# Objective 3.1 — Amplitude Modulation
| LO# | Description |
|----------|----------|
| 3.1 | I can determine the modulation index, output signal, and output signal bandwidth of an Amplitude Modulation (AM) system and assess whether the system is under-modulated, over-modulated, or fully modulated. |
By the end of this reading you should be able to:
1. Explain **why** we modulate rather than transmitting a message directly.
2. Determine the **modulation index** and classify a signal as under-, fully, or over-modulated.
3. Determine an AM system's **output signal** and its **bandwidth**.
4. Describe **frequency division multiplexing**.
## Modulation
Without any technological assistance, our ability to communicate over a
given distance is limited by how loud we can speak. If we wanted to talk
to someone next to us, we could use a normal conversational volume.
However, if we wanted to talk to someone on the other side of a crowded
room, we would need to yell fairly loud to get their attention. As the
communication distance increases, we would need to become louder and
louder in order to be heard. But, what if we wanted to talk to someone
many miles away, or deliver information without sending a messenger?
Whether you want to relax and listen to music from your favorite radio
station, or whether your life depends on maintaining a tactical
communication link during a battle with enemy forces, *modulation* plays
a key role. In fact, understanding how communication systems (radio,
satellites, optical fiber links, even the dial-up modem or network card
in your computer) operate requires a basic understanding of modulation.
Another advantage of modulation is the ability to transmit more than one
signal over a single communications *channel*. A single telephone line
can carry both voice and computer data because the two signals use
different frequencies for modulation. Similarly, many radio stations can
serve the same area because their signals use different frequencies for
modulation.
Although you most likely won't be designing or building transmitters any
time soon, even if you're only operating a communication system of some
kind, a basic understanding of *how* it works can be a great help.
Figure 1 shows a block diagram of a simplified communications system.
This block diagram serves to show where we are going with our discussion
of communications systems. We already discussed transducers and
transducer interfaces, and we will be covering modulation, antennas, and
the propagation of electromagnetic waves over the course of this block.
Figure 1: A simplified communications system.
## Types of Modulation
To modulate simply means to *change* something in some way. In
Electrical and Computer Engineering, we normally modulate a sinusoidal
signal called the *carrier signal*, using a *message signal*. In this
case, the information in the *message signal* is encoded within the
*carrier signal*. Consider a *carrier signal* of the form:
$$v_{c}(t) = A_{c}\cos\left( 2\pi f_{c} t + \phi_{c} \right)$$
With this type of signal, it is possible to change either the amplitude
(*Ac*), the frequency (*fc*), or the phase (*φc*) of the
sinusoidal carrier in such a way that the *information* (e.g. voice,
music, video, targeting coordinates, intel report, etc.) becomes part of
the modulated carrier, *vout*(*t*):
$$v_{out}(t) = A_{c}(t)\cos\left( 2\pi f_{c}(t)\, t + \phi_{c}(t) \right)$$
Note how the amplitude, frequency, and phase are now shown as
time-varying. The basic block diagram of a modulator is shown in Figure
2:
Figure 2: Block diagram of a modulator.
The basic modulation types are:
- Amplitude modulation (AM) -- information carried in the signal by
varying the amplitude of the carrier ($A_{c})$ in time.
- Frequency modulation (FM) -- information carried in the signal by
varying the frequency of the carrier ($f_{c})$ in time.
- Phase modulation (PM) -- information carried in the signal by
varying the phase of the carrier $(\phi_{c})$ in time.
We are most familiar with AM and FM terminology because of the radios in
our cars, but Figure 3 shows the difference between the two.
Figure 3: Amplitude modulation compared with Frequency modulation.
In Figure 3, the *message*, *vm*(*t*), is the voice or data signal we
want to send. The *carrier*, *vc*(*t*), will be modified by
*vm*(*t*)*,* depending on the type of modulation. The output of the
modulator is either *vFM*(*t*) (message carried in the frequency) in
the case of an FM modulator, or *vAM*(*t*) (message carried in the
amplitude) in the case of an AM modulator. The difference between the
outputs of the two different modulation schemes is evident. Of course,
to the listener the most noticeable difference when listening to an AM
radio versus an FM radio is the noise. This is because, in general,
noise affects the *amplitude* of a signal. Since the message in an AM
signal is contained entirely in the amplitude, AM signals are very
susceptible to noise. For FM signals, however, the message is contained
in the frequency and not the amplitude. Even if the amplitude of an FM
signal is affected by a noisy environment, such as a lightning storm,
this noise is effectively ignored during the demodulation process, since
the amplitude and frequency are independent.
Along with FM, Phase Modulation (PM) signals are also far more immune to
noise than AM. Although PM is typically much less familiar to most
people, it is used extensively with cell phones. PM is based on the
signal's *rate of change* (or slope). The highest frequency in the PM
signal occurs when the message has the steepest *positive* slope. The
lowest frequency occurs where the message has the steepest *negative*
slope. The graphs below compare the output of an FM modulator with that
of a PM modulator. As it appears, PM and FM are very similar.
Figure 4: Frequency modulation vs. phase modulation.
In each of these graphs, the message displayed is a sinusoid. However,
the message signal will vary depending on the application -- and does
not have to be a sinusoid. In general, there are two types of messages:
*analog* and *digital*. While the principles between the two are the
same, the message content will vary based on the scheme used. This class
will focus on analog modulation.
## Amplitude Modulation
Amplitude modulation makes use of a device called a *function
multiplier*, but allows for the addition of a DC bias voltage, *B*.
Adding the bias voltage can make demodulation more straightforward. To
add the DC bias voltage, we just use a summer before the function
multiplier, as shown in the block diagram in Figure 5. In these block
diagrams the Σ circle is the summer, the ⊗ circle is the function
multiplier, and the circle with a sine wave inside it is the *oscillator*
that generates the carrier.
The *function multiplier* is an electronic device that produces an
output signal equal to the product of the input signals. When radio
frequency (RF) signals are used, this device is also known as a *mixer*.
The function multiplier is different than the scalar multiplier
(amplifier) discussed earlier with transducer interfaces because the
output of the scalar multiplier is a scaled version of the input.
However, the output of the function multiplier is a signal whose shape
follows the product of the input functions. Keep in mind the units do
not multiply, so the product of two voltage signals will have units of
V, not of V2.
Figure 5: A generic AM block diagram.
Notice the output of the modulator has the same frequency as the
carrier; however, the amplitude varies as the message changes over time.
In fact, the message signal creates a kind of envelope, while the
carrier signal "colors in between the lines" of that envelope. Two
important facts to note with function multipliers:
1. The carrier frequency is ALWAYS much higher than the message
frequency.
2. The carrier is ALWAYS a sinusoid; however, the message could be any
analog or digital signal... it's the information you want to
communicate!
Mathematically speaking, we represent the result of the modulator as:
$$\begin{aligned}
v_{out}(t) &= A_{c}\cos\left( 2\pi f_{c}t \right) \cdot \left[ B + A_{m}\cos\left( 2\pi f_{m}t \right) \right] \\
&= \frac{1}{2}A_{c}A_{m}\cos\left( 2\pi\left[ f_{c} - f_{m} \right]t \right) + \frac{1}{2}A_{c}A_{m}\cos\left( 2\pi\left[ f_{c} + f_{m} \right]t \right) \\
&\quad + A_{c}B\cos\left( 2\pi f_{c}t \right)
\end{aligned}$$
where the "*m"* subscript denotes the message and the "*c"* subscript
denotes the carrier. This equation looks both scary and complicated, but
it really isn't. Remember, when dealing with sinusoidal terms, the two
key pieces of information needed are: amplitude and frequency. In
generic terms, the output of the function multiplier is:
$$\begin{aligned}
v_{out}(t) &= \frac{\textbf{Product of amplitudes}}{2}\cdot\cos\left( 2\pi\cdot\textbf{difference of frequencies}\cdot t \right) \\
&\quad + \frac{\textbf{Product of amplitudes}}{2}\cdot\cos\left( 2\pi\cdot\textbf{sum of frequencies}\cdot t \right) \\
&\quad + \textbf{Bias}\cdot\textbf{carrier amplitude}\cdot\cos\left( 2\pi\cdot\textbf{carrier frequency}\cdot t \right)
\end{aligned}$$
Therefore, you can think of the function multiplier as dividing the
combined energy contained in the message and carrier signals between two
new sinusoidal signals: one at the difference of the frequencies and one
at the sum of the frequencies. If there is a bias, it simply produces a
signal at the carrier frequency. It is perhaps more intuitive to
visualize this result in the frequency domain:
Figure 6: AM in the frequency domain.
The spectrum sketches in this reading are not drawn to scale. Read each
spike's frequency and amplitude from its printed labels, not from its
position or height on the page.
Now is a good time to introduce a few new terms. The first of these is
*modulation index* *(α)* which is given by the equation:
$$\alpha = \frac{\max(\text{message})}{B}$$
For a single-tone message the peak is just $A_{m}$, so this reduces to
$\alpha = A_{m}/B$ — that is the form you will see in most of the
worked examples below. Use the general form whenever the message is a sum of
tones, because then the peak is not any one amplitude. When the tones are
cosines that all peak at the same instant, the peak of the message is the
sum of their amplitudes.
We can determine the modulation index directly from the frequency plot,
if it is available. We simply add up signals in the sidebands and divide
them by the signal at the carrier frequency:
$$\frac{\frac{A_{c}A_{m}}{2} + \frac{A_{c}A_{m}}{2}}{A_{c}B} = \frac{A_{c}A_{m}}{A_{c}B} = \frac{A_{m}}{B} = \alpha$$
The equation sheet writes this as $2\cdot Amp_{SB}/Amp_{Carrier}$. It holds for
a single-tone message, which has one sideband on each side of the carrier.
AM signals can be characterized by their modulation index, where we call
the signals, over-, under-, or 100%-modulated:
1) α \> 1 means the signal is over-modulated.
2) α \< 1 means the signal is under-modulated.
3) α = 1 means the signal is 100%-modulated.
:::{admonition} Key Concept
:class: key-concept
$\alpha = \max(\text{message})/B$. The carrier amplitude $A_c$ scales both the envelope swing and the bias, so it **cancels** — $\alpha$ does not depend on it.
:::
### AM Signals in the Time Domain
While you will not be asked to draw time domain modulation plots on your
own, you do need to understand what they look like and why. The output
of the function multiplier can be drawn in the time domain by following
these steps:
1. The message is drawn on the graph using dashed lines.
2. The amplitude is set to (*Ac* \* *Am*), and the period of the
message signal is marked.
3. The message reflection is then represented about the x-axis using
dashed lines.
4. The carrier frequency is then used to "color in" between the two
dashed lines. The dashed lines are called the envelope, and are
used as a guideline to better distinguish the changes in amplitude
of the carrier.
We could also represent the output signal in the frequency domain, by
identifying the two key pieces of information -- amplitude and
frequency.
$$v_{out}(t) = \mathbf{3}\cos\left( 2\pi\cdot\mathbf{99{,}000}\,t \right) + \mathbf{3}\cos\left( 2\pi\cdot\mathbf{101{,}000}\,t \right)\ \text{V}$$
Therefore, we can plot *vout*(*t*) as:
Notice that the message is contained in frequencies symmetrical about the
carrier frequency, 100kHz. So, by changing the carrier frequency, we can
change the location of the output signal on the frequency spectrum. If,
for example, we wanted to broadcast at 1MHz, we would simply change the
carrier frequency to 1MHz.
### Modulation Example
To understand the effect of bias, consider the message signal
$v_{m}(t) = 2\cos(2\pi\cdot 2000t)$ V, which is input into the
following amplitude modulator. We will start with a bias of 0V, then
change it to 1V, 2V and finally to 3V. We will graph the output signal,
*vAM*, in both the *time* and *frequency* domain for each case.
It would be helpful at first to list the known quantities:
1) Am = 2V
2) Ac = 2V
3) fc = 590 kHz
4) *T*c = 1.7 µs
5) fm = 2 kHz
6) *T*m = 500 µs
***Bias = 0V***
We begin with the bias set to 0V.
| | |
|--------|---------|
|1) The graph of the message signal is: | |
|2) The amplitude of our message signal is 2V, which means it varies from 2V to -2V. Now, we can draw the graph in the time domain. First, the message signal and its reflection are represented about the x-axis as a dashed line. | |
|3) Finally, the carrier signal is drawn between these envelopes and we label the y-axis||
Notice the maximum and minimum values of the previous two figures are
different. To find out why, let's look at the maximum and minimum values
of the message as it flows through the modulation system:
When we start, our message has an amplitude of 2V, which means it varies
between 2V and -2V. Then, as we pass through the function multiplier,
the message is multiplied by a carrier signal. Since the amplitude of
the carrier signal is 2V, we simply multiply 2V and -2V by 2, giving us
4V and -4V.
In the equation for this signal, the amplitude of the message signal
(*Am*) is 2V and the bias voltage (*B*) is 0V; therefore, the
modulation index is:
$$\alpha = \frac{A_{m}}{B} = \frac{2V}{0V} = \infty$$
While this may not make much sense mathematically, by definition
whenever *α* \> 1, then our output signal will be *over-modulated*. It
is important to note that when we do not include bias in our design, the
output signal will always be over-modulated. Notice in the graph above
that part of the signal still drops below the x-axis, and the two
envelopes *over*lap. This is another indication the output signal will
be over-modulated.
Finally, we can easily plot the output signal in the frequency domain.
The equation of the output is:
$$\begin{aligned}
v_{out}(t) &= \frac{1}{2}A_{c}A_{m}\cos\left( 2\pi\left[ f_{c} - f_{m} \right]t \right) + \frac{1}{2}A_{c}A_{m}\cos\left( 2\pi\left[ f_{c} + f_{m} \right]t \right) \\
&\quad + A_{c}B\cos\left( 2\pi f_{c}t \right)
\end{aligned}$$
$$v_{out}(t) = 2\cos\left( 2\pi\cdot 588{,}000t \right) + 2\cos\left( 2\pi\cdot 592{,}000t \right)\ \text{V}$$
Therefore, we can draw the amplitude spectrum as:
***Bias = 1V***
|||
|--------------------------------------------|--------------|
|1)The amplitude of our message signal is 2V, which means it varies from 2V to -2V. By adding a bias of 1V, we shift the entire signal up, so that it alternates between 3V and -1V, as shown. Therefore, the graph of the message signal is:||
|2) Now, the message signal and its reflection are represented about the x-axis as a dashed line.||
|3) Finally, the carrier signal is drawn between these envelopes and we label the y-axis:||
Notice the maximum and minimum values of the previous two figures are
different. To find out why, let's look at the maximum and minimum values
of the message as it flows through the modulation system:
When we start, our message has an amplitude of 2V, which means it varies
between 2V and -2V. When we pass through the summer, however, we add 1V
of bias, so our signal now varies from 3V to -1V. Finally, we pass
through the function multiplier where our biased message is multiplied
by a carrier signal. Since the amplitude of the carrier signal is 2V, we
simply multiply 3V and -1V by 2 or -2 and pick the biggest and smallest
results, giving us 6V and -6V.
In the equation for this signal, the amplitude of the message signal
(*Am*) is 2V and the bias voltage (*B*) is 1V; therefore, the
modulation index is:
$$\alpha = \frac{A_{m}}{B} = \frac{2V}{1V} = 2$$
By definition, whenever *α* \> 1, then our output signal will be
*over-modulated*. Notice in the graph above that part of the signal
still drops below the x-axis, and the two envelopes *over*lap. This is
another indication the output signal will be over-modulated.
Finally, we can write the equation of the output signal as:
$$v_{out}(t) = 2\cos\left( 2\pi\cdot 588{,}000t \right) + 2\cos\left( 2\pi\cdot 592{,}000t \right) + 2\cos\left( 2\pi\cdot 590{,}000t \right)\ \text{V}$$
So we can plot the frequency amplitude spectrum as:
Notice how the bias voltage has generated a new "spike" at the carrier
frequency, and we can calculate the modulation index directly from the
amplitude spectrum:
$$\alpha = \ \frac{2V + 2V}{2V} = 2$$
Which agrees with our previous calculation.
***Bias = 2V***
|||
|-----------------------------------------|-----------------|
|1) We’ve now shifted the original curve up to the point where it just touches the x-axis. This will always occur when the bias equals the amplitude of the message signal. The message signal is:||
|2) Following the same steps for graphing, the output, vAM(t), looks as shown:||
We can examine the maximum and minimums for our 100% modulated case as
before:
The modulation index for this system is:
$$\alpha = \frac{A_{m}}{B} = \frac{2V}{2V} = 1$$
Whenever α = 1, we will have an output signal that is *100%-modulated*.
This case occurs when the message just barely touches the x-axis.
Finally, we can write the equation of the output signal as:
$$v_{out}(t) = 2\cos\left( 2\pi\cdot 588{,}000t \right) + 2\cos\left( 2\pi\cdot 592{,}000t \right) + 4\cos\left( 2\pi\cdot 590{,}000t \right)\ \text{V}$$
and, the amplitude spectrum is:
Notice how the spike at the carrier frequency is increasing based on the
increasing DC bias.
***Bias = 3V***
|||
|-------------------------------|---------------------------|
|1) Our signal now varies from 5V to 1V, with none of the signal dropping below or touching the x-axis. Therefore, the message signal is:||
|2) Following the same steps for graphing, the output, vAM(t), looks as shown:||
The maximum and minimums of the signals as they flow through the
modulator are:
Therefore, the message envelope varies from 10 V to 2 V and the mirror
image varies from -2 V to -10 V.
And, finally, our modulation index in this case is
$$\alpha = \frac{A_{m}}{B} = \frac{2V}{3V} = 0.667$$
When *α* \< 1, our output signal will be *under-modulated*.
Under-modulation occurs when the message signal is entirely above the
x-axis, and the two message envelopes do not overlap.
Finally, we can write the equation of the output signal as:
$$v_{out}(t) = 2\cos\left( 2\pi\cdot 588{,}000t \right) + 2\cos\left( 2\pi\cdot 592{,}000t \right) + 6\cos\left( 2\pi\cdot 590{,}000t \right)\ \text{V}$$
and we can plot the amplitude spectrum:
Again, we notice the increasing signal at the carrier frequency, due to
the higher bias voltage.
To summarize the three modulation cases (ignoring the bias = 0V case,
which is also over-modulated), the output signals for bias voltages of
1V, 2V, and 3V are given in Figure 7.
Notice the effect the bias has on the output signals. In the first
signal, the two envelopes overlap. As more bias is added, however, the
envelopes are pulled apart until by the last signal, they are no longer
touching. At 100%-modulation and for under-modulated signals, the phase
change of the carrier as it crosses the x-axis does not occur;
therefore, it is easier to demodulate and we can use a cheaper
demodulator.
|**Over-Modulated**| **100%-Modulated**|**Under-Modulated**|
|:----------------:|:-----------------:|:-----------------:|
|$\alpha = \frac{A_{m}}{B} = \frac{2V}{1V} = \mathbf{2}$|$\alpha = \frac{A_{m}}{B} = \frac{2V}{2V} = \mathbf{1}$|$\alpha = \frac{A_{m}}{B} = \frac{2V}{3V} = \mathbf{0.667}$|
||||
||||
Figure 7: Summary of the effect of various DC bias voltages on a
modulated signal.
## AM with More Than One Frequency
Although it is instructive to consider how to use a single frequency as
a message signal, it is not very practical. In fact, most systems want
to send information containing a bandwidth of frequencies. Any
communications system, for example, wants to ensure it can at least use
data in the audible range (20 Hz -- 22kHz) as its message signal.
Everything up to this point still applies to a message signal with more
than one frequency. For example, consider the following modulation
system:
Where *v*m(*t*) is now:
$$v_{m}(t) = 2\cos\left( 2\pi\cdot 1000t \right) + \cos\left( 2\pi\cdot 3000t \right) + 6\cos\left( 2\pi\cdot 5000t \right)\ \text{V}$$
In this case, determining the time domain representation of the output
signal (*v*AM) is overly complicated. Even using the distributive law
to write the equation of the output signal is overly tedious. However,
it is perfectly feasible to draw this signal in the frequency domain, if
we recall that each frequency in the message signal will produce *two*
frequencies in the output signal -- one at each frequency added to the
carrier frequency, and one at each frequency subtracted from the carrier
frequency. The amplitude of each frequency will be half of the product
of that frequency's amplitude (in the message signal) and the amplitude
of the carrier frequency. So, the example above produces **six**
frequencies in the output:
First, take notice how the spectrum is symmetric around the carrier
frequency, *fc.* The frequencies to the right of the carrier frequency
are simply the frequencies from the message signal added to the carrier
frequency:
590kHz + 1 kHz = 591 kHz
590 kHz + 3 kHz = 593 kHz
590 kHz + 5 kHz = 595 kHz
The amplitudes of those frequencies are calculated by
$\frac{A_{m}A_{c}}{2}$:
$$\text{at 591 kHz:}\quad \frac{2V \cdot 2V}{2} = 2V$$
$$\text{at 593 kHz:}\quad \frac{1V \cdot 2V}{2} = 1V$$
$$\text{at 595 kHz:}\quad \frac{6V \cdot 2V}{2} = 6V$$
Finally, observe how the frequencies to the left of the carrier
frequency are the higher frequencies reflected (as a mirror image)
around the carrier frequency. This represents the difference of the
message and carrier frequencies. The amplitudes are also the same as
those on the right of the carrier frequency. With this in mind, it does
not matter how many frequencies are contained in a signal, we can find
the amplitude spectrum by:
1) Adding the carrier frequency to each message frequency
2) Calculating the amplitude by $\frac{A_{m}A_{c}}{2}$ for each
message frequency
3) Reflecting the results around the carrier frequency on the graph.
Another concept that is useful to understand is the idea of *bandwidth*.
Bandwidth, when taken in context with modulation, is the range of
frequencies in a given band, particularly those used for transmitting a
signal. When modulating a signal, the transmitter is limited to a
certain bandwidth they are allowed to transmit over. To find the
bandwidth of a signal, the lowest frequency is subtracted from the
highest frequency.
BW = *fhigh -- flow*
Examining our modulated signal, we can determine the bandwidth to be:
BW = *fhigh -- flow* = 595kHz -- 585kHz = 10kHz
The bandwidth of a modulated signal is ALWAYS equal to twice the highest
frequency of the message signal.
:::{admonition} Key Concept
:class: key-concept
$BW = 2f_{m,\max}$. Bandwidth is set by the **message**, not the carrier — the carrier only decides *where* that block of spectrum sits. In the AM output the bias sits at $f_c$, not at $f = 0$, so $f_{low}$ is the lowest sideband.
:::
## Digital Modulation
One of the key principles of modulation is that the carrier
**must** be a sinusoidal signal, but the message data
can be anything -- analog or digital. So far, we have discussed the
modulation of analog signals (sinusoids), but it is just as easy to use
a digital signal as a message signal. In the case of amplitude
modulation, we choose one amplitude as a binary zero (usually 0V) and
another as a binary one (5V, for example). When the message signal is a
binary zero, the output signal is zero. When the message signal is a
one, the output signal's amplitude is 5V. We call this Amplitude Shift
Keying (ASK). We can do the same thing with frequency -- one frequency
represents a binary zero, while another represents a binary one. This is
called Frequency Shift Keying (FSK). The two methods are shown in
Figure 8. Many modern systems make use of both methods to transmit
information.
Figure 8: Amplitude Shift Keying and Frequency Shift Keying.
## Frequency Division Multiplexing (FDM) System
In order to send two or more signals through the same channel, we
consider two options. First, a signal can be sent at any frequency, but
only during specified time intervals. This is called Time Division
Multiplexing (TDM). Second, a signal can be sent at any time, but only
in a specified frequency range. This is Frequency Division Multiplexing
(FDM).
FDM is a multiplexing technique used in analog telephone communications,
FM stereo, space vehicle telemetry, and many other applications. In FDM,
several independent signals are simultaneously transmitted through one
channel by allocating each signal a unique band of the available
frequency spectrum. Many times the signal will need to be filtered to
restrict its bandwidth over the frequency spectrum to prevent
interference with other signals. The signals shown in the frequency
domain below are from three local Colorado Springs FM radio stations.
Notice each signal has its own unique center frequency, but the
frequency content is the same for all three stations (in the audible
range). This is an example of FDM.
Breaking this process down, FDM involves band-limiting the input
signals, moving their spectra through modulation, and combining these
into one composite signal. In general, an FDM signal is generated from
any number of input signals (i.e., two or more) by translating each one
to a unique frequency band. The translated signals are then added
together creating the composite FDM signal whose amplitude spectrum
consists of the non-overlapping, frequency translated spectrum of all
the signals. Your basic FDM system is shown below.
To guarantee no overlap of spectra at the output of the summer, the input
signals to the multipliers must not contain frequencies that would
overlap with any of the other channels. Low pass filters serve this
purpose, with cutoff frequencies set to the highest desired frequency of
the input signal.
### FDM Example Problem 1
Two music signals are multiplexed according to
the system below. Graph the output signal in the frequency domain.
**Understand**: We are multiplexing signals in the frequency domain (two
signals on the same graph).
**Identify Key Information**:
- **Knowns:** We know the frequency content of each input signal, the
cutoff frequencies of the band-limiting LPFs, and the amplitude and
frequency of each function multiplier.
- **Unknowns:** We want to graph the output spectrum.
- **Assumptions:** The transmission system can support the entire
bandwidth.
**Plan**: We will follow each signal through its own modulation path,
taking into account frequencies cut off by the LPFs, and determining the
final modulated signal. Since the signals are then added together, they
can be plotted on the same axis.
**Solve:** Consider the first signal and what happens when it passes
through the filter. Since $f_{\text{cutoff}}$ = 15 kHz, every frequency above 15 kHz
will be cut off. Therefore, the signal coming out of the filter will be:
For the second signal, the filter will also kill all frequencies greater
than 15 kHz:
Everything else we do is the same (mathematically) as previous problems.
We shift the first signal by the carrier frequency, 120 kHz (since this
represents the sum of the frequencies), then draw its mirror image
(since this represents the difference of the frequencies). Next, we
shift the second signal by 155 kHz and draw its mirror image. If the
LPFs were not in the signal chain, the output signals would interfere
with each other, however, because the filters enforce a channel
bandwidth of 30 kHz, though, the resulting signals no longer interfere
with each other. The unused 5 kHz between the two channels, from 135 kHz
to 140 kHz, is a *guard band*.
**Answer**: The output signal for the multiplexing system is shown in
the following graph:
::::{admonition} Type-along
:class: type-along
A message $v_m = 2\cos(2\pi\cdot 1000t)$ V is biased by $B = 4$ V and multiplied by a carrier $v_c = 5\cos(2\pi\cdot 50{,}000t)$ V.
1. What is the modulation index?
2. Is the signal under-, fully, or over-modulated?
3. What frequencies appear at the output, and with what amplitudes?
4. What is the bandwidth?
:::{admonition} Solution
:class: dropdown
**1.** $\alpha = \max(\text{message})/B = 2/4 = 0.5$.
**2.** $\alpha < 1$, so **under-modulated** — the envelope never reaches zero.
**3.** The carrier survives at 50 kHz because a bias is present, with amplitude $A_cB = (5)(4) = 20$ V. The sidebands sit at 49 kHz and 51 kHz, each with amplitude $A_cA_m/2 = (5)(2)/2 = 5$ V. Check: $2(5)/20 = 0.5$, the same $\alpha$.
**4.** $BW = 51 - 49 = 2$ kHz, which is $2f_m$.
:::
::::
## Wrap-Up
We modulate to choose a transmission frequency, shrink the antenna, and fit many channels into one band. Multiplying a message by a carrier produces **sum and difference sidebands** of amplitude $A_cA_m/2$ around $f_c$, and a bias $B$ adds a carrier term of amplitude $A_cB$. The **modulation index** $\alpha = \max(\text{message})/B$ decides whether the envelope stays above zero, and $BW = 2f_{m,\max}$.
Objective 3.2 takes the message back off the carrier — and $\alpha$ decides which demodulator you can use.