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82-428: FHT may refer to: Fast Hartley transform Feminizing hormone therapy Fetal heart tones First-hitting-time model Female hose thread, for the female garden hose thread Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title FHT . If an internal link led you here, you may wish to change

164-398: A 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n ] = ( a 11

246-830: A 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n ) . {\displaystyle \mathbf {A} ={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{bmatrix}}={\begin{pmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{pmatrix}}.} This may be abbreviated by writing only

328-401: A i , j ) 1 ≤ i , j ≤ n {\displaystyle \mathbf {A} =(a_{i,j})_{1\leq i,j\leq n}} in the case that n = m {\displaystyle n=m} . Matrices are usually symbolized using upper-case letters (such as A {\displaystyle {\mathbf {A} }} in the examples above), while

410-2469: A s ( 2 π N ( n 1 k 1 + n 2 k 2 + n 3 k 3 ) ) {\displaystyle X(k_{1},k_{2},...,k_{r})=\sum _{n_{1}=0}^{N-1}\sum _{n_{2}=0}^{N-1}\sum _{n_{r}=0}^{N-1}x(n_{1},n_{2},n_{3}){\rm {cas}}({\frac {2\pi }{N}}(n_{1}k_{1}+n_{2}k_{2}+n_{3}k_{3}))} = ∑ n 1 : e v e n ∑ n 2 : e v e n ∑ n 3 : e v e n + ∑ n 1 : e v e n ∑ n 2 : e v e n ∑ n 3 : o d d + ∑ n 1 : e v e n ∑ n 2 : o d d ∑ n 3 : e v e n {\displaystyle =\sum _{n_{1}:even}\sum _{n_{2}:even}\sum _{n_{3}:even}+\sum _{n_{1}:even}\sum _{n_{2}:even}\sum _{n_{3}:odd}+\sum _{n_{1}:even}\sum _{n_{2}:odd}\sum _{n_{3}:even}} + ∑ n 1 : e v e n ∑ n 2 : o d d ∑ n 3 : o d d + ∑ n 1 : o d d ∑ n 2 : e v e n ∑ n 3 : e v e n + ∑ n 1 : o d d ∑ n 2 : e v e n ∑ n 3 : o d d {\displaystyle +\sum _{n_{1}:even}\sum _{n_{2}:odd}\sum _{n_{3}:odd}+\sum _{n_{1}:odd}\sum _{n_{2}:even}\sum _{n_{3}:even}+\sum _{n_{1}:odd}\sum _{n_{2}:even}\sum _{n_{3}:odd}} + ∑ n 1 : o d d ∑ n 2 : o d d ∑ n 3 : e v e n + ∑ n 1 : o d d ∑ n 2 : o d d ∑ n 3 : o d d . {\displaystyle +\sum _{n_{1}:odd}\sum _{n_{2}:odd}\sum _{n_{3}:even}+\sum _{n_{1}:odd}\sum _{n_{2}:odd}\sum _{n_{3}:odd}.} It

492-512: A s ( 2 π n 1 k 1 N 1 ) … c a s ( 2 π n r k r N r ) . {\displaystyle {\hat {X}}(k_{1},k_{2},...,k_{r})=\sum _{n_{1}=0}^{N_{1}-1}\sum _{n_{2}=0}^{N_{2}-1}\dots \sum _{n_{r}=0}^{N_{r}-1}x(n_{1},n_{2},...,n_{r}){\rm {cas}}({\frac {2\pi n_{1}k_{1}}{N_{1}}})\dots {\rm {cas}}({\frac {2\pi n_{r}k_{r}}{N_{r}}}).} It

574-660: A s ( 2 π n 1 k 1 N 1 + ⋯ + 2 π n r k r N r ) , {\displaystyle X(k_{1},k_{2},...,k_{r})=\sum _{n_{1}=0}^{N_{1}-1}\sum _{n_{2}=0}^{N_{2}-1}\dots \sum _{n_{r}=0}^{N_{r}-1}x(n_{1},n_{2},...,n_{r}){\rm {cas}}({\frac {2\pi n_{1}k_{1}}{N_{1}}}+\dots +{\frac {2\pi n_{r}k_{r}}{N_{r}}}),} with k i = 0 , 1 , … , N i − 1 {\displaystyle k_{i}=0,1,\ldots ,N_{i}-1} and where c

656-545: A s ( 2 π m l M ) , {\displaystyle X_{2}(0,l)=\sum _{m=0}^{M-1}(\sum _{n=0}^{N-1}x(n,m)){\rm {cas}}({\frac {2\pi ml}{M}}),\;} l = 1 , 2 , … , M − 1 {\displaystyle l=1,2,\dots ,M-1} X 3 ( k , l ) = ∑ n = 0 N − 1 ∑ m = 0 M − 1 x ( n , m ) c

738-577: A s ( 2 π n k N ) + ∑ m = 1 M − 1 x ( 0 , m ) c a s ( 2 π m l M ) {\displaystyle X_{3}(k,l)=\sum _{n=0}^{N-1}x(n,0){\rm {cas}}({\frac {2\pi nk}{N}})+\sum _{m=1}^{M-1}x(0,m){\rm {cas}}({\frac {2\pi ml}{M}})} + ∑ n = 1 N − 1 ∑ m = 1 M − 1 x ( n , m ) c

820-556: A s ( 2 π n k N ) , {\displaystyle X_{1}(k,0)=\sum _{n=0}^{N-1}(\sum _{m=0}^{M-1}x(n,m)){\rm {cas}}({\frac {2\pi nk}{N}}),\;} k = 0 , 1 , … , N − 1 {\displaystyle k=0,1,\ldots ,N-1} X 2 ( 0 , l ) = ∑ m = 0 M − 1 ( ∑ n = 0 N − 1 x ( n , m ) ) c

902-769: A s ( 2 π n k N + 2 π m l M ) , {\displaystyle X_{3}(k,l)=\sum _{n=0}^{N-1}\sum _{m=0}^{M-1}x(n,m){\rm {cas}}({\frac {2\pi nk}{N}}+{\frac {2\pi ml}{M}})\;,} k = 1 , 2 , … , N − 1 {\displaystyle k=1,2,\ldots ,N-1} l = 1 , 2 , … , M − 1. {\displaystyle l=1,2,\ldots ,M-1.} Developing X 3 {\displaystyle X_{3}} further, X 3 ( k , l ) = ∑ n = 0 N − 1 x ( n , 0 ) c

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984-808: A s ( 2 π n k N + 2 π m l M ) . {\displaystyle +\sum _{n=1}^{N-1}\sum _{m=1}^{M-1}x(n,m){\rm {cas}}({\frac {2\pi nk}{N}}+{\frac {2\pi ml}{M}}).} At this point we present the Fermat number transform (FNT). The t Fermat number is given by F t = 2 b + 1 {\displaystyle F_{t}=2^{b}+1} , with b = 2 t {\displaystyle b=2^{t}} . The well known Fermat numbers are for t = 0 , 1 , 2 , 3 , 4 , 5 , 6 {\displaystyle t=0,1,2,3,4,5,6} ( F t {\displaystyle F_{t}}

1066-514: A s ( x ) = cos ⁡ ( x ) + sin ⁡ ( x ) . {\displaystyle {\rm {cas}}(x)=\cos(x)+\sin(x).} Similar to the 1-D case, as a real and symmetric transform, the MD-DHT is simpler than the MD-DFT. For one, the inverse DHT is identical to the forward transform, with the addition of a scaling factor; and second, since the kernel

1148-672: A field or a ring . In this section, it is supposed that matrix entries belong to a fixed ring, which is typically a field of numbers. The sum A + B of two m × n matrices A and B is calculated entrywise: ( A + B ) i , j = A i , j + B i , j , 1 ≤ i ≤ m , 1 ≤ j ≤ n . {\displaystyle ({\mathbf {A}}+{\mathbf {B}})_{i,j}={\mathbf {A}}_{i,j}+{\mathbf {B}}_{i,j},\quad 1\leq i\leq m,\quad 1\leq j\leq n.} For example, The product c A of

1230-700: A k -by- m matrix B represents another linear map g : R m → R k {\displaystyle g:\mathbb {R} ^{m}\to \mathbb {R} ^{k}} , then the composition g ∘ f is represented by BA since ( g ∘ f ) ( x ) = g ( f ( x ) ) = g ( A x ) = B ( A x ) = ( B A ) x . {\displaystyle (g\circ f)({\mathbf {x}})=g(f({\mathbf {x}}))=g({\mathbf {Ax}})={\mathbf {B}}({\mathbf {Ax}})=({\mathbf {BA}}){\mathbf {x}}.} The last equality follows from

1312-520: A " 2 × 3 {\displaystyle 2\times 3} matrix", or a matrix of dimension 2 × 3 {\displaystyle 2\times 3} . Matrices are commonly related to linear algebra . Notable exceptions include incidence matrices and adjacency matrices in graph theory . This article focuses on matrices related to linear algebra, and, unless otherwise specified, all matrices represent linear maps or may be viewed as such. Square matrices , matrices with

1394-614: A , b ) , ( a + c , b + d ) , and ( c , d ) . The parallelogram pictured at the right is obtained by multiplying A with each of the column vectors [ 0 0 ] , [ 1 0 ] , [ 1 1 ] {\displaystyle {\begin{bmatrix}0\\0\end{bmatrix}},{\begin{bmatrix}1\\0\end{bmatrix}},{\begin{bmatrix}1\\1\end{bmatrix}}} , and [ 0 1 ] {\displaystyle {\begin{bmatrix}0\\1\end{bmatrix}}} in turn. These vectors define

1476-406: A 2-by-3 submatrix by removing row 3 and column 2: The minors and cofactors of a matrix are found by computing the determinant of certain submatrices. A principal submatrix is a square submatrix obtained by removing certain rows and columns. The definition varies from author to author. According to some authors, a principal submatrix is a submatrix in which the set of row indices that remain

1558-427: A DHT of length N into a DHT of length N /2 and two real-input DFTs ( not DHTs) of length N /4. In this way, they argued that a DHT of power-of-two length can be computed with, at best, 2 more additions than the corresponding number of arithmetic operations for the real-input DFT. On present-day computers, performance is determined more by cache and CPU pipeline considerations than by strict operation counts, and

1640-407: A matrix are called rows and columns , respectively. The size of a matrix is defined by the number of rows and columns it contains. There is no limit to the number of rows and columns, that a matrix (in the usual sense) can have as long as they are positive integers. A matrix with m {\displaystyle {m}} rows and n {\displaystyle {n}} columns

1722-429: A matrix over a field F is a rectangular array of elements of F . A real matrix and a complex matrix are matrices whose entries are respectively real numbers or complex numbers . More general types of entries are discussed below . For instance, this is a real matrix: The numbers, symbols, or expressions in the matrix are called its entries or its elements . The horizontal and vertical lines of entries in

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1804-412: A matrix plus the rank equals the number of columns of the matrix. A square matrix is a matrix with the same number of rows and columns. An n -by- n matrix is known as a square matrix of order n . Any two square matrices of the same order can be added and multiplied. The entries a ii form the main diagonal of a square matrix. They lie on the imaginary line that runs from the top left corner to

1886-596: A nonzero determinant and the eigenvalues of a square matrix are the roots of a polynomial determinant. In geometry , matrices are widely used for specifying and representing geometric transformations (for example rotations ) and coordinate changes . In numerical analysis , many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimensions. Matrices are used in most areas of mathematics and scientific fields, either directly, or through their use in geometry and numerical analysis. Matrix theory

1968-464: A number c (also called a scalar in this context) and a matrix A is computed by multiplying every entry of A by c : ( c A ) i , j = c ⋅ A i , j {\displaystyle (c{\mathbf {A}})_{i,j}=c\cdot {\mathbf {A}}_{i,j}} This operation is called scalar multiplication , but its result is not named "scalar product" to avoid confusion, since "scalar product"

2050-507: A single generic term, possibly along with indices, as in A = ( a i j ) , [ a i j ] , or ( a i j ) 1 ≤ i ≤ m , 1 ≤ j ≤ n {\displaystyle \mathbf {A} =\left(a_{ij}\right),\quad \left[a_{ij}\right],\quad {\text{or}}\quad \left(a_{ij}\right)_{1\leq i\leq m,\;1\leq j\leq n}} or A = (

2132-465: A slight difference in arithmetic cost is unlikely to be significant. Since FHT and real-input FFT algorithms have similar computational structures, neither appears to have a substantial a priori speed advantage ( Popović  [ sr ] and Šević, 1994). As a practical matter, highly optimized real-input FFT libraries are available from many sources (e.g. from CPU vendors such as Intel ), whereas highly optimized DHT libraries are less common. On

2214-743: A subscript. For instance, the matrix A {\displaystyle \mathbf {A} } above is 3 × 4 {\displaystyle 3\times 4} , and can be defined as A = [ i − j ] ( i = 1 , 2 , 3 ; j = 1 , … , 4 ) {\displaystyle {\mathbf {A} }=[i-j](i=1,2,3;j=1,\dots ,4)} or A = [ i − j ] 3 × 4 {\displaystyle {\mathbf {A} }=[i-j]_{3\times 4}} . Some programming languages utilize doubly subscripted arrays (or arrays of arrays) to represent an m -by- n matrix. Some programming languages start

2296-399: Is commutative , that is, the matrix sum does not depend on the order of the summands: A + B = B + A . The transpose is compatible with addition and scalar multiplication, as expressed by ( c A ) = c ( A ) and ( A + B ) = A + B . Finally, ( A ) = A . Multiplication of two matrices is defined if and only if the number of columns of the left matrix is the same as

2378-446: Is a 3 × 2 {\displaystyle {3\times 2}} matrix. Matrices with a single row are called row vectors , and those with a single column are called column vectors . A matrix with the same number of rows and columns is called a square matrix . A matrix with an infinite number of rows or columns (or both) is called an infinite matrix . In some contexts, such as computer algebra programs , it

2460-1403: Is also covered in the stated reference), X ( k , l ) = ∑ n = 0 N − 1 ∑ m = 0 M − 1 x ( n , m ) c a s ( 2 π n k N + 2 π m l M ) , {\displaystyle X(k,l)=\sum _{n=0}^{N-1}\sum _{m=0}^{M-1}x(n,m){\rm {cas}}({\frac {2\pi nk}{N}}+{\frac {2\pi ml}{M}}),\;} k = 0 , 1 , … , N − 1 {\displaystyle k=0,1,\ldots ,N-1} , l = 0 , 1 , … , M − 1 {\displaystyle l=0,1,\ldots ,M-1} can be decomposed into 1-D and 2-D circular convolutions as follows, X ( k , l ) = { X 1 ( k , 0 ) X 2 ( 0 , l ) X 3 ( k , l ) {\displaystyle X(k,l)={\begin{cases}X_{1}(k,0)\\X_{2}(0,l)\\X_{3}(k,l)\end{cases}}} where X 1 ( k , 0 ) = ∑ n = 0 N − 1 ( ∑ m = 0 M − 1 x ( n , m ) ) c

2542-470: Is an m × n matrix, x designates a column vector (that is, n ×1 -matrix) of n variables x 1 , x 2 , ..., x n , and b is an m ×1 -column vector, then the matrix equation is equivalent to the system of linear equations Using matrices, this can be solved more compactly than would be possible by writing out all the equations separately. If n = m and the equations are independent , then this can be done by writing where A

FHT - Misplaced Pages Continue

2624-456: Is called an m × n {\displaystyle {m\times n}} matrix, or m {\displaystyle {m}} -by- n {\displaystyle {n}} matrix, where m {\displaystyle {m}} and n {\displaystyle {n}} are called its dimensions . For example, the matrix A {\displaystyle {\mathbf {A} }} above

2706-694: Is commonly used due to the simplicity of such R-C algorithms, but they are not optimized for general M-D spaces. Other fast algorithms have been developed, such as radix-2, radix-4, and split radix. For example, Boussakta (2000) developed the 3-D vector radix, X ( k 1 , k 2 , . . . , k r ) = ∑ n 1 = 0 N − 1 ∑ n 2 = 0 N − 1 ∑ n r = 0 N − 1 x ( n 1 , n 2 , n 3 ) c

2788-472: Is not commutative , in marked contrast to (rational, real, or complex) numbers, whose product is independent of the order of the factors. An example of two matrices not commuting with each other is: whereas Besides the ordinary matrix multiplication just described, other less frequently used operations on matrices that can be considered forms of multiplication also exist, such as the Hadamard product and

2870-668: Is often denoted M ( m , n ) , {\displaystyle {\mathcal {M}}(m,n),} or M m × n ( R ) . {\displaystyle {\mathcal {M}}_{m\times n}(\mathbb {R} ).} The set of all m -by- n matrices over another field , or over a ring R , is similarly denoted M ( m , n , R ) , {\displaystyle {\mathcal {M}}(m,n,R),} or M m × n ( R ) . {\displaystyle {\mathcal {M}}_{m\times n}(R).} If m   =   n , such as in

2952-666: Is often used as a synonym for " inner product ". For example: The subtraction of two m × n matrices is defined by composing matrix addition with scalar multiplication by –1 : The transpose of an m × n matrix A is the n × m matrix A (also denoted A or A ) formed by turning rows into columns and vice versa: ( A T ) i , j = A j , i . {\displaystyle \left({\mathbf {A}}^{\rm {T}}\right)_{i,j}={\mathbf {A}}_{j,i}.} For example: Familiar properties of numbers extend to these operations on matrices: for example, addition

3034-1330: Is prime for 0 ≤ t ≤ 4 {\displaystyle 0\leq t\leq 4} ), (Boussakta, 1988). The Fermat number transform is given by X ( k , l ) = ∑ n = 0 N − 1 ∑ m = 0 M − 1 x ( n , m ) α 1 n k α 2 m l mod F t {\displaystyle X(k,l)=\sum _{n=0}^{N-1}\sum _{m=0}^{M-1}x(n,m)\alpha _{1}^{nk}\alpha _{2}^{ml}\mod F_{t}} with k = 0 , … , N − 1 , l = 0 , … , M − 1 {\displaystyle k=0,\ldots ,N-1,l=0,\ldots ,M-1} . α 1 {\displaystyle \alpha _{1}} and α 2 {\displaystyle \alpha _{2}} are roots of unity of order N {\displaystyle N} and M {\displaystyle M} respectively ( α 1 N = α 2 M = 1 mod F t ) {\displaystyle (\alpha _{1}^{N}=\alpha _{2}^{M}=1\mod F_{t})} . Going back to

3116-626: Is real, it avoids the computational complexity of complex numbers . Additionally, the DFT is directly obtainable from the DHT by a simple additive operation (Bracewell, 1983). The MD-DHT is widely used in areas like image and optical signal processing. Specific applications include computer vision, high-definition television, and teleconferencing, areas that process or analyze motion images (Zeng, 2000). As computing speed keeps increasing, bigger multidimensional problems become computationally feasible, requiring

3198-423: Is sometimes denoted cas( z ) , and should not be confused with cis( z ) = e = cos( z ) + i  sin( z ) , or e = cis(− z ) which appears in the DFT definition (where i is the imaginary unit ). As with the DFT, the overall scale factor in front of the transform and the sign of the sine term are a matter of convention. Although these conventions occasionally vary between authors, they do not affect

3280-447: Is the branch of mathematics that focuses on the study of matrices. It was initially a sub-branch of linear algebra , but soon grew to include subjects related to graph theory , algebra , combinatorics and statistics . A matrix is a rectangular array of numbers (or other mathematical objects), called the entries of the matrix. Matrices are subject to standard operations such as addition and multiplication . Most commonly,

3362-418: Is the i th coordinate of f  ( e j ) , where e j = (0, ..., 0, 1, 0, ..., 0) is the unit vector with 1 in the j th position and 0 elsewhere. The matrix A is said to represent the linear map f , and A is called the transformation matrix of f . For example, the 2×2 matrix can be viewed as the transform of the unit square into a parallelogram with vertices at (0, 0) , (

FHT - Misplaced Pages Continue

3444-590: Is the inverse matrix of A . If A has no inverse, solutions—if any—can be found using its generalized inverse . Matrices and matrix multiplication reveal their essential features when related to linear transformations , also known as linear maps . A real m -by- n matrix A gives rise to a linear transformation R n → R m {\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{m}} mapping each vector x in ⁠ R n {\displaystyle \mathbb {R} ^{n}} ⁠ to

3526-413: Is the same as the set of column indices that remain. Other authors define a principal submatrix as one in which the first k rows and columns, for some number k , are the ones that remain; this type of submatrix has also been called a leading principal submatrix . Matrices can be used to compactly write and work with multiple linear equations, that is, systems of linear equations. For example, if A

3608-447: Is used in place of M . {\displaystyle {\mathcal {M}}.} Several basic operations can be applied to matrices. Some, such as transposition and submatrix do not depend on the nature of the entries. Others, such as matrix addition , scalar multiplication , matrix multiplication , and row operations involve operations on matrix entries and therefore require that matrix entries are numbers or belong to

3690-403: Is used to represent a mathematical object or property of such an object. For example, [ 1 9 − 13 20 5 − 6 ] {\displaystyle {\begin{bmatrix}1&9&-13\\20&5&-6\end{bmatrix}}} is a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix",

3772-442: Is useful to consider a matrix with no rows or no columns, called an empty matrix . The specifics of symbolic matrix notation vary widely, with some prevailing trends. Matrices are commonly written in square brackets or parentheses , so that an m × n {\displaystyle m\times n} matrix A {\displaystyle \mathbf {A} } is represented as A = [

3854-550: The ( 1 , 3 ) {\displaystyle (1,3)} entry of the following matrix A {\displaystyle \mathbf {A} } is 5 (also denoted a 13 {\displaystyle {a_{13}}} , a 1 , 3 {\displaystyle {a_{1,3}}} , A [ 1 , 3 ] {\displaystyle \mathbf {A} [1,3]} or A 1 , 3 {\displaystyle {{\mathbf {A} }_{1,3}}} ): Sometimes,

3936-533: The Kronecker product . They arise in solving matrix equations such as the Sylvester equation . There are three types of row operations: These operations are used in several ways, including solving linear equations and finding matrix inverses . A submatrix of a matrix is a matrix obtained by deleting any collection of rows and/or columns. For example, from the following 3-by-4 matrix, we can construct

4018-1050: The n -by- n matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, for example, I 1 = [ 1 ] , I 2 = [ 1 0 0 1 ] , ⋮ I n = [ 1 0 ⋯ 0 0 1 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ 1 ] {\displaystyle {\begin{aligned}\mathbf {I} _{1}&={\begin{bmatrix}1\end{bmatrix}},\\[4pt]\mathbf {I} _{2}&={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\\[4pt]\vdots &\\[4pt]\mathbf {I} _{n}&={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix}}\end{aligned}}} It

4100-421: The (matrix) product Ax , which is a vector in ⁠ R m . {\displaystyle \mathbb {R} ^{m}.} ⁠ Conversely, each linear transformation f : R n → R m {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}} arises from a unique m -by- n matrix A : explicitly, the ( i , j ) -entry of A

4182-598: The Cooley–Tukey algorithm is commonly known as the fast Hartley transform (FHT) algorithm, and was first described by Bracewell in 1984. This FHT algorithm, at least when applied to power-of-two sizes N , is the subject of the United States patent number 4,646,256, issued in 1987 to Stanford University . Stanford placed this patent in the public domain in 1994 (Bracewell, 1995). As mentioned above, DHT algorithms are typically slightly less efficient (in terms of

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4264-407: The DFT transforms a convolution into a pointwise multiplication of complex numbers ( pairs of real and imaginary parts), the DHT transforms a convolution into a simple combination of pairs of real frequency components. The inverse DHT then yields the desired vector z . In this way, a fast algorithm for the DHT (see below) yields a fast algorithm for convolution. (This is slightly more expensive than

4346-400: The DHT analogous to the fast Fourier transform (FFT), the DHT was originally proposed by Ronald N. Bracewell in 1983 as a more efficient computational tool in the common case where the data are purely real. It was subsequently argued, however, that specialized FFT algorithms for real inputs or outputs can ordinarily be found with slightly fewer operations than any corresponding algorithm for

4428-411: The DHT is its own inverse ( involutory ), up to an overall scale factor. The DHT can be used to compute the DFT, and vice versa. For real inputs x n , the DFT output X k has a real part ( H k + H N−k )/2 and an imaginary part ( H N−k − H k )/2. Conversely, the DHT is equivalent to computing the DFT of x n multiplied by 1 +  i , then taking the real part of

4510-822: The DHT of real data for roughly a factor of two less computation than that of the equivalent complex FFT (Frigo and Johnson, 2005). On the other hand, a non-DHT-based adaptation of Rader's algorithm for real-input DFTs is also possible (Chu & Burrus , 1982). The rD-DHT (MD-DHT with "r" dimensions) is given by X ( k 1 , k 2 , . . . , k r ) = ∑ n 1 = 0 N 1 − 1 ∑ n 2 = 0 N 2 − 1 … ∑ n r = 0 N r − 1 x ( n 1 , n 2 , . . . , n r ) c

4592-623: The DHT. Formally, the discrete Hartley transform is a linear, invertible function H : R → R (where R denotes the set of real numbers ). The N real numbers x 0 , ..., x N −1 are transformed into the N real numbers H 0 , ..., H N −1 according to the formula The combination cos ⁡ ( z ) + sin ⁡ ( z ) {\displaystyle \cos(z)+\sin(z)} = 2 cos ⁡ ( z − π 4 ) {\displaystyle ={\sqrt {2}}\cos \left(z-{\frac {\pi }{4}}\right)}

4674-509: The FFT, however, that compute the same result in only O( N log N ) operations. Nearly every FFT algorithm, from Cooley–Tukey to prime-factor to Winograd (1985) to Bruun's (1993), has a direct analogue for the discrete Hartley transform. (However, a few of the more exotic FFT algorithms, such as the QFT, have not yet been investigated in the context of the DHT.) In particular, the DHT analogue of

4756-409: The above-mentioned associativity of matrix multiplication. The rank of a matrix A is the maximum number of linearly independent row vectors of the matrix, which is the same as the maximum number of linearly independent column vectors. Equivalently it is the dimension of the image of the linear map represented by A . The rank–nullity theorem states that the dimension of the kernel of

4838-446: The above-mentioned formula f ( i , j ) {\displaystyle f(i,j)} is valid for any i = 1 , … , m {\displaystyle i=1,\dots ,m} and any j = 1 , … , n {\displaystyle j=1,\dots ,n} . This can be specified separately or indicated using m × n {\displaystyle m\times n} as

4920-400: The bottom right corner of the matrix. If all entries of A below the main diagonal are zero, A is called an upper triangular matrix . Similarly, if all entries of A above the main diagonal are zero, A is called a lower triangular matrix . If all entries outside the main diagonal are zero, A is called a diagonal matrix . The identity matrix I n of size n is

5002-406: The case of square matrices , one does not repeat the dimension: M ( n , R ) , {\displaystyle {\mathcal {M}}(n,R),} or M n ( R ) . {\displaystyle {\mathcal {M}}_{n}(R).} Often, M {\displaystyle M} , or Mat {\displaystyle \operatorname {Mat} } ,

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5084-503: The corresponding lower-case letters, with two subscript indices (e.g., a 11 {\displaystyle {a_{11}}} , or a 1 , 1 {\displaystyle {a_{1,1}}} ), represent the entries. In addition to using upper-case letters to symbolize matrices, many authors use a special typographical style , commonly boldface Roman (non-italic), to further distinguish matrices from other mathematical objects. An alternative notation involves

5166-503: The corresponding procedure for the DFT, not including the costs of the transforms below, because the pairwise operation above requires 8 real-arithmetic operations compared to the 6 of a complex multiplication. This count doesn't include the division by 2, which can be absorbed e.g. into the 1/ N normalization of the inverse DHT.) Just as for the DFT, evaluating the DHT definition directly would require O( N ) arithmetical operations (see Big O notation ). There are fast algorithms similar to

5248-464: The decomposition, the last term for X 3 ( k , l ) {\displaystyle X_{3}(k,l)} will be denoted as X 4 ( k , l ) {\displaystyle X_{4}(k,l)} , then Matrix (math) In mathematics , a matrix ( pl. : matrices ) is a rectangular array or table of numbers , symbols , or expressions , with elements or entries arranged in rows and columns, which

5330-442: The entries of a matrix can be defined by a formula such as a i , j = f ( i , j ) {\displaystyle a_{i,j}=f(i,j)} . For example, each of the entries of the following matrix A {\displaystyle \mathbf {A} } is determined by the formula a i j = i − j {\displaystyle a_{ij}=i-j} . In this case,

5412-412: The essential properties of the transform. The transform can be interpreted as the multiplication of the vector ( x 0 , ...., x N −1 ) by an N -by- N matrix ; therefore, the discrete Hartley transform is a linear operator . The matrix is invertible; the inverse transformation, which allows one to recover the x n from the H k , is simply the DHT of H k multiplied by 1/ N . That is,

5494-404: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=FHT&oldid=876230174 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Fast Hartley transform Because there are fast algorithms for

5576-474: The matrix itself is sometimes defined by that formula, within square brackets or double parentheses. For example, the matrix above is defined as A = [ i − j ] {\displaystyle {\mathbf {A} }=[i-j]} or A = ( ( i − j ) ) {\displaystyle {\mathbf {A} }=((i-j))} . If matrix size is m × n {\displaystyle m\times n} ,

5658-448: The matrix, and commonly denoted by a i , j {\displaystyle {a_{i,j}}} or a i j {\displaystyle {a_{ij}}} . Alternative notations for that entry are A [ i , j ] {\displaystyle {\mathbf {A} [i,j]}} and A i , j {\displaystyle {\mathbf {A} _{i,j}}} . For example,

5740-704: The need for fast multidimensional algorithms. Three such algorithms follow. In pursuit of separability for efficiency, we consider the following transform (Bracewell, 1983), X ^ ( k 1 , k 2 , . . . , k r ) = ∑ n 1 = 0 N 1 − 1 ∑ n 2 = 0 N 2 − 1 … ∑ n r = 0 N r − 1 x ( n 1 , n 2 , . . . , n r ) c

5822-414: The number of floating-point operations) than the corresponding DFT algorithm (FFT) specialized for real inputs (or outputs). This was first argued by Sorensen et al. (1987) and Duhamel & Vetterli (1987). The latter authors obtained what appears to be the lowest published operation count for the DHT of power-of-two sizes, employing a split-radix algorithm (similar to the split-radix FFT ) that breaks

5904-474: The number of rows of the right matrix. If A is an m × n matrix and B is an n × p matrix, then their matrix product AB is the m × p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B : where 1 ≤ i ≤ m and 1 ≤ j ≤ p . For example, the underlined entry 2340 in the product is calculated as (2 × 1000) + (3 × 100) + (4 × 10) = 2340: Matrix multiplication satisfies

5986-497: The numbering of array indexes at zero, in which case the entries of an m -by- n matrix are indexed by 0 ≤ i ≤ m − 1 {\displaystyle 0\leq i\leq m-1} and 0 ≤ j ≤ n − 1 {\displaystyle 0\leq j\leq n-1} . This article follows the more common convention in mathematical writing where enumeration starts from 1 . The set of all m -by- n real matrices

6068-463: The other hand, the redundant computations in FFTs due to real inputs are more difficult to eliminate for large prime N , despite the existence of O( N log N ) complex-data algorithms for such cases, because the redundancies are hidden behind intricate permutations and/or phase rotations in those algorithms. In contrast, a standard prime-size FFT algorithm, Rader's algorithm , can be directly applied to

6150-472: The result. As with the DFT, a cyclic convolution z = x ∗ y of two vectors x = ( x n ) and y = ( y n ) to produce a vector z = ( z n ), all of length N , becomes a simple operation after the DHT. In particular, suppose that the vectors X , Y , and Z denote the DHT of x , y , and z respectively. Then the elements of Z are given by: where we take all of the vectors to be periodic in N ( X N = X 0 , et cetera). Thus, just as

6232-425: The row-column approach. The drawback is that the implementation of these radix-type of algorithms is hard to generalize for signals of arbitrary dimensions. Number theoretic transforms have also been used for solving the MD-DHT, since they perform extremely fast convolutions. In Boussakta (1988), it was shown how to decompose the MD-DHT transform into a form consisting of convolutions: For the 2-D case (the 3-D case

6314-651: The rules ( AB ) C = A ( BC ) ( associativity ), and ( A + B ) C = AC + BC as well as C ( A + B ) = CA + CB (left and right distributivity ), whenever the size of the matrices is such that the various products are defined. The product AB may be defined without BA being defined, namely if A and B are m × n and n × k matrices, respectively, and m ≠ k . Even if both products are defined, they generally need not be equal, that is: A B ≠ B A . {\displaystyle {\mathbf {AB}}\neq {\mathbf {BA}}.} In other words, matrix multiplication

6396-405: The same number of rows and columns, play a major role in matrix theory. Square matrices of a given dimension form a noncommutative ring , which is one of the most common examples of a noncommutative ring. The determinant of a square matrix is a number associated with the matrix, which is fundamental for the study of a square matrix; for example, a square matrix is invertible if and only if it has

6478-458: The use of a double-underline with the variable name, with or without boldface style, as in A _ _ {\displaystyle {\underline {\underline {A}}}} . The entry in the i -th row and j -th column of a matrix A is sometimes referred to as the i , j {\displaystyle {i,j}} or ( i , j ) {\displaystyle {(i,j)}} entry of

6560-467: The vertices of the unit square. The following table shows several 2×2 real matrices with the associated linear maps of ⁠ R 2 . {\displaystyle \mathbb {R} ^{2}.} ⁠ The blue original is mapped to the green grid and shapes. The origin (0, 0) is marked with a black point. Under the 1-to-1 correspondence between matrices and linear maps, matrix multiplication corresponds to composition of maps: if

6642-834: Was also presented in Boussakta (2000) that this 3D-vector radix algorithm takes ( 7 4 ) N 3 log 2 ⁡ N {\displaystyle ({\frac {7}{4}})N^{3}\log _{2}N} multiplications and ( 31 8 ) N 3 log 2 ⁡ N {\displaystyle ({\frac {31}{8}})N^{3}\log _{2}N} additions compared to 3 N 3 log 2 ⁡ N {\displaystyle 3N^{3}\log _{2}N} multiplications and ( 9 2 ) N 3 log 2 ⁡ N + 3 N 2 {\displaystyle ({\frac {9}{2}})N^{3}\log _{2}N+3N^{2}} additions from

6724-1143: Was shown in Bortfeld (1995), that the two can be related by a few additions. For example, in 3-D, X ( k 1 , k 2 , k 3 ) = 1 2 [ X ^ ( k 1 , k 2 , − k 3 ) + X ^ ( k 1 , − k 2 , k 3 ) + X ^ ( − k 1 , k 2 , k 3 ) − X ^ ( − k 1 , − k 2 , − k 3 ) ] . {\displaystyle X(k_{1},k_{2},k_{3})={\frac {1}{2}}[{\hat {X}}(k_{1},k_{2},-k_{3})+{\hat {X}}(k_{1},-k_{2},k_{3})+{\hat {X}}(-k_{1},k_{2},k_{3})-{\hat {X}}(-k_{1},-k_{2},-k_{3})].} For X ^ {\displaystyle {\hat {X}}} , row-column algorithms can then be implemented. This technique

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