While sitting in ELEC 241 class, he falls asleep during
a critical time when an AM receiver is being described.The received signal has the form
where the phase
is unknown. The message signal is
;
it has a bandwidth of
Hz
and a magnitude less than 1(
).
The phase
is unknown. The instructor drew a
diagram for a receiver on the board;
Sammy slept through the description of what the unknownsystems where.
What are the signals
and
?
What would you put in for the unknown systems that
would guarantee that the final output contained themessage regardless of the phase?
Think of a trigonometric identity that would prove
useful.
Sammy may have been asleep, but he can think of a far
simpler receiver. What is it?
Jamming
Sid Richardson college decides to set up its own AM
radio station KSRR. The resident electrical engineerdecides that she can choose
any carrier frequency and message bandwidth for the station.
A rival college decides to
jam its
transmissions by transmitting a high-power signal thatinterferes with radios that try to receive KSRR. The
jamming signal
is what is known as a
sawtooth wave (depicted in
[link] ) having a period
known to KSRR's engineer.
Find the spectrum of the jamming signal.
Can KSRR entirely circumvent the attempt to jam it by
carefully choosing its carrier frequency andtransmission bandwidth? If so, find the station's
carrier frequency and transmission bandwidth in termsof
,
the period of the jamming signal; if not, showwhy not.
Am stereo
A stereophonic signal consists of a "left" signal
and a "right" signal
that conveys sounds coming from an orchestra's left andright sides, respectively. To transmit these two
signals simultaneously, the transmitter first forms thesum signal
and the difference signal
.
Then, the transmitter amplitude-modulates the differencesignal with a sinusoid having frequency
,
where
is the bandwidth of the left and right signals. The sum
signal and the modulated difference signal are added,the sum amplitude-modulated to the radio station's
carrier frequency
, and transmitted. Assume the spectra of the
left and right signals are
as shown .
What is the expression for the transmitted signal?
Sketch its spectrum.
Show the block diagram of a stereo AM receiver that
can yield the left and right signals as separateoutputs.
What signal would be produced by a conventionalcoherent AM receiver that expects to receive a
standard AM signal conveying a message signal havingbandwidth
?
Novel am stereo method
A clever engineer has submitted a patent for a new
method for transmitting two signals
simultaneously in the
same transmission bandwidth as
commercial AM radio.
As shown , her approach is to
modulate the positive portion of the carrier with onesignal and the negative portion with a second.
In detail the two message signals
and
are bandlimited to
Hz and have maximal amplitudes equal to 1. The carrier
has a frequency
much greater than
.
The transmitted signal
is given by
In all cases,
.
The plot shows the transmitted signal when the messagesare sinusoids:
and
where
.
You, as the patent examiner, must determine whether thescheme meets its claims and is useful.
Provide a more concise expression for the
transmitted signal
than given above.
What is the receiver for this scheme? It would yield
both
and
from
.
Find the spectrum of the positive portion of the
transmitted signal.
Determine whether this scheme satisfies the design
criteria, allowing you to grant the patent. Explainyour reasoning.
A radical radio idea
An ELEC 241 student has the bright idea of using a
square wave instead of a sinusoid as an AM carrier. Thetransmitted signal would have the form
where the message signal
would be amplitude-limited:
Assuming the message signal is lowpass and has a
bandwidth of
Hz,
what values for the square wave's period
are feasible. In other words, do some combinationsof
and
prevent reception?
Assuming reception is possible, can
standard radios receive this
innovative AM transmission? If so, show how acoherent receiver could demodulate it; if not, show
how the coherent receiver's output would becorrupted. Assume that the message bandwidth
kHz.
Secret communication
An amplitude-modulated secret message
has the following form.
The message signal has a bandwidth of
Hz and a magnitude less than 1
(
). The idea is to offset the carrier frequency by
Hz
from standard radio carrier frequencies. Thus,"off-the-shelf" coherent demodulators would assume the
carrier frequency has
Hz. Here,
.
Sketch the spectrum of the demodulated signal
produced by a coherent demodulator tuned to
Hz.
Will this demodulated signal be a “scrambled”
version of the original? If so, how so; if not, why not?
Can you develop a receiver that can demodulate the
message without knowing the offset frequency
?
Signal scrambling
An excited inventor announces the discovery
of a way of using analog technology to render musicunlistenable without knowing the secret recovery
method. The idea is to modulate the bandlimited message
by a special periodic signal
that is zero during half of its period, which
renders the message unlistenable and superficially, atleast, unrecoverable (
[link] ).
What is the Fourier series for the periodic
signal?
What are the restrictions on the period
so that the messagesignal can be recovered from
?
ELEC 241 students think they have "broken" the
inventor's scheme and are going to announce it to theworld. How would they recover the original message
without having detailed knowledge
of the modulating signal?