The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
s. It performs an
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
,
symmetric,
involutive,
linear operation on
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s (or
complex, or
hypercomplex numbers, although the Hadamard matrices themselves are purely real).
The Hadamard transform can be regarded as being built out of size-2
discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
s (DFTs), and is in fact equivalent to a multidimensional DFT of size .
It decomposes an arbitrary input vector into a superposition of
Walsh function
In mathematics, more specifically in harmonic analysis, Walsh functions form a complete orthogonal set of functions that can be used to represent any discrete function—just like trigonometric functions can be used to represent any continuous ...
s.
The transform is named for the
French
French (french: français(e), link=no) may refer to:
* Something of, from, or related to France
** French language, which originated in France, and its various dialects and accents
** French people, a nation and ethnic group identified with Franc ...
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
Jacques Hadamard (), the German-American mathematician
Hans Rademacher, and the American mathematician
Joseph L. Walsh.
Definition
The Hadamard transform ''H''
''m'' is a 2
''m'' × 2
''m'' matrix, the
Hadamard matrix
In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of row ...
(scaled by a normalization factor), that transforms 2
''m'' real numbers ''x''
''n'' into 2
''m'' real numbers ''X''
''k''. The Hadamard transform can be defined in two ways:
recursively, or by using the
binary (
base-2) representation of the indices ''n'' and ''k''.
Recursively, we define the 1 × 1 Hadamard transform ''H''
0 by the
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), an ...
''H''
0 = 1, and then define ''H''
''m'' for ''m'' > 0 by:
where the 1/ is a normalization that is sometimes omitted.
For ''m'' > 1, we can also define ''H''
''m'' by:
where
represents the
Kronecker product
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to ...
. Thus, other than this normalization factor, the Hadamard matrices are made up entirely of 1 and −1.
Equivalently, we can define the Hadamard matrix by its (''k'', ''n'')-th entry by writing
where the ''k''
''j'' and ''n''
''j'' are the bit elements (0 or 1) of ''k'' and ''n'', respectively. Note that for the element in the top left corner, we define:
. In this case, we have:
This is exactly the multidimensional
DFT, normalized to be
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation In mathematics, a unitary representation of a grou ...
, if the inputs and outputs are regarded as multidimensional arrays indexed by the ''n''
''j'' and ''k''
''j'', respectively.
Some examples of the Hadamard matrices follow.
where
is the bitwise dot product of the binary representations of the numbers i and j. For example, if
, then
agreeing with the above (ignoring the overall constant). Note that the first row, first column element of the matrix is denoted by
.
''H''
1 is precisely the size-2 DFT. It can also be regarded as the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
on the two-element ''additive'' group of Z/(2).
The rows of the Hadamard matrices are the
Walsh function
In mathematics, more specifically in harmonic analysis, Walsh functions form a complete orthogonal set of functions that can be used to represent any discrete function—just like trigonometric functions can be used to represent any continuous ...
s.
Advantages of the Walsh–Hadamard transform
Real
According to the above definition of matrix ''H'', here we let ''H'' = ''H''
'm'',''n''
In the Walsh transform, only 1 and −1 will appear in the matrix. The numbers 1 and −1 are real numbers so there is no need to perform a
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
calculation.
No multiplication is required
The DFT needs irrational multiplication, while the hadamard transform does not. Even rational multiplication is not needed, since sign flips is all it takes.
Some properties are similar to those of the DFT
In the Walsh transform matrix, all entries in the first row (and column) are equal to 1.
Discrete Fourier transform:
In discrete Fourier transform, when m equal to zeros (mean first row), the result of DFT also is 1.
At the second row, although it is different from the first row we can observe a characteristic of the matrix that the signal in the first raw matrix is low frequency and it will increase the frequency at second row, increase more frequency until the last row.
If we calculate zero crossing:
First row = 0 zero crossing
Second row = 1 zero crossing
Third row = 2 zero crossings
⋮
Eight row = 7 zero crossings
Relation to Fourier transform
The Hadamard transform is in fact equivalent to a multidimensional DFT of size .
Another approach is to view the Hadamard transform as a Fourier transform on the
Boolean group .
Using the
Fourier transform on finite (abelian) groups, the Fourier transform of a function
is the function
defined by
where
is a
character
Character or Characters may refer to:
Arts, entertainment, and media Literature
* ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk
* ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
of
. Each character has the form
for some
, where the multiplication is the boolean dot product on bit strings, so we can identify the input to
with
(
Pontryagin duality) and define
by
This is the Hadamard transform of
, considering the input to
and
as boolean strings.
In terms of the above formulation where the Hadamard transform multiplies a vector of
complex numbers
on the left by the Hadamard matrix
the equivalence is seen by taking
to take as input the bit string corresponding to the index of an element of
, and having
output the corresponding element of
.
Compare this to the usual
discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
which when applied to a vector
of
complex numbers instead uses characters of the
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
.
Computational complexity
In the classical domain, the Hadamard transform can be computed in
operations (
), using the
fast Hadamard transform
Fast or FAST may refer to:
* Fast (noun), high speed or velocity
* Fast (noun, verb), to practice fasting, abstaining from food and/or water for a certain period of time
Acronyms and coded Computing and software
* ''Faceted Application of Subje ...
algorithm.
In the quantum domain, the Hadamard transform can be computed in
time, as it is a
quantum logic gate
In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, ...
that can be
parallelized.
Quantum computing applications
The Hadamard transform is used extensively in
quantum computing
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
. The 2 × 2 Hadamard transforms
is the
quantum logic gate
In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, ...
known as the Hadamard gate, and the application of a Hadamard gate to each qubit of an n-qubit register in parallel is equivalent to the Hadamard transform
Hadamard gate
In quantum computing, the Hadamard gate is a one-
qubit rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
, mapping the qubit-basis states
and
to two superposition states with equal weight of the computational
basis states
and
. Usually the phases are chosen so that
in
Dirac notation. This corresponds to the
transformation matrix
in the
basis, also known as the
computational basis. The states
and
are known as
and
respectively, and together constitute the
polar basis in
quantum computing
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
.
Hadamard gate operations
One application of the Hadamard gate to either a 0 or 1 qubit will produce a quantum state that, if observed, will be a 0 or 1 with equal probability (as seen in the first two operations). This is exactly like flipping a fair coin in the standard
probabilistic model of computation. However, if the Hadamard gate is applied twice in succession (as is effectively being done in the last two operations), then the final state is always the same as the initial state.
Hadamard transform in quantum algorithms
Computing the quantum Hadamard transform is simply the application of a Hadamard gate to each qubit individually because of the tensor product structure of the Hadamard transform. This simple result means the quantum Hadamard transform requires log ''n'' operations, compared to the classical case of ''n'' log ''n'' operations.
Many
quantum algorithm
In quantum computing, a quantum algorithm is an algorithm which runs on a realistic model of quantum computation, the most commonly used model being the quantum circuit model of computation. A classical (or non-quantum) algorithm is a finite s ...
s use the Hadamard transform as an initial step, since it maps ''m'' qubits initialized with
to a superposition of all 2
''m'' orthogonal states in the
basis with equal weight. For example, this is used in the
Deutsch–Jozsa algorithm
The Deutsch–Jozsa algorithm is a deterministic quantum algorithm proposed by David Deutsch and Richard Jozsa in 1992 with improvements by Richard Cleve, Artur Ekert, Chiara Macchiavello, and Michele Mosca in 1998. Although of little current pra ...
,
Simon's algorithm
In computational complexity theory and quantum computing, Simon's problem is a computational problem that is proven to be solved exponentially faster on a quantum computer than on a classical (that is, traditional) computer. The quantum algorithm ...
, the
Bernstein–Vazirani algorithm
The Bernstein–Vazirani algorithm, which solves the Bernstein–Vazirani problem, is a quantum algorithm invented by Ethan Bernstein and Umesh Vazirani in 1992.
It is a restricted version of the Deutsch–Jozsa algorithm where instead of dis ...
, and in
Grover's algorithm. Note that
Shor's algorithm
Shor's algorithm is a quantum computer algorithm for finding the prime factors of an integer. It was developed in 1994 by the American mathematician Peter Shor.
On a quantum computer, to factor an integer N , Shor's algorithm runs in polynom ...
uses both an initial Hadamard transform, as well as the
quantum Fourier transform, which are both types of
Fourier transforms on finite groups; the first on
and the second on
.
Molecular phylogenetics (evolutionary biology) applications
The Hadamard transform can be used to estimate
phylogenetic tree
A phylogenetic tree (also phylogeny or evolutionary tree Felsenstein J. (2004). ''Inferring Phylogenies'' Sinauer Associates: Sunderland, MA.) is a branching diagram or a tree showing the evolutionary relationships among various biological spec ...
s from molecular data.
Phylogenetics
In biology, phylogenetics (; from Greek φυλή/ φῦλον [] "tribe, clan, race", and wikt:γενετικός, γενετικός [] "origin, source, birth") is the study of the evolutionary history and relationships among or within groups ...
is the subfield of
evolutionary biology
Evolutionary biology is the subfield of biology that studies the evolutionary processes (natural selection, common descent, speciation) that produced the diversity of life on Earth. It is also defined as the study of the history of life ...
focused on understanding the relationships among organisms. A Hadamard transform applied to a vector (or matrix) of site pattern frequencies obtained from a DNA
multiple sequence alignment can be used to generate another vector that carries information about the tree topology. The invertible nature of the phylogenetic Hadamard transform also allows the calculation of site likelihoods from a tree topology vector, allowing one to use the Hadamard transform for
maximum likelihood estimation
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
of phylogenetic trees. However, the latter application is less useful than the transformation from the site pattern vector to the tree vector because there are other ways to calculate site likelihoods that are much more efficient. However, the invertible nature of the phylogenetic Hadamard transform does provide an elegant tool for mathematic phylogenetics.
The mechanics of the phylogenetic Hadamard transform involve the calculation of a vector
that provides information about the topology and branch lengths for tree
using the site pattern vector or matrix
.
where
is the Hadamard matrix of the appropriate size. This equation can be rewritten as a series of three equations to simplify its interpretation:
The invertible nature of this equation allows one to calculate an expected site pattern vector (or matrix) as follows:
We can use the Cavender–Farris–
Neyman (CFN) two-state
substitution model
In biology, a substitution model, also called models of DNA sequence evolution, are Markov models that describe changes over evolutionary time. These models describe evolutionary changes in macromolecules (e.g., DNA sequences) represented as seque ...
for DNA by encoding the
nucleotide
Nucleotides are organic molecules consisting of a nucleoside and a phosphate. They serve as monomeric units of the nucleic acid polymers – deoxyribonucleic acid (DNA) and ribonucleic acid (RNA), both of which are essential biomolecu ...
s as binary characters (the
purine
Purine is a heterocyclic aromatic organic compound that consists of two rings ( pyrimidine and imidazole) fused together. It is water-soluble. Purine also gives its name to the wider class of molecules, purines, which include substituted purines ...
s A and G are encoded as R and the
pyrimidine
Pyrimidine (; ) is an aromatic, heterocyclic, organic compound similar to pyridine (). One of the three diazines (six-membered heterocyclics with two nitrogen atoms in the ring), it has nitrogen atoms at positions 1 and 3 in the ring. The othe ...
s C and T are encoded as Y). This makes it possible to encode the multiple sequence alignment as the site pattern vector
that can be converted to a tree vector
, as shown in the following example:
The example shown in this table uses the simplified three equation scheme and it is for a four taxon tree that can be written as ((A,B),(C,D)); in
newick format. The site patterns are written in the order ABCD. This particular tree has two long terminal branches (0.2
transversion substitutions per site), two short terminal branches (0.025 transversion substitutions per site), and a short internal branch (0.025 transversion substitutions per site); thus, it would be written as ((A:0.025,B:0.2):0.025,(C:0.025,D:0.2)); in newick format. This tree will exhibit
long branch attraction if the data are analyzed using the
maximum parsimony
In phylogenetics, maximum parsimony is an optimality criterion under which the phylogenetic tree that minimizes the total number of character-state changes (or miminizes the cost of differentially weighted character-state changes) is preferred. ...
criterion (assuming the sequence analyzed is long enough for the observed site pattern frequencies to be close to the expected frequencies shown in the
column). The long branch attraction reflects the fact that the expected number of site patterns with index 6 -- which support the tree ((A,C),(B,D)); -- exceed the expected number of site patterns that support the true tree (index 4). Obviously, the invertible nature of the phylogenetic Hadamard transform means that the tree vector means that the tree vector
corresponds to the correct tree. Parsimony analysis after the transformation is therefore
statistically consistent, as would be a standard maximum likelihood analysis using the correct model (in this case the CFN model).
Note that the site pattern with 0 corresponds to the sites that have not changed (after encoding the nucleotides as purines or pyrimidines). The indices with asterisks (3, 5, and 6) are "parsimony-informative", and. the remaining indices represent site patterns where a single taxon differs from the other three taxa (so they are the equivalent of terminal branch lengths in a standard maximum likelihood phylogenetic tree).
If one wishes to use nucleotide data without recoding as R and Y (and ultimately as 0 and 1) it is possible to encode the site patterns as a matrix. If we consider a four-taxon tree there are a total of 256 site patterns (four nucleotides to the 4th power). However, symmetries of the
Kimura three-parameter (or K81) model allow us to reduce the 256 possible site patterns for DNA to 64 patterns, making it possible to encode the nucleotide data for a four-taxon tree as an 8 × 8 matrix
in a manner similar to the vector of 8 elements used above for transversion (RY) site patterns. This is accomplished by recoding the data using the
Klein four-group
In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity)
and in which composing any two of the three non-identity elements produces the third one ...
:
As with RY data, site patterns are indexed relative to the base in the arbitrarily chosen first taxon with the bases in the subsequent taxa encoded relative to that first base. Thus, the first taxon receives the bit pair (0,0). Using those bit pairs one can produce two vectors similar to the RY vectors and then populate the matrix using those vectors. This can be illustrated using the example from Hendy et al. (1994),
which is based on a multiple sequence alignment of four primate hemoglobin pseudogenes:
The much larger number of site patterns in column 0 reflects the fact that column 0 corresponds to
transition differences, which accumulate more rapidly than transversion differences in virtually all comparisons of genomic regions (and definitely accumulate more rapidly in the hemoglobin pseudogenes used for this worked example
). If we consider the site pattern AAGG it would to binary pattern 0000 for the second element of the Klein group bit pair and 0011 for the first element. in this case binary pattern based on the first element first element corresponds to index 3 (so row 3 in column 0; indicated with a single asterisk in the table). The site patterns GGAA, CCTT, and TTCC would be encoded in the exact same way. The site pattern AACT would be encoded with binary pattern 0011 based on the second element and 0001 based on the first element; this yields index 1 for the first element and index 3 for the second. The index based on the second Klein group bit pair is multiplied by 8 to yield the column index (in this case it would be column 24) The cell that would include the count of AACT site patterns is indicated with two asterisks; however, the absence of a number in the example indicates that the sequence alignment include no AACT site patterns (likewise, CCAG, GGTC, and TTGA site patterns, which would be encoded in the same way, are absent).
Other applications
The Hadamard transform is also used in
data encryption, as well as many
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
and
data compression
In information theory, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression is either lossy or lossless. Lossless compressio ...
algorithms
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
, such as
JPEG XR and
MPEG-4 AVC
Advanced Video Coding (AVC), also referred to as H.264 or MPEG-4 Part 10, is a video compression standard based on block-oriented, motion-compensated coding. It is by far the most commonly used format for the recording, compression, and distr ...
. In
video compression
In information theory, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression is either lossy or lossless. Lossless compressio ...
applications, it is usually used in the form of the
sum of absolute transformed differences
The sum of absolute transformed differences (SATD) is a block matching criterion widely used in fractional motion estimation for video compression. It works by taking a frequency transform, usually a Hadamard transform, of the differences betwee ...
. It is also a crucial part of significant number of algorithms in quantum computing. The Hadamard transform is also applied in experimental techniques such as
NMR,
mass spectrometry
Mass spectrometry (MS) is an analytical technique that is used to measure the mass-to-charge ratio of ions. The results are presented as a '' mass spectrum'', a plot of intensity as a function of the mass-to-charge ratio. Mass spectrometry is u ...
and
crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics ( condensed matter physics). The wor ...
. It is additionally used in some versions of
locality-sensitive hashing, to obtain pseudo-random matrix rotations.
See also
*
Fast Walsh–Hadamard transform
In computational mathematics, the Hadamard ordered fast Walsh–Hadamard transform (FWHT''h'') is an efficient algorithm to compute the Walsh–Hadamard transform (WHT). A naive implementation of the WHT of order n = 2^m would have a Computation ...
*
Pseudo-Hadamard transform The pseudo-Hadamard transform is a reversible transformation of a bit string that provides cryptographic diffusion. See Hadamard transform.
The bit string must be of even length so that it can be split into two bit strings ''a'' and ''b'' of equ ...
*
Haar transform
*
Generalized distributive law The generalized distributive law (GDL) is a generalization of the distributive property which gives rise to a general message passing algorithm. It is a synthesis of the work of many authors in the information theory, digital communications, sign ...
External links
*
*
* Pan, Jeng-shyan
Data Encryption Method Using Discrete Fractional Hadamard Transformation(May 28, 2009)
* Lachowicz, Dr. Pawel
Walsh–Hadamard Transform and Tests for Randomness of Financial Return-Series(April 7, 2015)
*
*
References
{{DEFAULTSORT:Hadamard Transform
Quantum algorithms
Transforms