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<HTML><HEAD><TITLE>EECS451 Home Page</TITLE> </HEAD><BODY bgcolor=#ffffff text=#000000><center><B><H1>EECS 451<br> Digital Signal Processing<br>Fall '96</H1></B></center><CENTER><!-- IMG SRC="line.gif" ALT="------------------------------"></CENTER><UL><LI><!WA0><A HREF="#Announce">Announcements</A></LI><LI><!WA1><A HREF="http://www.eecs.umich.edu/courses/eecs451/feedback.html">Send me Feedback!</A></LI><LI><!WA2><A HREF="#Assign">Assignments</A></LI><LI><!WA3><A HREF="#Group">Group Projects</A></LI><LI><!WA4><A HREF="#Code">MATLAB Code</A></LI><LI><!WA5><A HREF="#Brain">Brain Teasers</A></LI><LI><!WA6><A HREF="#Details">Course Details</A></LI></UL><CENTER><!WA7><IMG SRC="http://www.eecs.umich.edu/courses/eecs451/line.gif" ALT="------------------------------"></CENTER><A NAME="Announce"><H1><B>Announcements</B></H1></A><UL><P><LI><!WA8><IMG SRC="http://www.eecs.umich.edu/courses/eecs451/new.gif" BORDER=0>Remember that exam 3 will be a take-home exam given out on Dec. 6, to be returned Dec. 10, and the<!WA9><A HREF="#Group">group projects</A> will be due on Dec. 13.</LI></UL><CENTER><!WA10><IMG SRC="http://www.eecs.umich.edu/courses/eecs451/line.gif" ALT="------------------------------"></CENTER><A NAME="Assign"><H1><B>Assignments</B></H1><UL><P><LI><!WA11><IMG SRC="http://www.eecs.umich.edu/courses/eecs451/new.gif" BORDER=0><B>Homework #8:</B> Chapter 8 problems: 1, 9, 12, 15, 19, 23<br><B>Due in class Wednesday, November 27</B></LI><P><LI><B>Exam Review Problems<br>Do not hand in for grading!</B><P>Chapter 4 problems: 8a),c), 10a),b), 17a)-d), 28, 45a)-c), 54<br>Chapter 5 problems: 5, 7, 11, 21, 22, 23</LI><P><LI><B>Homework #7:</B><br>Chapter 5 problems: 16, 18, 28<br>Chapter 6 problems: 3, 8, 37</LI><P><LI><B>Homework #6:</B> Chapter 4 problems: 15, 22, 25, 35, 48, 49</LI><P><LI><B>Homework #5:</B> see last two pages of Monday's handout (Oct. 21)</LI><P><LI><B>Exam review problems</B><br>Chapter 2 problems: 28, 32, 37, 41, 44, 46 (Use Z-Transforms as necessary)<br>Chapter 3 problems: 14 a) c) e) g) i), 35, 41, 45, 47, 49</LI><P><LI>Read Chapters 4 and 5.</LI><P><LI><B>Homework #4:</B>Chapter 3 problems: 24, 25, 26, 34, 39, 44<P><B>NOTE: </B>The solution to problem 34 a) has a slight mistake. Thepole is at -0.81, not 0.81.</LI><P><LI><B>Homework #3:</B>Chapter 3 problems: 3, 9, 13, 16, 22, 23</LI><P><LI><B>Homework #2:</B>Chapter 2 problems: 25, 30, 33, 35, 36, 39</LI><P><LI>Read Chapter 3.</LI><P><LI><B>Homework #1:</B>Chapter 2 problems: 3, 5, 7, 8, 11, 15, 18, 21, 23, 24<P>Note that the solutions to homework #1 havesome errors in them. Specifically, Problem 2.7 parts (d), (h), and (l)are time-VARIANT systems, not time-invariant as the solutions indicate.</LI><P><LI>Read Chapter 2.</LI><P><LI>Skim through Chapter 1, mainly for your own interest.</LI></UL><CENTER><!WA12><IMG SRC="http://www.eecs.umich.edu/courses/eecs451/line.gif" ALT="------------------------------"></CENTER><A NAME="Group"><H1><B>Group Projects</B></H1><P>Remember, the group projects will count for 15% of your grade and will bedue on December 13th.<P>For those of you doing data analysis on a PC, I suggest you considerusing the <!WA13><A HREF="http://web.cs.mun.ca/~chris3/goldwave/goldwave.html">GoldWave Digital Audio Editor</A>. This is a shareware package thatcan be found on the CAEN Pentium machines in the Media Union. The programis located on the S: drive in the \BIN\GOLDWAVE directory.<P>GoldWave does not replace MATLAB for doingFourier analysis, filtering, LPC modelling, etc. but it is a goodtool for manipulation of audio at the sample level and performingsimple transformations such as volume adjustments. Most importantly,GoldWave is an excellent tool for performing conversions betweenaudio formats and extracting short segments of your waveforms for inputto MATLAB. GoldWave will, for example, convert WAV files to/from MATLABformat.<P>If you'd like some data to play with, and don't have any of your ownyet, here are 2 sample files that you can use. They are recordingsof Prof. Wakefield's voice pronouncing 5 vowels, as heard in class.Use your browser's "Save Link As..." feature to save these to a filein your account (Shift-Left-Click instead of just Left-Click if youare using Netscape).<P><B>NOTE: </B> these sounds are currently corrupted due to a bad filetransfer. I will have fixed versions of these sounds available ASAP.<P><UL><P><LI>The <!WA14><A HREF="http://www.eecs.umich.edu/courses/eecs451/gwvowel.raw">gwvowel.raw</A> file is in raw (i.e.,headerless) 16-bit, signed, monophonic format. This file was createdon a PC so it is in little-endian format (meaning the least significant8 bits of each 16-bit sample comes first in the file). <P><LI>If you want to doyour experiments on a big-endian machine (such as a Sun or HP), you should usethis file, <!WA15><A HREF="http://www.eecs.umich.edu/courses/eecs451/gwbsw.raw">gwbsw.raw</A>, which is the same databut it has already been byte-swapped for you. If you use the wrongfile, there'll be no doubt about it when you play it over speakers orheadphones.</UL><CENTER><!WA16><IMG SRC="http://www.eecs.umich.edu/courses/eecs451/line.gif" ALT="------------------------------"></CENTER><A NAME="Code"><H1><B>MATLAB Code</B></H1></A><P>Here is some MATLAB code that may be of useor of interest. You can get the code by clicking on each link,copying the code, and pasting it into an editor window. Easier is touse your browser's "Save Link As..." function to save the code to a file.Using Netscape on a Unix workstation, Right-Click on the link andselect "Save Link As..." or Shift-Left-Click to achieve the same thing.<P><B><FONT SIZE=+2>PLEASE DO NOT PRINT OUT THIS CODE</FONT></B>. Rememberhow uptight CAEN is about printing things out. Too many people printingoff copies of the same file will lead to all of these files beingremoved.<P><UL><LI><!WA17><IMG SRC="http://www.eecs.umich.edu/courses/eecs451/new.gif" BORDER=0><!WA18><A HREF="http://www.eecs.umich.edu/courses/eecs451/p528.m">Problem 5.28 from the text</A></LI><LI><!WA19><IMG SRC="http://www.eecs.umich.edu/courses/eecs451/new.gif" BORDER=0><!WA20><A HREF="http://www.eecs.umich.edu/courses/eecs451/filtplot.m">Plot magnitude/phase of filter response (filtplot.m)</A></LI><LI><!WA21><A HREF="http://www.eecs.umich.edu/courses/eecs451/stft.m">Short-time Fourier Transform computation (stft.m)</A></LI><LI><!WA22><A HREF="http://www.eecs.umich.edu/courses/eecs451/star.m">Short-time Autoregressive modelling (star.m)</A></LI><LI><!WA23><A HREF="http://www.eecs.umich.edu/courses/eecs451/dftlec17.m">MATLAB examples from lecture 17</A></LI><LI><!WA24><A HREF="http://www.eecs.umich.edu/courses/eecs451/orthstem.m">Plot a sequence and its FFT as stems</A></LI><LI><!WA25><A HREF="http://www.eecs.umich.edu/courses/eecs451/overlay.m">Plot a sequence to overlay an orthstem plot</A></LI><LI><!WA26><A HREF="http://www.eecs.umich.edu/courses/eecs451/hw5p1.m">Homework #5 Problem #1</A> code to plot bk vectors</LI><LI><!WA27><A HREF="http://www.eecs.umich.edu/courses/eecs451/hw5p4a.m">Homework #5 Problem #4a</A> code to plot Hadamard vectors</LI><LI><!WA28><A HREF="http://www.eecs.umich.edu/courses/eecs451/hw5p4d.m">Homework #5 Problem #4d</A> code to plot expansion coefficients of a sinusoid on the Hadamard basis</LI></UL><CENTER><!WA29><IMG SRC="http://www.eecs.umich.edu/courses/eecs451/line.gif" ALT="------------------------------"></CENTER><A NAME="Brain"><H1><B>Brain Teasers</B></A><!WA30><IMG SRC="http://www.eecs.umich.edu/courses/eecs451/small-brain.gif" ALIGN=Middle></H1>Here is a compendium of the "brain teaser du jour" questions that havebeen sent out on the mailing list. Someone suggested they might makegood review questions or study aids (or even Quals questions, nudge nudge,wink wink, say no more).<UL><P><LI><!WA31><IMG SRC="http://www.eecs.umich.edu/courses/eecs451/new.gif" BORDER=0> Consider the analog unit step signal u(t)=1, t>=0, u(t)=0, t<0. Is there any sampling frequency we can use to sample u(t) so that the resulting digital signal, u(n), exhibits no frequency-domain aliasing?<P> Realizing that the answer to the above question is "no", now consider the sampled signal u(n). What does this signal look like, for *any* sampling rate? How can this be explained?</LI><P><LI>Show that an FIR system may be written as an IIR system with aninfinite number of poles.</LI><P><LI>Show that an antisymmetric impulse response (i.e., h(n) = -h(-n)) hasa zero in its frequency response at omega=pi (and, of course, omega=-pi).</LI><P><LI>Consider the discrete-time signal:<P><CODE><PRE> x(n) = cos(2*pi*f*n) for n in the range [0,15]</PRE></CODE><P>and the frequency 'f' is 5/16. What is the 16-point DFT of this signal? Nowdefine the signal:<P><CODE><PRE> y(n) = x(n), n in the range [0,15] = 0, n in the range [16,31]</PRE></CODE><P>What is the 32-point DFT of this signal? It is possible to answer allof the above questions without a computer and without doing any math.<P>Having done so, repeat this exercise using MATLAB. Display the magnitudeof your DFT's on a logarithmic scale to see the difference between theoryand reality. Also, do it again using f=5.5/16. Think about what you'dexpect to see both for the 16-point DFT and the 32-point DFT.</LI><P><LI>Let {bk} be one set of basis vectors for N-point sequences. Let {vk}be another set of basis vectors for the same space. Find the "change of basis" transformation. That is, find a matrix T such that:<PRE> h y = T x</PRE><P>transforms the vector of expansion coefficients x (expanded on the {bk}vectors) to the vector of expansion coefficients y (expanded on the {vk}vectors). As its name implies, this transformation maps a data vectorfrom one basis to another. Both x and y represent the same data vector,but written in terms of different bases.<P>What's the change of basis transformation for mapping a Fourier expansiononto a Hadamard expansion?</LI><P><LI>Consider the continuous function:<CENTER><PRE>f(w) = exp(-jw(N-1)/2)*sin(wN/2)/sin(w/2)</PRE></CENTER><P>where 'w' is a real-valued continuous variable, N is a positive integer (N>1),'j' is the square root of -1, and exp() is the standard exponentialfunction (i.e., "e to the power of...").<P> Compute the N-point circular convolution of f(w) with itself: f(w) ** f(w)where '**' indicates circular convolution. NOTE: If you can do this problemIN YOUR HEAD without doing ANY MATH, then you are in great shape (mentallyspeaking, of course).</LI><P><LI>Consider two NxN circulant matrices A and B (see page 4 of Lecture 20for an example of a circulant matrix).<P><OL><LI>Show that AB (this is matrix multiplication) is also a circulantmatrix.<LI>Show that AB=BA. Is this true in general (i.e., for ANY matrix)?<LI>Why are the above properties important when considering whichmatrices can represent linear circularly-shift-invariant operators?<LI>Prove that circular convolution is commutative (HINT: you've alreadydone it).</OL></LI><P><LI><!WA32><A HREF="http://www.eecs.umich.edu/courses/eecs451/brain.html">Brain teasers from the first third of the course</A>have been placed on a separate page.</LI></UL><P>So where are the answers? Come to <!WA33><A HREF="#Office">office hours</A>!<P>Want more? Tell me and I'll try to think of some.<CENTER><!WA34><IMG SRC="http://www.eecs.umich.edu/courses/eecs451/line.gif" ALT="------------------------------"></CENTER><A NAME="Details"><H1><B>Course Details</B></H1><DL><P><DT><B><FONT SIZE=+1>Meeting Times</FONT></B></DT><DD><TABLE BORDER=2 CELLPADDING=2> <TR> <TD><B>Lectures</B></TD> <TD>MWF 12:30-1:30</TD> <TD>GGB 1504 (Lee Iacocca Lecture Hall)</TD> </TR> <TR> <TH ROWSPAN=3><B>Recitation</B></TH> <TD>Tues 1:30-2:30</TD> <TD>EECS 1003</TD> </TR> <TR> <TH COLSPAN=2>OR</TH> </TR> <TR> <TD>Thurs 2:30-3:30</TD> <TD>GGB 1371</TD> </TR></TABLE><P>The two recitation sections will be identical so please come to onlyone or the other. We'd like to keep the recitations more informal andthat is hard to do if everyone comes to the same recitation.<P>GGB 1371 is hard to find. It is in the north-east section of G.G. Brown,in that heavy machinery area where they build the concrete canoe, etc.If you're coming to Thursday's recitation, plan on a few extra minutesfor finding the room. Use the building maps posted on the walls. Bring your hard-hat.</DD><P><DT><B><FONT SIZE=+1>Instructor</FONT></B></DT><DD>Prof. Gregory H. Wakefield <br> <!WA35><A HREF="mailto:ghw@eecs.umich.edu">ghw@eecs.umich.edu</A> <br> 4118 EECS <br> 3-9857</DD><P><DT><B><FONT SIZE=+1>Teaching Assistant</FONT></B></DT><DD>Andrew Sterian <br> <!WA36><A HREF="mailto:asterian@eecs.umich.edu">asterian@eecs.umich.edu</A></DD><A NAME="Office"><P><DT><B><FONT SIZE=+1>Office Hours</FONT></B></DT><DD><TABLE BORDER=4 CELLPADDING=4> <TR> <TH ROWSPAN=4>Andrew Sterian</TH> <TD>Monday 2:30-4:30</TD> <TD ROWSPAN=4>EECS 2420</TD> </TR> <TR> <TD>Tuesday 2:30-4:30</TD> </TR> <TR> <TD>Wednesday 1:30-3:30</TD> </TR> <TR> <TD>Thursday 1:30-2:30</TD> </TR> <TR> <TH>Prof. Wakefield</TH> <TD>Friday 1:30-3:30</TD> <TD>EECS 4118</TD> </TR> </TABLE></DD><P><DT><B><FONT SIZE=+1>Textbook</FONT></B></DT><DD><B>Digital Signal Processing: Principles, Algorithms, and Applications.</B> (3rd ed.) John Proakis and Dimitris Manolakis, Prentice-Hall.</DD><P><DT><B><FONT SIZE=+1>Workload</FONT></B></DT><DD> <UL> <LI>Homework (10%) <B>NOTE: </B>One homework will not be counted</LI> <LI>Group Project (15%)</LI> <LI>3 Exams (25% each)</LI> </UL> Computer environment will utilize MATLAB</DD><P><DT><B><FONT SIZE=+1>Syllabus</FONT></B></DT><DD> <UL> <LI>Introduction: What you have learned in the course... [3-5]</LI> <LI>Chapter 2: Discrete-Time Signals and Systems [3]</LI> <LI>Chapter 3: Z-Transform [6]</LI> <LI>Chapter 4: Frequency Analysis [6]</LI> <LI>Chapter 5: Discrete Fourier Transform [2]</LI> <LI>Chapter 6: FFT [2]</LI> <LI>Chapter 7: Implementation [6]</LI> <LI>Chapter 8: Design of Digital Filters [6]</LI> <LI>Chapter 9: Sampling and Reconstruction [2]</LI> <LI>Chapter 12: Power Spectrum Estimation [3]</LI> </UL></DD></DL></BODY></HTML>⌨️ 快捷键说明
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