On the Analysis of Mathematical Practices in Signal Theory Courses

Abstract

The contribution aims at subject-specific analyses of student solutions of an exercise from an electrical engineering signal theory course. The basis for the analyses is provided by praxeological studies (in the sense of the Anthropological Theory of Didactics) and the identification of two institutional mathematical discourses, one related to higher mathematics for engineers and one related to electrical engineering. Regarding the relationship between institutional observations and analyses of students’ solutions, we refer, among others, to Weber’s (1904) concept of ideal types. In the subject-specific analyses of student solutions we address in particular transitions and interrelations within single processing steps that refer to the two mathematical discourses and different forms of embedding of mathematics into the electrical engineering context. Finally, we present a few ideas for teaching.

Appendix: Exercise and sample solution

The exercise under consideration is structured in three items:

• 1. A message signal s(t) = cos(Ωt) has to be amplitude modulated. The result is x(t) = A[1 + m cos(Ωt)] cos(2πf₀t)

• 2. The result of item 1. Has to be written as the sum of three harmonics. The result is

$$ x(t)=A\cos \left(2\pi {f}_0t\right)+\frac{Am}{2}\cos \left(2\pi {f}_0t+\Omega t\right)+\frac{Am}{2}\cos \left(2\pi {f}_0t-\Omega t\right) $$

• 3. The result of item 2. Has then to be displayed graphically in the complex plane as a rotating phasor with varying amplitude.

Our analysis focusses item 3. of the exercise. The exact problem definition of item 3 is:

• 3. Graphically display x(t) in the complex plane as a rotating phasor with varying amplitude using the relationship $\cos \left(2\pi ft\right)=\mathfrak{R}\left\{\exp \left(j2\pi ft\right)\right\}$ and the result under item 2.

Sample solution:

One first writes

$$ x(t)=A\cos \left(2\pi {f}_0t\right)+\frac{Am}{2}\cos \left(2\pi {f}_0t+\Omega t\right)+\frac{Am}{2}\cos \left(2\pi {f}_0t-\Omega t\right)=A\mathfrak{R}\left\{\exp \left(j2\pi {f}_0t\right)\right\}+\frac{Am}{2}\mathfrak{R}\left\{\exp \left(j\left(2\pi {f}_0t+\Omega t\right)\right)\right\}+\frac{Am}{2}\mathfrak{R}\left\{\exp \left(j\left(2\pi {f}_0t-\Omega t\right)\right)\right\}=\mathfrak{R}\left\{\exp \left(j2\pi {f}_0t\right)\underset{A(t)}{\underbrace{\left[A+\frac{Am}{2}\exp \left(j\Omega t\right)+\frac{Am}{2}\exp \left(-j\Omega t\right)\right]}}\right\} $$

(1)

and interprets the expression in the square bracket as a real-valued time-dependent amplitude A(t), which modulates the carrier phasor exp(j2πf₀t) rotating at frequency f₀ in Fig. 13.

Fig. 13

Representation of x(t) = A[1 + m cos(Ωt)] cos(2πf₀t) as the real part of a rotating phasor A(t) exp(j2πf₀t) with ω₀ = 2πf₀