3.2 Power Spectral Density (PSD)
In the previous section, we defined the autocorrelation method for describing how a random process evolves in time (or frequency, or space). In this section, we present an alternative way of analyzing a random process by using Fourier analysis. It turns out that the autocorrelation method and the power spectral density method in this section are related by the Wiener-Khintchine theorem, providing different perspectives of the same information.
3.2.1 Correlation in Spectral Domains
Whenever a mathematical operation is performed on a random process, a new random process describes the result. This truth applies to the Fourier transform; the Fourier transform of a random process is itself a random process. Thus, transforming a random time-varying channel, , into a random frequency-varying channel produces a new random process, , that we can analyze with ensemble statistics.
Equation (3.2.1) must have a particular form when describing a random process that, in this form, is WSS in the time domain. The spectral components of must be uncorrelated. Mathematically, we can write this condition as
which clearly evaluates to 0 whenever ω1 ≠ ω2. A proof of this may be found in Theorem 3.1
Theorem 3.1: Uncorrelated Spectrum for WSS
Statement: A random process is WSS if and only if its spectral components are uncorrelated.
Now consider the result for uncorrelated spectral components: evaluates to zero for all ω1 ≠ ω2. For this case, we could then write the complex exponential function as exp (jω1[t1 – t2]). The entire right-hand side then satisfies the condition for WSS, since it depends only on the time difference t2 – t1.
The converse - WSS leads to uncorrelated spectral components - is necessarily true as well. There is no mathematical way to force exp (j[ω1t1 – ω2t2]) into a function expressible in terms of t2 – t1 unless its multiplier, , is zero for all ω1 ≠ ω2. This is only true if the spectral components are all uncorrelated.
Note the special form used in Equation (3.2.2), where the delta function forces the spectral correlation to zero for all ω1 ≠ ω2. The function in Equation (3.2.2) is called the power spectral density (PSD). It describes how the spectral power of the random channel is distributed in the Doppler domain. The PSD is the most important spectral-domain tool for analyzing WSS random processes.
In channel modeling circles, the condition of complete spectrum decorrelation in Equation (3.2.2) is often referred to as uncorrelated scattering (US) - especially when discussing the channel delay spectrum.
Fourier transforms are only rigidly defined for energy signals. All truly WSS random processes, however, are power signals. Thus, the spectrum autocorrelation, , has infinite values. We avoid this problem in the future by using the finite-valued PSD in all spectral analysis. Through a mathematical sleight-of-hand, the delta function in Equation (3.2.2) "absorbs" the infinite values, which is why is called a density.
3.2.2 The Wiener-Khintchine Theorem
The usefulness of a PSD arises from the Wiener-Khintchine theorem [[Cou93, p. 70]. This theorem states that the PSD and the autocorrelation of a WSS random process are Fourier transform pairs:
The Wiener-Khintchine theorem implies that studying autocorrelation properties in the base domain is equivalent to studying average signal power in the spectral domain. Thus, there are two ways to characterize the same second-order statistics of a WSS random process. Theorem 3.2 proves the Wiener-Khintchine theorem.
Theorem 3.2: Wiener-Khintchine Theorem
Statement: The PSD and autocorrelation of a WSS random process are Fourier transform pairs.
Proof: Recall from Theorem 3.1 that the autocorrelation function is written as
which, for a WSS random process, becomes
Making the cosmetic substitutions ω = ω2 and Δt = t1 – t2 produces the Fourier transform relationship of Equation (3.2.4).
3.2.3 Statistics with Three-Dimensional Space
In our discussion of spatial selectivity, we have dealt with only scalar space - movement along a linear spatial variable, r. Of course, reality dictates that wireless receivers are capable of operating in three-dimensional space, which requires adding dependencies to the spatial representation as well as its Fourier spectrum. The Fourier transform of a function of three-dimensional position is actually a threefold transform over scalar coordinates. Thus, we write the transform pairs as
where x, y, and z are the Cartesian position coordinates and kx, ky, and kz are their corresponding wavenumbers in the spectral domain. The three-fold Fourier transform therefore requires three integrations, which are written as
The notation of Equation (3.2.5) is cumbersome. Throughout this work, a set of vector shorthand notation is used to simplify the equations and emphasize the concepts rather than the raw mathematics.
First, the dependencies of both scalar positions and wavenumbers are collapsed into a three-dimensional position vector wavevector, respectively:
where the denotes a unit vector. Next, the threefold integrations over position variables or wavenumber variables is a abbreviated into one integration over a vector partial, or . These single integrations are defined to be
With these substitutions, the set of forward and inverse Fourier transforms for functions of vector position is given by the following:
The simple vector notation makes it almost trivial to define a 3D spatial autocorrelation function and a wavevector PSD, since the definitions follow logically from their scalar spatial computerparts. The 3D spatial autocorrelation function (assuming a WSS spatial channel) is defined as
The wavevector PSD for this random channel process is given by the Fourier transform:
Although it is easier to introduce spatial channel modeling concepts with scalar space dependencies r and k, a full 3D spatial channel model will be important in subsequent chapters.
3.2.4 Summary of Single-Dependency PSDs
The spectral definitions for time-Doppler channels are equally valid for use with other channel dependencies. Thus, the Doppler PSD is used for time-varying channels, the delay PSD is used for frequency-varying channels, and the wavenumber PSD is used for space-varying channels. Engineers often refer to these PSDs as simply the Doppler spectrum, the delay spectrum, and the wavenumber or wavevector spectrum (although this is, technically, a little ambiguous). Table 3.1 summarizes all of the relevant autocorrelation-spectrum relationships for the WSS complex baseband channel.
Table 3.1. Transform Definitions for Autocorrelation and Power Spectral Densities
SCALAR SPACE AUTOCORRELATION
VECTOR SPACE AUTOCORRELATION