Fall 2014 EE 445S Real-Time Digital Signal Processing Laboratory Prof. Evans Homework #3 Child of Filter Analysis, Simulation, and Design Assigned on Friday, October 3, 2014 Due on Friday, October 10, 2014, by 11:00 am sharp in class Homework submitted after 11:00 am will be subject to a penalty of 2 points per minute late. Reading: Johnson, Sethares & Klein, chapters 4-6, and Appendix B This assignment is intended to review frequency-domain analysis of LTI systems and provide an introduction to the simulation and design of infinite impulse response 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) Rao (UTC 1.130) 10:00 am Evans (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) The Matlab scripts that accompany the Software Receiver Design book also run in GNU Octave, which is free software. Please see the note on page vii of the SRD book for more information. 3.1. Using Finite Impulse Response Filtering to Improve Signal Quality. 27 points. Johnson, Sethares & Klein, exercise 4.21, on page 78. 9 points for each of the two parts. In addition, please complete the following part: c. for the narrowband interference occurring inside the transmission band that you simulated in part b, design a notch filter using a second-order infinite impulse response filter and write the Matlab code toapply the notch filter to the output of the bandpass filter used to reduce out-of-band interference and compute the change in SNR due to your notch filter. 9 points. 3.2. Amplitude Modulation. 27 points. Johnson, Sethares & Klein, exercise 5.10 on page 88. For the bandpass filter in Figure 5.8, design and implement an infinite impulse response filter. 3.3. Infinite Impulse Response Filter Design. 46 points. This problem asks you to design a discrete-time digital IIR filter for an electrocardiogram (ECG) signal. This problem is a sequel to homework problem 2.3. An ECG device “monitors the cardiac status of a patient by recording the heart’s electrical potential versus time. Such devices play a very important role to save life of patients who survive heart attack or suffer from serious heart diseases. The time to respond to a heart attack is very critical for these patients. An early detection of conditions that lead to the onset of cardiac arrest allows doctors to provide proper treatment on time and prevents death or disability from cardiac arrest." [1] "There exist three types of noise that contaminate the ECG signal: the baseline wander noise (BW), electromyographic interference (EMG), and the power line interference. The BW is induced by electrodes’ changes due to perspiration, movement and respiration, and is typically below 0.5 Hz. The power line interference either 50 Hz or 60 Hz and its harmonics are a significant source of noise." [1]. Our goal in designing the filter is to attenuate baseline wander noise and powerline interference. It would take more sophisticated processing to track and cancel the EMG noise because EMG noise appears in the same frequencies as the ECG signal. Here are the bandpass filter specifications for your design: - For frequencies 0 Hz to 1 Hz, the stopband attenuation should be at least 40 dB. - For frequencies 6 Hz to 40 Hz, the passband ripple should be no greater than 1 dB. - For frequencies above 45 Hz, the stopband attenuation should be at least 40 dB. These specifications would be compatible with the monitor mode in modern ECG monitors. Please use a sampling rate of 250 Hz. From a quick survey of commercial ECG systems, I found sampling rates that vary from 100 Hz to 1000 Hz. The PTB Diagnostic ECG Database uses a sampling rate of 1000 Hz and the QT ECG Database uses a sampling rate of 250 Hz. (a) Design IIR filters using the Butterworth, Chebyshev type I, Chebyshev type II, and Elliptic (Equiripple) design methods. For each design method, find the filter of smallest order to meet the specifications. The filter order is the number of poles. Turn in plots of the magnitude and phase responses for each IIR filter you have designed to meet the specifications. Describe the passband and stopband response for each filter design as either