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* A first-order filter, for example, will reduce the signal amplitude by half ( so power reduces by a factor of 4 ), or, every time the frequency doubles ( goes up one octave ); more precisely, the power rolloff approaches 20 dB per decade in the limit of high frequency.
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first-order and filter
The filter is usually a first-order lowpass filter ( as pictured ).
The gain-magnitude frequency response of a first-order ( one-pole ) low-pass filter.
The magnitude Bode plot for a first-order filter looks like a horizontal line below the cutoff frequency, and a diagonal line above the cutoff frequency.
If the transfer function of a first-order low-pass filter has a zero as well as a pole, the Bode plot will flatten out again, at some maximum attenuation of high frequencies ; such an effect is caused for example by a little bit of the input leaking around the one-pole filter ; this one-pole – one-zero filter is still a first-order low-pass.
The Bode plot for this type of filter resembles that of a first-order filter, except that it falls off more quickly.
The frequency response at the cutoff frequency in a first-order filter is 3 dB below the horizontal line.
For example, a first-order low-pass filter can be described in Laplace notation as
The gain in the passband is − R < sub > 2 </ sub >/ R < sub > 1 </ sub >, and the stopband drops off at − 6 dB per octave ( that is − 20 dB per decade ) as it is a first-order filter.
Figure 1: A passive, analog, first-order high-pass filter, realized by an RC circuit
The simple first-order electronic high-pass filter shown in Figure 1 is implemented by placing an input voltage across the series combination of a capacitor and a resistor and using the voltage across the resistor as an output.
Figure 2 shows an active electronic implementation of a first-order high-pass filter using an operational amplifier.
A short comparison with pulse-width modulation shows that a 1-bit DAC with a simple first-order integrator would have to run at 3 THz ( which is physically unrealizable ) to achieve 24 meaningful bits of resolution, requiring a higher-order low-pass filter in the noise-shaping loop.
All first-order filters have a Butterworth filter characteristic.
Figure 1 ( a ): The Bode plot for a first-order ( one-pole ) highpass filter ; the straight-line approximations are labeled " Bode pole "; phase varies from 90 ° at low frequencies ( due to the contribution of the numerator, which is 90 ° at all frequencies ) to 0 ° at high frequencies ( where the phase contribution of the denominator is − 90 ° and cancels the contribution of the numerator ).
Figure 1 ( b ): The Bode plot for a first-order ( one-pole ) lowpass filter ; the straight-line approximations are labeled " Bode pole "; phase is 90 ° lower than for Figure 1 ( a ) because the phase contribution of the numerator is 0 ° at all frequencies.
A second-order low-pass filter with a very low quality factor has a nearly first-order step response ; the system's output responds to a step input by slowly rising toward an asymptote.
He also showed that this filter can be well approximated by first-order derivatives of Gaussians.
first-order and for
** Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion.
The Löwenheim-Skolem theorems tell us that if we restrict ourselves to first-order logic, any axiom system for the reals admits other models, including both models that are smaller than the reals and models that are larger.
In contrast, other, more systematic algorithms achieved, at least theoretically, completeness for first-order logic.
The traditional test for the presence of first-order autocorrelation is the Durbin – Watson statistic or, if the explanatory variables include a lagged dependent variable, Durbin's h statistic.
* Tarski's axioms: Alfred Tarski ( 1902 – 1983 ) and his students defined elementary Euclidean geometry as the geometry that can be expressed in first-order logic and does not depend on set theory for its logical basis, in contrast to Hilbert's axioms, which involve point sets.
The asks for an algorithm that takes as input a statement of a first-order logic ( possibly with a finite number of axioms beyond the usual axioms of first-order logic ) and answers " Yes " or " No " according to whether the statement is universally valid, i. e., valid in every structure satisfying the axioms.
By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.
Suppose that we had a general decision algorithm for statements in a first-order language.
A theory about some topic is usually first-order logic together with: a specified domain of discourse over which the quantified variables range, finitely many functions which map from that domain into it, finitely many predicates defined on that domain, and a recursive set of axioms which are believed to hold for those things.
There are many deductive systems for first-order logic that are sound ( all provable statements are true ) and complete ( all true statements are provable ).
There are numerous deductive systems for first-order logic, including systems of natural deduction and Hilbert-style systems.
A first-order formula is called logically valid if it is true in every structure for the language of the formula.
It says that for any first-order theory T with a well-orderable language, and any sentence S in the language of the theory, there is a formal proof of S in T if and only if S is satisfied by every model of T ( S is a semantic consequence of T ).
The completeness theorem is a central property of first-order logic that does not hold for all logics.
Henkin's proof directly constructs a term model for any consistent first-order theory.
Historically, ML stands for metalanguage: it was conceived to develop proof tactics in the LCF theorem prover ( whose language, pplambda, a combination of the first-order predicate calculus and the simply typed polymorphic lambda calculus, had ML as its metalanguage ).
Its first-order Fresnel lens can be seen for about 19 miles ( 30 kilometers ), in good conditions.
* Natural language understanding: Convert chunks of text into more formal representations such as first-order logic structures that are easier for computer programs to manipulate.
The relational model for database management is a database model based on first-order predicate logic, first formulated and proposed in 1969 by Edgar F. Codd.
* November 1 – The current Cape Lookout, North Carolina, lighthouse is lighted for the first time ( its first-order Fresnel lens can be seen for 19 miles ).
Precise expressions for the transition probability, based on first-order perturbation Hamiltonians, can be found in Thompson and Baker.
A first-order emergent structure occurs as a result of shape interactions ( for example, hydrogen bonds in water molecules lead to surface tension ).
first-order and example
For example, if we can enumerate all such definable numbers by the Gödel numbers of their defining formulas then we can use Cantor's diagonal argument to find a particular real that is not first-order definable in the same language.
An example is the well-known calculation of the tiny first-order quantum-mechanical correction to the classical Newtonian gravitational potential between two masses.
Truth-functional propositional logic and first-order predicate logic are semantically complete, but not syntactically complete ( for example, the propositional logic statement consisting of a single variable " a " is not a theorem, and neither is its negation, but these are not tautologies ).
For example, the transcendental functions are defined in terms of infinite limiting processes, and therefore there is typically no way to define them in first-order logic.
These infinitesimals have different first-order properties than the reals because, for example, the basic infinitesimal x does not have a square root.
For example, while it was once assumed that catagenetic processes were first-order reactions, some research has shown that this may not be the case.
For example, in an interpretation of first-order logic, the domain of discourse is the set of individuals that the quantifiers range over.
where, for example, the first-order term is
For example, if the domain is the set of all real numbers, one can assert in first-order logic the existence of an additive inverse of each real number by writing ∀ x ∃ y ( x + y = 0 ) but one needs second-order logic to assert the least-upper-bound property for sets of real numbers, which states that every bounded, nonempty set of real numbers has a supremum.
When the model coefficients sum to one, as in this example, the production function is first-order homogeneous, which implies constant returns to scale, that is, if all inputs are scaled by a common factor greater than zero, output will be scaled by the same factor.
The center of U ( L ) is called Z ( L ) and consists of the left-and right-invariant differential operators ; this in the case of G not commutative will often not be generated by first-order operators ( see for example Casimir operator of a semi-simple Lie algebra ).
For example the first-order time derivative is often approximated using the following forward finite difference, where the subscripts indicate a discrete time location,
Truth-functional propositional logic and first-order predicate logic are semantically complete, but not syntactically complete ( for example the propositional logic statement consisting of a single variable " a " is not a theorem, and neither is its negation, but these are not tautologies ).
For example, HOL admits categorical axiomatizations of the natural numbers, and of the real numbers, which are impossible with first-order logic.
" It is possible to define a formula True ( n ) whose extension is T *, but only by drawing on a metalanguage whose expressive power goes beyond that of L. For example, a truth predicate for first-order arithmetic can be defined in second-order arithmetic.
For example, the set of ( codes for ) formulas of first-order Peano arithmetic that are true in N is definable by a formula in second order arithmetic.
The power of language to encode these preconceived " stereotypes " based solely on accent is an example of second-order indexicality ( representative of a more complex and subtle system of indexical form than that of first-order indexicality ).
This simple example gives a single-pole filter ( a first-order Butterworth filter ), or a filter that rolls off 6 dB per octave.
* The class of Hilbert systems, of which the most famous example is the 1928 Hilbert-Ackermann system of first-order logic ;
" For example, a general coupled set of first-order SDEs is often written in the form:
0.175 seconds.