Home > Articles

This chapter is from the book

5.7 Small-Scale Fading Selectivity

In this section, we discuss small-scale fading effects. These effects, which occur on the order of a wavelength distance, are caused by the constructive and destructive interference of multipath components. These effects manifest in the time domain, leading to time selectivity or, in the frequency domain, leading to frequency selectivity. These two forms of selectivity are independent of each other, under some assumptions. Before proceeding with the mathematical details, we first develop the intuition behind time- and frequency-selective channels. Then we present the foundations for determining whether a channel is frequency selective or time selective. We conclude with showing the potential system models used in each fading regime.

5.7.1 Introduction to Selectivity

In this section, we introduce frequency selectivity and time selectivity. We pursue these in further detail in subsequent sections.

Frequency selectivity refers to the variation of the channel amplitude with respect to frequency. To understand how multipath plays an important role in frequency selectivity, consider the following two examples. Figure 5.32(a) illustrates the frequency response with a single channel tap. In the frequency domain, the Fourier transform of the single impulse function yields a flat channel; that is, the channel amplitude does not vary with frequency. We explored the flat channel in Section 5.1. Alternatively, in Figure 5.32(b), the channel impulse response has a significant amount of multipath. In the frequency domain (taking the Fourier transform of the sum of shifted impulse functions), the resultant channel varies considerably with frequency.

FIGURE 5.32

Figure 5.32 A single path channel (a) and a multipath channel (b). The single path has a frequency-flat response whereas the multipath channel has a frequency-selective response.

The effect of frequency selectivity, though, depends critically on the bandwidth used to communicate in that channel. For example, illustrated in Figure 5.33 are the discrete-time complex baseband equivalent channels for two different choices of bandwidth, 1MHz and 10MHz. With the smaller bandwidth, the impulse response looks nearly like a discrete-time delta function, corresponding to a flat channel. With the larger bandwidth, there are more taps (since the symbol period is smaller) and there are many more significant taps, leading to substantial intersymbol interference. Therefore, the same channel can appear frequency flat or frequency selective depending on the signal bandwidth. The range of frequencies over which the channel amplitude remains fairly constant is the coherence bandwidth.

FIGURE 5.33

Figure 5.33 (a) Discrete-time complex baseband equivalent channel for a five-path channel with a narrow bandwidth; (b) discrete-time complex baseband equivalent channel for a five-path channel with a bandwidth ten times larger

Time selectivity refers to the variation in the channel amplitude as a function of time. The degree or extent of time selectivity of a channel is measured using the coherence time of the channel. This simply refers to the time duration over which the channel remains fairly constant. The coherence time provides guidance on when the received signal model can be assumed to be LTI. Time selectivity is a function of the mobility present in the channel and is usually measured using the Doppler spread or the maximum Doppler shift.

For the coherence time to be useful, it must be compared with some property of the signal. For example, suppose that the receiver processes data in packets of length Ntot. For the LTI assumption to be valid, the channel should be roughly constant during the entire packet. Then, if TNtot is less than the coherence time, we might say that the LTI assumption is good and the channel is time invariant or slow fading. This does not mean that the channel does not vary at all; rather it means that during the window of interest (in this case Ntot symbols with period T) the channel can be considered to be time invariant. If TNtot is greater than the coherence time, then we might say that the LTI assumption is not good and that the channel is time selective or fast fading.

Note that the bandwidth also plays a role in determining the coherence time of a signal. Assuming a sinc pulse-shaping filter, the symbol period T = 1/B. As a result, making the bandwidth larger reduces the duration of the Ntot symbols. Of course, increasing the bandwidth makes it more likely that the channel is frequency selective. This shows how the time and frequency variability of a channel are intertwined.

The time and frequency selectivity are coupled together. For example, if the receiver is moving and the multipaths come from different directions, each has a different Doppler shift. As a result, the time variability with several multipaths is more severe than with a single multipath. Fortunately, making some statistical assumptions about the channel allows the time selectivity to be decoupled from the frequency selectivity. Specifically, if the channel is assumed to satisfy the wide-sense stationary uncorrelated scattering assumption (WSSUS) [33], then the time-selective and frequency-selective parts of the correlation function can be decoupled, and the decision about a channel’s selectivity can be made separately based on certain correlation functions, discussed in the next sections. Quantities computed from these correlation functions are compared with the bandwidth of the signal of interest, the symbol period, and the block length Ntot to determine the effective selectivity.

To explain this mathematically, suppose that the continuous-time complex baseband equivalent channel impulse response is h(t, τ) where t is the time and τ is the lag. This is a doubly selective channel, in that it varies both with time and with lag. The channel is measured assuming a large bandwidth, typically larger than the eventual bandwidth of the signal that will be used in this channel. For example, measurements might be made in a 500MHz channel and the results of those measurements used to decide how to partition the bandwidth among different signals. We use continuous time as this is historically how the WSSUS description was developed [33], but interpretations are also available in discrete time [152, 301].

Under the WSSUS model, a statistical description of the channel correlations is made through two functions: the power delay profile Rdelay(τ), and the Doppler spectrum SDoppler(f). The power delay profile gives a measure of how the energy of the channel varies in the lag domain for two signals of very short duration (so Doppler is neglected). The Doppler spectrum measures essentially how much power is present in different Doppler shifts of two narrowband signals (so lag is neglected). Frequency selectivity is decided based on Rdelay(τ), and time selectivity is decided based on SDoppler(f). In the next sections, we explore further the idea of determining the selectivity of a channel based on the power delay profile and the Doppler spectrum, or their Fourier transforms.

5.7.2 Frequency-Selective Fading

We use the power delay profile Rdelay(τ) to determine whether a channel is frequency-selective fading. The power delay profile is typically determined from measurements; common power delay profiles can be found in different textbooks and in many standards. For example, the GSM standard specifies several different profiles, including parameters like typical urban, rural, bad urban, and others. Intuitively, in a flat channel (in the bandwidth where the power delay profile was measured) Rdelay(τ) should be close to a delta function.

The typical way to measure the severity of a power delay profile is based on the root mean square (RMS) delay spread. Define the mean excess delay as

05equ317.jpg

and the second moment as

05equ318.jpg

Then the RMS delay spread is the difference:

05equ319.jpg

With this definition, a channel is said to be frequency flat if the symbol period satisfies T gt.jpg σRMS,delay. This means that the effective spread is much smaller than a symbol, so there will be little ISI between adjacent symbols.

Example 5.30 Consider the exponential power delay profile Rdelay(τ) = e−τ/γ. Determine the RMS delay spread.

Answer: The mean excess delay is

05equ320.jpg

The second moment is

05equ323.jpg

Therefore, the RMS delay spread is

05equ326.jpg

Therefore, the value γ is the RMS delay spread.

The Fourier transform of the power delay profile is known as the spaced-frequency correlation function:

05equ328.jpg

It measures the correlation as a function of the difference Δlag = f2f1 between sinusoids sent on two different carrier frequencies. The spaced-frequency correlation function is used to define the coherence bandwidth of the channel. One definition of coherence bandwidth is the smallest value of Δlag such that |Sdelay(Bcoh)| = 0.5Sdelay(0). Essentially this is the first point where the channel becomes decorrelated by 0.5.

It is common to define the coherence bandwidth based on the RMS delay spread. For example,

05equ329.jpg

The coherence bandwidth is interpreted like traditional bandwidth and is meant to give a measure over which the channel is (statistically speaking) reasonably flat. In particular, a channel is flat if the bandwidth B lt.jpg Bcoh. There are several different definitions of coherence bandwidth in the literature; all have an inverse relationship with the RMS delay spread [191, 270].

Example 5.31 Determine the spaced-frequency correlation function, the coherence bandwidth from the spaced-frequency correlation function, and the coherence bandwidth from the RMS delay spread for the exponential power delay profile in Example 5.30.

Answer: The spaced-frequency correlation function is

05equ330.jpg

To find the coherence bandwidth from the spaced-frequency correlation function:

05equ333.jpg

and

05equ334.jpg

The smallest nonnegative value of Δlag that determines the coherence bandwidth is

05equ335.jpg

Based on the RMS delay spread,

05equ336.jpg

Since e298-1.jpg and 1/5 = 0.2, these two expressions differ by a factor of 2. The coherence bandwidth based on the RMS delay spread is more conservative between the two measures in this case.

In practice, the power delay profile or spaced-frequency correlation function is determined from measurements. For example, suppose that a training signal is used to generate channel estimate h[n, ℓ] at time n. Then the discrete-time power delay profile can be estimated from M observations as e298-2.jpg. The spaced-frequency correlation function Sdelaylag) could be estimated by sending sinusoids at Δf = f2f1 and estimating the correlation between their respective channels at several different times. Or it could be computed in discrete time using an OFDM system by taking the DFT of each channel estimate H[n, k] for a given n, assuming K total subcarriers, then estimating the spaced-frequency correlation as a function of subcarriers as e298-3.jpg.

5.7.3 Time-Selective Fading

The Doppler spectrum SDoppler(f) is used to determine if a channel is time-selective fading. The Doppler spectrum can be estimated via measurements or more commonly is based on certain analytical models. Given a general Doppler spectrum, a common approach for determining the severity of the Doppler is to define an RMS Doppler spread σRMS,doppler in the same way the RMS delay spread is defined. Then a signal is considered to be time invariant if B gt.jpg σRMS,doppler. In mobile channels, the maximum Doppler shift may be used instead of the RMS Doppler spread. The maximum Doppler shift is fm = fcν/c where ν is the maximum velocity and c is the speed of light. The maximum shift occurs when the transmitter is moving either straight to or straight from the receiver with velocity ν. For many systems, the maximum Doppler shift gives a reasonable approximation of the RMS Doppler spread, typically leading to a more conservative definition of time selective.

The maximum Doppler shift varies as a function of mobility. When there is higher mobility, the velocity is higher. From a system design perspective, we often use the maximum velocity to determine the coherence time. For example, a fixed wireless system might assume only pedestrian speeds of 2mph, whereas a mobile cellular system might be designed for high-speed trains that travel at hundreds of miles per hour.

Example 5.32 Determine the maximum Doppler shift for a cellular system with fc = 1.9GHz that serves high-speed trains with a velocity of 300km/h.

Answer: The velocity of 300km/h in meters per second is 83.3m/s. The maximum Doppler shift is then

05equ337.jpg

Analytical models can also be used to determine the Doppler spectrum. Perhaps the most common is the Clarke-Jakes model. In this model, the transmitter is stationary while the receiver is moving with a velocity of ν directly toward the transmitter. There is a ring of isotropic scatterers around the receiver, meaning that multipaths arrive from all different directions and with different corresponding Doppler shifts. The Doppler spectrum under this assumption is

05equ339.jpg

Plotting the Clarke-Jakes spectrum in Figure 5.34, we see what is known as the horned Doppler spectrum.

FIGURE 5.34

Figure 5.34 The Clarke-Jakes spectrum plotted for fm = 533Hz

The time selectivity of the channel can also be determined from the spaced-time correlation function, which is the inverse Fourier transform of the Doppler spectrum:

05equ340.jpg

The spaced-time correlation function essentially gives the correlation between narrowband signals (so delay spread can be neglected) at two different points in time Δtime = t2t1. The coherence time of the channel is defined as the smallest value of Δtime such that |RDopplertime)| = 0.5RDoppler(0). The resulting value is Tcoh. It is common to define the coherence time based on either the RMS Doppler spread or the maximum Doppler shift, in the same way that the coherence bandwidth is defined. A channel is said to be LTI over block Ntot if TNtot lt.jpg Tcoh. For the Clarke-Jakes spectrum, it is common to take Tcoh = 0.423/fm [270].

Example 5.33 Determine the coherence time using the maximum Doppler shift for the same situation as in Example 5.32. Also, if a single-carrier system with 1MHz of bandwidth is used, and packets of length Ntot = 100 are employed, determine if the channel is time selective.

Answer: The coherence time is Tcoh = 1/(5fm) = 0.375ms. With a bandwidth of 1MHz, T is at most 1µs, assuming sinc pulse shaping, less if other forms of pulse shaping are used. Comparing NtotT = 100µs with Tcoh, we can conclude that the channel will be time invariant during the packet.

For the Clarke-Jakes spectrum, the spaced-time correlation function can be computed as

05equ341.jpg

where J0(·) is the zeroth-order Bessel function. The spaced-time correlation function is plotted in Figure 5.35. The ripples in the temporal correlation function lead to rapid decorrelation but do show some longer-term correlations over time.

FIGURE 5.35

Figure 5.35 The spaced-time correlation function corresponding to the Clarke-Jakes spectrum plotted for fm = 533Hz

An interesting aspect of time-selective fading is that it depends on the carrier. Normally we use the baseband equivalent channel model and forget about the carrier fc. Here is one place where it is important. Note that the higher the carrier, the smaller the coherence time for a fixed velocity. This means that higher-frequency signals suffer more from time variations than lower-frequency signals.

In practice, it is common to determine the spaced-time correlation function from measurements. For example, suppose that short training sequences are used to generate channel h[n, ℓ] at time n over N measurements. The spaced-time correlation function may then be estimated as e300-1.jpg. The Doppler spectrum could also be measured through estimating the power spectrum.

5.7.4 Signal Models for Channel Selectivity

The selectivity of a channel determines which signal processing channel model is appropriate. As a result of the decomposition of time and frequency selectivity, there are four regions of selectivity. In this section, we present typical models for each region and comment on the signal processing required at the receiver in each case.

Time Invariant/Frequency Flat

In this case, the equivalent system, including the channel, carrier frequency offset, and frame delay (supposing symbol synchronization has been performed), can be written as

05equ342.jpg

for n = 0, 1, . . . , Ntot − 1. The signal processing steps required at the receiver were dealt with extensively in Section 5.1.

Time Invariant/Frequency Selective

In this case, the equivalent system, including the channel, carrier frequency offset, and delay, is

05equ343.jpg

for n = 0, 1, . . . , Ntot − 1 where the impulse response e301-1.jpg includes the effects of multipath, the transmit pulse shape, and receive matched filter, as well as any symbol synchronization errors. The signal processing steps required at the receiver were dealt with extensively in Section 5.2 through Section 5.4.

Time Variant/Frequency Flat

Assuming that the channel changes slowly with respect to the symbol period but changes faster than TNtot,

05equ344.jpg

for n = 0, 1, . . . , Ntot − 1. If the channel changes too fast relative to T, then the transmit pulse shape will be significantly distorted and a more complex linear time-varying system model will be required. We have incorporated the presence of a small carrier frequency offset into the time-varying channel; the presence of larger offsets would require a different model.

One way to change the channel estimation and equalization algorithms in Section 5.1 is to include a tracking loop. The idea is to use periodically inserted pilots (at less than the coherence time) to exploit the correlation between the samples of h[n]. In this way, an estimate of e302-1.jpg can be generated by using concepts of prediction and estimation, for example, a Wiener or Kalman filter [172, 143]. Another approach is to avoid channel estimation altogether and resort to differential modulation techniques like DQPSK. Generally, these methods have an SNR penalty over coherent modulations but have relaxed or no channel estimation requirements [156, 287].

Time Variant/Frequency Selective

Assuming that the channel changes slowly with respect to the symbol period but changes faster than TNtot,

05equ345.jpg

for n = 0, 1, . . . , Ntot − 1. The channel is described by a two-dimensional linear time-varying system with impulse response e302-2.jpg, which is often called a doubly selective channel. We have again incorporated the presence of carrier frequency offset into the time-varying channel. If the channel changes a lot, then some additional Nyquist assumptions may need to be made and the discrete-time model would become much more complex (typically this is required only for extremely high Dopplers).

The time- and frequency-selective channel is the most challenging from a signal processing perspective because the channel taps change over time. Operating under this kind of channel is challenging for two reasons. First, estimating the channel coefficients is difficult. Second, even with the estimated coefficients, the equalizer design is also challenging. One way to approach this problem is to use basis expansion methods [213] to represent the doubly selective channel as a function of a smaller number of more slowly varying coefficients. This parametric approach can help with channel estimation. Then modifications of OFDM modulations can be used that are more suitable for time-varying channels [320, 359]. Operating in the time- and frequency-selective region is common for underwater communication [92]. It is not yet common for terrestrial systems, though there is now growing interest by companies like Cohere Technologies [230].

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.

Overview


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.

Surveys

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.

Newsletters

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.

Security


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

Children


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

Marketing


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.

Choice/Opt-out


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.

Links


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