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This chapter is from the book

This chapter is from the book

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, 031fig01.gif, into a random frequency-varying channel produces a new random process, 047fig03.gif, that we can analyze with ensemble statistics.

One possible analysis is to define an autocorrelation, 047fig04.gif, for characterizing the development of 047fig03.gif with respect to Doppler. The definition of this would be

Equation 3.2.1

03equ21.gif


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 047fig03.gif must be uncorrelated. Mathematically, we can write this condition as

Equation 3.2.2

03equ22.gif


which clearly evaluates to 0 whenever ω1ω2. 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.

Proof: Start with the definition of temporal autocorrelation in the time domain - Equation (3.1.2) - and substitute the inverse Fourier transforms for 048fig01.gif and 048fig02.gif:

048equ01.gif


Now consider the result for uncorrelated spectral components: 047fig04.gif evaluates to zero for all ω1ω2. For this case, we could then write the complex exponential function as exp (1[t1 – t2]). The entire right-hand side then satisfies the condition for WSS, since it depends only on the time difference t2t1.

The converse - WSS leads to uncorrelated spectral components - is necessarily true as well. There is no mathematical way to force exp (j1t1ω2t2]) into a function expressible in terms of t2t1 unless its multiplier, 047fig04.gif, is zero for all ω1ω2. 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 ω1ω2. The function 048fig04.gif in Equation (3.2.2) is called the power spectral density (PSD). It describes how the spectral power of the random channel 031fig01.gif 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, 047fig04.gif, has infinite values. We avoid this problem in the future by using the finite-valued PSD 048fig06.gif 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 048fig06.gif 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:

Equation 3.2.3

03equ23.gif


Equation 3.2.4

03equ24.gif


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

049equ01.gif


which, for a WSS random process, becomes

049equ02.gif


Making the cosmetic substitutions ω = ω2 and Δt = t1t2 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

050equ01.gif


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

Equation 3.2.5

03equ25.gif


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:

050equ02.gif


where the 050fig02.gif denotes a unit vector. Next, the threefold integrations over position variables or wavenumber variables is a abbreviated into one integration over a vector partial, 050fig01.gif or 050fig03.gif. These single integrations are defined to be

050equ03.gif


With these substitutions, the set of forward and inverse Fourier transforms for functions of vector position is given by the following:

Equation 3.2.6

03equ26.gif


Equation 3.2.7

03equ27.gif


where (·) is a scalar product 050fig04.gif.

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

Equation 3.2.8

03equ28.gif


The wavevector PSD for this random channel process is given by the Fourier transform:

Equation 3.2.9

03equ29.gif


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

TIME AUTOCORRELATION

051equ01.gif

DOPPLER SPECTRUM

051equ02.gif

FREQUENCY AUTOCORRELATION

051equ03.gif

DELAY SPECTRUM

051equ04.gif

SCALAR SPACE AUTOCORRELATION

051equ05.gif

WAVENUMBER SPECTRUM

051equ06.gif

VECTOR SPACE AUTOCORRELATION

051equ07.gif

WAVEVECTOR SPECTRUM

051equ08.gif

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