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# Introduction to Engineering Analysis Assignment 8

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ENSC 180: Introduction to Engineering Analysis
Assignment 8

Note: MATLAB codes should include definition of all variables; headings to identify
the program structure plan; and appropriate captions and labels for tables and
figures. Submit a .pdf report documenting your inputs and outputs in addition to a
separate zip-file containing all M-files. Marks will be deducted for solutions that are
unrealistic/impractical (as future engineers student should learn to think practically)
and poorly documented.
1. When broadcasting radio waves, different signals can intermix during transmission, and the
receiver will need to know how to differentiate between them. Amplitude Modulation (AM) and
Frequency Modulation (FM) are two methods of differentiating the signals so that they can be
transmitted and then recovered. In both, a sinusoidal carrier signal is used to modify the message.
The carrier signal is of the form;
𝑆(𝑡) = 𝐴𝑐cos(2𝜋𝑓𝑐
𝑡)
with Ac =carrier amplitude and fc = carrier frequency (Hz)
The carrier will modulate a modulating/message signal m(t);
𝑚(𝑡) = 𝐴𝑚 cos(2𝜋𝑓𝑚𝑡)
with Am = message amplitude and fm = message frequency (Hz) << fc
AM modulation is defined by;
𝑆𝐴𝑀(𝑡) = 𝐴𝑐
[1 + 𝑚(t)]cos(2𝜋𝑓𝑐
𝑡)
while FM modulation is defined by;
𝑆𝐹𝑀(𝑡) = 𝐴𝑐cos(2𝜋𝑓𝑐
𝑡 + 2𝜋𝑘𝑓𝐴𝑚 ∫ 𝑚(𝜏)𝑑𝜏 𝑡
0
)
where kf = the frequency sensitivity.
Write a MATLAB program that takes a carrier frequency fc fm in Hz (assume the amplitudes
and kf are all 1). Create a message signal with frequency fm = 0.02 Hz and modulate it via each
method, then output a figure with four separate subplots stacked on top of each other; the first is
the message, the second is the carrier signal, the third is the AM wave and the fourth the FM
wave. Comment on the results. (Hint: Keep the frequency values in the mHz or Hz ranges to
better display the signal features) (20 marks)
2. Diode is a two-terminal component widely used in electronic design. The relationship between
diode forward current ID and diode forward voltage VD can be modeled as:
𝐼𝐷 = 𝐼𝑆(𝑒
𝑉𝐷
𝑛𝑉𝑇 − 1)
where IS is the reverse current, n is a quality factor and VT is thermal voltage. Thermal voltage
VT is given by:
VT =
kT
q
where k = 1.38×10−23 J/K and q = 1.6×10−19 C. T is the temperature in Kelvin (0 C = 273K).
Consider a case where n = 1, IS = 10-12 A and plot the forward diode current (ID) vs diode voltage
(VD) at -75C, 25C and 75C. For plotting, use voltage as the horizontal axis and current as the
vertical axis with VD from 0 to 0.8V. Plot the 3 curves corresponding to the 3 temperatures in
separate figures aligned vertically in one page. Comment on the behavior current-voltage
relationship at different temperatures. (20 marks)
3. In electronics, a MOSFET is a voltage-controlled field effect transistor. A MOSFET is a threeterminal device containing a drain (D), a gate (G) and a source (S), body is always connected to
the lowest voltage in the circuit. Consider a N-channel MOS as shown in the following diagram:
The Current-Voltage curve of a MOSFET has three regions: cut-off, linear and saturation, and
they are defined by:
cut-off: ID = 0 for VGS < VT
For VGS VT, the gate can be turned on which allows the MOSFET enters into linear and
saturation regions defined by:
linear: ID =
1
2
Kn
W
L
[2(VGS − VT
)VDS − VDS
2
] for VDS ≤ VGS − VT
Saturation: ID=
1
2
Kn
W
L
(VGS − VT
)
2
(1 + λVDS) for VDS VT
where Kn is the transconductance parameter, W/L is the aspect ratio between width and length,
VT is the threshold voltage, VGS is the voltage difference between Gate and Source, VDS is the
voltage difference between Drain and Source.
Given: Kn = 100u (unit: A/V2
), W/L = 5 (unitless), VT = 1V, lambda = 0.01
Plot ID vs. VDS for different gate to source voltages VGS. Vary VGS starting from 0V with
increments of 0.5V each time to generate 10 characteristic curves. (20 marks)
4. The following model describes the motion of a 2-DOF system described by the generalized
coordinates x1 and x2 under an external excitation f(t).
[
𝑥̇1
𝑥̇2
] = [
0 1
−5 −2
][
𝑥1
𝑥2
]+ [
0
1
] 𝑓(𝑡)
a. Use the initial function to plot the position 𝑥1 of the mass if the initial position is 5 and the
initial velocity is 3.
b. Use the step function to plot the response of the system for zero initial conditions and a
step function excitation with magnitude equal to 10.
Note: use the control system toolbox to solve this problem. (20 marks)
5. When images are scaled or resized, the values between pixels need to be guessed/interpolated in
order to fill in the missing gaps. Write a function that takes a grayscale image and a scaling
factor (any positive decimal value) as input and resizes the image using the following methods;
a) Nearest neighbouring pixel
b) Linear interpolation
c) Spline interpolation
Use the test image provided (a8_img.jpg), read in the image and output the resulting image for
each of the three methods. Test out your function with 1.5 scaling, 3.1 scaling, and for 0.3
scaling (9 images total). You cannot use the built-in MATLAB function imresize(), but any
other MATLAB built-in functions are acceptable.
(Hint: the image is in uint8 format, with values between 0 and 255, but in order to process and
display properly, MATALB requires images to be in double format with values between 0 and 1)
(20 marks)

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