Review of Linear Systems and Signals MaterialFall 2014 EE 445S Real-Time Digital Signal Processing Laboratory Prof. EvansHomework #0Review of Linear Systems and Signals MaterialAssigned on Friday, August 29, 2014Due on Friday, September 5, 2014, by 11:00am sharp in classLate homework is subject to a penalty of two points per minute late.Reading: Johnson, Sethares and Klein, chapters 1-2 and Appendices A and FThis assignment is intended to review key concepts from Linear Systems and Signals.Moreover, all of the problems on homework #0 directly relate to lab #2. This homework willbe graded and be counted towards the overall homework grade for the course. Here are key sections from Lathi’s Linear Systems and Signals book (2nd ed) and Oppenheim &Willsky’s Signals and Systems book (2nd ed) with respect to material in EE 445S:O&W Lathi Topic1.6 1.7 System properties1.3 – 1.4 1.4 Basic continuous-time signals3.2 ## 2.4-4 Fundamental theorem for continuous-time linear systems **1.3 – 1.4 3.3 Basic discrete-time signals3.2 ## 3.8-3 Fundamental theorem for discrete-time linear systems **9.7.2 2.6 Stability of continuous-time filters10.7.2 3.10 Stability of discrete-time filters10.1 – 10.3 5.1 Z transforms10.5 5.2 Properties of the z-transform10.7.3 – 10.7.4 5.3 Transfer functions10.8 5.4 Realizations of transfer functions4.3 – 4.4 7.3 Fourier transform properties7.1 8.1 Sampling theorem** Please see Appendix F and slide 5-13 in the course reader for the fundamental theorem.## O&W covers a slightly different version of the fundamental theorem in which a complex exponential is the input to a linear time-invariant system. Lathi also has that version as well.Other signals and systems textbooks should contain equivalent material.The MATLAB code in Johnson, Sethares & Klein also runs in LabVIEW Mathscript and GNUOctave (see footnote on JSK page viii about Octave). Feel free to use these environments.Please submit any MATLAB code that you have written with the homework solution. In thecourse reader, Appendix D gives a brief introduction to MATLAB. For quick review ofcommands in MATLAB for generating and plotting signals, please seehttp://www.ece.utexas.edu/~bevans/courses/ee313/lectures/02_Signals/lecture2.pptAs stated on the course descriptor, “Discussion of homework questions is encouraged.Please be sure to submit your own independent homework solution.”Office hours for the teaching assistants and Prof. Evans; bold indicates a 30-minute timeslot.Time Slot Monday Tuesday Wednesday Thursday Friday9:30 am Evans(UTC 1.130)Evans(UTC 1.130)10:00 am Evans(UTC 1.130)Evans(UTC 1.130)10:30 am11:00 am Labor DayHolidayEvans(UTC 1.130)Evans(UTC 1.130)12:00 pm Labor DayHolidayEvans(UTC 1.130)Evans(cafe)12:30 pm Evans(cafe)1:00 pm Evans(cafe)2:00 pm3:00 pm3:30 pm Debarati(ACA 111)4:00 pm Debarati(ACA 111)4:30 pm Debarati(ACA 111)5:00 pm Rao(ACA 111)Debarati(ACA 111)5:30 pm Rao(ACA 111)Debarati(ACA 111)6:00 pm Rao(ACA 111)Debarati(ACA 111)6:30 pm Rao(ACA 111)7:00 pm Rao(ACA 111)1. Continuous-Time Sinusoidal Generation. 27 points.In practice, we cannot generate a two-sided sinusoid sin(2  fc t), nor can we wait until the endof time to observe a one-sided sinusoid sin(2  fc t) u(t).In the lab, we can turn on a signal generator for a short time and observe the output in the timedomain on an oscilloscope or in the frequency domain using a spectrum analyzer. Consider a finite-duration cosine that is on from 0 sec to 1 sec given by the equationc(t) = sin(2  fc t) rect(t – ½)where fc is the carrier frequency (in Hz).(a) Using MATLAB, LabVIEW Mathscript or GNU Octave, plot c(t) for -0.5 < t < 1.5 for fc = 10 Hz. Turn in your code and plot. You may find the rectpuls command useful. 6 points.Give a formula for the Fourier transform of c(t) for a general value of fc. 6 points.(b) Sketch by hand the magnitude of the Fourier transform of c(t) for a general value of fc. Using MATLAB, LabVIEW Mathscript or GNU Octave, plot the magnitude of the Fourier transform of c(t) for fc = 5 Hz. Turn in your code and plot. 9 points.(c) Describe the differences between the magnitude of the Fourier transforms of c(t) and atwo-sided cosine of the same frequency. What is the bandwidth of each signal? 6points.Please read homework hints at http://users.ece.utexas.edu/~bevans/courses/realtime/homework2. Downconversion. 19 points.A signal x(t) is input to a mixer to produce the output y(t) wherey(t) = x(t) cos(0 t)where 0 = 2  f0 and f0 = 5 kHz. A block diagram of the mixer is shown below on the left.The Fourier transform of x(t) is shown below on the right.(a) Using Fourier transform properties, derive an expression for Y(f) in terms of X(f). 6 points.(b) Sketch Y(f) vs. f. Label all important points on the horizontal and vertical axes. 6points.(c) What operation would you apply to the signal y(t) in part (b) to obtain a basebandsignal? The process of extracting the baseband signal from a bandpass signal is knownas downconversion. 7 points.Please read homework hints at http://users.ece.utexas.edu/~bevans/courses/realtime/homework3. Sampling in Continuous Time. 24 points.Sampling the amplitude of an analog, continuous-time signal f(t) every Ts seconds can bemodeled in continuous time asy(t) = f(t) p(t)where p(t) is the impulse train defined by nsnTttp)(Ts is known as the sampling duration. The Fourier series expansion of the impulse train iswhere s = 2  / Ts is the sampling rate in units of radians per second.(a) Plot the impulse train nsnTttp)(. 6 points.(b) Note that in part (a), p(t) is periodic. What is the period? 6 points.(c) Using the Fourier series representation of p(t) given above, please give a formula forP(), which is the Fourier transform of p(t). Express your answer for P() as animpulse train in the Fourier domain. 6 points.(d) What is the spacing of adjacent impulses in the impulse train in P() with respect to frequency  in rad/s? 6 points.Please read homework hints at http://users.ece.utexas.edu/~bevans/courses/realtime/homework4. Discrete-Time Sinusoidal Generation. 30 points.Consider a causal discrete-time linear time-invariant system with input x[n] and output y[n]being governed by the following difference equation:y[n] = (2 cos w0) y[n-1] - y[n-2] + x[n] - (cos w0) x[n-1]The