Nonlinear Transformations of Random Processes

By Ralph Deutsch

This concise treatment of nonlinear noise techniques encountered in system applications is suitable for advanced undergraduates and graduate students. The book is also a valuable reference for systems analysts and communication engineers, as it discusses the basic mathematical theories of nonlinear transformations applied to random processes encountered in communications and control systems. Prerequisites include a familiarity with statistics, probability, complex variables, and Fourier and Laplace transforms.
The first five chapters present specific classes of nonlinear devices and random processes that in combination lead to closed form solutions for the statistical properties of the transformed process. Subsequent chapters address techniques based on the use of series representations, general systematic approaches to the subject of nonlinear transformations of random processes, and sampling and quantizing a random process. A helpful Appendix features notes on hypergeometric functions.

• Language: English
• Publisher: Dover Publications
• Release date: Nov 8, 2017
• ISBN: 9780486826035

Book preview

INDEX

1.1NOTION OF AN ENVELOPE

The notion of the envelope of a time series is usually an intuitive concept arising from elementary studies of signal modulation. In Figure 1-1, one easily recognizes the dotted lines to be the envelope of the modulated signal. If the signal is the result of modulation of one frequency by another, then we can write

Figure 1-1. Modulated Voltage

where ω0 denotes 2π times the carrier frequency f0 and ωm = 2πfm corresponds to a frequency such that ω0 ≫ ωm. Under these conditions the envelope function is denned to be Am cos ωmt. The definition corresponds to what one would expect from an examination of Figure 1-1 and suffices for many discussions of modulation theory. However, more complicated signal forms are encountered when noise sources corrupt the signal. The simple concept of an envelope function is soon found to suffer from several defects.

Suppose we formally try to extend the simple definition of a signal envelope to a signal consisting of a wide sense stationary random process. Adopting the simple series representation of the process, for illustrative purposes, consider the signal

⍺n are random phase angles uniformly distributed in the range (0, 2πis the signal power associated with frequency is Ωn. It is semantically convenient to call both Ω = 2πf and f the frequency.

Arbitrarily select some frequency too to be designated as the midband frequency. All other frequencies ton are then written as

Restrict the midband frequency choice so that ω0 ≫ ωn. The {ωn} will constitute a set of frequencies about ω0 associated with random phase angles, ⍺n. Intuitively, we can see that the simple notion of an envelope function for the random process has little meaning unless the condition ω0 ≫ ωn .

Rewrite Eq. (1-2) in the equivalent form

Eq. (1-4) can be put into the form of Eq. (1-1) by writing

By analogy with Eq. (1-1) the envelope function for the random process is defined as

Several difficulties are immediately encountered in this loose would lead to the same envelope function V(t). If the condition ω0 ≫ ωn , what becomes of the intuitive notion of the envelope? Finally, because of the nonlinear relation in Eq. (1-6), it is a formidable task to compute V(t) and is generally difficult, if not entirely impossible, to obtain the statistical characteristics of the random process V(t).

1.2ENVELOPES AND PRE-ENVELOPES

If g(t) is a real-valued function on the interval –∞ < t < ∞, its Hilbert transform ĝ(t) is defined as the principal value of the integral

Without further statement, all the functions considered in this chapter are assumed to possess Hilbert transforms. The following properties follow at once from the transform definition, [1].

(i)If g(t) = cos (ωt + ϕ), its Hilbert transform is ĝ(t) = sin (ωt + ϕ).

(ii)Under rather general conditions if f = ĝ.

(iii)If

is the Fourier transform of h(t), then the Fourier transform Q(fis

(iv) The convolution of two functions

has the Hilbert transform

With these tools we are now in a position to present Dugundji’s [2] approach to an unambiguous definition for the envelope of a real-time series. Although the use of Hilbert transforms to define envelopes had been used before, [3, 4] Dugundji’s work is the first extensive exposition on the subject.

Definition 1-1: If h(t) is a real time series, its corresponding pre-envelope function z(t) is defined as

Definition 1-2: The envelope of h(t) is defined as the absolute value of the pre-envelope function z(t).

The important point is to ascertain that these definitions reduce to the intuitive notion of an envelope for simple waveforms and still do not suffer the ambiguous defects of the loose definition. For example, consider the same time series E(t) defined in Eq. (1-2). Its pre-envelope function is

Choose an arbitrary frequency Ω and write

This can be written as

where

Using Definition 1-2, the desired envelope function is

Notice that since z(t) is independent of the arbitrary choice of the midband frequency Ω, V(t) is also completely independent of this choice.

Although it has been shown that the definition of the envelope corresponds to the intuitive concept, the motivation and general applicability of the definition needs further amplification. We first note two properties of the pre-envelope function.

(a)Consider the Fourier transform of

Applying property (iii), the result is

(b)If z(t) is a complex valued function whose Fourier transform, Z(f), vanishes for all f < 0, then z .

Property b can be demonstrated in the following manner. Define

where the asterisk denotes the complex conjugate, and H(f) is the Fourier transform of h(t). That is,

Employing the definition of Eq. (1-16), it follows that h(t) = h*(f), or that as defined, h(tcorresponding to h(t). From property α it follows that the Fourier transform of y(t) is z(f), except possibly at the point f = 0. This implies that y = z and establishes property b.

For most noise theory analyses of physical situations, the usual practice is to consider only positive frequencies in the frequency spectrum of a random time series. This is often accomplished in the mathematical analysis by doubling the positive frequency terms and neglecting the negative frequency terms. The common justifying argument is that the negative frequencies merely reflect the positive ones in complex conjugate form and, therefore, can be reflected into positive frequency terms. Notice that property a of the pre-envelope is nothing more than a statement that only positive frequency terms are considered with a doubling of coefficients.

1.3CORRELATION AND SPECTRUM RELATIONS FOR ENVELOPE FUNCTIONS

In this section it is tacitly assumed that all the functions have the required properties that give meaning to all integrals and operations. These assumptions permit us to investigate some formal mathematical relations without considering all the exceptional cases for which the statements may fail to hold.

The important results developed in this section were obtained by Dugundji under the implied restriction that the random process was stationary and ergodic. Zakai has pointed out that these results remain true when the time averages employed by Dugundji are replaced by ensemble averages, regardless of ergodicity, the only requirement being that the process be wide sense stationary, [5].

Setting aside Zakai’s generalization for the moment, let x(t) and y(t) be two wide sense stationary ergodic processes. Their cross-covariance function is denned as

Throughout this book, E{·} represents the expected value of the quantity within the braces.

The Fourier transform of Rxy is often called the cross-power density spectrum junction and is denoted by the symbol Wxy(f). Thus,

We now derive some useful relations concerning the spectrum of the envelopes and pre-envelopes of real valued wide sense stationary ergodic random processes cognizant of the fact that it is known that the ergodic hypothesis can be omitted.

is the Hilbert transform of Rx = Rxx.

By definition

Assume that the order of integration can be interchanged and write

or

Let x(t) be a real valued process, then property c can be used to show that x and x are uncorrelated. It follows directly from definitions that

and we have established that the time series and its Hilbert transformed time series are completely uncorrelated.

is

This statement follows at once from property c and property iii for Hilbert transforms.

(e)The original time series, x(t, have identical covariance functions.

From property c, we have

Repeating the steps used to demonstrate property c, one can show that

or, using Eq. (1-23),

Therefore,

which demonstrates that x have the same covariance function and hence the same power spectral density function.

(f) Let

then using properties d and e it follows that

and

From Definition 1-2, the envelope function of a real-valued time series x(t) can be written as

Squaring V(t) and taking expected values produces the result

where property e has been employed. Hence, one concludes that the ensemble average of the square of the envelope function is equal to twice the ensemble average of the square of the original time series, [5, 6].

(h)Karr has shown an inequality concerning the envelope of a correlation function which follows immediately from our previous results, [7].

The covariance function Rx(σ) corresponding to a real valued time random process x(t), can in itself be considered to be a time series in the σ – domain. The envelope of the covariance time function is obtained by writing

or

is

From property iii for Hilbert transforms

Therefore,

Moreover,

Adding Eqs. (1-38) and (1-39) shows that

Now,