EE 350 Problem Set 8 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 4:00 pm on Tuesday November 11 in the homework slot outside 121 EE East.Problem Weight Score37 2038 2039 2040 2041 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 4: sections 4.5 through 4.7• Lathi Chapter 5: section 5.1• Priemer Chapter 13Exam III is scheduled for Thursday, November 13 from 8:15 pm to 10:15 pm in 108 Forum (all sections). The thirdexam covers material from Problem Sets 6 through 8, and Recitations 8 through 11. The date and location of areview session will be announced on the EE 350 web page.Problem 37: (20 points)In the first problem you will prove a series of Fourier transform properties that will be used extensively in theremainder of this problem set. Gi ven thatf(t) ⇔ F (ω)g(t) ⇔ G(ω)and toand ωoare real- valued constants, derive the fol lowing Fouri er Transform properties:1. (3 points) Time Shift Propertyf(t − to) ⇔ F (ω)e−ωt02. (3 points) Frequency Shift Propertyf(t)eωot⇔ F (ω − ωo)3. (5 points) Time Convolutionf(t) ∗ g(t) ⇔ F (ω)G(ω)4. (5 points) Frequency Convolutionf(t)g(t) ⇔12πF (ω) ∗ G(ω)5. (4 points) Time Differentiationdnfdtn⇔ (ω)nF (ω)Problem 38: (20 points)A key goal of EE 350 is to insure that you have a solid understanding of the relationship between the ODE, impulseresponse, and frequency respo nse function representation of a LTI system. Consider a linear time-invariant causal(LTIC) system with input f(t), impul se response function representation h(t), and zero-state response y(t).1. (5 points) Using the appropriate property from Problem 37, show that the Fourier transform of the zero-stateresponse y(t) of the system to an arbitrary input f(t) isY (ω ) = H(ω)F (ω),where Y (ω), H(ω), and F (ω) are the Fourier transforms of y(t), h(t), a nd f(t), respectively. The Fouriertransform of the impul se response function h(t) is identical to the frequency response function H(ω) of thesystem.2. (10 points) As a specific example, consider a LTIC system with the impulse response functionh(t) =ω2nωde−ζωntsin(ωdt)u(t),where ωn> 0, 0 ≤ ζ < 1, andwd = ωnp1 − ζ2.By direct integration, determine the frequency response f unction of the system by computing the Fouriertransform of the impulse response f unction. Express you answer in the standard formH(ω) =˜Y˜F=bm(ω)m+ bm−1(ω)m−1+ · · · b1(ω) + b0(ω)n+ an−1(ω)n−1+ · · · a1(ω) + a0.3. (5 points) Using the time differentiatio n property and the results from parts 1 and 2, find the ODE represen-tation of the system. Express your answer in the formdnydtn+ an−1dn−1ydtn−1+ · · · + a1dydt+ aoy(t) = bmdmfdtm+ bm−1dm−1fdtm−1+ · · · + b1dfdt+ bof(t).Problem 39: (20 points)This problem shows how to calculate the Fourier transform of periodic signals.1. (1 point) Find the Fourier transform of δ(t).2. (2 point) In problem set 4 problem 17 you showed that δ(at) = δ(t)/|a|. Using this result and the dualityproperty (also know known as the Symm etry problem, see section 4.3-2 in the text), determine the Fouriertransform of f(t) = 1.3. (2 points) Using the result from part 2 and the frequency shift property from Problem 38, determine the Fouriertransform of eωot.4. (4 points) Find the Fourier transform of the periodic signals sin(ωot) and cos(ωot) given the result in part 3and the fact that ωois a real- valued constant.5. (6 points) Suppose that f(t) is a periodic si gnal with period T and has the Fourier series representationf(t) =∞Xn=−∞Dnenωot.Use the result from part 3 to show thatF (ω) = 2π∞Xn=−∞Dnδω −2πTn.6. (5 p oints) Find the Fourier transform of the periodi c signal f(t) appearing in Recitation 10 Problem 2 on pageRec 10.7.Problem 40: (20 points)A filter is a system that manipulates the frequency spectra of a signal in a desired fashion. For example, a low-passfilter will pass allow low frequency components to pass through and remove (or filter out) high frequency components.Filters play an important role in many areas of electrical engineering, including communication and control systems.Consider the frequency response functions for a set of four filters specified by their frequency response functionsH1(ω) = A rectω2Be−ωtoH2(ω) = Ah1 − rectω2Bie−ωtoH3(ω) = Arectω − ωo2B+ rectω + ωo2Be−ωtoH4(ω) =Aω/B + 1,where A, B, and ωoare positive real constants.1. (8 points) Sketch the magnitude ( |H(ω)| ) and phase (6H(ω)) for each of the four filters.2. (4 points) Identify each of the filters a s either a hi gh-pass filter, low-pass filter, or band-pass filter.3. (6 p oints) Find the i mpulse response functions for the frequency response functions H1(ω) and H4(ω) You mayuse the Fourier transform properties and elementary Fourier transform pairs derived in either lecture or in theproblems sets.4. (2 points) If the impulse response function h(t) of the filter is a causal signal, then the filter is said to berealizable. Which of the filters H1(ω) and H4(ω), if either, are realizable?Problem 41: (20 points)This problem considers the application of Fourier transform m ethods to communication systems.1. (10 points) An important operation f requently arising in communication systems is the modulation of a signalf(t) by a sinusoidal signal. Using an appropriate property from Problem 42 and the results from Problem 39part 4, show thatF{f(t) cos(ωct)} =12[F (ω + ωc) + F (ω − ωc)]F{f(t) sin(ωct)} =12[F (ω − ωc) − F (ω + ωc)] ,where ωcis a constant real-valued frequency. In m ost situations f(t) is band limited far below the carrierfrequency ωc, a nd is often called the baseband signal2. (10 points) The system in Figure 1, wherey(t) = x(t) + cos(ωot)andw(t) = y2(t),has been proposed for amplitude modulation.• (5 points) The spectrum of the input x(t) is shown in Figure 2, where ω1= ωo/100. Sketch and l abel thespectrum W (ω) of the signal w(t).• (5 points) It is desired to transmit the input signal x(t) using double-sideband, suppressed carrier amplitudemodula tion (DSB/SC-AM). The band-pass
View Full Document
Unlocking...