The bilinear transform ( also known as Tustin's method ) is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa.

The bilinear transform is a special case of a conformal mapping ( namely, the Möbius transformation ), often used to convert a transfer function of a linear, time-invariant ( LTI ) filter in the continuous-time domain ( often called an analog filter ) to a transfer function of a linear, shift-invariant filter in the discrete-time domain ( often called a digital filter although there are analog filters constructed with switched capacitors that are discrete-time filters ).

The bilinear transform is a first-order approximation of the natural logarithm function that is an exact mapping of the z-plane to the s-plane.

where is the numerical integration step size of the trapezoidal rule used in the bilinear transform derivation.

The bilinear transform maps the left half of the complex s-plane to the interior of the unit circle in the z-plane.

If we wish to implement this filter as a digital filter, we can apply the bilinear transform by substituting for the formula above ; after some reworking, we get the following filter representation:

where Γ depends on the coordinate system φ and is bilinear in u and v. In particular, Γ does not involve any derivatives on u or v. In this approach, Γ must transform in a prescribed manner when the coordinate system φ is changed to a different coordinate system.

As with most analog filters, the Chebyshev may be converted to a digital ( discrete-time ) recursive form via the bilinear transform.

The last one is known as the bilinear transform, or Tustin transform, and preserves the ( in ) stability of the continuous-time system.

Various generalizations of the Hilbert transform, such as the bilinear and trilinear Hilbert transforms are still active areas of research today.

The point where the necessity to perform a Bogoliubov transform becomes obvious is that in mean-field approximation the Hamiltonian of the system can be written in both cases as a sum of bilinear terms in the original creation and destruction operators, involving finite-terms, i. e. one must go beyond the usual Hartree-Fock method (-> Hartree-Fock-Bogoliubov method ).

* Tustin's method, also known as the bilinear transform, a digital control approximation

As a consequence, over a field of characteristic not equal to 2, the theories of symmetric bilinear forms and of quadratic forms in n variables are essentially the same.

The software renderer also uses nearest-neighbor texture filtering, as opposed to bilinear filtering or trilinear filtering used by modern video cards.

Unlike other interpolation techniques such as nearest neighbor interpolation and bicubic interpolation, bilinear interpolation uses only the 4 nearest pixel values which are located in diagonal directions from a given pixel in order to find the appropriate color intensity values of that pixel.

Bilinear filtering uses these points to perform bilinear interpolation between the four texels nearest to the point that the pixel represents ( in the middle or upper left of the pixel, usually ).

Some standards, such as VC-1, use bicubic interpolation ; H. 264 / AVC uses a 6-tap filter for half-pixel interpolation and then simple bilinear sampling to achieve quarter-pixel precision from the half-pixel data.

In other words, if we hold the first entry of the bilinear map fixed, while letting the second entry vary, the result is a linear operator, and similarly if we hold the second entry fixed.

Other bilinear transforms can be used to warp the frequency response of any discrete-time linear system ( for example to approximate the non-linear frequency resolution of the human auditory system ) and are implementable in the discrete domain by replacing a system's unit delays with first order all-pass filters.

The result of bilinear interpolation is independent of the order ( order here meaning which axis is interpolated first and which second ) of interpolation.

symmetric bilinear form with orthonormal basis v < sub > i </ sub >, the map sending a lattice to its dual lattice gives an automorphism with square the identity, giving the permutation σ that sends each label to its negative modulo n. The image of the above homomorphism is generated by σ and τ and is isomorphic to the dihedral group D < sub > n </ sub > of order 2n ; when n = 3, it gives the whole of S < sub > 3 </ sub >.

In a standard, low order FEM in 2D, for quadrilateral elements the most typical choice is the bilinear test or interpolating function of the form.

Analogues of Heisenberg groups over finite fields of odd prime order p are called extra special groups, or more properly, extra special groups of exponent p. More generally, if the derived subgroup of a group G is contained in the center Z of G, then the map from G / Z × G / Z → Z is a skew-symmetric bilinear operator on abelian groups.

In mathematics, the Weil pairing is a pairing ( bilinear form, though with multiplicative notation ) on the points of order dividing n of an elliptic curve E, taking values in nth roots of unity.

The above bilinear approximation can be solved for or a similar approximation for can be performed.

The inverse of this mapping ( and its first-order bilinear approximation ) is

A line through two of these points, Af and Af, will be transformed into the entire bilinear congruence having the tangents to **zg at Af and Af as directrices.

Older techniques, such as bilinear and trilinear filtering, do not take into account the angle a surface is viewed from, which can result in aliasing or blurring of textures.

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.

) through known user gamma setting into a 48 bits per pixel photometrically linear space where they are resampled with bilinear resampling to the target size, possibly taking aspect ratio correction into account.

The fiber of E has a bilinear form induced by cup product, which is antisymmetric if d is even, and makes E into a symplectic space.

In differential geometry a Poisson supermanifold is a differential supermanifold M such that the supercommutative algebra of smooth functions over it ( to clarify this: M is not a point set space and so, doesn't " really " exist, and really, this algebra is all we have ), is equipped with a bilinear map called the Poisson superbracket turning it into a Poisson superalgebra.

In contrast to bilinear interpolation, which only takes 4 pixels ( 2x2 ) into account, bicubic interpolation considers 16 pixels ( 4x4 ).

Using the bilinear form, semi norms can be constructed to define a polar topology on the vector spaces and turn them into locally convex spaces, generalizations of normed vector spaces.

In linear algebra, a bilinear transformation is a binary function where the sets X, Y, and Z are all vector spaces and the derived functions f < sup > x </ sup > and f < sub > y </ sub > are all linear transformations.

A bilinear transformation, like any binary function, can be interpreted as a function from X × Y to Z, but this function in general won't be linear.

In mathematics, a bilinear operator is a function combining elements of two vector spaces to yield an element of a third vector space that is linear in each of its arguments.

Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function

A function of two vector arguments is bilinear if it is linear separately in each argument.

A bilinear function is nondegenerate provided that, for every tangent vector X < sub > p </ sub > ≠ 0, the function

For n > 1, the only n-linear map which is also a linear map is the zero function, see bilinear map # Examples.

That is, a bilinear form is a function B: V × V → F which is linear in each argument separately:

For instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the duality between distributions and the associated test functions corresponds to the pairing in which one integrates a distribution against a test function, and Poincaré duality corresponds similarly to intersection number, viewed as a pairing between submanifolds of a given manifold.

The interpolated function should not use the term of or, but, which is the bilinear form of and.

In a mathematical context, bilinear interpolation is the problem of finding a function f ( x, y ) of the form