EE 353 Problem Set 5 Cover Sheet Spring 2020Last Name (Print
EE 353 Problem Set 5 Cover Sheet Spring 2020Last Name (Print):First Name (Print):User ID number (eg. xyz1234):Section:Submission deadlines:• Turn in the written solutions by 4:00 pm on Wednesday April 1 in the homework slot outside 121 EE East.Problem Weight Score38 2039 2040 2041 2542 30Total 115The 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 collaborated with on this assignment.Reading assignment:• Lathi Chapter 4: sections 4.5 through 4.7• Lathi Chapter 5: section 5.1Problem 38: (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. Given thatf(t) ⇔ F(ω)g(t) ⇔ G(ω)and to and ωo are real-valued constants, derive the following Fourier Transform properties using the integral definitionof the Fourier Transform and the Inverse Fourier Transform: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 Convolution Propertyf(t) ∗ g(t) ⇔ F(ω)G(ω)4. (5 points) Modulation Property (Frequency Convolution Property)f(t)g(t) ⇔12πF(ω) ∗ G(ω)5. (4 points) Time Differentiation Propertydnfdtn ⇔ (ω)nF(ω)Problem 39: (20 points)A key goal of EE 353 is to insure that you have a thorough understanding of the relationship between the ODE,impulse response, and frequency response function representation of a LTI system. Consider a linear time-invariantcausal (LTIC) system with input f(t), impulse 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), and f(t), respectively. The Fouriertransform of the impulse 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) = ωn e−ζωntcos(ωdt)u(t),where ωn > 0, 0 ≤ ζ < 1, andwd = ωnp1 − ζ2.By direct integration, determine the frequency response function of the system by computing the Fouriertransform of the impulse response function. 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 differentiation property and the results from parts 1 and 2, find the ODE representation 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 40: (20 points)This problem shows how to calculate the Fourier transform of periodic signals.1. (1 point) Use the integral definition of the Fourier transform to find the Fourier transform of δ(t).2. (2 point) In problem set 2 problem 20 you showed that δ(at) = δ(t)/|a|. Using this result and the dualityproperty (also know known as the Symmetry 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 ωo is a real-valued constant.5. (6 points) Suppose that f(t) is a periodic signal with period To and has the Fourier series representationf(t) = X∞n=−∞Dnenωot.Use the result from part 3 to show thatF(ω) = 2πX∞n=−∞Dnδω −2πTon.6. (5 points) Find the Fourier transform of the full wave rectified signal, f(t), from Problem 35, part 2.Problem 41: (25 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 engineering, including communication and control systems. Considerthe frequency response functions for a set of four filters specified by their frequency response functionsH1(ω) = A rect ω2Be−ωtoH2(ω) = Ah1 − rect ω2Bi e−ωtoH3(ω) = Arect ω − ωo2B+ rect ω + ωo2B e−ωtoH4(ω) = Aω/B + 1,where A, B, and ωo are positive, real constants.1. (12 points) Sketch the magnitude ( |H(ω)| ) and phase (6 H(ω)) for each of the four filters.2. (4 points) Identify each of the filters as either a high-pass filter, low-pass filter, or band-pass filter.3. (6 points) Find the impulse response functions for the frequency response functions H1(ω) and H4(ω). Youmay use the Fourier transform properties and elementary Fourier transform pairs derived in either lecture orin the problems sets.4. (3 points) If the impulse response function of the filter is a causal signal, the system it represents is also causaland the filter is said to be realizable. Which of the filters H1(ω) and H4(ω), if either, are realizable?Problem 42: (30 points)This problem considers the application of Fourier transform methods to sampled data and communication systems.1. (16 points) This problem shows how Fourier transforms can be used to analyze the operations of a sampleddata system. Consider a signal f(t) that has the spectrum F(ω) shown in Figure 1. The signal f(t) is said tobe band limited because the spectrum is zero except for a finite band of frequencies −2πβ ≤ ω ≤ 2πβ. Thesignal is applied to an ideal sampler shown in Figure 2; in practice we use the less than ideal analog to digitalconverter in place of the ideal sampler.F(ω)ω−2πβ 2πβ1Figure 1: Spectrum of the signal f(t).δT(t)f(t) f(t)Figure 2: Mathematical representation of an ideal sampling system.The signal δT (t) is a train of impulse functionsδT (t) = X∞n=−∞δ(t − nTs)where Ts defines the sample period.(a) (5 points) The sampler captures the value of the input signal f(t) at the sampling instants nTs. Theresulting output of the sampler is represented by¯f(t) = X∞n=−∞f(n)δ(t − nTs)wheref(n) = f(t)|t=nTsis the value of the input captured at each sample time t = nTs. Find the Fourier transform of the sampleroutput ¯f(t) and show that it can be expressed asF¯(ω) = 1TsX∞n=−∞F(ω − nωs)where ωs = 2πTs.(b) (6 points) Suppose that the sample period is given by Ts =14β. Sketch the spectrum F¯(ω). Repeat forTs =23β.(c) (5 points) Figure 3 shows the sampled signal followed by an ideal lowpass filter. Can this filter be used toexactly recover f(t) from either version of ¯f(t), sketched in the previous part? If so, specify the value ofthe filter cutoff frequency ωc and passband amplitude A in terms of the parameters Ts and β.δT(t)f(t) f(t)Ideal Lowpass Filterωωc-ωcAg(t)Figure 3: Reconstruction of f(t) using an ideal lowpass filter.2. (14 points) This problem shows an example of using the Fourier transform to analyze communication systems.The system in Figure 4, wherex(t) = f(t) + sin(ωot)andy(t) = x2(t),has been proposed for amplitude modulation.sin(ωot)f(t)Σ y = xx(t) y(t) z(t) 2 H(ω)Figure 4: System proposed for amplitude modulation.(a) (7 points) The spectrum of the input f(t) is shown in Figure 1, where 2πβ = ωo/100. Sketch and labelthe spectrum Y (ω) of the signal y(t). Hint: You will need to use the frequency convolution property of theFourier Transform to compute one of the terms of the spectrum you are to sketch.(b) (7 points) It is desired to transmit the input signal f(t) using double-sideband, suppressed carrier amplitude modulation (DSB/SC-AM). The system represented by the frequency response H(ω) is a band-passfilter with the spectrum shown in Figure 5. Determine the range of ω1 and the range of ω2 such that z(t)is the desired DSB/SC-AM transmission signal.H(ω)1-ω2-ω1ω1 ω2Figure 5: Ideal bandpass filter, H(ω).
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