EE 350 Problem Set 2 Cover Sheet Fall 2014Last Name (Print):First Name (Print):ID number (Last 4 digits):Sectio n:Submission deadl ines:• Turn in the written solutions by 4:0 0 pm on Tuesday September 16 in the homework slot outside 121 EE East.Problem Weight Score6 207 208 209 2010 20Total 100The solution submitted for grading represents my own analysis of the problem, and not that of another student.Signature:Neatly print the name(s) of the students you coll ab orated with on this assignment.Reading assignment:• Lathi Chapter 1 and Background, Sections B.2 and B.3• Priemer Chapters 1 and 2Problem 6:(20 points)1. (9 points) Determine if the following signals are odd, even, or neither.(a) (3 points) f(t) = eωt(b) (3 points) f(t) = e−|t|cos(ωt)(c) (3 points) f(t) = sin(t)/t2. (5 points) Find and sketch the odd, fo(t), and even, fe(t), components off(t) = cos(πt)[u(t) −u(t −1)].3. (6 points) For an even function fe(t) and an odd function fo(t), show thatZT /2−T /2fe(t) dt = 2ZT /20fe(t) dtZT /2−T /2fo(t) dt = 0.Problem 7: (20 points)1. (5 points) Determine if the signalf(t) = e|t|/τ[u(t + τ) − u(t −τ)],where τ i s a real-valued finite positi ve constant, is a power signal, an energy signal, or neither. If it is a poweror energy signal, determine the measure (Efor Pf) of the signal.2. (5 points) Determine if the signalf(t) = u(t − 10)is a power signal, an energy signal, or neither. If i t is a power or energy signal, determine the measure (EforPf) of the signal.3. (6 points) The root mean square (r m s) value of a signal f(t) is defined as the square-root of the signal powerPffrms=pPf=vuutlimT →∞1TZT2−T2f2(t)dt.The RMS value is the equivalent DC quantity, that, when applied to a purely resistive load, yields the sameaverage power as the time varying signal f(t). As an example, consider the voltage waveformf(t) = A cos2πTt + θwhere A > 0, T > 0, and θ are constants. Show that f(t) is a p ower signal and determine its rms value.4. (4 points) The definitions for power and energy sig nals can be generalized to complex-valued signals asEf=Z∞−∞|f(t)|2dtPf= limT →∞1TZT2−T2|f(t)|2dtwhere |f(t)|2= f(t)f∗(t) and f∗(t) is the complex-conjugate of f(t). Determine if the signalf(t) = (2 − 2) sin(t)u(−t),is a power signal, an energy signal, or neither. If i t is a power or energy signal, determine the measure (EforPf) of the signal.Problem 8: (20 points)For the systems described by the equatio ns below, where the input is f(t) and the zero-state response is y(t),determine if the system is• zero-state linear,• time-invariant or time varying,• causal or noncausal, and• instantaneous (memoryless) or dynami c.Justify your answers.1. y(t) = |f(t + 1)|2. y(t) = e−|t|f(t −1)3. y(t) = cos(f(2t))4. y(t) =Rt−∞f(τ)e(t−τ)dτProblem 9: (20 points)When a system is linear and ti me invariant (LTI), the task of computing the zero-state response i s si mplified. As anexample, suppose that we have already found the zero-state response y(t) of a LTI system for some input f(t), and itbecomes necessary to find the zero-state resp onse y1(t) for some new input, f1(t). If the input f1(t) can be expressedas a sum of terms, where each term is a scaled and time shifted version of f(t), then it is p ossible to use the systemLTI properties to express y1(t) as a sum of terms, where each term is a scaled and time shifted version of y(t). Todemonstrate this concept, consider Figure 1 which shows the zero-state response y(t) of a linear time-invariant (LTI)system to a particular input f(t).Figure 1: Zero-state response y(t) of a LTI system to an input f(t).1. (6 points) An input f1(t) = 2f(t−1)+ 2f(t−0.5) is applied to the LTI system. Express the resulting zero-stateresponse y1(t) in terms of y(t).2. (7 points) Using the graph of y(t) in Figure 1 and your result from part 1, carefully sketch the zero-stateresponse y1(t).3. (7 points) A new input f2(t) is applied to the LT I system and the resulting zero-state response y2(t) is shownin Figure 2. Express the input f2(t) as a summation of terms involving f(t) scaled in amplitude and shifted intime.Figure 2: Zero-state response y2(t) of the LTI system to an input f2(t).Problem 10: (20 points)The ability to manipulate compl ex numbers is important in the analysis of circuits and systems. In EE 210 you usedcomplex numbers known as phasors to quickly determine the sinusoidal steady-state response of an electric circuit.Please carefully read Background Section B.1 in Lathi before attempting the following review problems. Except f orthe MATLAB questions, solve the equations by hand, do not use a calculator. To receive cre dit for the MATLABquestions, include an m-file with your name and section that shows the MATLAB commands used to answers parts4 and 5.1. (2 points) Express the compl ex number 1/4in rectangular form a + b, where a and b are real.2. (2 points) Without using a calculator, express the following complex number2 + j2√3(1 − )√3 + in polar form reθ, where r and θ (in degrees) are real numbers (calculators are not permitted on the exams;you must b e able to work with complex numbers by hand).3. (8 points) Given s, u, v, and w are arbitrary complex numbers with nonzero magnitude, establish the followingresults:(a) (2 points) s s∗= |s|2(b) (2 points) s + s∗= 2 Re(s)(c) (2 points) s − s∗= 2 Im(s)(d) (2 points) su/vw = (|s||u|/|v||w|)e(6s+6u−6v−6w)4. (4 points) You can determine the roots of the equationxn+ an−1xn−1+ ···+ a1x + ao= 0using the MATLAB command roots. For example, to find the roots ofx3+ 4x2+ x − 3 = 0,enter>> roots([1, 4, 1, −3])and MATLAB will return the roots. First determine the roots of the equationx4+ 1 = 0by hand and sketch their location in the complex plane. Verify your results using the MATLAB commandroots.5. (4 points) The MATLAB command conv finds the product of two polynomials. For example, suppose thatp = x4+ 3x3− 2x2+ x − 5,q = x5+ 4x2+ x + 10,and we wish to determine the product s = pq. Use the MATLAB commands>> p = [1, 3, −2, 1, −5];>> q = [1, 0, 0, 4, 1, 10];>> s = conv(p, q)In this cases = [1, 3, −2, 5, 8, 5, 32, −39, 5, −50]and sos = x9+ 3x8−2x7+ 5x6+ 8x5+ 5x4+ 32x3−39x2+ 5x − 50.Determine the productx3+ 2x2+ x − 1x2− 1x4−x2+ 3using the MATLAB command
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