1 LINEAR TIME INVARIANCE
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Linear Time Invariance
1. For each system below, determine if it is linear or non-linear, and determine if it is
time-invariant or not time-invariant (adapted from Siebert 1986).
(a) y (t) = u(t + 1)
The system is linear tim
6 CONVOLUTION OF SINE AND UNIT STEP
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Convolution of Sine and Unit Step
The sine function q (t) has a zero value before zero time, and then is a unit sine wave
afterwards:
q (t) =
0 if t < 0
sin(t) if t 0
For the LTI systems whose impulse responses h(t)
5 LTI MACHINE?
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LTI Machine?
If you put a sequence of ve numbers into a certain machine, it responds with a ve-number
sequence; this exchange constitutes one experiment. For each possible machine characteris
tic below, give enough sets of input-output
3 FOURIER SERIES
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Fourier Series
Compute the Fourier series co ecients A0 , An , and Bn for the following signals on the
interval t = [0, 2 ]:
1. f (t) = 4 sin(t + /3) + cos(3t)
First, write this in a fully expanded form: y (t) = 4 sin(t) cos(/3) + 4
2 CONVOLUTION
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Convolution
The step function s(t) is dened as zero when the argument is negative, and one when the
argument is zero or positive:
s(t) =
0 if t < 0
1 if t 0
For the LTI systems whose impulse responses are given below, use convolution to
7 FOURIER SERIES CALCULATIONS
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Fourier Series Calculations
Compute the Fourier series coecients A0 , An , and Bn for the following signals on the
interval T = [0, 2 ]:
1. f (t) = 2 sin(t + /4) + cos(5t + /3)
Solution: use trigonometric identities to r
8 PROBABILITY PRIMER WITH DICE
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Probability Primer with Dice
You are given two fair dice.
1. Make a plot of the possible outcomes of one toss of one die, versus the likeliho od
(probability) of that outcome. Clearly there are six possible outcomes,
12 RANGING MEASUREMENTS IN THREE-SPACE
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Ranging Measurements in Three-Space
The global positioning system (GPS) and some acoustic instruments provide long-baseline
navigation - wherein a number of very long range measurements can be used to triangul
11 SEA SPECTRUM AND MARINE VEHICLE PITCH RESPONSE
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Sea Spectrum and Marine Vehicle Pitch Response
1. Make a plot of the spectrum for about one hundred frequencies from zero to 4 rad/s,
with modal frequency m = 1 rad/s, and signicant wave height H1/3
10 SIMULATION OF A SYSTEM DRIVEN BY A RANDOM DISTURBANCE
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10 Simulation of a System Driven by a Random Dis
turbance
1. Simulate the second-order system:
x + ax + bx = d(t)
with a = 0.4 and b = 2.25.
Figure 1 below shows responses to the same disturbanc
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4 BRETSCHNEIDER SPECTRUM DEFINITION
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Bretschneider Spectrum Denition
The formula for the Bretschneider (one-sided) ocean wave spectrum is
S ( ) =
4
5 m 2 5m /44
4
H1/3 e
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where is frequency in radians per second, m is the modal (most likely) frequ
9 AUTONOMOUS VEHICLE MISSION DESIGN, WITH A SIMPLE BATTERY MODEL13
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Autonomous Vehicle Mission Design, with a Simple
Battery Model
An autonomous land robot carries its own energy in the form of a E (t = 0) = 700W h
(Watt-hour) battery. The robot can stay