Phase detectorA phase detector or phase comparator is a frequency mixer, analog multiplier or logic circuit that generates a signal which represents the difference in phase between two signal inputs. The phase detector is an essential element of the phase-locked loop (PLL). Detecting phase difference is important in other applications, such as motor control, radar and telecommunication systems, servo mechanisms, and demodulators. Phase detectors for phase-locked loop circuits may be classified in two types.
Hilbert spaceIn mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.
Riesz potentialIn mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines an inverse for a power of the Laplace operator on Euclidean space. They generalize to several variables the Riemann–Liouville integrals of one variable. If 0 < α < n, then the Riesz potential Iαf of a locally integrable function f on Rn is the function defined by where the constant is given by This singular integral is well-defined provided f decays sufficiently rapidly at infinity, specifically if f ∈ Lp(Rn) with 1 ≤ p < n/α.
Copeland's methodCopeland's method is a ranked voting method based on a scoring system of pairwise "wins", "losses", and "ties". The method has a long history: Ramon Llull described the system in 1299, so it is sometimes referred to as "Llull's method" The Marquis de Condorcet described a similar system in the 1780s, so the method could be referred to as "Condorcet's method", but instead other systems were subsequently devised that choose the Condorcet winner. Arthur Herbert Copeland described the system in the 1950s, so it has been frequently been called "Copeland's method".
