Ecet 345 Success Begins / snaptutorial.com

ECET 345 Week 1 Homework
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1.Express the following numbers in Cartesian (rectangular) form.
2.Express the following numbers in polar form. Describe the quadrant of
the complex plane, in which the complex number is located.
3.(a) A continuous-time sine wave has a frequency of 60 Hz, an
amplitude of 117 V, and an initial phase of π/4 radians. Describe this
signal in a mathematical form using the Sin function.
4. A sinusoidal signal described by 50 Cos (20πt + π/4) passes through a
linear time invariant (LTI) system that applies a gain of 1.5 and a phase
lag of π/2 radians to the signal. Write the mathematical expression that
describes the signal that will come out of the LTI system.
5.A sinusoidal signal described by 20 Cos (2πt + π/4) passes through a
linear time invariant (LTI) system that applies a gain of 2 and a time
delay of 0.125 seconds to the signal. Write the mathematical expression
that describes the signal that will come out of the LTI system.
6. Apply the principle of superposition to determine whether the
following systems are linear. Sketch what the plot of the function looks
like.

7. A continuous time system, described by y(t) = 5 Cos (2*π*20*t +
π/2), is sampled at a rate 320 Hz.
8. Sketch the odd and even part of the following discrete signal. (See
pages 13–14 of the text.)
9. Express the signal given in Problem 8 as the sum of the following
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ECET 345 Week 1 iLab Observation of Wave-
Shapes and Their Spectrum
Objective of the lab experiment:
The objective of this experiment is to observe the shapes of different
kinds of signals such as sine waves, square waves, and so on and to
study how the shape of a signal alters its spectrum.

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ECET 345 Week 1 Lab Signal Observation And
Recreation
Objective:
Using a Tower system and supplied HCS12-based program,
experimentally observe the closest equivalent of four key signals
(impulse, sinusoidal, exponential, and square wave) on the oscilloscope
and then create them in MATLAB.
Equipment list:
• Tower System with ADCDAC board
• Oscilloscope
• Three BNC to alligator
• One PC running CodeWarrior 5.9
• MATLAB
• 2.0 mm flathead screwdriver

ECET 345 Week 1 Lab
1. How does an experimental approximation of an impulse differ from
an ideal impulse?
2. A step input is applied to the following circuit at time t = 0. What
will the output waveform look like? What is the significance of the time
constant for the circuit? How will the observed waveform change as the
value of capacitor is increased?
3. How does a sine wave differ from a cosine wave?
4. With the knowledge you gained in the theory section, how can an
exponentially growing sinusoid be generated? Can a physical system
generate an exponentially growing sinusoid?
5. In finance, what does the growth curve of a compound interest
savings account look like over time
6. Create a MATLAB code that generates an amplitude modulated (AM)
signal with a 1 Hz information frequency and a 100 Hz carrier
frequency, with the carrier amplitude equal to 1.0 and the information
frequency amplitude equal to 0.9. The general form of an AM signal is
where Ac is the amplitude of the carrier, Ai is the amplitude of the
information signal, ωi is the frequency of the information signal in
radians/second, ωc is the frequency of the carrier signal in
radians/second, and t is time. Plot the graph over the time period of 0 to
2 seconds. Give meaningful labels to the X and Y axis as well as a title
to the graph
7. Take the signal generated in the previous question and find the signal
spectrum using the tools you have learned in earlier labs. Paste the signal
spectrum and code below. (Hint: Use the method of finding the signal

spectrum of a signal that was shown in Lab 1.) Give meaningful labels
to the X and Y axis as well as a title to the graph.
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1.Redraw the following schematics with the impedance of each of the
element shown in Laplace domain. Then determine the overall
impedance of the entire circuit between the two ends of the shown
circuit and express it in Laplace domain as a ratio of two polynomials in
s, with the coefficients of the highest power if s in the numerator and
denominator are made unity. (Follow the method outlined in the lecture
to determine the impedances of elements in Laplace domain and then
use the formulas for combining impedances in series and parallel.)
2. (a) Apply Laplace transform to the following differential equation and
express it as an algebraic equation in s.

3. An RC circuit with an initial condition is shown below. The toggle
switch is closed at t = 0. Assuming that a current i(t) flows clockwise in
the circuit, Write the integral equation that governs the behavior of the
circuit current and solve it for the current in the circuit i(t) and voltage
across the capacitor as a function of time using Laplace transforms. Note
the polarity of the initial condition as marked in the figure. (Take help
from the document “Solving RC, RLC, and RL Circuits Using Laplace
Transforms” (located in Doc Sharing) and the Week 2 Lecture to see
how initial conditions are entered in Laplace domain.)
4. The voltage in a circuit, expressed in Laplace domain, is given by the
questions below.
5.An RLC circuit is shown below. There is an initial voltage of 5 V on
the capacitor, with polarity as marked in the circuit. The switch is closed
at t = 0 and a current i(t) is assumed to flow clockwise. Write the
integral-differential equation of this circuit using Kirchoff’s method
(sum of all voltages around a loop is zero). Apply Laplace transform as
outlined in the lecture for Week 2 and in the document “Solving RC,
RLC, and RL Circuits Using Laplace Transforms” (located in Doc
Sharing) and write i(s) in Laplace transform notation. Express the
denominator with the coefficient of the highest power of s unity. Then
invert to obtain the current in time domain, i(t).
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ECET 345 Week 2 iLab Response of RC circuits

The objective of this experiment is to experimentally measure the
impulse and step response of an RC circuit and compare it to theoretical
results using Laplace transform.
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The transfer function of a circuit is given by
Express the transfer function in a form in which the coefficients of the
highest power ofs are unity in both numerator and denominator.

What is the characteristic equation of the system? (Hint: see this week’s
lecture for a definition of characteristic equation.)
Determine the order of the transfer function.
Determine where the poles and zeroes of the system are located.
____________
Using MATLAB, plot the pole zero map and the Bode plot of the two
transfer functions and paste the graphs below. Identify and briefly
discuss the differences between the Bode plot of the two transfer
functions.
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ECET 345 Week 3 Lab Transfer Function
Analysis Of Continuous Systems
ECET 345 Week 3 Lab Transfer Function Analysis of Continuous
Systems

The objective of this experiment is to create continuous (s domain)
transfer functions in MATLAB and explore how they can be
manipulated to extract relevant data.
We shall first present an example of how MATLAB is used for s
(Laplace) domain analysis, and then the student shall be required to
perform specified analysis on a given circuit.
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1. A shiny metal disk with a dark spot on it, as shown in figure below,
is rotating clockwise at 100 revolutions/second in a dark room. A human
observer uses a strobe that flashes 99 times/second to observe the spot
on the metal disk (a strobe is a flashing light whose rate of flashing can
be varied). The spot appears to the human observer as if it is rotating
slowly

2. (a) A system samples a sinusoid of frequency 480 Hz at a rate of 100
Hz and writes the sampled signal to its output without further
modification. Determine the frequency that the sampling system will
generate in its output.
3. The spectrum of an analog signal is shown below, containing . Such a
signal is sampled by an ideal impulse sampler at a 100 Hz rate. List the
first 10 positive frequencies that will be produced by the replication.
(Hint: Follow the method outlined in the lecture for spectrum replication
of sampled signals.)
4. The spectrum of an analog signal is shown below. It is sampled, with
an ideal impulse sampler, at a rate of 200 Hz
5. Determine the Z transform of the signal,, shown below using the basic
definition of Z transform . All values not shown can be assumed to be
zero.
6. a) A simulation diagram is shown below. Determine the difference
equation associated with the diagram.
7. An analog signal is given by f(t) = t (i.e., it increases linearly with
time and is thus is a unit ramp.) It is convolved with a second signal,
g(t), which is of the form g(t) = 1 (i.e., it has a constant value of 1 or is a
unit step function). The two signals are shown below.
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ECET 345 Week 4 iLab Part 1 RC Circuit
Frequency Response
Objective of the lab experiment: The objective of this experiment is to
experimentally measure the frequency response of a simple RC circuit
using Multisim and observe how changing R and C will affect the
outcome.
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ECET 345 Week 4 iLab Part 2 Experimental
Observation of Aliasing

The objective of this experiment is to observe the effect of aliasing in a
discrete sampling system and to measure how aliasing alters the
frequency of an input signal that is beyond the Nyquist limit. This lab
can also be used to quantitatively and qualitatively observe the effect of
an antialiasing filter, even though we do not do so in this exercise.
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1.Using z-transform tables (page 776 of text or equivalent), find the z-
transform of
2.Find the inverse z-transform, x(n), of the following functions by
bringing them into a form such that you can look up the inverse z-
transform from the tables. This will require some algebraic and /or
trigonometric manipulation/calculation. You will also need a table of z-
transforms (page 776 of text or equivalent). When computing the value

of trigonometric functions, keep in mind that the arguments are always
in radians and not in degrees.
3.Find the first seven values (i.e., x(n) for n = 0 to 6) of the function
given below.
Hint: Manually calculate the three parts separately for various values of
n and add or subtract them point by point for various values of n. For
example, for n = 2 equals 2 * 2 * 1 (or 4); for n = 5 equals 2 * u(2) or 2
* 1 = 2; and so on. Also keep in mind that u(n - k) is a unit step function
delayed byksamples, and hence it will be zero for all values of (n - k),
which are negative and 1 otherwise.
4.The simulation diagram of a discrete time system is shown below.
Find the first six output (y(0) to y(6)) of the system when an input x(n) ,
as computed in problem 3, is applied to the discrete time system.
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ECET 345 Week 5 iLab Convolution of Signals

The objective of this experiment is to demonstrate how the convolution
is used to process signals entering a system.
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1.Find the z-transform x(z) of x(n) = . Hint: Follow the method used in
the lecture for Week 6. Also, when evaluating the numerical value of a
trig function, keep in mind that the arguments of trig functions are
always in radians and not in degrees.
2. Find the system transfer function of a causal LSI system whose
impulse response is given by and express the result in positive powers of
z. Hint: The transfer function is just the z-transform of impulse response.
However, we must first convert the power of -0.5 from (n - 1) to (n - 2)
by suitable algebraic manipulation.
3. Express the following signal, x(n), in a form such that z-transform
tables can be applied directly. In other words, write it in a form such that

the power of 0.25 is (n-1) and the argument of sin is also expressed with
a (n-1) multiplier.
4. The transfer function of a system is given below. Find its impulse
response in n-domain. Hint: First expand using partial fraction
expansion and then perform its inversion using z-transform tables
5. The transfer function of a system is given by
6. A simulation diagram is shown below. We apply a unit impulse to
such a system. Determine the numerical values of the first three outputs.
You are free to use MATLAB where appropriate or do it entirely by
hand.
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ECET 345 Week 6 iLab Z-Domain Analysis of
Discrete Systems

The objective of this experiment is to perform z domain analysis of
discrete (sampled) signals and systems and extract useful information
(such as impulse and step response, pole zero constellation, frequency
response, etc.) from a z domain description of the system, such as its
transfer function. We shall also study conversion of analog transfer
functions (in s domain) into equivalent z domain transfer functions using
bilinear transform.
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1.A sine wave of 60 Hz, amplitude of 117 V, and initial phase of zero
(or 117 sin(2π*60t) is full wave rectified and sampled at 2,048 samples
per second after full wave rectification. Research the Fourier series for a
full wave rectified sine wave (on the Internet or in circuit theory books,
such as Linear Circuits by Ronald E. Scott) and write it below.
Then write a MATLAB program that samples and stores 4,096 points of
full wave rectified sine wave and performs Fourier analysis (FFT) of the
full wave rectified sine wave on the stored points.

Plot the results in both linear and log scale (in two separate figures) and
extract the amplitude of the DC component and the first four harmonics
(first , second, third, and fourth multiple of the fundamental frequency)
of the Fourier analysis, then enter them in the table given below. The DC
component is given by the first number in the Fourier analysis. Hint:
Full wave rectification can be achieved in MATLAB simply by taking
the absolute value (abs command) of the sine wave.
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ECET 345 Week 7 iLab Fourier Analysis of
Time Domain Signals
The objective of this experiment is to perform Fourier analysis to obtain
frequency domain signature of signals and systems that are measured or
whose characteristics are known in time domain. Towards this end, we
shall learn how to use Fourier transform to obtain Bode plots of systems

from time domain data passing through the system. We shall also learn
the equivalence of convolution operation in time domain with
multiplication operation in frequency domain.
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