Find the complex Fourier series representations of the following signals without explicitly calculating Fourier integrals.
What is the signal's period in each case?
Find the Fourier series representation for the following periodic
signals. For the third signal, find the complex
Fourier series for the triangle wave
without performing the usual
Fourier integrals. Hint: How is this signal related to
one for which you already have the series?
Phase Distortion
We can learn about phase distortion by returning to
circuits and investigate the
following circuit.
Find this filter's transfer function.
Find the magnitude and phase of this transfer
function. How would you characterize this
circuit?
Let
be a square-wave of period
.
What is the Fourier series for the output voltage?
Use Matlab to find the output's waveform for the cases
and
.
What value of
delineates the two kinds of results you found? The
software in fourier2.m might be
useful.
Instead of the depicted circuit, the square wave is
passed through a system that delays its input, which
applies a linear phase shift to the signal's
spectrum. Let the delay
be
.
Use the transfer function of a delay to compute using
Matlab the Fourier series of the output. Show that the
square wave is indeed delayed.
Approximating Periodic Signals
Often, we want to approximate a reference signal by a
somewhat simpler signal. To assess the quality of an
approximation, the most frequently used error measure is
the mean-squared error. For a periodic signal
,
where
is the reference signal and
its approximation. One convenient way of finding
approximations for periodic signals is to truncate their
Fourier series.
The point of this problem is to analyze whether this
approach is the best (i.e., always
minimizes the mean-squared error).
Find a frequency-domain expression for the
approximation error when we use the truncated
Fourier series as the approximation.
Instead of truncating the series, let's generalize
the nature of the approximation to including any set
of
terms: We'll always include the
and the negative indexed term corresponding to
.
What selection of terms minimizes the mean-squared
error? Find an expression for the mean-squared error
resulting from your choice.
Find the Fourier series for the
depicted signal. Use
Matlab to find the truncated approximation and best
approximation involving two terms. Plot the
mean-squared error as a function of
for both
approximations.
Long, Hot Days
The daily temperature is a consequence of several
effects, one of them being the sun's heating. If this
were the dominant effect, then daily temperatures would
be proportional to the number of daylight hours. The
plot shows
that the average daily high temperature does
not behave that way.
In this problem, we want to understand the temperature
component of our environment using Fourier series and
linear system theory. The file
temperature.mat contains these data
(daylight hours in the first row, corresponding average
daily highs in the second) for Houston, Texas.
Let the length of day serve as the sole input to a
system having an output equal to the average daily
temperature. Examining the plots of input and
output, would you say that the system is linear or
not? How did you reach you conclusion?
Find the first five terms
(,
... ,
)
of the complex Fourier series for each signal.
Use the following formula that approximates the integral required to find the Fourier coefficients.
What is the harmonic distortion in the two signals?
Exclude
from this calculation.
Because the harmonic distortion is small, let's
concentrate only on the first harmonic. What is the
phase shift between input and output signals?
Find the transfer function of the simplest possible
linear model that would describe the data.
Characterize and interpret the structure of this
model. In particular, give a physical explanation
for the phase shift.
Predict what the output would be if the model had no
phase shift. Would days be hotter? If so, by how
much?
Fourier Transform Pairs
Find the Fourier or inverse Fourier transform of the
following.
Duality in Fourier Transforms
"Duality" means that the Fourier transform and the inverse Fourier transform are very similar.
Consequently, the waveform
in the time domain and the spectrum
have a Fourier transform and an inverse Fourier transform, respectively, that are very similar.
Calculate the Fourier transform of the signal shown
below.
Calculate the inverse Fourier transform of the spectrum shown
below.
How are these answers related?
What is the general relationship between the Fourier transform of
and the inverse transform of
?
Spectra of Pulse Sequences
Pulse sequences occur often in digital communication and in other fields as well.
What are their spectral properties?
Calculate the Fourier transform of the single pulse shown
below.
Calculate the Fourier transform of the two-pulse sequence shown
below.
Calculate the Fourier transform for the ten-pulse sequence shown in below.
You should look for a general expression that holds for sequences of any length.
Using Matlab, plot the magnitudes of the three spectra.
Describe how the spectra change as the number of repeated pulses increases.
Spectra of Digital Communication Signals
One way to represent bits with signals is shown in [link].
If the value of a bit is a “1”, it is represented by a positive pulse of duration
.
If it is a “0”, it is represented by a negative pulse of the same duration.
To represent a sequence of bits, the appropriately chosen pulses are placed one after the other.
What is the spectrum of the waveform that represents the alternating bit sequence “...01010101...”?
This signal's bandwidth is defined to be the frequency range over which 90% of the power is contained.
What is this signal's bandwidth?
Suppose the bit sequence becomes “...00110011...”.
Now what is the bandwidth?
Lowpass Filtering a Square Wave
Let a square wave (period
)
serve as the input to a first-order lowpass system
constructed as a RC filter. We want to derive an
expression for the time-domain response of the filter to
this input.
First, consider the response of the filter to a
simple pulse, having unit amplitude and width
.
Derive an expression for the filter's output to this
pulse.
Noting that the square wave is a superposition of a
sequence of these pulses, what is the filter's
response to the square wave?
The nature of this response should change as the
relation between the square wave's period and the
filter's cutoff frequency change. How long must the
period be so that the response does
not achieve a relatively
constant value between transitions in the square
wave? What is the relation of the filter's cutoff
frequency to the square wave's spectrum in this
case?
Mathematics with Circuits
Simple circuits can implement simple mathematical
operations, such as integration and differentiation. We
want to develop an active circuit (it contains an
op-amp) having an output that is proportional to the
integral of its input. For example, you could use an
integrator in a car to determine distance traveled from
the speedometer.
What is the transfer function of an integrator?
Find an op-amp circuit so that its voltage output is
proportional to the integral of its input for all
signals.
Where is that sound coming from?
We determine where sound is coming from because we have
two ears and a brain. Sound travels at a relatively
slow speed and our brain uses the fact that sound will
arrive at one ear before the other. As shown here, a
sound coming from the right arrives at the left ear
seconds after it
arrives at the right ear.
Once the brain finds this propagation delay, it can
determine the sound direction. In an attempt to model
what the brain might do, RU signal processors want to
design an optimal system that
delays each ear's signal by some amount then adds them
together.
and
are the delays applied to the left and right signals
respectively. The idea is to determine the delay values
according to some criterion that is based on what is
measured by the two ears.
What is the transfer function between the sound signal
and the processor output
?
One way of determining the delay
is to choose
and
to maximize the power in
.
How are these maximum-power processing delays
related to ?
Arrangements of Systems
Architecting a system of modular components means
arranging them in various configurations to achieve some
overall input-output relation. For each of the following, determine
the overall transfer function between
and
.
system a
system b
system c
The overall transfer function for the cascade (first
depicted system) is particularly interesting. What does
it say about the effect of the ordering of linear,
time-invariant systems in a cascade?
Filtering
Let the signal
be the input to a linear, time-invariant filter having the transfer function shown below.
Find the expression for
, the filter's output.
Circuits Filter!
A unit-amplitude pulse with duration of one second serves as the input to an RC-circuit having transfer function
How would you categorize this transfer function:
lowpass, highpass, bandpass, other?
Find a circuit that corresponds to this transfer function.
Find an expression for the filter's output.
Reverberation
Reverberation corresponds to adding to a signal its
delayed version.
Assuming
represents the delay, what is the input-output
relation for a reverberation system? Is the system
linear and time-invariant? If so, find the transfer
function; if not, what linearity or time-invariance
criterion does reverberation violate.
A music group known as the ROwls is having trouble
selling its recordings. The record company's
engineer gets the idea of applying different delay
to the low and high frequencies and adding the
result to create a new musical effect. Thus, the
ROwls' audio would be separated into two parts (one
less than the frequency
,
the other greater than
),
these would be delayed by
and
respectively, and the resulting signals added. Draw
a block diagram for this new audio processing
system, showing its various components.
How does the magnitude of the system's transfer
function depend on the two delays?
Echoes in Telephone Systems
A frequently encountered problem in telephones is echo.
Here, because of acoustic coupling between the ear piece
and microphone in the handset, what you hear is also
sent to the person talking. That person thus not only
hears you, but also hears her own speech delayed
(because of propagation delay over the telephone
network) and attenuated (the acoustic coupling gain is
less than one). Furthermore, the same problem applies
to you as well: The acoustic coupling occurs in her
handset as well as yours.
Develop a block diagram that describes this
situation.
Find the transfer function between your voice and
what the listener hears.
Each telephone contains a system for reducing echoes
using electrical means. What simple system could
null the echoes?
Effective Drug Delivery
In most patients, it takes time for the concentration of an administered drug to achieve a constant level in the blood stream.
Typically, if the drug concentration in the patient's intravenous line is
, the concentration in the patient's blood stream is
.
Assuming the relationship between drug concentration in the patient's drug and the delivered concentration can be described as a linear, time-invariant system, what is the transfer function?
Sometimes, the drug delivery system goes awry and delivers drugs with little control.
What would the patient's drug concentration be if the delivered concentration were a ramp?
More precisely, if it were
?
A clever doctor wants to have the flexibility to slow down or speed up the patient's drug concentration.
In other words, the concentration is to be
, with
bigger or smaller than
.
How should the delivered drug concentration signal be changed to achieve this concentration profile?
Catching Speeders with Radar
RU Electronics has been contracted to design a Doppler radar system.
Radar transmitters emit a signal that bounces off any conducting object.
Signal differences between what is sent and the radar return is processed and features of interest extracted.
In Doppler systems, the object's speed along the direction of the radar beam is the feature the design must extract.
The transmitted signal is a sinsusoid:
.
The measured return signal equals
, where the Doppler offset frequency
equals
, where
is the car's velocity coming toward the transmitter.
Design a system that uses the transmitted and return signals as inputs and produces .
One problem with designs based on overly simplistic design goals is that they are sensitive to unmodeled assumptions.
How would you change your design, if at all, so that whether the car is going away or toward the transmitter could be determined?
Suppose two objects traveling different speeds provide returns.
How would you change your design, if at all, to accomodate multiple returns?
Demodulating an AM Signal
Let
denote the signal that has been amplitude modulated.
Radio stations try to restrict the amplitude of the
signal
so that it is less than one in magnitude. The frequency
is very large compared to the frequency content of the
signal. What we are concerned about here is not
transmission, but reception.
The so-called coherent demodulator simply multiplies
the signal
by a sinusoid having the same frequency as the
carrier and lowpass filters the result. Analyze
this receiver and show that it works. Assume the
lowpass filter is ideal.
One issue in coherent reception is the phase of the
sinusoid used by the receiver relative to that used
by the transmitter. Assuming that the sinusoid of
the receiver has a phase
, how does the
output depend on
? What is the
worst possible value for this phase?
The incoherent receiver is more commonly used
because of the phase sensitivity problem inherent in
coherent reception. Here, the receiver full-wave
rectifies the received signal and lowpass filters
the result (again ideally). Analyze this receiver.
Does its output differ from that of the coherent
receiver in a significant way?
Unusual Amplitude Modulation
We want to send a band-limited signal having the
depicted spectrum with amplitude modulation in the usual way.
I.B. Different suggests using the square-wave carrier shown below.
Well, it is different, but his friends wonder if any technique can demodulate it.
Find an expression for
, the Fourier transform of the modulated signal.
Sketch the magnitude of
, being careful to label important magnitudes and frequencies.
What demodulation technique obviously works?
I.B. challenges three of his friends to demodulate
some other way.
One friend suggests modulating
with
, another wants to try modulating with
and the third thinks
will work.
Sketch the magnitude of the Fourier transform of the signal each student's approach produces.
Which student comes closest to recovering the original signal?
Why?
Sammy Falls Asleep...
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 unknown
systems where.
What are the signals
and
?
What would you put in for the unknown systems that
would guarantee that the final output contained the
message 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 engineer
decides 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 that
interferes 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 and
transmission bandwidth? If so, find the station's
carrier frequency and transmission bandwidth in terms
of
,
the period of the jamming signal; if not, show
why not.
AM Stereo
A stereophonic signal consists of a "left" signal
and a "right" signal
that conveys sounds coming from an orchestra's left and
right sides, respectively. To transmit these two
signals simultaneously, the transmitter first forms the
sum signal
and the difference signal
.
Then, the transmitter amplitude-modulates the difference
signal 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 separate
outputs.
What signal would be produced by a conventional
coherent AM receiver that expects to receive a
standard AM signal conveying a message signal having
bandwidth
?
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 one
signal 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 messages
are sinusoids:
and
where
.
You, as the patent examiner, must determine whether the
scheme 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. Explain
your 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. The
transmitted 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 combinations
of
and
prevent reception?
Assuming reception is possible, can
standard radios receive this
innovative AM transmission? If so, show how a
coherent receiver could demodulate it; if not, show
how the coherent receiver's output would be
corrupted. 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 music
unlistenable 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, at
least, unrecoverable ([link]).
What is the Fourier series for the periodic
signal?
What are the restrictions on the period
so that the message
signal can be recovered from
?
ELEC 241 students think they have "broken" the
inventor's scheme and are going to announce it to the
world. How would they recover the original message
without having detailed knowledge
of the modulating signal?