Sinusoids, Transforms and Transfer FunctionsFall 2014 EE 445S Real-Time Digital Signal Processing Laboratory Prof. EvansHomework #1Sinusoids, Transforms and Transfer FunctionsAssigned on Saturday, September 6, 2014Due on Friday, September 19, 2014, by 11:00am sharp in classLate homework will be subject to a penalty of 2 points per minute late.Reading: Johnson, Sethares & Klein, Software Receiver Design, chap. 1-3, Appendices A & FThis assignment is intended to continue our review of key concepts from Linear Systems and Signals.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.You may use any computer program to help you solve these problems, check answers, etc. Pleasesubmit any MATLAB code that you have written for the homework solution. In the course reader,Appendix D gives a brief introduction to MATLAB. The MATLAB code in the Johnson, Sethares andKlein book also runs in LabVIEW Mathscript and GNU Octave. Another option is Python via itsscientific and numerical extensions.As stated on the course descriptor, “Discussion of homework questions is encouraged. Please besure to submit your own independent homework solution.”Office hours for the teaching assistants Ms. Kundu and Mr. Rao, 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 Evans(UTC 1.130)Evans(UTC 1.130)Evans(UTC 1.130)12:00 pm Evans(UTC 1.130)Evans(UTC 1.130)Evans(cafe)12:30 pm Evans(cafe)1:00 pm Evans(cafe)2:00 pm3:00 pm3:30 pm Kundu(ACA 111)4:00 pm Kundu(ACA 111)4:30 pm Kundu(ACA 111)5:00 pm Rao(ACA 111)Kundu(ACA 111)5:30 pm Rao(ACA 111)Kundu(ACA 111)6:00 pm Rao(ACA 111)Kundu(ACA 111)6:30 pm Rao(ACA 111)7:00 pm Rao(ACA 111)The points below add up to 99. Everyone who submits homework #1 will receive the extra point.1. Transfer Functions. 48 points.With x[n] denoting the input signal and y[n] denoting the output signal, give the difference equationrelating the input signal to the output signal in the discrete-time domain, give the initial conditions andtheir values, and find the transfer function in the z-domain and the associated region of convergencefor the z-transform function, for the following linear time-invariant discrete-time systems:(a) Causal averaging filter with two coefficients. See lecture slide 3-10. 12 points.(b) Causal discrete-time approximation to first-order differentiator. See lecture slide 3-20.12 points.(c) Causal discrete-time approximation to first-order integrator. See online hints. 12 points.(d) Causal bandpass filter with center frequency 0 given by the input-output relationshipy[n] = (2 cos w0) r y[n-1] – r2 y[n-2] + x[n] - (cos w0) x[n-1]where 0 < r < 1. Here, r is the radius of the two pole locations. 12 points.The following sections might be helpful: Appendix F in Johnson, Sethares & Klein’s Software Defined Radio book Sections 5.1 and 5.2 in Lathi's book Linear Systems and Signals, or Sections 11.2 and 11.3 in Roberts’ Signals and Systems bookRecall that transfer functions of the form H(z) = Y(z) / X(z) only apply for linear time-invariantsystems. A linear time-invariant system is uniquely defined by its impulse response. The generalizedtransform of the impulse response is a way to compute the transfer function.Comment: The linear time-invariant (LTI) system in (d) whose input-output relationship isy[n] = (2 cos w0) r y[n-1] – r2 y[n-2] + x[n] – (cos w0) x[n-1] has several applications. When r = 1, the impulse response of the LTI system iscos(w0 n) u[n]Hence, the LTI system can be used as a sinusoidal generator. For r = 1, the system is not bounded-input bounded-output (BIBO) stable. If cos(w0 n) u[n] were the input signal, resonance would lead tounbounded amplitude on the output. (Resonance does not always lead to an unbounded output.)The unbounded response to input cos(w0 n) u[n] can be used to our advantage. If the filter outputwere to grow very large in absolute value, then we know that the input signal would have a componentequal or at least approximately equal to cos(w0 n) u[n]. The BIBO instability would allow us to detecta sinusoid. Applications of detecting sinusoidal tones in a signal include identification of notes inmusic, tracking of frequency hopping (e.g. in Bluetooth) and touchtone telephone signal decoding. Inpractice, we use r  1 (e.g. r = 0.95) to have good frequency selectivity (i.e. a narrow passband).Please read homework hints at http://users.ece.utexas.edu/~bevans/courses/realtime/homework2. Spectral Analysis. 27 points. Johnson, Sethares & Klein, Exercise 3.3 on page 43, but use the following signals (9 points each):(a) A rectangular pulse s(t) = rect(t/8) which has an amplitude of 1 from -4 (inclusive) to 4 (non-inclusive). Plot the signal in the time domain for -8 < t < 8. Give the formula for the Fourier transform of s(t). Estimate fmax. Plot the spectrum. 9 points.(b) A truncated sinc pulse s(t) = sinc(t) rect(t/8) where sinc(x) = sin(x) / (x). Plot the signal in the time domain for -8 < t < 8. Estimate fmax. Plot the spectrum. You do not have to give the Fourier transform of s(t). 9 points.(c) A decaying exponential s(t) = exp(-t) u(t). Plot the signal in the time domain for -8 < t < 8. Estimate fmax. Plot the spectrum. You don’t have to give the formula for S(f). 9