Teaching Mathematics and Statistics for students in Electronics

Christos P. Kitsos

T.E.I. – Athens, Dept of Mathematics

Abstract: The aim of this paper is to propose a new framework for teaching Mathematics and Statistics to those students who are attending a course leading to a BSc degree in Electronics. Both courses of Mathematics and Statistics have to be revised and computer oriented, i.e. the appropriate software has to be adopted so that the students to be aware how can be evaluated the different approaches, that theory proposes. Statistical packages appear to be friendlier with Minitab being the most appropriate for teaching. Excel, appears to be also useful. The traditional use of FORTRAN for Mathematical problems seems not to be so popular nowadays MATLAB or Mathematica or C++ appear, to be friendlier to an engineering student. Two main proposals are in this paper: to enlarge the Mathematics and Statistics courses appropriately, as there is a necessity for this, due to the various Electronics courses.

Key-Words: Packages, Stochastic Process, Density Function, Filter

1 Introduction

Teaching Mathematics and Statistics have been accepted the last decade many essential changes. This happened mainly on the improvements of appropriate software: Mathematica, MATLAB, SPSS, Minitab, Excel, etc. In this paper we are focusing more on the covered material in association with the existing software, than to the software itself.

It is good to remind that the first non-military application of the computers was on the 1951, not that many years ago. It was the USA census performed from the USA Census Bureau. Even at 60’s a histogram, or a variance was a real computational task, while in late 70’s, some progress was made. The statistical oriented language COUNT, was a pioneering one on evaluating descriptive statistics, in late 70’s. At that time there was no statistical package at the market. No question to present a graph, or the evaluation of the roots of an equation, the use of FORTRAN subroutines and Library was important. But the user needed the appropriate knowledge of the code and the subroutine, so usually was a mathematician or an engineer. The WATFOR (the FORTRAN language of the Waterloo University, Canada) and the APL (A Programming Language) provided to the mathematical and statistical community a guide for solving their problems, [1], while FORTRAN even in ‘90s, provided solutions, even nowadays, for solving differential equations, and not only. The APL was extremely simple, but associated with a “huge” keyboard system. The results were coming “slowly”, and while the inverse o matrix was easy to be evaluated, the matrix was needed not to be more than 10x10 in dimension. But there was a progress, something knew was coming.

2 Teaching Mathematics

Industrial Mathematics and Statistics, in the beginning of 3rd millennium, see [2], provide a strong theoretical support to various industrial problems and therefore to Electronics. The fact that the student can nowadays easily evaluate even integrals provide evidence that the teaching orientation should not be on calculating; therefore “Calculus” has to change background and targets. The students must be more oriented towards and electronic approach than to a theoretical approach. They have to be aware what a Z-transformation is, what are the state equations, either for discrete or continuous case, what is an operator, so that to absorb easier Laplace or Fourier transformations. Knowing Laplace transformation and the convolution it is easy to realize the transfer function, as the Laplace transformations of the impulse function in continuous case. The Z-transformation helps in discrete case. This is a mathematical example and the student does expect and deserves to know this in advance, when is attending control systems. The calculation of the appropriate Lablace transformation, like the resolvant matrix, certainly is needed. The solution of differential equations cannot the only target, of a course in Mathematics with examples from electronics. So, if the course in Mathematics cover all the subjects the Electronics need, the lectures of the main electronic courses they do not spend time solving mathematical problems.

Solving differential equations the state-transition equation for example, and others possible knowledge from the field of electronics should be clarified.

The Complex Analysis provides support to stability tests of Discrete-Data System, while steady-state error of a system with a step-function input can be also clarified to a third semester mathematical course. We believe that Complex Analysis might be more useful, in some cases, than Real Analysis.

2.1 Teaching Linear Algebra

As a Vector-Matrix Representations of State Equations is needed as well as the eigenvalue problem the subject of Linear Algebra should be covered at a second semester. The subject of Linear Algebra needs a careful consideration: It is strongly related with Geometry. With a nice foundation, based on the homogeneous coordinating system, the second-degree equation with 2 or 3 variables represent the seventeen varieties of standard forms of equations. That is the second-degree equation:

Ax2 + By2 + Cz2 + 2Dx + 2Ey + 2Fz + G = 0

Represents, with certain choice of the coefficients, the following:

Ellipsoid (real or imaginary)

Imaginary elliptic cone

Hyperboloid (of one or two sheets)

Real elliptic cone

Paraboloid (elliptic or hyperbolic)

Elliptic cylinder (real or imaginary)

Cylinder (parabolic or hyperbolic)

Intersecting planes (real or imaginary)

Parallel planes (real or imaginary)

Coincident planes

All these might not be seen so useful to practical electronically oriented applications. But from teaching purposes can be a simple example of what an orthogonal transformation is, and how the inner product, under an orthogonal transformation, remains invariant while this is not a case under a non- orthogonal transformation. This is essential in a Physics course, where the student learns that changing the coordinating system to polar or spherical that has an influence to the measure of distance, volume, grand and to the Laplacian, among other characteristic influences. Moreover the student learns and gets the insight understanding of what a graph is. This is so essential not only to the real analysis but to the complex analysis as well.

2.2 Teaching Complex Analysis

The student has to be aware of the Complex analysis. Even in elementary Physics the complex resistance is so useful idea for student to learn and understand. Even more the Laplace transformation provide food for thought: The Nyquist Stability Criterion, [3], is based on a combination of a graph and a transfer function. In principle the Nyquist criterion is a semigraphical method that determines the stability of a closed-loop system by investigating the properties of the frequency-domain plot, of the loop transfer function, L(s) say. Practically in a polar coordinating system the graph of L(jω) is presented in the Im{L(jω)} versus Re{L(jω)} system, while ω varies from zero to infinity. It is not out target to present the Nyquist criterion. It is only our purpose to be clear that a nice course in Math can be helpful to the appropriate course discussing Automatic Control Systems. So, the student has to be aware what a (closed) curve is and, more important, to understand the difference of two particular points which play an important role to this scenario:

  • Encircled: A point or region in a complex function plane is said to be encircled by a closed curve (path) if it is located inside the curve.
  • Enclosed: A point or region is said to be enclosed a closed curve (path) if it is encircled in the counter-clock-wise direction, or the point or region lies to the left of the curve (path), when the path is traversed in the prescribed direction.

The above two definitions might be also examples of no rigorous mathematical definitions. But the student needs to understand, rather than to be exercised to the strict (and beautiful!) mathematical language of symbols. Therefore the Nyquist criterion, originated as an application to complex analysis theory provide evidence that Complex Analysis can be, appropriately taught, part of a course in Math. Usually there is a sort section on complex numbers, that is certainly useful, but some points of Complex Analysis, like, trivially, one can mention the poles of a complex function are necessary for the interesting in his studies student in a Electronics department. As the student clarifies some points on Complex Analysis can understand the Principle of Argument:

Let D(s) be a single valued function, with a finite number of poles. Suppose that an arbitrary closed curve C is considered, so that does not go through any one of the poles or zeros of D(s). The corresponding CD locus (graph) in the D(s)-plane will encircle the origin N=Z-P times, were Z is the number of zeros and P is the number of poles of D(s) encircled by the s-plane locus C.

Thus the student can understand the three cases for N: negative, zero, positive in Math, and so to communicate better on an analysis concerning the Nyquist path, and the result to obtain closed-loop stability (Z=0), open-loop stability (P=0).

So Complex Analysis I do believe is essential for the students to an Electronics department. The term “Applied Mathematics” is so vague and so spread over different areas of Mathematics – but not too many would include the subject of Complex Analysis within such a term, despite the practical problems can face.

Usually Complex Analysis is considered as a special branch to Mathematical Analysis. Needless to say, not that many are thinking that elements of Complex Analysis, see [4], are really so helpful to applications for the students in an Electronics department.

2.3 Course work

Therefore a generous and tight first semester course should cover integrations of any kind, functions with more than one variable, among the elementary analytic geometry approaches. etc. A second semester course needs to cover differential equations and Linear Algebra, Laplace transformations, etc. A third course might cover, particular kind of differential equations, complex analysis, and Fourier analysis, among other subjects, in discussion with the needs of the department. The student has to learn to work and think in a mathematical term and not to memorize techniques that might never use in an electronics classroom. This certainly needs communication skills and industrial experience from the lecturer and not an “on the blackboard’ practiced mathematician. The adoption of the appropriate parts of different packages should be an exercise for a student in such a field.

Therefore the traditional course in Mathematics, for students in Electronics, certainly needs adjustments, and is in honor to those who have already decided some.

3 Teaching Statistics

Teaching Stat is a more difficult task: not because the subject might be considered as more difficult, no such an approach – all the subjects are equally difficult. The students only classified, according to their background the courses to easy or difficult ones. The problem with a Statistical course is that even the international experience and therefore the references on such a subject is restricted. It is difficult to know were statistics is hidden to some electronically oriented problems.

The answer, I believe, is that Statistics appears when the physical problem is not anymore deterministic, but stochastic. That means practically everywhere! That is why Prof. Smith, at Southampton, UK, defines “Statistics is the science of Sciences”! This is of course too much, but indicates that Statistics is involved to all fields of Science and activities. So we are going back to Lord Kelvin, who used to say, “Only if you measure something, you can say that you know about it, otherwise your knowledge is obscure about it”. As all the descriptive statistics course is now covered in SPSS, Minitab, etc., with the DESCRIPTIVES command, people in electronics avoid an introductory course in statistics, and they are right – needles to say.

3.1Industrial Statistics

We already discussed that Descriptive Statistics is not needed to a student in a BSc course in Electronics. But should be one on Engineering Statistics course as Industrial statistics covers usually the Quality Control problem, so essential to all fields of engineering or industry. Therefore more interesting statistical problems might be adopted, so that to attract the attention of a student in Electronics. Here are some: The Experiment Design, oriented back to 1920’s by Sir R.A. Fisher has as an initial task to cover agricultural problems but today is applied to any industrial (or medical biological or toxicological) experiment, see [5] and the corresponding discussion at[6]. The optimal design theory, has been extended to dynamic systems, see [7], and therefore I do believe the optimal experimental design theory is a useful statistical tool to anyone who performs experiments under economical constrains. The introduction of the optimal design theory in dynamic systems, as in [7] provides to the experimenter the appropriate framework to “design” an experiment, than to perform an experiment Therefore the main ideas have to be adopted in a Statistical course, addressed to the students in Electronics. The problem of Reliabilityand survival analysis is essential as well. Students in electronics can understand easier the reliability and maintenance problem than a mathematician or a chemist. Therefore here it comes the problem of a statistical background: you need the sense of probability to cover the reliability subject.

There is no question that regression or (practically equivalent) least square method is essential, but certainly the appropriate software is needed: Minitab is suggested for teaching purposes. The main reason is that Minitab is huddling the calculations with matrices with a better way, than a lot of other packages. The inverse of a matrix (with the command INVERSE M1 M2, were the inverse of M1 is placed to M2) is a very strong tool, as the inversion of a matrix needs many calculations (of order n! for an nxn matrix). The same package can be friendly for solving Quality Control problems, while the Statgraphics, so useful some years ago for the graphical presentation of a function, can be adopted for the Response Surface Techniques, see the pioneering book as in [8].

In Taquchi experimental design approach, in his very well known paper as in [9], or quality improvement approach the design factors are separated into variability control factors, target control factors and neutral factors. Moreover the problem of accuracy of statistical distributions, see [10] (which among the most frequently download papers in Computation Statistics and Data Analysis journal) might be interesting from an electronically or software point of view. But in practice to calculate statistical tests with a significant level 0.00001, say, are, needless to say, mean less.

3.2 Stochastic Processes

The stochastic processes theory is a hidden subject to signal processing. The weak law of stationary, the linear filtering of a random process, the power spectal density (in second-order theory) is very important subjects to understand the signal theory from an electronic point of view. The mean square (either error-MSE or convergence in m.s.) is a useful statistical tool and deserves time to be devoted on it, so that to be clarified to the electronics theory students.

Here is some discussion, as an example, on it:

If we let Y(t) to be a real-valued weakly stationary process, continuous in time, this can have a physical meaning of a desired signal X(t) plus the noise N(t), ie X(t) = Y(t) + N(t). As the X(t) is an information –bearing term, we would like to reduce the noise N(t), under the assumption that is independent of X(t). The well-known compromise is to consider the output of the linear system as the approximation Z*(t), while the theoretically ideal is Z(t) = h(t)*Y(t), with h(t) being the “appropriate chosen” function and * is the convolution. The target is the mean square error MSE=E{|Z*(t) – Z(t)|2} to be minimized, for a particular function h. A filter h is said to be the optimal, if it yields to the minimum MSE, over the class of filters under investigation.

Here are the most commonly chosen filters :

-Causal filters: integration for convolution from zero to infinity.

-With physical delay T: integration from –T to infinity.

-To operate on past data: integration from zero to T.

-Non-causal filters: over o given time domain

So the student has to be aware for the main points of stochastic process, for attending a course on the signal processing.

Differentiating a random process and the problem of sampling a continuous time process is out of question very important subjects. The student who tries to understand the “Mean square sampling Theory” or “Karhunen-Loeve expansion” without an appropriate statistical background, needless to say, has to try very hard. How he can explain why if a stochastic process X(t), with a spectral density function S(f) = No/2 for all f and hence the corresponding autocorrelation R(τ)=δ(τ), with δ(τ) the Dirac function is yielding to the strange behaviour that all eigenvalues are identical, so that all terms in the Karhumen-Loeve representation have the same second moment. The knowledge of stochastic processes is essential in electronics. Typical example cn be the Markov chains theory, which is applied to Reliability problems, when the system of components under investigation is supposed to be “parallel” or in “series”, or if can assume that we can or we cannot repair it, etc. Therefore the knowledge of the Markov stochastic process, although it is not so clear useful, it appears to be necessary eventually, see [11].

This is a very difficult approach for a student. He needs the appropriate input so that to expect some output, from him. Even the knowledge of software cannot be helpful. Therefore: strong tights between theory and practice are asked, and a skilled lecturer who can connect the practical application with the theoretical problems, giving more emphasis to the practice.

4 Conclusions

While we are interesting on evaluating the accuracy of statistical distributions in different packages, [5], there is not an international experience of what a student in electronics, of undergraduate level should be taught in Stat, while in Math is rather clear and internationally accepted. This is due, in principal, on the different number of semesters, in different European countries, were students have also a different background on Mathematics. Modern information technology society, permits a long distance teaching, to those who are interesting, through the World Wide Web site. A very informative site is the Statistics Site of Glasgow University:

Of course, yet, nobody has suggested that the course work in class has to be eliminated! We have introduced the distance learning courses in the world education system, through the Open Universities, but there is no course, at least in EE, without course-material and work in an undergraduate level.