This chapter is from the book
This chapter is from the book
This chapter is from the book
3.4 Width of the PSD
It is a basic principle in Fourier analysis that as a function becomes narrower in one domain, its Fourier transform pair becomes wider in the other domain. Therefore, the width of a PSD can provide insight into the coherence of a channel: As the PSD widens in the transform domain, its autocorrelation becomes narrower and coherence decreases. The most common (and useful) definition of a channel’s PSD width is the root-mean-square (RMS) width. RMS widths may be defined for PSDs as a function of delay, Doppler, or wavenumber.
3.4.1 RMS Delay Spread
The RMS delay spread is defined from the delay spectrum of a stochastic channel. Mathematically, this delay spread is the second centered moment of this spectrum, defined as
)
A larger delay spread implies increased frequency selectivity and a smaller coherence bandwidth. Many wireless engineers calculate
the coherence bandwidth using the rule of thumb )
.
Example 3.2: Exponential Delay Spectrum
Problem: The received power of multipath components typically decreases exponentially as a function of time delay. As a result, many engineers approximate the delay spectrum as an exponential-shaped function. Develop expressions for delay spectrum and autocorrelation as a function of RMS delay spread, στ.
Solution: If the RMS delay spread of an exponential-shaped spectrum is στ, then the delay spectrum must take the form
)
where S0 is an arbitrary constant and u(τ) is the unit step function. Computing the inverse Fourier transform of this function produces the frequency autocorrelation:
)
The shape and widths of the delay spectrum and frequency autocorrelation are shown below:
)
Note the inverse proportionality between the widths of the spectrum and the autocorrelation as the RMS delay spread, στ, increases.
3.4.2 RMS Doppler Spread
In definition, the RMS Doppler spread is nearly identical to RMS delay spread. This delay spread is calculated as the second centered moment of the Doppler spectrum:
)
An increased Doppler spread implies a channel with faster temporal fluctuations and a smaller coherence time.
Example 3.3: Gaussian Doppler Spectrum
Problem: The power in the Doppler spectrum of many types of time-varying channels is concentrated heavily at ω = 0 and quickly diminishes for larger values of |ω|. One model for this type of behavior is a Gaussian-shaped Doppler spectrum. Develop expressions for Doppler spectrum and autocorrelation as a function of Doppler spread, σω.
Solution: If the RMS Doppler spread of a Gaussian-shaped spectrum is σω, then the Doppler spectrum must take the following form:
)
where S0 is some arbitrary constant. The temporal autocorrelation, )
, is the inverse Fourier transform:
)
The shape and relative widths of the Doppler spectrum and temporal autocorrelation are shown below:
)
Note the inverse proportionality between the widths of the spectrum and the autocorrelation as the RMS Doppler spread, σω, increases.
3.4.3 RMS Wavenumber Spread
The definition for RMS wavenumber spread follows immediately from the definitions of the previous RMS spreads. The definition for wavenumber spread is then given by
)
Increased wavenumber spread implies a channel with more rapid spatial fluctuations and a smaller coherence distance.
Example 3.4: Omnidirectional Wavenumber Spectrum
Problem: A common model for wavenumber spectrum is the Clarke omnidirectional spectrum, which arises in cluttered outdoor environments when multipath power arrives from the horizon in every direction [Cla68]. The wavenumber spectrum for this case is
)
where k is wavenumber, S0 is an arbitrary constant, and k0 is the maximum free-space wavenumber. Find expressions for the RMS wavenumber spread and spatial autocorrelation.
Solution: The wavenumber spread may be calculated in terms of k0 using Equation (3.4.3):
)
The spatial autocorrelation of this wavenumber spectrum is
)
where J0(·) is a zero-order Bessel function.
The U-shaped spectrum in Example 3.4 is a famous classical result in mobile radio propagation, although it is usually presented in textbooks as a Doppler spectrum instead of a wavenumber spectrum. These textbooks also characterize spatial fading, although there is an implied distance = velocity × time (r = vt) substitution. This substitution converts a space-varying channel to a time-varying channel, as discussed in Chapter 2, Section 2.3.4.
3.4.4 Channel Duality Principle
By now it is obvious that a tremendous amount of similarity exists in the stochastic characterization of time, frequency, and space. An analytical technique developed for one dependency is equally applicable to the others, whether it is an autocorrelation function, a PSD, an RMS width, or other quantity. This is the principle of duality in channel modeling [Bel64].
The time, frequency, and spatial duals of the basic stochastic concepts discussed in this section are summarized by Table 3.3. Despite that each dependency characterizes a completely different aspect of the wireless channel, the concepts and terminology used to study each are the same. A generic summary of the random channel characterization presented thus far might go something like this:
The stochastic wireless channel as a function of (time, frequency, space) displacement may be characterized by an autocorrelation function. The Fourier transform of the autocorrelation produces a (Doppler, delay, wavenumber) spectrum. The width of this spectrum is characterized by its RMS (Doppler, delay, wavenumber) spread. As this spread increases, the (temporal, frequency, spatial) selectivity of the channel increases and the coherence (time, bandwidth, distance) decreases.
The above statement may be read several times, substituting the first, second, and third words in each set with each respective reading.
Table 3.3. Channel Duality Relationships Between Time, Frequency, and Space.
|
TIME |
FREQUENCY |
SPACE |
|
|---|---|---|---|
|
Dependency |
time, t |
frequency, f |
position, r |
|
Coherence |
time, Tc |
bandwidth, Bc |
distance, Dc |
|
Spectral Domain |
Doppler, ω |
delay, τ |
wavenumber, k |
|
Spectral Width |
Doppler spread, σω |
delay spread, στ |
w.n. spread, σk |
Example 3.5: Coherence Time Duality
Problem: The RMS Doppler spread, σω, of a time-varying, narrowband channel is known. Estimate the coherence time, Tc.
Solution: We have yet to formally define coherence time. However, it was stated in Section 3.4 that the coherence bandwidth, Bc, for a dispersive channel is inversely proportional to 5στ. It follows by duality that a similar rule of thumb should work for time-varying channels. The following substitutions are
made: Bc ⇒ Tc and )
(the 2π is the difference between
Fourier transform definitions of the Doppler and delay domains). The "new" coherence time rule of thumb is now
)
which agrees with rule-of-thumb estimates presented in [Rap02a].
3.4.5 Definition of a Rate Variance
Before continuing the discussion on angle spectra, it is important to introduce the concept of a fading rate variance. The fading rate variance is a key second-order statistic that determines how quickly a channel fluctuates as a function of time, space, or frequency.
Consider the snapshots of two stochastic processes in Figure 3.5. Both processes represent time-varying voltages and both have magnitudes that follow an identical Rayleigh PDF (which are introduced in Chapter 5). Clearly, they are not the same process, as Signal 1 fluctuates much faster than the process on the bottom. Their second-order statistics, which relate to development over time, are quite different.
)
Figure 3.5. Time-varying, Rayleigh-distributed stochastic processes with different second-order statistics.
Although WSS second-order statistics are best characterized with either an autocorrelation function or a PSD, there are simpler
measures of how rapidly a process evolves over time, space, or frequency. One such measure may be based on the derivative of the channel. Consider the narrowband, fixed, time-varying channel, )
. A useful measure of fluctuations could be based on )
. Simply taking the ensemble average of this quantity is not useful, since the mean derivative of all WSS processes is 0.
Rather, taking the ensemble average of )
produces a result which, true to our intentions, measures the amount of fluctuation of the stochastic time-varying process,
.
There is one final and subtle adjustment that must be made to our fluctuation measure. For a complex-valued process such as a baseband channel, we would like to measure the fluctuations of a process that has stationary phase. For example, a WSS stochastic channel has a phase that is itself a stochastic process. This process may be written as
)
If the centroid of the Doppler spectrum, )
, is not zero, then the stochastic phase process will not be mean-stationary. Instead, the mean phase will be a function of time:
)
The constant )
is the centroid of the Doppler spectrum. The solution to removing this nonstationary phase is to multiply the channel )
, by a complex exponential of the form, exp)
.
Keep in mind that performance in a communication link depends on envelope - or, equivalently, power - which is independent of phase. Multiplication of any complex process by a linear phase taper does nothing to the envelope of the process. We use statistics based on a phase-centered complex channel because (as we will see in Chapter 7) they can be linked directly to average changes in the envelope process.
After making the final adjustment for nonstationary phase, the final measure of temporal channel fluctuation becomes
)
The measure, )
, is called a fading rate variance because the time-derivative of the channel describes the rate of fading, and the mean-squared value represents the variance
of that process [Dur98a]. Since the complex exponential term in Equation (3.4.4) corrects the nonstationary phase, the fluctuations measured by the temporal fading rate variance is related to the envelope
fluctuations of , and not just the steady-state progression of phase.
Of course fading rate variances exist for other dependencies as well. For a measure of channel fluctuations as a function
of frequency, it is possible to define a frequency fading rate variance, , for a static, fixed channel )
:
)
where )
is the centroid of the delay spectrum. By duality, a spatial fading rate variance, )
, can be defined for a static, narrowband channel )
, where r represents displacement in a fixed direction in space:
)
The value of )
is the centroid of the wavenumber spectrum calculated for the direction of displacement in space. These fading rate variances
are more than just an intuitive academic measure of fading fluctuations - there is a powerful relationship between fading
rate variance and the RMS spread of a PSD.
3.4.6 Fundamental Spectral Spread Theorem
It is also possible to calculate the mean-squared derivative of a time-varying channel given its Doppler spectrum, )
. There is a basic theorem in stochastic process theory that relates the mean-squared derivatives of complex processes to
PSDs [Pap91]:
)
For the case of fading rate variance, we are interested in the case of n = 1:
)
To account for the modulation of the process by the factor exp, it is possible to shift the spectrum in Doppler frequency by an amount – and adjust the limits of integration:
)
This equation may be rearranged into the following form:
)
where )
. Now the fading rate variance has become a simple function of two familiar measures: average power, E {P(t)}, and RMS Doppler spread, σω.
The relationship in Equation (3.4.10) is a fundamental result and holds for any fading rate variance and spectral spread. For stochastic wireless channels, the following results hold:
)
)
)
It becomes clear from equations (3.4.11), (3.4.12), and (3.4.13) why RMS spectral spreads provide such accurate measures of channel coherence. The RMS spectral spread, due to its direct relationship to fading rate variance, is an excellent measure of channel fluctuation in the base domain. Equations (3.4.11), (3.4.12), and (3.4.13) are valuable starting points when applying stochastic channel theory to real-world problems.