Instructor Office Hours: WED 2-3:15, after class, or by appointment.
See
what we did in Fall 1998. Fall 2005
will be quite similar.
TAs:Pam Limpiti [tlimpiti (at) wisc (dot) edu]
and Tariq Qureshi [tqureshi (at) wisc (dot) edu]
TA Office Hours:
| Day | Time | Location | TA |
|---|---|---|---|
| MON | 2:30-4:00 | B640 | Pam |
| WED | 10:00-12:00 | B630 | Tariq |
| 1:00-2:00 | B640 | Pam | |
| 5:00-6:30 | B640 | Pam | |
| THU | 9:00-10:00 | B630 | Tariq |
| 10:00-11:00 | B640 | Pam |
Syllabus
Signal Processing Demos: Convolution,
Fourier Series, Transforms, etc.
(Note: Be sure to unzip into a new directory.)
Homework Solutions and Notes from Discussions
Class Schedule for Fall 2005
9/6 Tuesday. Overview of the course. Review of some mathematical background
on complex numbers, exponentials, and sinusoids. Integration by parts.
Assigned HW #1 Ch 1: 48(a), 49(c)(f)(i), 51(a)(e), 53(d), 56(e) DUE THU 9/15.
NO DISCUSSION THIS EVENING.
9/8 Covered the geometric series formula (see Problem 1.54 on p. 73)
Section 1.1.2: Signal Energy and Power. Section 1.2.1: Transformations
of the Independent Variable.
Assigned HW #1 (continued) Ch 1: 21(b)(c)(d), 22(a)(d)(h) DUE THU 9/15.
9/13 Section 1.2.2 Periodic Signals, Section 1.2.3 Even & Odd Signals,
Section 1.4 The Unit Impulse & Unit Step Functions (finished
discrete time only).
9/15 Section 1.4 The Unit Impulse & Unit Step Functions (continuous time).
Section 1.5 focused on block diagrams; series, parallel, feedback.
Assigned HW #2 Ch 1: 21(e)(f), 22(e)(f), 23(a)(b), 24(a), 25(a)(c)(e),
26(a)(b)(c) DUE THU 9/22.
9/20 Gave a little background on signals/waveforms. Started Section 1.6
System Properties. Covered memory, causality, stability, and
time invariance. Download Supplementary Lecture Notes (2-page pdf file).
9/22 Linearity. Worked lots of examples.
Assigned HW #3 Ch 1: 27(a)(c)(e), 28(c)(d), 42(a)(b)(c), 43(a)(b),
55(a)(f) DUE THU 9/29.
9/27 Showed that every discrete-time linear system has the representation
(Ax)[n] = \sum_{k=-\infty}^\infty x[k] \hat h[n,k], where
\hat h[n,k] := (A\delta_k)[n] is the response at time n of the
system to the input waveform \delta_k, which is a unit impulse
at time k. If the system is also time-invariant then
\hat h[n,k] = (A\delta_k)[n] = (A\delta)[n-k] = \hat h[n-k,0].
In this case, we put h[n] := (A\delta)[n]. We call h the system
impulse response. In other words, h is the response
of the system when the input is the unit impulse \delta.
Worked a couple of examples. Started showing how to do
graphical convolution to evaluate
(h*x)[n] := \sum_{k=-\infty}^\infty h[n-k]x[k].
Reference: Section 2.1.
9/29 Worked Examples 2.3 and 2.4 on discrete-time convolution.
Started Example 2.5.
Assigned HW #4 Ch 2: 21, 24 DUE THU 10/6.
10/4 Finished Example 2.5. Section 2.2 Continuous-Time LTI Systems.
Showed that every linear system can be expressed as an integral
using the time-varying impulse response \hat h(t,\tau). Linear
systems that are also time invariant (LTI) can be expressed
as the convolution of the impulse response h and the input signal x.
Worked Examples 2.7 and 2.8.
10/6 Characterization of memoryless, causality, and stability for LTI
systems in terms of properties of their impulse response.
Discussed several examples. Briefly mentioned that convolution
is associative, commutative, and distributes over addition.
Assigned HW #5 Ch 2: 22(a)(b)(c)(d), 28(a)(c)(e)(g),
29(a)(c)(e)(g), 44(a)(b)(c)(d), 48(a),(c)-(h) DUE THU 10/13.
10/11 Unit Step Response. Section 2.3 Properties of LTI Systems.
Convolution is commutative, associative, and distributes over
addition; i.e., h*x=x*h, g*(h*x) = (g*h)*x, and h*(x+y) = h*x + h*y.
Section 2.4 Causal LTI Systems Described by Difference Equations;
did a simple discrete-time example.
10/13 Worked some review problems.
The exam will cover everything from the beginning of the semester
through the lecture on Thur. 10/6.
Suggested Review:
Ch 1: 24(b), 25(d), 26(d), 27(b)(d)(f), 28(a)(b), 55(e).
Ch 2: 23(a)(b)(d), 25(a), 28(b)(d)(f), 29(b)(d)(f), 47.
Re-work as many of the HW problems as you can.
Note that Problems 23 and 47 in Chapter 2 are a little different
from the HW that has been assigned.
10/17 MON. Exam 1 in 1800 EH at 7:15 pm
10/18 Started Ch. 3, Section 3.3 Fourier Series Representation of
Continuous-Time Periodic Signals. Worked Examples 3.3 and 3.5.
Also worked several other examples. Derived the analysis eq. (3.37).
10/20 Used the Fourier series representation to discuss how LTI systems
process periodic signals. Briefly mentioned Laplace and Fourier
transforms. Spent most of the lecture going over the properties
of Fourier series; see Section 3.5 of the text. Note also Table 3.1
on p. 206 that summarizes these properties.
Assigned HW #6 Ch 3: 22(a; graphs a,b,d,f)(b), 23(a)(c),
26(a)(b)(c) DUE THU 10/27.
10/25 Returned and went over exam. Reviewed Examples 3.5 and 3.6.
Started Example 3.7 using integration by parts.
10/27 Worked Example 3.7 three different ways. Emphasized use of Fourier
Series properties to avoid integration by parts. Started Chapter 4:
Developed the Fourier Transform and inversion formula as limiting
case of Fourier Series. Reference: Section 4.1.1.
Assigned HW #7 Use Fourier Series Properties, not integration
by parts to solve: Ch 3: 22(a; graphs a,b) 24, 25 DUE THU 11/3.
11/1 Worked many examples of computing Fourier transforms and inverse
Fourier transforms. In particular, worked Examples 4.1, 4.3, 4.4,
4.5, and 4.8. Also worked an example of computing Fourier Series
coefficients using the "derivative method." Tariq distributed
midsemester TA and course evaluations.
11/3 Section 4.3 Properties of Continuous-Time Fourier Transforms.
Assigned HW #8 Ch 4: 21(a)(b)(d)(g)(i), 22(b)(c)(d) DUE THU 11/10.
11/8 Discussed signal separation and filtering. Section 4.5 The
Multiplication Property and its use in sending multiple
signals over the same wire channel or wireless channel. Worked an
example using Parseval's equation. Worked an example using the
differentiation formula. Worked Examples 4.22 and 4.23.
11/10 Discussed the sampling theorem [Ref. Sections 7.1-7.3].
Key concepts: For a signal bandlimited to w_M, for perfect
reconstruction, you must sample at w_s > 2 w_M, where 2 w_M
is the Nyquist rate. In other words, the sampling period
or sampling interval, T:=2 pi/w_s, must satisfy T < pi/w_M.
If w_s is less than the Nyquist rate, you will have aliasing.
Assigned HW #9 Ch 4: 23(a)(b)(c), 24(a)(b), 25(b)-(f)
NOTE: in 25 do NOT evaluate X(jw) explicitly. Also, on (f),
consult the table on p. 328 of transform properties. DUE THU 11/17.
11/15 Discussed Tomography and its connection to Fourier transforms.
Very briefly discussed the computation of Fourier transforms on
the computer. Mentioned fast Fourier transform (FFT). Started
Chapter 5, The Discrete-Time Fourier Transform (DTFT).
11/17 Went over properties of the DTFT and worked several examples.
Assigned HW #10 Ch 5: 21(a)(d)(j), 22(a)(c)(f) DUE WED 11/23.
11/22 Started Laplace Transforms. Worked many examples.
Exam 2 will focus on Fourier series and Fourier transforms of
continuous-time functions. The exam is closed book and closed notes;
however, the exam will provide the Fourier Series Table 3.1 on p. 206
as well as the Fourier Transform Tables 4.1 and 4.2 on pp. 328-329.
Suggested Review:
Ch 3: 34, 35, 40(a)(b)(c), 47
Other: Find the Fourier series coefficients a_k of the waveforms
x(t) and z(t) graphed on p. 263; be clever - do a minimum amount
of work by using properties of Fourier series!
Ch 4: 21(c)(e; table on p. 329 may be helpful to avoid int by parts)
21(h)(j), 22(a; use properties, tables)(b), 26(a; use table!),
27, 28(a)(b; part (vi)).
11/23 WED HW #10 DUE TODAY at 3:30 pm in my office, 3615 EH.
If you want to drop it off earlier on WED, you can slide it under
my office door if I am not there.
11/24 THANKSGIVING -- No class.
11/29 Worked Review and other problems.
11/30 WED. Exam 2 in 1800 EH at 7:15 pm
12/1 Went over Exam 2. Discussed partial fraction expansion. Laplace
transforms of causal signals. Laplace transforms of derivatives
of causal signals. Laplace transforms of differential equations
with causal inputs.
Assigned HW #11 Ch 5: 23(a),(c)-(f) 24(a)(g)(h)(i) DUE THU 12/8.
12/6 Partial fraction expansion of 1/[(s-p)(s-q)^2]. Properties of Laplace
transforms. For causal waveforms without impulses note in particular
the initial and final value properties. See tables on p. 691 and 717.
Started Section 9.7 LTI Systems and Laplace Transforms.
12/8 Finished discussion of Laplace transforms. Mentioned Slepian's
paper, "On Bandwidth." TA evaluations.
Assigned HW #12 Ch 9: 21(a)(c)(f)(i), 22(a)(c)(e)(g) DUE THU 12/15.
Old final exams: Sp 93, Fall 93, Sp 94, Sp 98.
12/13 Worked problems from old final exams. Distributed teaching evaluations.
12/15 Last Class Day.
12/20 TUESDAY: FINAL EXAM 2:45-4:45 PM in 1800 EH.