ee 353 problem set 5

EE 353 Problem Set 5 Cover Sheet Spring 2020 Last 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 Score 38 20 39 20 40 20 41 25 42 30 Total 115 The 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 5.1 Problem 38: (20 points) In the first problem you will prove a series of Fourier transform properties that will be used extensively in the remainder of this problem set. Given that f(t) ⇔ F(ω) g(t) ⇔ G(ω) and to and ωo are real-valued constants, derive the following Fourier Transform properties using the integral definition of the Fourier Transform and the Inverse Fourier Transform: 1. (3 points) Time Shift Property f(t − to) ⇔ F(ω)e −ωt0 2. (3 points) Frequency Shift Property f(t)e ωot ⇔ F(ω − ωo) 3. (5 points) Time Convolution Property f(t) ∗ g(t) ⇔ F(ω)G(ω) 4. (5 points) Modulation Property (Frequency Convolution Property) f(t)g(t) ⇔ 1 2π F(ω) ∗ G(ω) 5. (4 points) Time Differentiation Property d n f dtn ⇔ (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-invariant causal (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-state response y(t) of the system to an arbitrary input f(t) is Y (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 Fourier transform of the impulse response function h(t) is identical to the frequency response function H(ω) of the system. 2. (10 points) As a specific example, consider a LTIC system with the impulse response function h(t) = ωn e −ζωnt cos(ωdt)u(t), where ωn > 0, 0 ≤ ζ < 1, and wd = ωn p 1 − ζ 2. By direct integration, determine the frequency response function of the system by computing the Fourier transform of the impulse response function. Express you answer in the standard form H(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 form d ny dtn + an−1 d n−1y dtn−1 + · · · + a1 dy dt + aoy(t) = bm d mf dtm + bm−1 d m−1f dtm−1 + · · · + b1 df dt + 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 duality property (also know known as the Symmetry problem, see section 4.3-2 in the text), determine the Fourier transform of f(t) = 1. 3. (2 points) Using the result from part 2 and the frequency shift property from Problem 38, determine the Fourier transform of e ωot . 4. (4 points) Find the Fourier transform of the periodic signals sin(ωot) and cos(ωot) given the result in part 3 and 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 representation f(t) = X∞ n=−∞ Dne nωot . Use the result from part 3 to show that F(ω) = 2π X∞ n=−∞ Dnδ ω − 2π To n . 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-pass filter 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. Consider the frequency response functions for a set of four filters specified by their frequency response functions H1(ω) = A rect ω 2B e −ωto H2(ω) = A h 1 − rect ω 2B i e −ωto H3(ω) = A rect ω − ωo 2B + rect ω + ωo 2B e −ωto H4(ω) = A ω/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). You may use the Fourier transform properties and elementary Fourier transform pairs derived in either lecture or in the problems sets. 4. (3 points) If the impulse response function of the filter is a causal signal, the system it represents is also causal and 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 sampled data system. Consider a signal f(t) that has the spectrum F(ω) shown in Figure 1. The signal f(t) is said to be band limited because the spectrum is zero except for a finite band of frequencies −2πβ ≤ ω ≤ 2πβ. The signal is applied to an ideal sampler shown in Figure 2; in practice we use the less than ideal analog to digital converter in place of the ideal sampler. F(ω) ω −2πβ 2πβ 1 Figure 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 functions δT (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. The resulting output of the sampler is represented by ¯f(t) = X∞ n=−∞ f(n)δ(t − nTs) where f(n) = f(t)|t=nTs is the value of the input captured at each sample time t = nTs. Find the Fourier transform of the sampler output ¯f(t) and show that it can be expressed as F¯(ω) = 1 Ts X∞ n=−∞ F(ω − nωs) where ωs = 2π Ts . (b) (6 points) Suppose that the sample period is given by Ts = 1 4β . Sketch the spectrum F¯(ω). Repeat for Ts = 2 3β . (c) (5 points) Figure 3 shows the sampled signal followed by an ideal lowpass filter. Can this filter be used to exactly recover f(t) from either version of ¯f(t), sketched in the previous part? If so, specify the value of the filter cutoff frequency ωc and passband amplitude A in terms of the parameters Ts and β. δT (t) f(t) f(t) Ideal Lowpass Filter ω ωc -ωc A g(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, where x(t) = f(t) + sin(ωot) and y(t) = x 2 (t), has been proposed for amplitude modulation. sin(ωo t) f(t) Σ y = x x(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 label the spectrum Y (ω) of the signal y(t). Hint: You will need to use the frequency convolution property of the Fourier 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-pass filter 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 -ω1 ω1 ω2 Figure 5: Ideal bandpass filter, H(ω).

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