Abstract
We add fuzziness to the signals sent through a continuous Gaussian transmission channel: fuzzy signals are modeled by means of triangular fuzzy numbers. Our approach is mixed, fuzzy/stochastic: below we do not call into question the probabilistic nature of the channel, and fuzziness will concern only the channel inputs. We argue that fuzziness is an adequate choice when one cannot control crisply each signal fed to the channel. Using the fuzzy arithmetic of interactive fuzzy numbers, we explore the impact that a fuzziness constraint has on channel capacity; in our model we are ready to tolerate a given fuzziness error F. We take the chance to put forward a remarkable case of “irrelevance” in fuzzy arithmetic.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Actually \(f(x_1, \ldots , x_n)=g(x_1, \ldots , x_n)\) is an identity for crisp numbers \(x_1, \ldots , x_n\) if and only if the two fuzzy quantities \(Z_1 \dot{\,=\,} f (X_1, \ldots , X_n)\) and \(Z_2 \dot{\,=\,}g(X_1, \ldots , X_n)\) are deterministically equal whatever the joint distribution of the n fuzzy quantities \(X_1, \ldots , X_n\); deterministic equality is formally defined in this Section. Cf. e.g. [9] for a discussion of this straightforward but important result, called there the Montecatini lemma.
- 2.
Our functions below are quite well-behaved, and so we write directly maxima rather than suprema. With a slight imprecision, when the support of a function (the set where the function is positive) is an interval, the interval will be closed anyway.
- 3.
Else, one might re-scale each \(S_i\) of fuzziness c, \(c>0\), to \(S_i / c\) of fuzziness 1, and consequently re-scale P and N (cf. Sect. 4) dividing them by \(c^2\).
- 4.
This “forced linearization” is not new in soft computing, when one replaces genuine products of triangular numbers by pseudo-products which “force” linearity on the result; in practice, going to \(W^*\) amounts to replace genuine squares as in (3) by pseudo-squares.
- 5.
Just as a hint, using the random coding technique to construct optimal codes, in a codeword \(s_1 \ldots s_n\), each \(s_i\) can be seen as the output of a Gaussian distribution \(\mathcal{N}(0,\varPi )\), where the expected normalized sum of squares of the random outputs is constrained to be \(\varPi \); expectation is unconditionally additive, and so the expectation of the normalized sum of the absolute values is soon computed to be \( \sqrt{(2 / \pi ) \varPi }\), a value approximated with high probability by the actual outputs if n is large; cf. [4] for details.
- 6.
Note that we are assuming triangular fuzzy signals of unit fuzziness: were it not so, the last bound would be on \({ P \over c^2}\) rather than P, cf. footnote 2.
- 7.
As a hint to computations, from the second equation obtain \(u \dot{\,=\,}1 + 2\sqrt{(2 /\pi )\varPi }\) as a function of \(\zeta \) and F to be replaced in the first equation re-written as \( 1 + 2\sqrt{(2 /\pi ) (P - \zeta ) } =u\); for given F and P deal numerically with the resulting fourth-degree equation in \(\zeta \).
References
Bortolussi L, Sgarro A (2010) Possibilistic coding: error detection vs. error correction. In: Borgelt Ch et al (ed) Combining soft computing and statistical methods in data analysis. Advances in intelligent and soft computing, vol 77. Springer, pp 41–48
Bortolussi L, Dinu LP, Franzoi L, Sgarro A (2015) Coding theory: a general framework and two inverse problems. Fundam Inf 141:297–310
Carlsson C, Fullér R, Majlender P (2004) Additions of completely correlated fuzzy numbers. In: IEEE international conference on fuzzy systems, pp 535–539
Coroianu L, Fullér R (2013) On multiplication of interactive fuzzy numbers. In: IEEE International symposium on intelligent systems and informatics, SISY, pp 181–185
Cover TM, Thomas JA (2006) Elements of information theory. Willey series in telecommunications. Wiley, New York
Dubois D, Prade H (1981) Additions of interactive fuzzy numbers. IEEE Trans Autom Control AC-26 4:926–930
Dubois D, Prade H (2000) Fundamentals of fuzzy sets. Kluwer Academic Publishers
Franzoi L, Sgarro A (2015) (Ir)relevance of interactivity in fuzzy arithmetic. Math Pannonica 25(1):93–103
Fullér R, Majlender P (2004) On interactive fuzzy numbers. Fuzzy Sets Syst 143(3):353–369
Klir GJ, Folger TA (1988) Fuzzy Sets. Uncertainty and information. Prentice Hall
Sgarro A (2002) Possibilistic information theory: a coding-theoretic approach. Fuzzy Sets Syst 132(2):11–32
Sgarro A, Franzoi L (2012) Fuzzy arithmetic for fuzzy n-poles. In: Proceedings of IPMU, vol 3, pp 1–8
Sgarro A, Franzoi L (2016) (Ir)relevant T-norm joint distributions in the arithmetic of fuzzy quantities. Accepted by IPMU 2016, Eindhoven, Netherlands, 20–24 June 2016
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing Switzerland
About this paper
Cite this paper
Franzoi, L., Sgarro, A. (2017). Fuzzy Signals Fed to Gaussian Channels. In: Ferraro, M., et al. Soft Methods for Data Science. SMPS 2016. Advances in Intelligent Systems and Computing, vol 456. Springer, Cham. https://doi.org/10.1007/978-3-319-42972-4_28
Download citation
DOI: https://doi.org/10.1007/978-3-319-42972-4_28
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-42971-7
Online ISBN: 978-3-319-42972-4
eBook Packages: EngineeringEngineering (R0)