Rayleigh signal fading due to multipath propagation in wireless channels is widely modeled using sum-of-sinusoids simulators. In particular, Jakes' (1994) simulator and derivatives of Jakes' simulator have gained widespread acceptance. Despite this, few in-depth studies of the simulators' statistical behavior have been reported in the literature. Here, the extent to which Jakes' simulator adequately models the multipath Rayleigh fading propagation environment is examined. The results show that Jakes' simulator does not reproduce some important properties of the physical fading channel. Some possible improvements to Jakes' simulator are examined. The significances of the number and the symmetries of the Doppler frequency shifts on the validity of the simulator's reproduction of the physical fading channel are elucidated.

Limitations of sum-of-sinusoids fading channel simulators

https://www.dtc.umn.edu/s/resources/gg-se.pdf

A bibliography on nonlinear system identification

There are various cases in physics and engineering sciences (especially communications) where one requires the envelope probability density function (PDF) of the sum of several random sinusoidal signals. According to the correspondence between a random sinusoidal signal and a random vector, the sum of random vectors can be considered as an abstract mathematical model for the above sum. Now it is desired to obtain the PDF of the length of the resulting vector. Considering the common and reasonable assumption of uniform distributions for the angles of the vectors, many researchers have obtained the PDF of the length of the resulting vector only for special cases. However in this paper, the PDF is obtained for the most general case in which the lengths of vectors are arbitrary dependent random variables. This PDF is in the form of a definite integral, which may be inappropriate for analytic manipulations and numerical computations. So an appropriate infinite Laguerre expansion is also derived. Finally, the results are applied to solve a typical example in computing the scattering cross section of random scatterers.

/pdf/on-the-pdf-of-the-sum-of-random-vectors-4sq6bic8jj.pdf

On the PDF of the sum of random vectors

A simple model to describe a nonlinear device or system is proposed which extends the power series expansion, conventionally restricted to amplitude nonlinearities, to include phase nonlinearities as well. Four different test methods are selected for which the experimentally observed nonlinearity parameters are related to the "gain" and "phase" coefficients of the extended series. A set of simplified relationships is derived where the "1-dB gain compression point" represents gain contributions only while phase nonlinearities are included in the "intercept point," the "third-order intermodulation (IM) coefficients," and the "noise-power-ratio (npr)." For a TWT amplifier in which phase nonlinearities dominate, the third-order IM coefficient was measured. The results are compared with those calculated from single-tone and noise-loading tests using the relationships derived from the model. Agreement to /spl plusmn/ 1 dB is found over a 15-dB power range.

Characterization of Nonlinearities in Microwave Devices and Systems

A random n -vector X=(X_{1}, \cdots , X_{n}) is said to be spherically symmetric (SS) if its joint characteristic function (CF) can be expressed as a function of the quadratic form u \rho u \prime , where u = (u_{1}, \cdots , u_{n}) and \rho is an n \times n positive definite matrix. The investigation in this paper is concerned with the properties of such vectors and some detection problems involving them. We first prove a theorem characterizing the form of SS random vectors X and use it to find the form of the probability density functions (pdf's) of X and of X + N , where N \sim {\cal N (0, \sigma^{2}I) is an independent Gaussian vector and I is the identity matrix. Applying these results we look at the problem of detecting a known signal vector in the presence of X + N when \rho =I . For the k -ary detection problem we present two conditions under which the "minimum distance" receiver is optimum. Lastly, we discuss an application of our findings to the problem of coherent detection of binary phase-shift keyed (PSK) signals in the presence of multiple co-channel interferences and white Gaussian noise.

Detection in the presence of spherically symmetric random vectors

This paper investigates the statistical properties of the sum, S, of an n-dimensional Gaussian random vector, N, plus the sum of M vectors, X 1 , …, X M , having random amplitudes and independent arbitrary orientations in n-dimensional space. We derive expressions for the probability density function (p.d.f.) and distribution function (d.f.) of S and of its length, |S|, as series expansions involving only the moments of |X i |, i = 1, …, M. In addition, we find the p.d.f. and d.f. of the projection of S onto 1-dimensional space. Our results are generalizations of the n = 2-dimensional problem of finding the statistical properties of a sum of constant-amplitude sinusoids having independent uniformly distributed phase angles plus Gaussian noise. The latter problem has been treated by Rice1 and Esposito and Wilson,2 but our results can also deal with sinusoids having random amplitudes. When n = 3, our findings treat, in the presence of a Gaussian vector, the classical problem of “random flights” dating back to Rayleigh. Some calculations for the 2- and 3-dimensional problem are presented, and an application to coherent phase-shift-keying communications systems is discussed.

Statistical properties of a sum of sinusoids and Gaussian noise and its generalization to higher dimensions

This paper studies a general description of interchannel and intra-channel crosstalk interference created in a communications system. This description is in the form of a Volterra series expansion of the interference signal in terms of the signal which produced the interference. From it we are able to precisely define the “intelligible” part of the crosstalk. This description also provides us with quantitative measures of the amount of crosstalk created in some communications channel by signals in another channel, as well as a measure (intelligible crosstalk ratio) of the amount of intelligible crosstalk produced. We then consider a particular model for the generation of intelligible crosstalk [or direct adjacent channel interference (DACI)] between two neighboring angle-modulated channels in which the signal in one channel adds to the signal of the second channel, the sum is filtered, and the filter output then passes through an AM-PM conversion device. Using our definition, a simple expression for the intelligible crosstalk ratio for this model is derived in terms of the filter characteristic. We observe that this crosstalk ratio exhibits a number of properties usually associated with DACI.

http://ieeexplore.ieee.org/iel7/6731005/6771672/06771675.pdf

Joel Goldman

Papers

Statistical properties of a sum of sinusoids and Gaussian noise and its generalization to higher dimensions

A volterra series description of crosstalk interference in communications systems