Home > Articles > Networking > Wireless/High Speed/Optical

Wireless Channel Modeling

The emergence of wireless systems that require more and more bandwidth means the task of conquering the wireless channel is becoming more difficult. This chapter presents an overview of channel modeling history in the context of wireless communications and demonstrates the need for studying the full space–time wireless channel. Topics covered include the historical context of wireless channel modeling, importance of the spatial channel interface, and trends in wireless influencing channel modeling.
This chapter is from the book

This chapter is from the book

There are few things in nature more unwieldy than the power-limited, space-varying, time-varying, frequency-varying wireless channel. Yet there is great reward for engineers who can overcome these limitations and transmit data through such harsh environments. The explosive worldwide growth of personal communications services through the 1990s is a testament to the business opportunities that result from conquering the wireless channel. However, given the emergence of newer wireless systems that require more and more bandwidth, the task of conquering the wireless channel is becoming more difficult. This task requires a thorough background in wireless channel modeling.

Furthermore, understanding and modeling the wireless channel provides beautiful insight into a number of other problems in the physical sciences. This book presents the basic principles for describing the random fading that plagues space–time wireless channels. Although most of the examples and discussions are in the context of commercial radio applications, it is possible to apply the theory to a wide range of problems in any field that involves dynamic wave propagation.

This chapter presents an overview of channel modeling history in the context of wireless communications and motivates the need for studying the full space–time wireless channel. This chapter includes Section 1.1: Historical context of wireless channel modeling. Section 1.2: Importance of the spatial channel interface. Section 1.3: Trends in wireless influencing channel modeling. Section 1.4: Content summary of this book.

Indeed, the current state of wireless communications points to a coming epoch when understanding the space–time channel is not a luxury, but an absolute necessity.

1.1 Perspectives in Propagation

To understand the importance of radio channel modeling, it helps to understand some history and background in the development of wireless communications. This section shows how the material presented in Space–Time Wireless Channels fits into the historical context of wireless technology development.

1.1.1 Early Years of Radio

The world's first digital radio system was actually the world's first radio system. Guglielmo Marconi's first wireless transmission in 1897 used Morse code (a digital representation of text) to communicate from ship to shore. He soon commercialized his technology by installing wireless systems in transatlantic ocean vessels. These Marconi wireless systems were first used to send distress calls to other nearby boats or shoreline stations, even in the famous luxury liner Titanic.

This first wireless system used a spark-gap transmitter, a glorified spark plug that sprayed electromagnetic waves in all directions at all frequencies. The spark-gap transmitter could be wired to send simple Morse code sequences, but the real challenge of the system was to receive the radio signal. For that, Marconi used a coherer, a device that could only detect the presence or absence of strong radio waves. This form of detection - coupled with the fact that only mechanical switching forms of signal amplification existed - meant that Marconi's wireless was only capable of digital transmission.


What Is a Coherer?

A coherer is a glass tube that contains loose metal filings resting on the bottom, as illustrated in Figure 1.1. Two contact wires are placed on opposite ends of the tube, allowing an external apparatus to measure the overall resistance to electrical current through the tube. Normally, resistance across loose metal filings is large due to the loose, jagged contact points between the small shards of metal. If a strong electromagnetic wave (i.e., from a nearby radio transmitter) travels across the filings, they cohere; the overall resistance drops and the radio wave is detected. To repeat detection, a coherer must be shaken mechanically to return the filings to their uncohered state.

Figure 1.1Figure 1.1 In the presence of a strong electric field, metal filings cohere and their overall electrical resistance drops.

The Marconi wireless was heavily limited in range and data speed by the power required to send and receive signals. However, radio communications - as well as every other electronic technology - changed in 1906 when Lee de Forest invented the first vacuum tube. The vacuum tube amplifies analog waveforms, so radio communication was liberated from its low-rate, on-and-oA keying. It was now possible to transmit high-fidelity analog signals, such as voice and music, over amplitude modulation (AM). Commercial AM radio stations proliferated across the world in the 1920s, as marked on the timeline in Figure 1.2.

Figure 1.2Figure 1.2 Some important milestones in radio communications.

The next great milestone in radio came in 1933, when E. H. Armstrong invented frequency modulation (FM). FM radio was the first example of signal-processing used to overcome the noisy, deleterious radio channel. In this case, the nonlinear modulation scheme of FM was capable of trading usable bandwidth for signal fi-delity. For once, engineers could design radio links with a degree of freedom other than transmit power.

Many other wireless devices followed (television, military radios, radar, etc.), but perhaps the most important and sublime milestone occurred in 1948 with Claude E. Shannon's publication of his famous "A Mathematical Theory of Communications" [Sha48]. There are two extremely important principles outlined in this paper that revolutionized the design of communication links:

All analog signals can be represented by sets of discrete digital symbols to a controllable degree of precision.

The fundamental rate at which digital symbols may be sent through any channel is a function of bandwidth, signal power, and noise power.

In essence, Shannon's theory predicted that digital communications, rather than analog communications, was the best way to send data through any link. It was only a matter of time before most radio communications would use digital modulation. It turned out to be a long time, however.

1.1.2 Cellular on the Scene

Digital communications may have been preferable to analog communications, but with the technology of 1948 it was still not possible. Any type of digital communications requires discrete signal-processing operations that simply were not possible (commercially) with the vacuum tube technology of the day. Only with the advent of solid-state devices, such as the transistor in 1947, could engineers implement the signal processing required for digital communications. Even this was not possible overnight, as solid-state electronics had to undergo years of research and development before producing integrated circuits (IC) that were fast enough and cheap enough to implement signal processing. In the meantime, the wireless industry continued developing analog radios.

The modern cellular phone and paging industry was birthed in the post-Shannon period using analog radio technology. In 1949, Al Gross (the inventor of the walkie-talkie) introduced the first mobile pager, for use by hospital doctors. In 1979, the first commercial cellular phone market opened in Tokyo using a type of analog FM modulation to send voice signals to users. Similar systems in North America and Europe followed. By the late 1980s, analog cellular communications was a commercial success, and companies were pressing government regulatory agencies to open up new radio spectrum for more voice services.

Cellular telephony presented wireless engineers with a uniquely diAerent design challenge. Previously, most radio systems operated in a noise-limited radio channel, where thermal noise was the sole source of signal degradation. As a mobile receiver moved in such a channel, fading would cause the signal power to fluctuate in space as the average noise power remained nearly constant; performance of the radio link depends on maintaining an adequate signal-to-noise ratio. Cellular networks, however, have interference-limited channels, because nearby cells reuse the same frequency spectrum. As illustrated in Figure 1.3, the interfering signal fluctuates in space along with the desired information signal. Careful spatial modeling becomes much more crucial for the noise-limited case.

Figure 1.3Figure 1.3 Fading for mobile communications causes sporadic moments of poor signal-to-interference+noise ratio (SINR) levels.

In 1993, the second generation of cellular telephone networks, called Personal Communications Services (PCS), were launched and rapidly spread throughout the world in just several years. Unlike their predecessors, these PCS networks were true digital communications systems, enabled by the cheap, fast, solid-state devices for signal-processing and radio-frequency (RF) electronics that became available at the end of the 1980s. Finally, the prophetic ideas of Shannon and digital information transmission had become a reality in the commercial wireless industry. In fact, the whole story of wireless communications is a story of great ideas followed by decades of incubation until the intellectual and industrial forces governing hardware development enabled implementation. Indeed, one must only look at recent vintages of proposed signaling techniques to understand the future of commercial wireless.

1.1.3 Origins of Channel Modeling

Most engineers are aware of the great inventions made at Bell Laboratories (the transistor, the laser, the communications satellite, to name just a few). Less appreciated, however, is the laboratory's enormous contribution to the theory of channel modeling and statistical communications analysis. The stochastic modeling work by Bell Laboratories researcher Stephen O. Rice stands out as one of the crucial achievements in describing radio communications. In 1944, Rice published a theory of random noise, which has since become a foundation of statistical analysis in communications [Ric44], [Ric45]. This work, originally used to characterize the noise in large-carrier AM and some FM signals, had extensive application for the description of fading signals, including level crossing rates, fade duration statistics, and the Rician envelope distribution that bears his name.

Still, the type of random processes described by Rice were signals with a single time dependency. In the early post-war period, there was little need for other types of analysis in radio communications. However, work began on the concept of mobile cellular telephony in the mid 1940s. As work progressed, it became obvious that newer channel descriptions were necessary - models that described the concept of spatial multipath fading. Unlike point-to-point microwave or satellite links, the cellular telephone user would operate receiver terminals buried within the clutter of a dense scattering environment, such as a cityscape or a neighborhood. In these types of propagation environments, objects such as terrain, buildings, and trees would block direct contact between a user terminal and a tower-top base station, but would instead provide a link between the two by scattering numerous low-powered radio waves from one to the other. These multipath waves arrive at the receiver from many diAerent directions and, as illustrated in Figure 1.4, create pockets of constructive and destructive interference in space. As a receiver moves through space, it rapidly experiences the peaks and nulls of multipath fading, often losing a signal momentarily, even though there is a large amount of average signal power propagating through the immediate area.

Figure 1.4Figure 1.4 Small-scale fading for a mobile receiver in a multipath environment.

Spatial fading is, perhaps, not a novel concept to most engineers. The first experience with spatial fading is usually encountered in a basic engineering or physics course on electromagnetism. The classic example is the transmission line, as shown in Figure 1.5. Recall that, unless the load is matched perfectly to the impedance of the transmission line, a wave sent down the line will partially reflect so that the net wave propagation is the superposition of two waves traveling in opposite directions. If we used a "mobile" power meter to measure the power along the transmission line, we would observe a standing wave interference pattern with peaks and nulls, each occurring at half-wavelength intervals.

Figure 1.5Figure 1.5 Signal fluctuation across a time-harmonic transmission line.

The transmission line example is a simple one. There are only two discrete waves. These waves, which are also scalar in the analysis, create a standing wave interference pattern that is regular and predictable. Contrast this to the case of multipath fading for a mobile wireless receiver. The multipath channel may have numerous waves of vastly diAering magnitudes that obey vector wave propagation laws.

Yet the transmission line example is still a useful analogy. For example, as the frequency of the voltage source is increased, the wavelength of radiation decreases and the distance between peaks and nulls across the transmission line shrinks proportionally. The same eAect is true of spatial multipath fading for a wireless mobile receiver. Modern cellular telephones operate in the upper UHF and microwave bands. In these bands, the wavelength of radiation is less than 1 m. At these wavelengths it is possible for a mobile receiver to receive a signal with a high-powered peak in one region of space and then, with just a few centimeters of movement, to receive virtually nothing.

As research on mobile telephony accelerated at Bell Laboratories during the 1960s, some important innovations in channel modeling were made. Researchers proposed the sum-of-waves model for spatial multipath [Oss64]. Clarke later extended this work to several basic scattering distributions, applying much of the Rician random process theory to the spatial fading for mobile receivers [Cla68]. Gans published a method for constructing a Doppler spectrum from the angles-of-arrival of multipath waves [Gan72]. Jakes published seminal work on the concept of space diversity - using multiple antennas to avoid deep signal fades [Jak71], [Jak74].

An important result in all of this research was the emergence of the omnidirectional Rayleigh fading model. This model assumes that radio waves arrive at the mobile receiver with equal power from all directions. This spatial channel models a fluctuating received signal strength with Rayleigh statistics. When a constant velocity is assumed by the mobile receiver, the model provides useful analytical expressions for channel coherence and fading statistics. This simple model is still a de facto standard in mobile radio system design.

1.1.4 Rayleigh Pessimism

The omnidirectional Rayleigh model is also unrealistic, but it is unrealistic in a way that endears it to engineers. It is a pessimistic channel model for conventional mobile receivers. There are two important characteristics of a mobile wireless receiver aAected by small-scale channel fading: fade margin and update rate. The omnidirectional Rayleigh is useful for calculating both of these parameters.

Fade Margin

Fade margin is a critical parameter in the site design of a cellular radio system. The mobile handset requires a minimum signal-to-interference+noise ratio (SINR) to maintain the specified data rate and not drop calls. This minimum SINR is most diLcult to achieve when the handset is operating on the fringe of a cell, where the distance between mobile unit and base station is the largest and received power is the weakest.

Because of small-scale fading, it is not enough to simply design a cellular network based on average power at the fringe of cells. If this were the case, a handset would drop a call at even the slightest signal fade, since the received signal power fluctuates over space. Instead, wireless engineers add anywhere from 12 dB to 18 dB of extra fade margin to the radio power link budget to ensure that small-scale fading does not drive the received signal below an acceptable level. On the business side of wireless, dB's translate into dollars, so the fade margin cannot be made arbitrarily high. The actual amount depends on the modulation scheme and the distribution of signal fades.

At this point, fading with Rayleigh statistics becomes a useful benchmark for a link design. Rayleigh statistics are considered by many to be a worst-case scenario of signal fading because the received signal strength experiences such deep fades (as we will see in Chapter 5). If a wireless receiver works in a Rayleigh fading channel, then it is likely to work in other types of channels. Thus, the fade margins of all systems are based on Rayleigh spatial fading statistics.

Update Rate

A narrowband receiver in a fading channel must apply some type of automatic gain control (AGC) to counter the unpredictable fading in received signal strength. An AGC unit in a handset is basically an amplifier with variable gain that increases when signal strength is low and decreases when signal strength is high. The key issue for an engineer designing AGC is update rate, the maximum speed that the receiver must change the gain in a realistic fading channel. To calculate this, we have to consider the worst-case scenario where fading is changing the fastest. The most pessimistic estimate would be the highest possible mobile user velocity in the propagation with the multipath arriving from all directions in space.

Once again, the omnidirectional Rayleigh model provides the pessimistic result. Since the angle-of-arrival is omnidirectional, it is impossible to generate fading with closer peaks and nulls in space (the reason for this becomes clearer in Chapter 6). A user talking in a car on the highway in an omnidirectional Rayleigh propagation environment will experience the fastest signal fading and will require the highest update rate. The update rate of most adaptive equalizers (of which the AGC is a simple example) are designed with this philosophy.

Changing Paradigm

Of course, Rayleigh pessimism is predicated on a purely passive design philosophy; the wireless engineer designs a communications system for worst-case fading scenarios and accepts whatever penalties are imposed upon a receiver by spatial fading. Emerging digital radios take more proactive eAorts to overcome channel fading with dynamic modulation, diversity, and channel coding schemes. Furthermore, wireless systems of the future may employ space–time processing to further combat the fading channel. In fact, as we will see in Chapter 9, these wireless systems exploit the presence of multipath to enhance the transmission of data through a wireless link - a far cry from the design philosophy of early mobile radio. Ironically, Rayleigh fading becomes an optimistic, best-case scenario of operation for these future systems -and it is very bad engineering practice to design with optimism.

1.1.5 Channels with Multiple Dependencies

Despite the advances in spatial channel modeling in the 1960s, the treatment of the wireless channel was still usually restricted to a single scalar dependency. Since the analysis of fading channels requires some elegant random process theory, there was diLculty in characterizing channels with full space, time, and frequency dependencies. The seminal contribution to this part of the puzzle comes from work performed at MIT's Lincoln Laboratories during the 1950s. A research group performed groundbreaking research in the field of radio astronomy, where a theoretical framework for studying stochastic signals of multiple dependencies was first developed [Gre62].

Out of this work came a brilliant piece of scholarship by P. A. Bello in 1963, which was the first research to describe stochastic communication channels that were combinations of time-varying and frequency-varying random processes [Bel63]. The work itself may have been a little ahead of its time, however, since broadband communications were not in widespread use (thus, no need for the frequency-varying aspects of channel models). As time marched onward, bandwidths became larger and joint channel dependencies became more important. The work by H. Hashemi in the late 1970s was some of the first to study truly random wideband temporal channels in the context of wireless communications [Has79].

Extensions of the original Lincoln Lab theory to joint spatial modeling of wireless channels were absent as late as the year 2000. R. G. Vaughan presented joint-dependency Fourier analysis techniques for space in [Vau00], and B. H. Fleury presented the first description of joint space–time–frequency wireless channels in [Fle00]. Clearly, the field of space–time wireless channel modeling is still in its infancy, with a great deal of work and innovation left to be done.

InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information

To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.


Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.


If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information

Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.


This site is not directed to children under the age of 13.


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information

If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.


Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

Sale of Personal Information

Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents

California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure

Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact

Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice

We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020