Wireless communication and mathematics
Wireless communication has become integral to modern life, facilitating everyday activities such as texting, voice calls, and internet access through the use of radio waves. At its core, wireless communication involves encoding information onto these waves, allowing for efficient transmission through the atmosphere. This complex process heavily relies on mathematical principles, particularly those rooted in information theory, which was pioneered by mathematician Claude Shannon. His work has significantly influenced various wireless technologies, including modems and multi-antenna systems.
Mathematical methods such as stochastic modeling, signal processing, and graph theory play essential roles in the development and optimization of communication networks. They help quantify performance metrics, design protocols for effective signal transmission, and address challenges associated with signal quality. Factors like absorption, diffraction, and scattering are critical considerations for engineers, as they affect how radio waves propagate and can impact overall communication quality.
Frequency, a key parameter in wireless communication, measures the rate of wave cycles and is expressed in hertz (Hz). Modern wireless networks typically operate within specific frequency ranges, such as 2.4 GHz and 5 GHz for consumer devices, while other applications like satellite communication may utilize significantly higher frequencies. Understanding these mathematical and engineering principles is essential for the ongoing advancement of wireless communication technologies.
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Wireless communication and mathematics
Wireless communication has become ubiquitous in the twenty-first century. Consider all of the aspects of one’s life that are impacted by wireless communications, including text messaging and voice calls over a cellular network, and e-mail and Web surfing over a wireless Internet connection. Wireless communication consists of encoding information onto radio waves and passing them through the atmosphere—not unlike how an amplitude modulation (AM) or frequency modulation (FM) radio signal is sent and received. Wireless communication would not be possible without mathematics, and mathematicians contribute in many ways to creating, sustaining, and studying wireless processes and technologies.
Information theory plays a central role in wireless communications; its origins are attributed to mathematician Claude Shannon in the mid-twentieth century. Sergio Verdu, who is cited as a world-renown researcher in wireless communications noted, “Claude Shannon was the archetypical seamless combination of mathematician and engineer.… Shannon’s theory has been instrumental in anything that has to do with modems, wireless communications, multi-antenna and so on.”
Many other theoretical and applied mathematical methods have also been fundamental in wireless communication. For example, methods like stochastic calculus, stochastic modeling, control theory, graph theory, game theory, signal processing, wavelets, simulation and optimization, and multivariate statistical analysis have been used to develop communication networks, quantify or predict performance characteristics like network traffic, and to create protocols for signal transmission, encryption, and compression. Some mathematical models have been used by developers to quantify and compare wired versus wireless communication systems.
Mathematicians and engineers working in wireless communications must consider the properties of the waves and how the information is encoded. Information, whether an e-mail, telephone, video, or other data, is encoded onto the sinusoidal waveform by combining changes in frequency, amplitude, and phase. This encoding is accomplished by modifying various properties of a periodic sinusoidal function—the carrier wave—to embed information or message wave on the carrier. Figure 1 shows a simple example for the case of AM. The height or amplitude of the carrier wave is modified to represent or information or modulating wave.
Researchers also consider the variety of factors that can affect the strength and quality of the signal. A communications engineer or technician is most often concerned with behaviors that will affect the propagation of the radio wave through the air. These include absorption, attenuation, diffraction, free space path loss, gain, reflection, refraction, and scattering. A combination of these factors will impact the signal quality and determine the likelihood of a successful transmission.
One common number associated with a wireless signal is the frequency. Frequency is a measure of how many cycles occur for a given time period. A signal cycle occurs every time a waveform repeats. Frequency is measured in cycles per second, which are also called “hertz” (Hz) after German physicist Heinrich Hertz. A waveform that repeats once every second has a frequency of 1 hertz. Waves used in communications are at much higher frequencies, so some prefixes must be used to measure radio frequencies. The wireless networks used for laptops and smartphones at the beginning of the twenty-first century often operate at the 2.4 GHz and 5 GHz frequencies of the spectrum. AM and FM radio are in the kHz or MHz frequencies, while satellites operate at very high frequencies—often in the hundreds of GHz.
Bibliography
Agrawal, Prathima, Daniel Andrews, Philip Fleming, George Yin, and Lisa Zhang. “Wireless Communications.” In IMA Volumes in Mathematics and its Applications Series. Vol. 143. New York: Springer, 2007.
Boche, Holger, and Andreas Eisenblatter. “Mathematics in Wireless Communications.” In Production Factor Mathematics. Edited by Martin Grotschel, Klaus Lucas, and Volker Mehrmann. Berlin: Springer, 2010.
Leong, Y. K. “Mathematical Conversations—Sergio Verdu: Wireless Communciations, at the Shannon Limit.” National University of Singapore Newsletter of the Institute for Mathematical Sciences 11 (September 2007). http://www.princeton.edu/~verdu/singapore.pdf.