From jsb119 at msstate.edu Mon Mar 3 00:32:59 2008 From: jsb119 at msstate.edu (Jeff Brantley) Date: Mon Mar 3 08:15:49 2008 Subject: [ece3163] Test 2 #3 Solution Message-ID: <7323836.1204525979286.JavaMail.jsb119@msstate.edu> I've been looking over the posted solution to the old Exam #2, and I'm not comfortable with the solution in Problem #3. A similar problem for comparison is HW6, #1(f). The function is split into two functions, each multiplied by a form of the unit step. In the HW6 problem, we compensate for the "crossover" of the functions at n = 0 by the use of a unit pulse delta[n]. It appears the Exam #3 problem would require a similar compensation, although delta(t) is not useful since it is not the same as delta[n]. The following demonstrates (what I believe to be) the problem with the solution provided: http://www.ece.msstate.edu/~jsb119/junk/test2-3.pdf If this is the case, how should this problem be approached? From fowler at ece.msstate.edu Mon Mar 3 10:33:31 2008 From: fowler at ece.msstate.edu (James E. Fowler) Date: Mon Mar 3 10:33:41 2008 Subject: [ece3163] Test 2 #3 Solution In-Reply-To: <7323836.1204525979286.JavaMail.jsb119@msstate.edu> References: <7323836.1204525979286.JavaMail.jsb119@msstate.edu> Message-ID: <47CC285B.8060000@ece.msstate.edu> > I've been looking over the posted solution to the old Exam #2, and I'm > not comfortable with the solution in Problem #3. > > A similar problem for comparison is HW6, #1(f). > > The function is split into two functions, each multiplied by a form of > the unit step. In the HW6 problem, we compensate for the "crossover" > of the functions at n = 0 by the use of a unit pulse delta[n]. > > It appears the Exam #3 problem would require a similar compensation, > although delta(t) is not useful since it is not the same as delta[n]. > > The following demonstrates (what I believe to be) the problem with the > solution provided: > > http://www.ece.msstate.edu/~jsb119/junk/test2-3.pdf > > If this is the case, how should this problem be approached? You are correct in that there appears to be a discrepancy. This is, in essence, due the somewhat sloppy mathematics that we, as engineers, like to employ. It might make sense to look at it in a little simpler form. Consider x(t) = exp(-at)u(u) which we determined in lecture to have the Fourier transform X(w) = 1/(a+jw) by working through the integration. This was based on our definition of the unit step: u(t) = 1 for t >= 0 and 0 else. Now consider an alternative definition of the unit step, call it u'(t): u'(t) = 1 for t > 0 and 0 else. Clearly, u(t) and u'(t) are different since they define the step differently at t = 0. However, if you take the Fourier transform of x'(t) = exp(-at)u'(t), you get X'(w) = 1/(a+jw) since the integration works out exactly the same as for X(w) (the difference between integrating from 0 and 0+ is nothing in this case). Thus, it would appear that we have two distinct functions that have the same Fourier transform, which is problematic, since the Fourier transform is supposed to be one-to-one. However, if we "measure" the difference between x(t) and x'(t) using the integral squared error, E(x, x') = \int (x(t) - x'(t))^2 dt where the integration is from -infinity to +infinity, we get E(x, x') = 0. Thus, x(t) and x'(t) are different when we define "equals" to mean pointwise equality, but the same when we define "equals" to mean zero integral squared error. If we adopt this latter measure, then your two x(t) functions, which differ only for t=0, are the same, and therefore have the same Fourier transform. So, we'd really need to talk about squared-error equality when dealing with the Fourier transform to be mathematically correct. The situation is different in discrete time when we define a similar measure, the sum squared error E(x, x') = \sum (x[n] - x'[n])^2, because a difference in one sample value causes the sum to be non-zero. Thus, in Homework #5 prob. 1(f) we have to "compensate" with a \delta[n] term so that we maintain a zero E(x, x'). Hope this clears things up. -- ======================================================================= | Dr. James E. Fowler | | Professor | | Department of Electrical & Computer Engineering | | GeoResources Institute (GRI) | | Mississippi State University | | | | fowler@ece.msstate.edu | | http://www.ece.msstate.edu/~fowler | | | ---------- Postal Address ---------------------- Contact Info --------- | Box 9571, 406 Hardy Rd. | HPC2: Rm. 208, +1 (662)325-2578 | | Mississippi State, MS 39762 | ECE: Rm. 406, +1 (662)325-3640 | | USA | Fax: +1 (662)325-7692 | ======================================================================= From fowler at ece.msstate.edu Mon Mar 3 11:59:20 2008 From: fowler at ece.msstate.edu (James E. Fowler) Date: Mon Mar 3 12:00:17 2008 Subject: [ece3163] Exam II Solutions Message-ID: <47CC3C78.1000206@ece.msstate.edu> Solutions to Exam II have been posted to the class website. -- ======================================================================= | Dr. James E. Fowler | | Professor | | Department of Electrical & Computer Engineering | | GeoResources Institute (GRI) | | Mississippi State University | | | | fowler@ece.msstate.edu | | http://www.ece.msstate.edu/~fowler | | | ---------- Postal Address ---------------------- Contact Info --------- | Box 9571, 406 Hardy Rd. | HPC2: Rm. 208, +1 (662)325-2578 | | Mississippi State, MS 39762 | ECE: Rm. 406, +1 (662)325-3640 | | USA | Fax: +1 (662)325-7692 | ======================================================================= From fowler at ece.msstate.edu Wed Mar 5 08:38:32 2008 From: fowler at ece.msstate.edu (James E. Fowler) Date: Wed Mar 5 08:38:38 2008 Subject: [ece3163] Upcoming Schedule Message-ID: <47CEB068.90106@ece.msstate.edu> Some things to note about the upcoming weeks: - Spring break is next week (Mar. 10-14) - there will be no class - The university has a holiday Friday, Mar. 21 - there will be no class - Homework #7 will be due Wednesday Mar. 19 (a departure from our usual pattern due to the holiday that Friday) - I will be out of town Mar. 24-28 - Naveen will be handling lecture that week -- ======================================================================= | Dr. James E. Fowler | | Professor | | Department of Electrical & Computer Engineering | | GeoResources Institute (GRI) | | Mississippi State University | | | | fowler@ece.msstate.edu | | http://www.ece.msstate.edu/~fowler | | | ---------- Postal Address ---------------------- Contact Info --------- | Box 9571, 406 Hardy Rd. | HPC2: Rm. 208, +1 (662)325-2578 | | Mississippi State, MS 39762 | ECE: Rm. 406, +1 (662)325-3640 | | USA | Fax: +1 (662)325-7692 | ======================================================================= From fowler at ece.msstate.edu Wed Mar 5 14:08:30 2008 From: fowler at ece.msstate.edu (James E. Fowler) Date: Wed Mar 5 14:08:39 2008 Subject: [ece3163] Exam II - Revised Solutions Message-ID: <47CEFDBE.409@ece.msstate.edu> There was a small error in the solution to Prob. 3 in the solutions to Exam II as original posted. This error has been corrected and the revised solutions are now on the class website. -- ======================================================================= | Dr. James E. Fowler | | Professor | | Department of Electrical & Computer Engineering | | GeoResources Institute (GRI) | | Mississippi State University | | | | fowler@ece.msstate.edu | | http://www.ece.msstate.edu/~fowler | | | ---------- Postal Address ---------------------- Contact Info --------- | Box 9571, 406 Hardy Rd. | HPC2: Rm. 208, +1 (662)325-2578 | | Mississippi State, MS 39762 | ECE: Rm. 406, +1 (662)325-3640 | | USA | Fax: +1 (662)325-7692 | ======================================================================= From fowler at ece.msstate.edu Mon Mar 17 10:14:45 2008 From: fowler at ece.msstate.edu (James E. Fowler) Date: Mon Mar 17 10:15:01 2008 Subject: [ece3163] Upcoming Schedule Message-ID: <47DE8AE5.7000802@ece.msstate.edu> Recall that this Friday, Mar. 21, is a holiday, so there will not be classes. Homework #7 is thus due Wednesday, Mar. 19, this week. Next week, Mar. 24-28, I will be out of town. Naveen will be handling the lectures during that week. Homework #8, to be assigned this Wednesday, will be due Mar. 28 while I am gone. On that day, Naveen will assign Homework #9 which will be due Apr. 4. Note that Homework #9 will be the last homework before Exam III. Exam III will take place on Monday Apr. 7; the preceding Friday (Apr. 4) will be devoted to a review session. Finally, Naveen will be out of town Apr. 1-5; thus, there will not be a recitation Apr. 3. -- ======================================================================= | Dr. James E. Fowler | | Professor | | Department of Electrical & Computer Engineering | | GeoResources Institute (GRI) | | Mississippi State University | | | | fowler@ece.msstate.edu | | http://www.ece.msstate.edu/~fowler | | | ---------- Postal Address ---------------------- Contact Info --------- | Box 9571, 406 Hardy Rd. | HPC2: Rm. 208, +1 (662)325-2578 | | Mississippi State, MS 39762 | ECE: Rm. 406, +1 (662)325-3640 | | USA | Fax: +1 (662)325-7692 | ======================================================================= _______________________________________________ ece3163 mailing list ece3163@ece.msstate.edu http://www.ece.msstate.edu/mailman/listinfo/ece3163 From fowler at ece.msstate.edu Wed Mar 19 12:00:05 2008 From: fowler at ece.msstate.edu (James E. Fowler) Date: Wed Mar 19 12:00:21 2008 Subject: [ece3163] Homework #7 Solutions Message-ID: <47E14695.7000108@ece.msstate.edu> Solutions to Homework #7 are now on the class website. -- ======================================================================= | Dr. James E. Fowler | | Professor | | Department of Electrical & Computer Engineering | | GeoResources Institute (GRI) | | Mississippi State University | | | | fowler@ece.msstate.edu | | http://www.ece.msstate.edu/~fowler | | | ---------- Postal Address ---------------------- Contact Info --------- | Box 9571, 406 Hardy Rd. | HPC2: Rm. 208, +1 (662)325-2578 | | Mississippi State, MS 39762 | ECE: Rm. 406, +1 (662)325-3640 | | USA | Fax: +1 (662)325-7692 | ======================================================================= From fowler at ece.msstate.edu Mon Mar 31 08:32:16 2008 From: fowler at ece.msstate.edu (James E. Fowler) Date: Mon Mar 31 08:32:32 2008 Subject: [ece3163] Homework #8 Solutions Message-ID: <47F0E7E0.7070604@ece.msstate.edu> Solutions to Homework #8 have been posted to the class website. -- ======================================================================= | Dr. James E. Fowler | | Professor | | Department of Electrical & Computer Engineering | | GeoResources Institute (GRI) | | Mississippi State University | | | | fowler@ece.msstate.edu | | http://www.ece.msstate.edu/~fowler | | | ---------- Postal Address ---------------------- Contact Info --------- | Box 9571, 406 Hardy Rd. | HPC2: Rm. 208, +1 (662)325-2578 | | Mississippi State, MS 39762 | ECE: Rm. 406, +1 (662)325-3640 | | USA | Fax: +1 (662)325-7692 | =======================================================================