ECET 345 Exceptional Education / 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

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 iLab Signal Observation
and Recreation
Objective:
Using Multisim, create virtual circuits and experimentally observe the
closest equivalent of four key signals (impulse, sinusoidal,
exponential, and square wave) on the oscilloscope.

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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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ECET 345 Week 2 Lab Response OfRc Circuits
(100% Score)

The objective of this experiment is to experimentally measure the
step response of an RC circuit and compare it to response prediced
using MATLAB
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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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ECET 345 Week 4 Lab Experimental
Observation Of Aliasing (100% Score)

ECET 345 Week 4 Lab
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
Solution (100% Score)
The objective of this experiment is to demonstrate how the
convolution is used to process signals entering a system.
1. Convolution in the time domain is equivalent to what mathematical
operation in the frequency domain?
2. When we convolve the triangular 10 Hz input with the impulse
response of the 50 Hz low-pass filter, why is it that the peaks of

output become rounded and not a sharp point as in the input triangular
function?
3. Why is it that we get no (or very little) output when we convolve
the 60 Hz sinusoid with the impulse response of the filter?
4. When we apply the 10 Hz output, which is within the pass band of
the filter, we see that we get nearly the same sinusoid in the output
except for a time delay. How is the time delay a signal experiences as
it passes through a system related to the phase characteristic of the
system response?
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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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ECET 345 Week 6 Lab Z-Domain Analysis Of
Discrete Systems (100% Score)
ECET 345 Week 6 Lab 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.
Equipment list:
• MATLAB
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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 Solution (100% Score)

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.
(a) Application of Fourier transform to time domain signals
(a) What are the frequencies of the first seven peaks in the FFT?
(b) Does the spectrum contain only even, only odd, or both even and
odd harmonic peaks?
(c) Research the Fourier series expansion of a triangular wave using
the Internet. From the formula you come up with, compare the
amplitudes and frequencies of the harmonics that you found with
what the theory says they should be. Explain any differences.
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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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