Showing papers on "Complex normal distribution published in 1979"

Polar Quantization of a Complex Gaussian Random Variable

Abstract: We solve numerically the optimum fixed-level non-uniform and uniform quantization of a circularly symmetric complex (or bivariate) Gaussian random variable for the mean absolute squared error criterion. For a given number of total levels, we determine its factorization into the product of numbers of magnitude and phase levels that produces the minimum distortion. We tabulate the results for numbers of "useful" output levels up to 1024, giving their optimal factorizations, minimum distortion, and entropy. For uncoded quantizer outputs, we find that the optimal splitting of rate between magnitude and phase, averaging to 1.52 and 1.47 bits more in the phase angle than magnitude for optimum and uniform quantization, respectively, compares well with the optimal polar coding formula Of 1.376 bits of Pearlman and Gray [3]. We also compare the performance of polar to rectangular quantization by real and imaginary parts for both uncoded and coded output levels. We find that, for coded outputs, both polar quantizers are outperformed by the rectangular ones, whose distortion-rate curves nearly coincide with Pearlman and Gray's polar coding bound. For uncoded outputs, however, we determine that the polar quantizers surpass in performance their rectangular counterparts for all useful rates above 6.0 bits for both optimum and uniform quantization. Below this rate, the respective polar quantizers are either slightly inferior or comparable.

Calculation of Univariate and Bivariate Normal Probability Functions

Abstract: Mill's ratio is expressed as a convergent series in orthogonal polynomials. Truncation of the series provides an approximation for the complemented normal distribution function $Q(x)$, with its maximum error at a finite value of $x$. The analogous approximation for $xQ(x)$ is used to obtain a new method of calculating the bivariate normal probability function.

A note on the bivariate normal distribution

Abstract: In this note, sufficient conditions for a two dimensional random vector to have a bivariate normal distribution will be given in terms of the conditional distributions of the two components.

The distribution of the characteristic roots ofS 1 S 2 −1 under violations in the complex case and power comparisons of four tests

Abstract: The joint density function of the latent roots ofS 1 S 2 −1 under violations is obtained whereS 1 has a complex non-central Wishart distributionW c (p,n 1,Σ 1,Ω) andS 2, an independent complex central Wishart,W c (p,n 2,Σ 2, 0). The density and moments of Hotelling's trace are also derived under violations. Further, the non-null distributions of the following four criteria in the two-roots case are studied for tests of three hypotheses: Hotelling's trace, Pillai's trace, Wilks' criterion and Roy's largest root. In addition, tabulations of powers are carried out and power comparisons for tests of each of three hypotheses based on the four criteria are made in the complex case extending such work of Pillai and Jayachandran in the classical Gaussian case. The findings in the complex Gaussian are generally similar to those in the classical.