# Signals 315 - Lecture 22 Modulators and Amplitude Modulation by olliegoblue23

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```									         Systems & Signals 315
Lecture 22: Modulators and Amplitude Modulation

Dr. Herman A. Engelbrecht

Stellenbosch University
Dept. E & E Engineering

23 March 2009

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Outline

1   Modulators
Multiplier Modulators
Nonlinear Modulators
Switching Modulators

2   Double-sideband Modulation (DSB)
DSB Modulation Recap
DSB Demodulation Recap

3   Amplitude Modulation (AM)
AM Modulation
Modulation Index
AM Modulation

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Multiplier Modulators

m(t)                  xDSB (t)

Ac cos(ωct)

Multiply m(t) and cos(ωc t) using a analog
multiplier.
Could use two logarithmic ampliﬁers, a summing
ampliﬁer and an antilogarithmic ampliﬁer.
Balanced modulator (ex. 1496 IC) can be used
as a general multiplier but issues arise with
linearity, especially at high frequencies.
True multipliers are expensive and complex.
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Nonlinear Modulators
Use nonlinear devices (diode, transistor) to
achieve modulation.

y (t) ≈ ax(t) + bx 2 (t)

x1(t)        y1 (t)
m(t)                          NL

+       z(t)
BPF
−                   4bm(t) cos(ωc t)

−
cos(ωc t)                      NL
x2(t)        y2 (t)

2                   2
z(t) = y1 (t)−y2 (t) = ax1 (t) + bx1 (t) − ax2 (t) + bx2 (t)

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Nonlinear Modulators
Let x1 (t) = cos(ωc (t) + m(t) and
x2 (t) = cos(ωc (t) − m(t) then

y1 (t) = a cos(ωc t) + am(t) + b cos2 (ωc t)
+2bm(t) cos(ωc t) + bm2 (t)
y2 (t) = a cos(ωc t) − am(t) + b cos2 (ωc t)
−2bm(t) cos(ωc t) + bm2 (t)
z(t) = 2am(t) + 4bm(t) cos(ωc t)

Bandpass ﬁlter (BPF) can be used to remove
2am(t).

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Balanced Modulators
z(t) only contains one of the inputs (i.e. m(t)).
Circuit acts as a balanced bridge to the carrier
input.
Example of a class of balanced modulators.
BPF must reject one of the inputs, thus the
circuit is a single balanced modulator.
Circuits which are balanced w.r.t. both inputs are
called double balanced modulators.
A ring modulator is an example of a double
balanced modulator.

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Switching (Chopper) Modulators
Multiply m(t) with a function other than the
cos-function.
Periodic functions have impulses in the
spectrum.
Use a rectangular pulse train - an on/off switch.
Chopper modulator:
m(t)                   c(t)
c(t) = m(t) × pulse train
t
= m(t) Π τ ∗ ∆Tc (t)
τ = Tc 50% duty cycle
2                                       control signal
Tc = f1c

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Low-Cost Mixer

m(t)               c(t)            xc (t)       m(t)       xc (t)
BPF             ≡
center = fc                  Ac cos(ωc t)
control signal            bandwidth = 2W

The switch is non-linear, but the total system is
linear.
Can be used instead of a mixer (for messages
that needs to be mixed up to fc ).

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DSB Modulation Recap
xc (t) = Ac m(t) cos(ωc t)

m(t)                  xDSB (t)

Ac cos(ωct)
Mixer (“menger”) - Mix m(t) up to fc frequency.

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DSB Demodulation Recap
xr (t)                                         Ac m(t)
LPF

SYNC
2 cos(ωct)
Need a cos-function at the receiver, with the
same frequency and phase as the original
carrier. Requires coherent or sinchronous
demodulation.
Is there a way to demodulate DSB without
sinchronisation?
Remove the phase-reversal of xc (t) by keeping
m(t) from becoming negative i.e. add a DC
component to m(t).

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The DSB spectrum does not contain a
component at the carrier frequency fc , unless the
message m(t) has a DC component.
DSB systems with no spectral component at the
carrier frequency is referred to as suppresed
carrier systems (DSB-SC).
By adding a DC component to the message m(t)
we can ensure that xc (t) has a spectral
component at the carrier frequency fc .
Demodulation then becomes extracting the
envelope of xr (t), which leads to Amplitude
Modulation (AM).

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AM Modulation

xc (t) = Ac [A + m(t)] cos(ωc t), A + m(t) ≥ 0
= Ac [1 + µmp (t)] cos(ωc t) where
Ac = Ac A
m(t)
mp (t) =
|min m(t)|
|min m(t)|
µ =
A

µ - the modulation index (or the percentage
modulation)
0 ≤ µ ≤ 1 - required condition for AM to be
demodulated using an envelope detector

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Modulation Index
The modulation index is a measure of the
percentage modulation in the AM signal.
µ = 0: no modulation - only the carrier is
transmitted.
µ = 1: 100% modulation
µ > 1: overmodulation - causes distortion in
µ < 1: undermodulation - causes some
transmitted power to be wasted.

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AM Modulation
Types of DSB modulation:
Large carrier component: DSB-LC ≡ AM
Suppressed carrier: DSB-SC ≡ DSB
m(t)                            xc (t)

A                    ′
Ac cos(ωc t)

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