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CMU 18396 Signals and Systems - Problem

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PROBLEM SET 8PROBLEM SET 8Issued: 3/21/09Due: 3/25/09Reminder: The second quiz will be in class on Wednesday, April 8. It will be closed book, but you may bring in two sheets of notes.Reading: This problem set deals with linear modulation, frequency-division multiplexing, and FIR filter design using window functions. The topics on modulation and multiplexing correspond to OWN Secs. 8.0 through 8.5. FIR filtering using window design is covered by the Section 7.2 in the text by Oppenheim, Schafer, and Buck (OSB, which will be the text for 18-491 in the fall). This section is available on the course Website, along with an introduction written for this class.Next week we will begin a discussion of LaPlace transforms, and how they are used to describe signals and systems. This material will follow Chapter 9 in the text.This problem set has been shortened somewhat because it is appearing late.Problem 8.1: Problem 8.25 in OWN.Problem 8.2: Problem 8.36 in OWN. This is the basic operation of the superheterodyne receiver is the basis for all commercial broadcast radios. Despite the elaborate setup, this is nothing more than an applica-tion of Fourier transform properties and is not a difficult problem. Problem 8.3: As we discussed in class, an amplitude modulated waveform may be written in the form ofWe will compare this waveform with the very similar signalThe modulated waveform is sometimes referred to as narrow-band frequency modulation (NBFM) or quasi frequency modulation (QFM). In any case, it is clear that the only difference between between the functions and is that in the case of AM, the modulated signal (i.e. the term with ) is in phase with the injected carrier (i.e. the other term) and in the case of NBFM/QFM the two S&S S&S S&S S&S S&S S&S S&S S&S S&S S&S S&S S&S S&S S&S S&S S&S S&S S&S S&S S&S S&S S&S S&S S&S S&SSignals and Systems (18-396)Spring Semester, 2009Department of Electrical and Computer EngineeringxAMt() Ac1 βmt()+()ωct()cos Acωct()cos Acβmt() ωct()cos+==xQFMt() Acωct()cos Acβmt() ωct()sin+=xQFMt()xAMt() xQFMt()mt()18-396 Problem Set 8 -2- Spring 2009terms are 90 degrees out of phase.In order to understand how these signals work (and how to demodulate them), it is convenient to use the phasor representation that you first were exposed to in 18-220 (and perhaps 18-202):The function is referred to as the complex amplitude or phasor of and in general varies both with respect to time and with respect to the frequency components of . The magnitude of is the instantaneous amplitude of at any particular moment in time, and the phase of is the instantaneous phase of .(a) Show that the complex amplitude of can be represented by (b) A representation of the complex amplitude for the modulated signal is given in the fig-ure below. Obtain a similar diagram for the complex amplitude of the modulated signal .(c) Suppose that the message signal is where and . Obtain approximate expressions for the instantaneous magnitude and phase of for the two modulated waveforms and . Ignore and constant (or DC) components of your results as they convey no informa-tion and can be easily discarded. (d) Suggest a means by which the message waveform can be recovered (or demodulated) from the two modulated waveforms and .Problem 8.4:You are asked to design a causal and linear-phase bandpass FIR filter that has the following characteristics using the window method:• Passband frequencies of • Stopband frequencies of and • Maximum filter gain in the stopbands is dB relative to signal amplitude in the passbandxAMt() Ac1 βmt()+()ωct()cos Re Ac1 βmt()+()ejωctRe Γ t ω,()ejωct== =Γ t ω,() xAMt()mt()Γ t ω,() xAMt()Γ t ω,() xAMt()xQFMt()Γ t ω,()Ac1 jβmt()–()=Γ t ω,() xAMt()AcAcm(t)xQFMt()mt() ωmt()cos= ωmωc« β 1«Γ t ω,()xAMt() xQFMt()mt()xAMt() xQFMt()0.3πω0.6π≤≤ω 0.25π≤ 0.65πωπ≤≤60–18-396 Problem Set 8 -3- Spring 2009As we discussed in class, the filter is obtained by multiplying the unit sample response of an ideal bandpass filter (which is infinite in duration) by a finite-duration window function :(a) What is the type of window that would be needed to fulfill the specifications above? You must consider both the stopband attenuation and the width of the transition bands to obtain this answer. The window type should be one of the five “classic” windows discussed in class and specified in OSB Eq. 7.47. Be sure to identify both the window type and the length that is needed to meet the specifications. (b) What is the unit sample response of the ideal filter that is the basis (some would say the “proto-type”) for your filter design? Consider the fact that the cutoff frequencies should lie in the center of the transition bands and that the filter must be linear phase and causal. (c) Write the expression for the complete but nonideal filter that satisfies the specifications. Verify that the coefficients of the filter satisfy the symmetry constraint that ensures that the filter is causal.(d) Verify that your filter satisfies the specifications by checking it with the MATLAB function freqz. Specifically, the command freqz(h,1) should display the magnitude and phase of your filter. See the help file for freqz for more information about this useful function.hn[]hdn[] wn[]hn[]


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