ECE 901 Wireless Multipath Channel Models, Lec. 1, Spring 2008

1/23	Wednesday. Introduction to the course.

1/25	Distributed notes for introductory lecture on 1/23 and for Temporal
	Poisson Processes.  Constructed the homogeneous Poisson process by
	summing i.i.d. exp(lambda) interarrival times X_i to produce the
	arrival times T_k, which turn out to be Erlang RVs.  The T_k form
	a Markov chain.  The counting process N_t is then defined in terms
	of the arrival times.
	Assigned HW#1 (handout in class) DUE WED 1/30.

1/28	Covered the Inhomogeneous Poisson Process.  Derived joint conditional
	pdf of the arrival times.

1/30	Related joint conditional pdf of arrival times to the order statistics
	of iid RVs.  Derived expectation of shot noise (see link to part 4 of
	notes above).
	Assigned HW#2.  The M-file lincmb.m may be helpful in Problem 4.
	DUE WED 2/6.

2/1	Reviewed real-valued Gaussian random variables and vectors.
	(See link to part 5 of notes above.)

2/4	Reviewed complex-valued Gaussian random variables and vectors.
	Reviewed integrals of Gaussian processes, correlation functions,
	covariance functions.

2/6	Class postponed due to weather.

2/8	Introduced white noise, related correlation functions.
	Applied to a correlator.  Craig's formula and its connection to
	Chebyshev-Gauss quadrature.
	Assigned HW#3 (handout in class) DUE FRI 2/15.

2/11	M-ary detection & the MAP test (part 6 of notes).  Application
	to real Gaussian random vectors.  Also discussed definition of
	conditional probability when conditioning on a continuous RV.

2/13	Application to complex Gaussian random vectors.  Outlined how we
	will analyze the detection of deterministic waveforms in AWGN and
	reduce it to the finite-dimensional vector problem.

2/15	Showed how to convert the problem of detecting a known waveform
	in AWGN into the column-vector problem we have already solved.
	Then expressed the solution in terms of the orginal waveforms.

2/18	Expressed probability of error for optimum detection of known
	binary signals in AWGN as a function of the distance between the
	two waveforms.  Generalized the results to signals subject to
	multipath (see part 7 of notes).

2/20	Applied Craig's formula.  Ran demo comparing it with simulation.
	Introduced Marked Temporal Poisson Processes (see part 8 of notes).
	Showed that the mgf of a shot noise driven by such a process is
	the same as what would result from a shot noise driven by a
	two-dimensional Poisson process with an appropriate intensity function.
	Assigned HW#4 DUE WED 2/27.

2/22	In a shot-noise RV driven by a marked temporal Poisson process in
	which h(t,g) = a(t)g2, a critical quantity to compute is Mt(θ)=Et[exp(θG2)].
	If G is Rayleigh, Nakagami-m, or Rice, then Mt(θ) can be computed
	in closed form.  If G is lognormal, then Mt(θ) must be computed
	numerically.  Discussed Gaussian quadrature in Matlab.  My Matlab
	scripts for generating the nodes and weights discussed in class are:
	hermitequad.m and legendrequad01.m.  If you are interested in more
	background on Gaussian quadrature, you may read gaussquad.pdf.

2/25	Power delay profile, mean excess delay, mean square excess delay,
	and delay spread.  Introduced the notion of cluster point process.
	Trees and seeds example.  Distinguishing between trees and seeds
	and not doing so.  Compared the clustered delays of the Saleh-
	Valenzuela model.  Started introduction of measures (part 9 of notes).

2/27	Finished introduction of measures.  Started the Lebesgue Integral
	(part 10 of notes).
	Assigned HW#5 DUE WED 3/5.

2/29	Gave examples when Riemann and Lebesgue integrals do not exist.
	Showed that the Lebesgue measure of a countable subset of the
	real line must be zero.  Showed how to approximate any nonnegative
	function by a simple function.  Showed an example of how LMCT
	can be applied.

3/3	More examples of applying LMCT.  Computed expectation of shot noise.
	Computed mgf of Poisson shot noise.  Introduced Laplace functional.

3/5	Showed that the Laplace functional characterizes a Poisson random
	measure.  Constructed a Poisson process with a finite mean measure
	(see parts 11 and 12 of notes).
	Assigned HW#6 (handout in class) DUE WED 3/12.

3/7	Used the Laplace functional to show that the process constructed in
	the previous lecture was in fact a Poisson process.  Extended result
	to processes with a sigma-finite mean measure (see part 13 of notes,
	which can completely replace parts 11 and 12).

3/10	Discussed problems 3 and 4 on HW#5.  Discussed how to simulate
	random vectors and sequences.  Discussed how to simulate a
	single random variable.  See part 14 of notes.

3/12	Continued part 14 of notes on simulation of random variables.
	For information on how Matlab generates uniform and normal random
	numbers, see C. Moler, Numerical Computing with Matlab, Chapter 9.
	See also the papers by G. Marsaglia and W. W. Tsang,
	"The ziggurat method for generating random variables," J. Statist.
	Software, vol. 5, no. 8, Oct. 2000.
	"A fast, easily implementable method for sampling from decreasing
	or symmetric unimodal density functions," SIAM J. Sci. Statist.
	Comput., vol. 5, no. 2, pp. 349-359, June 1984.
	Thomas et al., "Gaussian random number generators," ACM Computing
	Surveys (CSUR), vol. 39, no. 4, Oct. 2007.
	G. Marsaglia, "Evaluating the normal distribution," J. Statist.
	Software, vol. 11, no. 5, July 2004.
	G. Marsaglia, W. W. Tsang, and J. Wang,
	"Fast generation of discrete random variables," J. Statist. Software,
	vol. 11, no. 3, July 2004.
	For information on ziggurats, including pictures see Wikipedia
	and the links there.  Great online book:
	Luc Devroye, Non-Uniform Random Variate Generation, 1986.
	Assigned HW#7 DUE WED 3/26.

3/14	Showed that YN has the desired pdf.  Briefly discussed ziggurat
	algorithms.


3/17	No class --- SPRING BREAK
3/19	No class --- SPRING BREAK
3/21	No class --- SPRING BREAK

3/24    Badri will speak.

3/26	Showed that marked Poisson processes can be viewed as a Poisson process
	in a product space (part 15 of notes).  Also derived the mean measure
	of the marked process.  Started Cluster Processes (part 16 of notes).
	Added derivation of the mean measure.
	Assigned HW#8 DUE WED 4/2.

3/28	Reviewed solutions of HW#7.  Continued analysis of cluster processes.

3/31	Finished part 16 of notes on cluster processes.

4/2	Started part 17 of notes on augmented cluster processes.

4/4	Finished Augmented Cluster Processes.

4/7	Started part 18 Multipath-Cluster Channels.

4/9	Continued Multipath-Cluster Channels.
	Assigned HW#9 DUE WED 4/16.

4/11	Continued Multipath-Cluster Channels.

4/14	Showed how the SV, 3a, and Spencer models are special cases.

4/16	Showed how the 4a model is a special case.

4/18	Derived density of the mean measure of the cluster-start process
	of the 4a model.

4/21	Derived formulas for m1(t) and m(τ|t).

4/23	Ran demos to plot m1(t).
	Handed out revised notes on Multipath-Cluster Channel Models.
	Introduce power-delay-angle profile.
	Assigned HW#10 (handout in class) DUE WED 4/30.

4/25	More details on power-delay-angle profile.

4/28	Went over a careful derivation of the power-delay-angle profile.
	Links to the papers by Spencer et al. and Chong et al. have been
	added to the "Links to key papers."

4/30	Started analysis of intersymbol interference (ISI).  See pp. 5-6 of
	part 19 of notes.

5/2	Discussed Time-Hopped Direct-Sequence Pulses.  See pp. 1-3 of
	part 19 of notes.

5/5	Completed derivation on pp. 3-4.  Covered pp. 5-6.

5/7	Covered pp. 7-8.
5/9