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### HW3Sol (1)

Course: ELEN 314, Fall 2011
School: Texas A&M
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Texas A&M - ELEN - 314
Texas A&M - ELEN - 314
Texas A&M - ELEN - 314
ECEN 314: Signals and SystemsPractice MidtermOctober 5, 2011Problem 1 (DT LTI System): Consider a serial interconnection of two DT LTI systemsas shown below.x[n]y[n]h [n]h [n]12The unit impulse responses of the LTI systems are given by h1 [n] =
Texas A&M - ELEN - 314
Lecture 2: Linear Transformations on the IndependentVariable of SignalsTie LiuAugust 30, 20111Transformation of signals in CTThe are many ways of transforming a CT signal into another. For instance, we can scaleit, shift it in time, dierentiate it,
Texas A&M - ELEN - 314
Lecture 3: Systems Described by LinearConstant-Coecient Dierential/Dierence EquationsTie LiuSeptember 5, 2011The input-output relationship of a system can be either explicit or implicit. An important example for an implicit description of a system is
Texas A&M - ELEN - 314
Lecture 4: Basic System PropertiesTie LiuSeptember 6, 20111CausalityA system is causal if the output does not anticipate future values of the input, i.e., if theoutput at any time depends only on values of the input up to that time. All real-time p
Texas A&M - ELEN - 314
Lecture 5: DT ConvolutionTie LiuSeptember 13, 20111LTI systemsRecall that a CT system is TI ifx(t) y (t)=x(t t0 ) y (t t0 ),t0 Rand that it is linearx1 (t) y1 (t) and x2 (t) y2 (t)=ax1 (t) + bx2 (t) ay1 (t) + by2 (t),a, b RFor ECEN 314, we
Texas A&M - ELEN - 314
Lecture 6: Properties of DT ConvolutionTie LiuSeptember 13, 2011Property 1 (The sifting property): x[n] [n n0 ] = x[n n0 ]. In particular, x[n] [n] =x[n].Proof. By denition,x[k ] [n k n0 ]x[n] [n n0 ] =k =Note that1, k = n n00, k = n n0 [n k n
Texas A&M - ELEN - 314
Lecture 7: CT ConvolutionTie LiuSeptember 19, 20111CT ConvolutionDT LTI systems:x[k ] [n k ]x[n] ==x[k ]h[n k ]y [n] =k =k =CT LTI systems: Need to express x(t) as a linear combination of a basic signal and itstime shifts.Can we use1, t =
Texas A&M - ELEN - 314
Lecture 8: More on the CT Unit Impulse FunctionTie LiuSeptember 20, 20111Operational Denition of the CT Unit Impulse functionThe CT unit impulse function (t): Idealization of a unit-area pulse that is so short that,for any physical systems of intere
Texas A&M - ELEN - 314
Lecture 9: Eigenfunctions of LTI SystemsTie LiuOctober 3, 20111Signal decompositions using eigenfunctions of LTIsystemsSignal decompositions using a set of basic signals cfw_k :ak k [n]x[n] =kx(t) =ak k (t)kFor LTI systems, knowing the respon
Texas A&M - ELEN - 314
Lecture 10: CT Fourier SeriesTie LiuOctober 4, 20111Periodic CT signalsA CT signal x(t) is said to be periodic if there exists T &gt; 0 such thatx(t + T ) = x(t),t Rwhere T is called a period. The smallest such T , if it exists, is called the fundame
Folsom Lake College - ECON - 101
file:/E|/Documents%20and%20Settings/Princess%20Delilah/Desktop/Isaac%20Asimov/Interview%20with%20Isaac%20Asimov.txt08/27/1988DR ISAAC ASIMOVTALKS WITHSLAWEK WOJTOWICZSlawek Wojtowicz 1988SW: If you ask any SF fan in Poland &quot;who is Isaac Asimov?&quot; he
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Mini-Case: Cathys CollectiblesMini-Case: Cathys CollectiblesHusameddin AlnajjarKeller Graduate School of Management.AbstractYour cousin Cathy runs a part-time business out of her apartment. She buys and sells collectiblessuch as antique prints, base
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Mini-Case: Central UniversityMini-Case: Central UniversityHusameddin AlnajjarKeller Graduate School of Management.AbstractSuppose you are the network manager for Central University, a medium-size university with13,000 students. The university has 10
Waterloo - MATH - 138
Math 138Assignment 1 Solutions1. If we are trying to make this true for all x then we are essentially trying to create anidentity between the two functionsxf ( x) = 8 +ag (t)dtt2h(x) = 8 3 xandthat is, we are trying to solve for a and g (t) so
Waterloo - MATH - 138
Math 138 Fall 2011Assignment 1Due Friday September 23 (in the drop box by 12 noon)Hand in the following:1. Find the value of a that makes the equationx8+ag (t)dt = 8 3 xt2true for all x by rst solving for g (t). (assume x and a are positive)2.
Waterloo - MATH - 138
Math 138Assignment 2 Solutions1.x3 + 1dxx2 2x + 5Use long division to getx3 + 1x+9=x+2 22 2x + 5xx 2x + 5The rst two terms are easy to deal with so we focus on the third term. Complete thesquare to getx+9x+9=x2 2x + 5(x 1)2 + 4then let
Waterloo - MATH - 138
Math 138 Fall 2011Assignment 2Due Friday September 30 (in the drop box by 12 noon)Hand in the following:1. Computex3 + 1dxx2 2x + 51dx1 x2(b) Use the result in (a) to derive2. (a) Findsec(x) dx = ln | sec(x) + tan(x)| + CHint: Note that sec(
Waterloo - MATH - 138
Math 138 Fall 2011Assignment 3Due Friday October 7 (in the drop box by 12 noon)Hand in the following:1. The curve f (u) = eu u4 1 has two roots u = 0 and u = b (b &gt; 0).Find the volume obtained when the region in the rst quadrant boundby the curve x
Waterloo - MATH - 138
Math 138Assignment 3 Solutions1. The plot of x = ey y 4 1 is shown below.b1-2-1012-1Figure 1: x = ey y 4 1To rotate this about the x-axis we must use cylindrical shells as we cant solve for y interms of x. Note that b is a root of ey y 4 1. T
Waterloo - MATH - 138
Math 138 Fall 2011Assignment 4Due Friday October 14 (in the drop box by 12 noon)Hand in the following:1. Consider the complex number z = x + iy where i is the imaginaryunit (a constant). Recall that we can rewrite this in polar form asz () = r cos +
Waterloo - MATH - 138
Math 138Assignment 4 Solutions1. (a) Using z = r cos + ir sin , we getdz= r sin + ir cos d(b) Note thatz = r cos + ir sin iz = ir cos r sin Thus, comparing with part a) we havedz= izdi.e. f (z ) = iz .(c) Using z () = r cos + ir sin we see t
Waterloo - MATH - 138
Math 138 Fall 2011Assignment 5Due : Not to be handed inFor practice:1. Do the recommended problems as per schedule:Section 9.5 #7, 9, 13, 19, 212. Find the solutions of the following initial value problems:dy+ y = x2 e1/x , y (1) = 3edxdy(b) y
Waterloo - MATH - 138
Math 138Assignment 5 Solutions1. See answers in textbook2. Find the solutions of the following initial value problems:dy+ y = x2 e1/x , y (1) = 3edxThis isnt in quite the correct form, so we divide through by x2 , noting that x = 0.Then the DE rea
Waterloo - MATH - 138
Math 138 Fall 2011Assignment 6Due Monday October 31 (in the drop box by 12 noon)Hand in the following:1. Consider the sequencea1 =5,a2 =5 5,a3 =5 5 5,a4 =5 5 5 5,(a) Write this as a recursive sequence(b) Does your sequence in (a) converge? I
Waterloo - MATH - 138
Math 138Assignment 6 Solutions1. (a) Based on the given terms we see thata1 =55 5 = 5a1a2 =5 5 5 = 5a2a3 =...from which we deduce the patternan+1 =5an ,a1 =5(b) This sequence converges. To prove this we show that it is increasing and boun
Waterloo - MATH - 138
Math 138 Fall 2011Assignment 7Due Monday Nov 7 (in the drop box by 12 noon)Hand in the following:1. Determine whether or not the following series are convergent. If so,nd the sum. If not, explain why.(a)n=1(b)n=11 + 2n4n11n(n + 2)e1/n(c)n=
Waterloo - MATH - 138
Math 138Assignment 7 Solutions1. Determine whether or not the following series are convergent. If so, nd the sum. If not,explain why.(a)n=1n=11 + 2n4n11 + 2n=4n1n=11+4n1n=12n=4n1144n=1n+4n=124n1These are both geometric seri
Waterloo - MATH - 138
Math 138 Fall 2011Assignment 8Due Monday Nov 14 (in the drop box by 12 noon)Hand in the following:1. Determine whether or not the following series converge or diverge bymaking a suitable comparison.n(a)n2 + 2 n + 2n=11nsin(b)n=1(c)n=1(d)
Waterloo - MATH - 138
Math 138 Fall 2011Assignment 9Due Monday Nov 21 (in the drop box by 12 noon)Hand in the following:1. Determine whether the series is absolutely convergent, conditionallyconvergent, or divergent.(a)n=1(b)n=1(1)n1n4n!en(1)n (c)n=1nn3 + 22
Waterloo - MATH - 138
Lecture 1: MATH 138-W12-003January 4, 2012I) Review of the Fundamental Theorem of Calculus (FTC), Section 5.3The rst theorem helps us to evaluate the following,xf (t) dt.g (x) =a1) If f (t) is positive throughout the whole interval than this can b
Waterloo - MATH - 138
Lecture 1: MATH 138-W12-003January 6, 2012I) Integration by Parts, Section 7.1I never met an integral I couldnt integrate by partsRecall that the product rule states,d[f (x)g (x)] = f (x)g (x) + f (x)g (x).dxIf we rearrange the terms by isolating
Waterloo - MATH - 138
Lecture 3 and 4: MATH 138-W12-003January 9 and 11, 2012I) Trigonometric integrals, Section 7.2Example: How would we evaluate sin3 x dx?Based on our previous example we could try and separate it as u = sin x and use the identity2sin x = 1 cos2 x. Thi
Waterloo - MATH - 138
Lecture 5: MATH 138-W12-003January 13, 2012I) Trigonometric substitution, Section 7.3In this lecture we focus on computing integrals involving functions that include a2 x2 , x2 a2and x2 + a2 using trigonometric substitutions. For example the rst occur
Waterloo - MATH - 138
Lecture 6: MATH 138-W12-003January 16, 2012I) Integration by Rational Functions by Partial Fractions, Section 7.4We know very well that to sum two fractions that are polynomials in x we can get a commondenominator. For example,212(x b) + (x a)3x (
Waterloo - MATH - 138
Lecture 8: MATH 138-W12-003January 20, 2012I) Volumes by disks, Section 6.2Now that weve established a multitude of techniques of integration we are going to learnhow to apply them to the very physical problem of computing volumes. Since we only know
Waterloo - MATH - 138
Lecture 9: MATH 138-W12-003January 23, 2012I) Volumes by shells, Section 6.3In the previous lecture (or set of notes) we should how to revolve a quadratic curve aboutthe y -axis we needed to complete the square to describe the curve as x(y ). This was
Waterloo - MATH - 138
Lecture 10: MATH 138-W12-003January 25, 2012I) Approximate (Numerical) Integration, Section 7.7We have learned a plethora of analytical techniques to integrate functions of a single variable.This allows us to evaluate a lot of integrals exactly, which
Waterloo - MATH - 138
Waterloo - MATH - 138
Lecture 11: MATH 138-W12-003January 27, 2012I) Improper Integrals, Section 7.8bPreviously, when we discussed integration we had something like a f (x) dx we assumedtwo things implicitly: 1) the bounds a, b were both nite and 2) the function f (x) doe
Waterloo - MATH - 138
Lecture 12: MATH 138-W12-003January 30, 2012I) Modelling with Differential Equations, Section 9.1Denition: A differential equation (DE) is an equation (or a system of equations) that involvethe derivatives of a function. In this lecture we introduce s
Waterloo - MATH - 137
Waterloo - MATH - 137
Math 137,Lecture 3,Friday September 12, 2008Topic: Functions (see pages 6-20 of course notes &amp;Chapter 1, pages 11-31 of Stewart)Exercise for Student:Answer:1
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Math 137, Lecture 5, Wednesday September 17, 2008Topic: Exponential &amp; Logarithmic Functions(Stewart, pages 52-63)Exercises for Student:1. Solve the equation x = ln(y +1Answer: y = (ex + ex )22. Solve log3 |3 2x| = 2.Answer: x = 6 or x = 3y 2 1),
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Math 137,Lecture 6,Friday September 19, 2008Topic: Transcendental FunctionsExercises for Student:Solve ln |x2 + x + 1| = 0. Answer: x = 0 or x = 1Answer: (A) (III), (B) (I), (C) (VI), (D) (V), (E) (II), (F) (IV)1
Waterloo - MATH - 137
Math 137,Lecture 7,Monday September 22, 2008Topic: Trigonometric Functions(see pages 21-31 of course notes)Exercises for Student:1. Using a sketch and the denition of sin and sin on the unit circle, show that| sin sin | | |.2. Prove the identity c
Waterloo - MATH - 137
Math 137,Lecture 9,Friday September 26, 2008Topic: Limits(pages 35-37 of course notes &amp; Stewart, pages 88-96)Exercises for Student:3x|x 2|evaluate lim f (x) and lim f (x). Hence, deduce if lim f (x) exists.x2x2+x2x2Answer: lim f (x) = 6, lim f
Waterloo - MATH - 137
Math 137, Lecture 10, Monday September 29, 2008Topic: Limit Laws &amp; Limits at Innity(pages 38-40 of course notes andStewart, pages 99-104 &amp; 130-137)Exercises for Student:tan xsin x= 1, use the limit laws to show that lim= 1.x0xx2. Evaluate lim
Waterloo - MATH - 137
Math 137,Lecture 12,Friday October 3, 2008Topic: Continuity(pages 42-46 of course notes &amp; Stewart, pages 119-126)Exercise for Student:x2 9continuous?x2 25Answer: (, 5) (5, 3] [3, 5) (5, )On what interval is the function g (x) =1
Waterloo - MATH - 137
Math 137,Lecture 13,Monday October 6, 2008Topics: Intermediate Value Theorem andBisection Method(pages 47-49 of course notes &amp; Stewart, pages 126-127)Exercise for Student:Using a sketch and the Intermediate Value Theorem, show that the equation ln
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CS 116 Winter 2012Assignment 01Due at 10:00am on Wednesday, January 18This assignment consists entirely of review material from CS 115 as covered in Module 01. You mustuse local and abstract list functions (map, filter, foldr) where appropriate. All h
Broward College - ENV - 1009