All-Pole Recursive Digital Filters Design Based on Ultraspherical Polynomials

A simple method for approximation of all-pole recursive digital filters, directly in digital domain, is described. Transfer function of these filters, referred to as Ultraspherical filters, is controlled by order of the Ultraspherical polynomial, nu. Parameter nu, restricted to be a nonnegative real number (nu ≥ 0), controls ripple peaks in the passband of the magnitude response and enables a trade-off between the passband loss and the group delay response of the resulting filter. Chebyshev filters of the first and of the second kind, and also Legendre and Butterworth filters are shown to be special cases of these allpole recursive digital filters. Closed form equations for the computation of the filter coefficients are provided. The design technique is illustrated with examples

Metaethical Relativism

Although relativism may be said to be one of the oldest doctrines in philosophy, dating back to the teachings of Protagoras in the 5th century B.C., when it comes to contemporary philosophy, there is no consensus on what makes a view qualify as "relativist". The problem is particularly accute in metaethics, since most of the views that up to a decade ago were described as “relativist” would be more accurately described as "contextualist" or even “expressivist” in light of the distinctions currenty drawn in philosophy of language and semantics. In this chapter, we distinguish two construals of relativism, developed in sections 2 and 3 respectively: the “metaphysical” construal, based on the idea that there is no single, absolute, universal morality, and the “semantic” construal, based on the idea that the truth value of moral claims is relative to a set of moral standards, or moral practices, or some other suitable parameter. Section 1 introduces the core relativist ideas in an informal way, and warns against possible misinterpretations

On the Status of Natural Divination in Stoicism

Cicero’s De divinatione portrays the Stoics as unanimous in advocating both natural and technical divination. I argue that, contrary to this, the earlier leaders of the school like Chrysippus had reasons to consider natural divination to be significantly epistemically inferior to its technical counterpart. The much more favorable treatment of natural divination in De divinatione is likely the result of changes introduced later, probably by Posidonius

On U-Statistics and Compressed Sensing II: Non-Asymptotic Worst-Case Analysis

In another related work, U-statistics were used for non-asymptotic "average-case" analysis of random compressed sensing matrices. In this companion paper the same analytical tool is adopted differently - here we perform non-asymptotic "worst-case" analysis. Simple union bounds are a natural choice for "worst-case" analyses, however their tightness is an issue (and questioned in previous works). Here we focus on a theoretical U-statistical result, which potentially allows us to prove that these union bounds are tight. To our knowledge, this kind of (powerful) result is completely new in the context of CS. This general result applies to a wide variety of parameters, and is related to (Stein-Chen) Poisson approximation. In this paper, we consider i) restricted isometries, and ii) mutual coherence. For the bounded case, we show that k-th order restricted isometry constants have tight union bounds, when the measurements m = \mathcal{O}(k (1 + \log(n/k))). Here we require the restricted isometries to grow linearly in k, however we conjecture that this result can be improved to allow them to be fixed. Also, we show that mutual coherence (with the standard estimate \sqrt{(4\log n)/m}) have very tight union bounds. For coherence, the normalization complicates general discussion, and we consider only Gaussian and Bernoulli cases here.Comment: 12 pages. Submitted to IEEE Transactions on Signal Processin