Abstract
The basic mathematical techniques that are relevant to the user of this book, are summarized. The main equations for trigonometric operations, Taylor series expansions, Fourier, Laplace and z-transforms are given. Statistics play an important role in today’s engineering practice. Some common probability distribution functions are presented and the properties summarized. The emphasis of this section is on practical application of statistics with introducing robust parameter estimation and avoiding various pitfalls.
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Notes
- 1.
This confusion is related to the use of the “long scale” numbering system in continental Europe and the “short scale” numbering system in the USA and UK.
- 2.
Paul Dirac (1902–1984) was a British physicist who made significant contributions to the development of quantum mechanics. In 1933 he received with Schrödinger the Nobel Prize.
- 3.
Strange in the sense that many normal mathematical operations cannot be performed, for instance δ2(t) does not exist.
- 4.
Integrand is the function being integrated.
- 5.
Obviously the Fourier transform can be used in many more domains, for instance the spatial domain. In this book the dominant use of Fourier transforms is in the time domain.
- 6.
This is an engineering book after all, see the math textbooks for lengthy and formal analyses.
- 7.
In modern communication theory, single side-band modulation coins its two signal components with a 90∘ phase relation (I/Q) as the “real part” and an “imaginary part.” This is a form of notation on a higher level than straight-forward Fourier analysis of a physical signal.
- 8.
THD = −43.8 dB is a short hand for 10−43.8∕10 = 4.167 × 10−5 power ratio. Use the exponential notation in complex calculations.
- 9.
The signal-to-noise ratio is defined in the opposite way: signal power divided by noise power. Therefore the minus sign that normally precedes the THD number is sometimes omitted.
- 10.
Using goniometric equivalences is an engineering short cut. The Fourier series expansion gives the same results.
- 11.
Similar to the Fourier analysis, an exception is the (real) I and Q signals in modern communication systems.
- 12.
“Stochastic” is the term to describe that in a process or equation random variables are used.
- 13.
Another term for a Gaussian distribution is a normal distribution (without the word “standard”). Social scientists call this the “bell-curve.”
- 14.
There are other definitions for autocorrelation. Some authors subtract the mean value x(t) → (x(t) − E(x)), others normalize the result: Rxx(τ)∕Rxx(0) or do both.
References
R.D. Carmichael, E.R. Smith, Mathematical Tables and Formulas (Dover Publications, New York, 1931). ISBN: 486-60111-0
M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions (Dover publications, New York, 1965). ISBN: 0-486-61272-4
W.H. Beyer, CRC Standard Mathematical Tables, 28th edn. (CRC Press, Boca Raton, 1987). ISBN: 0-8493-0628-0
W. Sansen, Distortion in elementary transistor circuits. IEEE Trans. Circuits Syst. II 46, 315–325 (1999)
W.J.J. Rey, Introduction to Robust and Quasi-Robust Statistical Methods (Springer, Berlin, 1983). ISBN: 0-387-12866-2
A. Papoulis, Probability, Random Variables, and Stochastic Processes. McGrawHill New York student edition 1965. 4th edition ISBN: 0-073-66011-6, McGrawHill, 2001
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Pelgrom, M.J.M. (2022). Mathematics. In: Analog-to-Digital Conversion. Springer, Cham. https://doi.org/10.1007/978-3-030-90808-9_2
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