A binomial QMF – properly an orthonormal binomial quadrature mirror filter – is an
orthogonal wavelet
An orthogonal wavelet is a wavelet whose associated wavelet transform is orthogonal.
That is, the inverse wavelet transform is the adjoint of the wavelet transform.
If this condition is weakened one may end up with biorthogonal wavelets.
Basics ...
developed in 1990.
The binomial QMF bank with perfect reconstruction (PR) was designed by
Ali Akansu
Ali Naci Akansu (born May 6, 1958) is a Turkish-American Professor of electrical & computer engineering and scientist in applied mathematics.
He is best known for his seminal contributions to the theory and applications of linear subspace meth ...
, and published in 1990, using the family of binomial polynomials for subband decomposition of discrete-time signals. Akansu and his fellow authors also showed that these binomial-QMF filters are identical to the
wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
filters designed independently by
Ingrid Daubechies
Baroness Ingrid Daubechies ( ; ; born 17 August 1954) is a Belgian physicist and mathematician. She is best known for her work with wavelets in image compression.
Daubechies is recognized for her study of the mathematical methods that enhance ...
from compactly supported orthonormal
wavelet transform
In mathematics, a wavelet series is a representation of a square-integrable ( real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal ...
perspective in 1988 (
Daubechies wavelet
The Daubechies wavelets, based on the work of Ingrid Daubechies, are a family of orthogonal wavelets defining a discrete wavelet transform and characterized by a maximal number of vanishing moments for some given support. With each wavelet type ...
). It was an extension of Akansu's prior work on
Binomial coefficient and
Hermite polynomials wherein he developed the Modified Hermite Transformation (MHT) in 1987.
Later, it was shown that the magnitude square functions of low-pass and high-pass binomial-QMF filters are the unique maximally flat functions in a two-band PR-QMF design framework.
[O. Herrmann]
On the Approximation Problem in Nonrecursive Digital Filter Design
IEEE Trans. Circuit Theory, vol CT-18, no. 3, pp. 411–413, May 1971.
References
{{reflist
External links
(First Wavelets Conference in USA)
Orthogonal wavelets
Turkish inventions