This chapter is from the book
This chapter is from the book
This chapter is from the book
3.3 Joint Statistics
Autocorrelation functions and PSDs are not just for characterizing random channels with a single dependency; these techniques are every bit as useful for multiple-dependency channels. The only difficult part in analyzing a random channel with multiple dependencies is visualizing the distinct behavior. To help with this, transform maps are presented in this section along with the definitions of joint autocorrelation and spectrum.
3.3.1 Joint Autocorrelation and Spectrum
To accommodate all the random dependencies of a channel, it is possible to define a joint PSD as a function of Doppler, delay, and wavenumber. Specifically, a random process describing the full channel with respect to time, frequency, and space has the following Fourier transform pair:
)
Just as in the single-dependency case, we can study the correlation of spectral components by writing
)
Once again, if the spectral components are uncorrelated with one another, we may write Equation (3.3.1) as
)
The multidependency random process described by Equation (3.3.2) is called a wide-sense stationary, uncorrelated scattering (WSS-US) process.
In seminal work on WSS-US time-varying channels, the term WSS refers to wide-sense stationary with respect to time, and the term US refers to wide-sense stationary with respect to frequency [Bel63]. This terminology gets confusing as we add dependencies to the random space–time channel. In this text, US emphasizes the ability to write the PSD in the form of Equation (3.3.2) - spectral components are jointly uncorrelated across all spectral domains.
Logically, we may also define an autocorrelation function for the multidependent WSS-US channel and apply the Wiener-Khintchine theorem. The definition follows from the single-dependency autocorrelation of Equation (3.1.4):
)
The Wiener-Khintchine theorem for WSS-US processes then leads to the following Fourier transform relationship between autocorrelation function and PSD:
)
See Example 3.1 for a demonstration of how to apply these definitions to a WSS-US channel with multiple dependencies.
Example 3.1: WSS-US Autocorrelation
Problem: Given the following autocorrelation function for a WSS-US channel,
)
derive an expression for the PSD, )
Solution: The solution follows mechanically by taking the Fourier transform of the autocorrelation function:
)
It is possible for a random channel to be WSS with respect to each dependency individually, but not jointly (i.e. WSS-US as used by this text). To understand how this can happen, consider the following example of joint spectral correlation with respect to Doppler and delay dependencies:
)
This spectral correlation results in an autocorrelation function that depends only on t1 – t2 and ω1 – ω2. Yet the cross-dependent term causes correlated behavior between Doppler and delay, which adds complexity to a second-order analysis.
3.3.2 Time–Frequency Transform Map
To truly understand the concept of joint spectra and autocorrelation functions, it is best to eliminate some of the dependencies
from the full space–time–frequency channel model. Consider the case of a fixed, single-antenna receiver operating in a channel
that is only a function of time, t, and frequency, f; this channel has no spatial dependency. For this type of channel, Figure 3.2 is a roadmap through the various stochastic relationships [Bel63], [Gre62]. The stochastic, second-order behavior of this channel may be characterized by its joint autocorrelation, )
, or its PSD, )
. In a WSS channel, knowledge of only one is necessary, since they are two-dimensional Fourier transform pairs.
)
Figure 3.2. Autocorrelation function and PSD relationships for time and frequency.
If the channel is narrowband (signal) bandwidth is less than the coherence bandwidth), then the fading is flat about the carrier frequency, and it suffices to characterize the stochastic channel with an autocorrelation that is only a function of time or a PSD that is only a function of Doppler. These one-dimensional functions may be calculated from their two-dimensional counterparts as
)
)
This calculation is shown on the left side of Figure 3.2. The remaining one-dimensional functions - the temporal autocorrelation and Doppler spectrum - are Fourier transform pairs.
The right-hand side of Figure 3.2 represents characterization of a static channel that does not vary with time. A one-dimensional frequency autocorrelation is calculated from the joint time-frequency autocorrelation by setting Δt equal to 0. A one-dimensional delay spectrum is calculated from the joint Doppler–delay spectrum by integrating out the Doppler frequency, ω. And, of course, the resulting frequency autocorrelation and delay spectrum are Fourier transform pairs.
Many researchers have used Figure 3.2 to understand and characterize the stochastic behavior of linear, time-varying channels. It was first presented by [Gre62] in the context of radio astronomy measurements. For this application, an antenna dish is located in a fixed position and receives signals that vary with time and frequency from the heavens. The concept may be extended to include spatial relationships as well.
The real impetus for the original work in [Gre62] had more to do with spying on radio transmissions by the former Soviet Union using moon bounce measurements. Soviet radio signals originating from the other side of the earth would bounce off the revolving and rotating moon, producing a radio signal with both dispersion (frequency-selective fading) and Doppler spreading (temporal fading). Spy satellites eventually led to a much more effective way to monitor the radio signals of other countries.
3.3.3 Space–Frequency Transform Map
For a static channel, we may extend the time-frequency transform map in Figure 3.2 for the space–frequency relationships shown in Figure 3.3. In this diagram, joint space-frequency autocorrelation and joint wavenumber–delay spectrum are shown to be two-dimensional Fourier transform pairs. One-dimensional spatial or frequency autocorrelations may be calculated by zeroing out Δf or Δr respectively in the joint autocorrelation. One-dimensional wavenumber or delay spectra may be calculated by integrating out τ or k respectively in the joint PSD.
)
Figure 3.3. Autocorrelation function and PSD relationships for space and frequency.
3.3.4 Complete Transform Map
Of course, all three dependencies - space, time, and frequency - may be studied jointly. The combined relationships of Figure 3.2 and Figure 3.3 are shown in one complete transform map, Figure 3.4. This figure is best viewed as a single "tile" in a transform map that continues with identical tiles in all directions. For example, the conversion from a joint space–time autocorrelation to a one-dimensional temporal autocorrelation is made by wrapping around from the right edge of Figure 3.4 to the left edge. Likewise, the complete autocorrelation of the stochastic channel is shown to be the three-domain Fourier transform pair of the complete PSD by wrapping around from the top edge of Figure 3.4 to the bottom edge.
)
Figure 3.4. Autocorrelation function and power spectrum relationships for space, time, and frequency.
Although Figure 3.4 contains numerous channel dependencies and definitions, navigation through the relationships is crucial for understanding the stochastic characterization of the multivariable wireless channel. When one considers vector space instead of scalar space, the characterization is even more complicated. Remember, however, that each dependency can be removed to isolate and understand the fading behaviors in a space–time channel. Use Table 3.2, which summarizes how to remove the dependencies, along with the transform map of Figure 3.4 to analyze the random space–time channel.
Table 3.2. How to Remove a Dependency in a Random Space–Time Channel Model.
|
DEPENDENCY TO REMOVE |
FROM AUTOCORRELATION |
FROM PSD |
|---|---|---|
|
Time |
Δt = 0 |
|
|
Frequency |
Δf = 0 |
|
|
Scalar Space |
Δr = 0 |
|
|
Vector Space |
|
|
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