Fall 2014 EE 445S Real-Time Digital Signal Processing Laboratory Prof. Evans Homework #2 Filter Analysis, Simulation, and Design Assigned on Friday, September 26, 2014 Due on Friday, October 3, 2014, 11:00am sharp in lecture Homework submitted after 11:00am will be subject to a penalty of 2 points per minute late. Reading: Johnson, Sethares and Klein, chapters 1, 2, 3 and 7, and Appendices A and F This assignment is intended to continue our review of key concepts from Linear Systems and Signals, and introduce the simulation and design of discrete-time linear time-invariant filters. 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 Topic 1.6 1.7 System properties 1.3 – 1.4 1.4 Basic continuous-time signals 3.2 ## 2.4-4 Fundamental theorem for continuous-time linear systems ** 1.3 – 1.4 3.3 Basic discrete-time signals 3.2 ## 3.8-3 Fundamental theorem for discrete-time linear systems ** 9.7.2 2.6 Stability of continuous-time filters 10.7.2 3.10 Stability of discrete-time filters 10.1 – 10.3 5.1 Z transforms 10.5 5.2 Properties of the z-transform 10.7.3 – 10.7.4 5.3 Transfer functions 10.8 5.4 Realizations of transfer functions 4.3 – 4.4 7.3 Fourier transform properties 7.1 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. Please submit 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 and Klein book also runs in LabVIEW Mathscript and GNU Octave. As 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 Friday 9:30 am Evans (UTC 1.130) Evans (UTC 1.130) 10:00 am Kundu (UTC 1.130) Evans (UTC 1.130) 10:30 am 11:00 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 pm 3:00 pm 3:30 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) 2.1. Frequency Responses. 24 points. For each LTI system in problem 1.1 on homework assignment #1, a) plot the pole-zero diagram for the transfer function. 3 points. b) is the filter bounded-input bounded-output (BIBO) stable? why or why not? 3 points. c) give a formula for the frequency response. 9 points. d) plot the magnitude response. 6 points. e) if the system is BIBO stable, pick the best one of the following choices to describe the frequency selectivity of the filter: lowpass, highpass, bandpass, or bandstop. 3 points.You may use the solution for problem 1.1 in your solution for this problem. If you do, please cite the solution set for homework #1 as needed. Please read the homework hints that are available on the homework Web page. 2.2. Finite Impulse Response Filter Design for Audio Signals. 30 points. In Exercise 7.10 on page 139 of Johnson, Sethares & Klein, the program specgong.m was used to analyze the sound of an Indonesian gong. The Matlab program can be simplified to the following: [gongsignal, fs] = wavread('gong.wav'); % Read the wave file and set sampling rate fs plotspec(gongsignal, 1/fs); % Plot time and frequency responses Zoom into the frequency response to see the peaks at about 520, 630 and 660 Hz. These are the prominent partials (or narrowband components) in the original gong signal. (a) Apply a 44-tap normalized averaging filter to the gong signal, plot the resulting signal in the time and frequency domains, and play the resulting signal and the original gong signal: fir44 = ones(1, 44) / 44; gongsignal44 = filter(fir44, 1, gongsignal); plotspec(gongsignal44, 1/fs); sound(gongsignal44, fs); sound(gongsignal, fs); Which of the original prominent partials are still prominent in the spectrum of the resulting signal? Which of the original prominent partials can you hear in the resulting signal? If one or more of the original prominent partials is not still prominent/audible, please explain why. (b) Apply a 70-tap normalized averaging filter to the gong signal, plot the resulting signal in the time and frequency domains, and play the resulting signal and the original gong signal: fir70 = ones(1, 70) / 70; gongsignal70 = filter(fir70, 1, gongsignal); plotspec(gongsignal70, 1/fs); sound(gongsignal70, fs); sound(gongsignal, fs); Which of the original prominent partials are still prominent in the spectrum of the resulting signal? Which of the original prominent partials can you hear in the resulting signal? If one or more of the original prominent partials is not still prominent/audible, please explain why. (c) Apply a 81-tap normalized averaging filter to the gong signal, play the resulting signal, and plot the resulting signal in the time and frequency domains: fir81 = ones(1, 81) / 81; gongsignal81 = filter(fir81, 1, gongsignal); plotspec(gongsignal81, 1/fs); sound(gongsignal81, fs); sound(gongsignal, fs); Which of the original prominent partials are still prominent in the spectrum of the resulting signal? Which of the original prominent partials can you hear in the resulting signal? If one or more of the original prominent partials is not still prominent/audible, please explain why.(d) Design an FIR filter using firpm that reduces the lowest partial from the original gong wave file without affecting the highest two partials. Apply the FIR filter to the gong signal, plot the resulting signal in the time and frequency domains, and play resulting and original gong signals. Which of the original prominent partials are still