Sns mid term-test2-solution
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This document contains a mid-term test for a signals and systems engineering diploma assessment. The test contains 4 questions that assess topics such as determining if signals are periodic or aperiodic, sketching signals, evaluating convolution sums and integrals, defining Laplace and z-transforms, and using properties like shifting to find impulse responses of linear time-invariant systems. The test aims to evaluate students' understanding of fundamental concepts in signals and systems.
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![PSDC ENGINEERING DIPLOMA ASSESSMENT
Signals And Systems ¨C Mid-Term Test 2 (B16 CE/B17 EEA&EEB)
7th January 2009, Wednesday 2:00pm ¨C 5:00pm (3 hours)
Question 1 (Signals and Systems)
(a) Determine whether the following signals are periodic or aperiodic.
If it is a periodic signal, determine its power.
(i) x[n] = 2cos(?¦Ðn) + sin(?¦Ðn)
(ii) x(t) = Odd{cos(2¦Ðt)}
(b) If x(t) = (t2 ¨C 1) [u(t) ¨C u(t¨C4)],
(i) Sketch x(t)
(ii) Sketch x(t ¨C 2)
(iii) Sketch x(¨C3t ¨C 2)
(iv) Sketch Even{x(t ¨C 2)}
(c) Determine the following systems are casual, linear and time-invariant.
(i) y[n] = 3x2[¨C2n]
(ii) y(t) = cos (t+ ? ¦Ð) x(t)
Question 2 (Convolution)
(a) Evaluate the convolution sum for y[n] = x[n]?h[n], where x[n] and h[n] are
shown in Fig. Q2 (a).
Fig. Q2 (a)
(b) Define the convolution of two signals x(t) and h(t).
Then, compute the convolution integral for y(t) = x(t)?h(t) of the following
signals:
h(t) = e¨C3tu(t) and x(t) = e¨C3tu(t¨C1).
1](https://image.slidesharecdn.com/snsmid-term-test2solution-110406043236-phpapp01/85/Sns-mid-term-test2-solution-1-320.jpg)
![PSDC ENGINEERING DIPLOMA ASSESSMENT
Question 3 (Laplace Transform)
(a) Define Bilateral (two-sided) Laplace Transform.
(b) Consider the signal, x(t) = e¨C3tu(t). Find the Laplace transform of x(t) with
the associated region of convergence (ROC).
(c) Consider a continuous-time LTI system described by
d 2 y( t ) dy( t ) dx( t )
2
?3 ? 2 y( t ) = ?3 + x( t )
dt dt dt
Using the shifting property, find the unit impulse response h(t).
Question 4 (z-Transform)
(a) Define Bilateral (two-sided) z-Transform.
(b) Consider the signal, x[k] = aku[k]. Find the z-transform of x[k] with the
associated region of convergence (ROC).
(c) Consider a discrete-time LTI system described by
y[k] ¨C y[k¨C1] ¨C 2 y[k¨C2] = x[k] + 2 x[k¨C1] + 2 x[k¨C2]
Using the shifting property, find the unit impulse response h[k].
"Learning without thought is useless; thought without learning is dangerous."
ѧ¶ø²»Ë¼ÔòØè,˼¶ø²»Ñ§Ôò´ù.
2](https://image.slidesharecdn.com/snsmid-term-test2solution-110406043236-phpapp01/85/Sns-mid-term-test2-solution-2-320.jpg)
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Problem 1
Recall that 1
1?x =
¡Æ¡Þ
n=0 x
n for |x| < 1.
Find a power series representation for the following functions and state the radius of
convergence for the power series
a) f(x) = x
2
(1+x)2
.
b) f(x) = 2
1+4x2
.
c) f(x) = x
4
2?x.
d) f(x) = x
1+x2
.
e) f(x) = 1
6+x
.
f) f(x) = x
2
27?x3 .
Problem 2
Find a Taylor series with a = 0 for the given function and state the radius of conver-
gence. You may use either the direct method (definition of a Taylor series) or known
series.
a) f(x) = ln(1 + x)
b) f(x) = sin x
x
c) f(x) = x sin(3x).
Problem 3
Find the radius of convergence and interval of convergence for the series
¡Æ¡Þ
n=1
(x+2)n
n4n
.
Ans. Radius r=2,
¡Ì
2 ? 2 < x <
¡Ì
2 + 2. Problem 4
Find the interval of convergence of the following power series. You must justify your
answers.
¡Æ¡Þ
n=0
n2(x+4)n
23n
.
Ans. ?12 < x < 4.
Problem 5
For the function f(x) = 1/
¡Ì
x, find the fourth order Taylor polynomial with a=1.
Problem 6
A curve has the parametric equations
x = cos t, y = 1 + sin t, 0 ¡Ü t ¡Ü 2¦Ð
a) Find dy
dx
when t = ¦Ð
4
.
b) Find the equation of the line tangent to the curve at t = ¦Ð/4. Write it in y = mx+b
form.
c) Eliminate the parameter t to find a cartesian (x, y) equation of the curve.
d) Using (c), or otherwise identify the curve.
Problem 7
State whether the given series converges or diverges
a)
¡Æ¡Þ
n=0 (?1)
n+1 n22n
n!
.
b)
¡Æ¡Þ
n=0
n(?3)n
4n?1
.
c)
¡Æ¡Þ
n=1
sin n
2n2+n
.
Problem 8
1
Approximate the value of the integral
¡Ò 1
0
e?x
2
dx with an error no greater than 5¡Á10?4.
Ans.
¡Ò 1
0
e?x
2
dx = 1 ? 1
3
+ 1
5.2!
? 1
7.3!
+ ... +
(?1)n
(2n+1)n!
+ .... n ¡Ý 5,
for n=5
¡Ò 1
0
e?x
2
dx ¡Ö 1 ? 1
3
+ 1
5.2!
? 1
7.3!
+ 1
9.4!
? 1
11.5!
¡Ö 0.747.
Problem 9
Find the radius of convergence for the series
¡Æ¡Þ
n=1
nn(x?2)2n
n!
.
Ans. R = 1¡Ì
e
.
Problem 10
Let f(x) =
¡Æ¡Þ
n=0
(x?1)n
n2+1
.
a) Calculate the domain of f.
b) Calculate f ¡ä(x).
c) Calculate the domain of f ¡ä.
Problem 11
Let f(x) =
¡Æ¡Þ
n=0
cos n
n!
xn.
a) Calculate the domain of f.
b) Calculate f ¡ä(x).
c) Calculate
¡Ò
f(x)dx.
Problem 12
Using properties of series, known Maclaurin expansions of familiar functions and their
arithmetic, calculate Maclaurin series for the following.
a) ex
2
b) sin 2x
c)
¡Ò
x5 sin xdx
d) cos x?1
x2
e)
d((x+1) tan?1(x))
dx
Problem 13
Calculate the Taylor polynomial T5(x), expanded at a=0, for
f(x) =
¡Ò x
0
ln |sect + tan t|dt.
Ans. T5(x) =
x2
2
+ x
4
4!
.
Problem 14
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you can for¨O¨O¨Osin x ? (x ? x33! + x55! )¨O¨O¨O
Same question: If
(
x ? x
3
3!
+ x
5
5!
)
is used to approximate sin x for |x| ¡Ü 0.8. What is
the maximum error? Explain what method you are using.
Problem 15
The Taylor polynomial T5(x) of degree 5 for (4 + x)
3/2 is
(4 + x)3/2 ¡Ö 8 + 3x + 3
16
x2 ? 1
128
x3 + 3
4096
x4 ? 3
32768
x5.
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The document discusses the Laplace transform and its properties. It begins by introducing Laplace transform as a tool to transform signals from the time domain to the complex frequency (s-domain). It then provides the Laplace transforms of some elementary signals like impulse, step, ramp functions. It discusses properties like linearity, time shifting, frequency shifting. It also covers the region of convergence, causality, stability analysis using poles in the s-plane. The document provides examples of finding the Laplace transform and analyzing signals based on properties like time shifting and frequency shifting. In the end, it summarizes the convolution property and the initial and final value theorems.Es400 fall 2012_lecuture_2_transformation_of_continuous_time_signal.pptx
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This document contains the mid-term exam for the Communication Networks course EE 333. The exam consists of 3 questions covering topics related to communication networks including:
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1) The document discusses solving partial differential equations using variable separation. It presents the one-dimensional wave equation and solves it using variable separation.
2) It derives three cases for the solution based on whether k is positive, negative, or zero. It then presents the general solution as a summation involving sines and cosines.
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This document contains 8 questions related to mathematics. Question 2 asks students to (a) form a differential equation by eliminating arbitrary constants from an expression involving logarithms, (b) solve a differential equation, and (c) calculate the temperature of a body cooling over time. Question 3 asks students to (a) find the Laplace transform of a periodic function and (b) solve a differential equation using Laplace transforms.8fbf4451c622e6efbcf7452222d21ea5_MITRES_6_007S11_hw02.pdf
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This document provides recommended problems related to signals and systems. Problem P2.1 involves determining the frequency and period of cosine signals with different phase shifts. Problem P2.2 is similar but with discrete-time cosine signals. Problem P2.3 involves sketching shifted and scaled versions of a sample discrete-time signal. The remaining problems involve determining whether signals are even, odd, or neither; evaluating sums; exploring properties of periodic signals; and analyzing sampled continuous-time signals.Signal and Systems part i
Signal and Systems part iPatrickMumba7
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This document provides recommended problems related to signals and systems. Problem P2.1 involves determining the frequency and period of cosine signals with different phase shifts. Problem P2.2 is similar but with discrete-time cosine signals. Problem P2.3 involves sketching shifted and scaled versions of a sample discrete-time signal. The remaining problems involve determining whether signals are even, odd, or neither; evaluating sums; exploring periodicity of signal sums; properties of even and odd signals; and sampling a continuous-time signal.Engr 213 final 2009
Engr 213 final 2009akabaka12
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This document contains a 9 question final exam for an applied ordinary differential equations engineering course. The questions cover a range of topics including: finding general and particular solutions to 1st and 2nd order differential equations using various methods; solving initial value problems; solving systems of differential equations; power series solutions; and modeling an LRC circuit with a differential equation. Students are given 3 hours to complete the exam worth a total of 100 points.
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The document discusses key concepts in amplifiers including:
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This document provides information on various topics related to semiconductor devices and transistor amplifiers, including:
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