# UT EE 445S - Homework 2 Filter Analysis, Simulation, and Design (6 pages)

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## Homework 2 Filter Analysis, Simulation, and Design

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- Pages:
- 6
- School:
- University of Texas at Austin
- Course:
- Ee 445s - Real-Time Digital Signal Processing Laboratory

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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 9 30 am Monday Tuesday 10 00 am Wednesday Evans UTC 1 130 Kundu UTC 1 130 Thursday Friday Evans UTC 1 130 Evans UTC 1 130 10 30 am 11 00 am 12 00 pm Evans UTC 1 130 Evans UTC 1 130 Evans UTC 1 130 Evans UTC 1 130 Evans UTC 1 130 Evans cafe Evans cafe Evans cafe 12 30 pm 1 00 pm 2 00 pm 3 00 pm 3 30 pm 4 00 pm 4 30 pm 5 00 pm 5 30 pm 6 00 pm Rao ACA 111 Rao ACA 111 Rao ACA 111 6 30 pm 7 00 pm Kundu ACA 111 Kundu ACA 111 Kundu ACA 111 Kundu ACA 111 Kundu ACA 111 Kundu ACA 111 Rao ACA 111 Rao ACA 111 2 1 Frequency Responses 24 points For each LTI system in problem 1 1 on homework assignment 1 a b c d e plot the pole zero diagram for the transfer function 3 points is the filter bounded input bounded output BIBO stable why or why not 3 points give a formula for the frequency response 9 points plot the magnitude response 6 points 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 plotspec gongsignal 1 fs Read the wave file and set sampling rate 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 prominent in the spectrum of the resulting signal Which of the original prominent partials can you hear in the resulting signal e Take the filtered gong signal from part a and perform downsampling by 2 Downsampling by 2 keeps every other sample and discards the rest Here s Matlab code for downsampling vector vec by 2 vecDownsampledBy2 vec 1 2 length vec Play the downsampled filtered gong signal at the same playback rate as the

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