a) Compute what the output bandpass spectrum, |X c (f )| , should be for an
ideal quadrature upconverter.
(b) A common design issue in quadrature modulators is that the quadrature
carrier has a phase offset compared to the in-phase carrier, i.e.,
√ √
x c (t) = x I (t) 2 cos(2π f c t) − x Q (t) 2 sin(2π f c t + θ)
For the test signal in a single sideband rejection test what will be the output
bandpass spectrum as a function of θ.