A Generalised Hadamard Transform for multi-phase or multilevel signals is
introduced, which includes the Fourier, Generalised, Discrete Fourier,
Walsh-Hadamard and Reverse Jacket Transforms. The jacket construction is
formalised and shown to admit a tensor product decomposition. Primary matrices
under this decomposition are identified. New examples of primary jacket
matrices of orders 8 and 12 are presented.Comment: To appear in the proceedings of the 2005 IEEE International Symposium
  on Information Theory, Adelaide, Australia, September 4-9, 200

Horadam, K. J.

English

arXiv

A generalised Hadamard transform for multiphase or multilevel signals is introduced, which includes the Fourier, generalised, discrete Fourier, Walsh-Hadamard and reverse jacket transforms. The jacket construction is formalised and shown to admit a tensor product decomposition. Primary matrices under this decomposition are identified. New examples of primary jacket matrices of orders 8 and 12 are presented

Horadam, K

RMIT Research Repository

A Generalised Hadamard Transform
K. J. Horadam
Mathematics, SMGS
RMIT University
Melbourne VIC 3000, Australia
Email: kathy.horadam@ems.rmit.edu.au
Abstract—A Generalised Hadamard Transform for multi-
phase or multilevel signals is introduced, which includes the
Fourier, Generalised, Discrete Fourier, Walsh-Hadamard and
Reverse Jacket Transforms. The jacket construction is formalised
and shown to admit a tensor product decomposition. Primary
matrices under this decomposition are identified. New examples
of primary jacket matrices of orders 8 and 12 are presented.
I. INTRODUCTION
There are many discrete signal transforms whose transform
matrices have entries on the unit circle. For instance, the
Discrete Fourier Transform (DFT) for signals of length n
and the Walsh-Hadamard Transform (WHT) are both special
cases of the Fourier Transform (FT) found by interpreting
the Cooley-Tukey Fast Fourier Transform in terms of abelian
group characters [6]. The family of discrete Generalised Trans-
forms {(GT )r, 0 ≤ r ≤ m − 1} for signals of length 2
m [3,
10.2] includes the WHT as case r = 0, the complex BIFORE
transform (CBT) as case r = 1 and the 2m-point DFT as case
r = m− 1. Both the WHT and DFT are suboptimal discrete
orthogonal transforms, but each has wide application.
These transform matrices all fall into the class of Butson
Hadamard matrices [2].
For signals where entries outside the complex nth roots
of unity are needed, Lee has introduced multi-phase or mul-
tilevel generalisations of the WHT and of the even-length
DFT under the name Reverse Jacket Transforms (RJT), see
[7]. These are so-called because the border (“jacket”) and
the centre of the unitary matrix representing the transform
change independently under inversion. The former admit a
recursive factorisation into tensor products so represent a fast
transform similar to that of the WHT. (Note that the formula
[7, Definition 5] is not a reverse jacket transform generalising
both the WHT and even-length DFT, as claimed, because it is
not unitary.)
The class of Butson Hadamard matrices can be generalised
to include all these transform matrices.
II. A GENERALISED HADAMARD TRANSFORM
In this most general situation, we work in a ring R with
unity 1. This includes R, C and Galois Field alphabets
GF (pa), though if we need to distinguish signal values x and
−x, the ring must have characteristic = 2.
Definition 2.1: Suppose R is a ring with unity 1, group of
units R∗ and that charR does not divide v. A square matrix
M of order v ≥ 2, with entries from a subgroup N ≤ R∗ is
a Generalised Butson Hadamard (GBH) matrix, if
MM∗ = M∗M = vIv,
where M∗ is the transpose of the matrix of inverse elements of
M : m∗ij = (mji)
−1. It is denoted GBH(N, v), or GBH(w, v)
if N is finite of order w.
A GBH matrix is always equivalent to a normalised GBH
matrix, which has first row and column consisting of all 1s. By
taking the inner product of any non-initial row of a normalised
GBH matrix M with the all-1s first column of M∗, we see
that the sum of the entries in any row of M , apart from the
first, must equal 0, and similarly for rows of M∗ (columns of
the matrix of inverses M (−1) = [m−1ij ]). The tensor product
of two GBH matrices over the same group N is a GBH matrix
over N .
Definition 2.2: Let x be a signal of length n from R∗,
where n ∈ R∗, let N ≤ R∗ and let B be a GBH(N,n).
A Generalised Hadamard Transform (GHT) of x is
xˆ = B x (1)
and an Inverse Generalised Hadamard Transform (IGHT) of
xˆ is
x = n−1B∗ xˆ. (2)
The next section describes a construction for GBH matrices
of even order and additional internal structure, containing the
WHT, even-length DFT and reverse jacket transform matrices.
III. THE JACKET MATRIX CONSTRUCTION
Throughout this section, let G be an indexing set of even
order 2n (sometimes G is a group such as Z2n or Z
a
2 but G
may be non-abelian or the group structure may be irrelevant).
Definition 3.1: Let R be a ring with unity 1. A normalised
GBH(N, 2n) matrix K indexed by G = {1, . . . , 2n} with
entries from N ≤ R∗ is a jacket matrix if it is of the form
K =


1 1 . . . 1 1
1 ∗ . . . ∗ ±1
...
...
...
...
...
1 ∗ . . . ∗ ±1
1 ±1 . . . ±1 ±1

 (3)
where the central entries ∗ are from N . The jacket width
of K is m ≥ 2 if K is permutation equivalent to a jacket
matrix K˜ in which rows 2, . . . ,m, 2n − m + 1, . . . , 2n and
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columns 2, . . . ,m, 2n−m + 1, . . . , 2n all consist of ±1 and
m is maximal for this property. Otherwise the jacket width is
m = 1.
All non-initial ±1 rows and columns of a jacket matrix K
necessarily sum to 0 in R. If K is a width m jacket matrix
it follows that K∗ is itself a width m jacket matrix. If also
2n ∈ R∗, then K has an inverse K−1 = (2n)−1K∗ over R.
If (2n)
1
2 ∈ R∗ then K̂ = (2n)−
1
2K is unitary (K̂K̂∗ = I2n)
and we will (by a slight abuse of terminology) also say K is
unitary.
Example 3.2: The matrix St of the WHT of length 2
t
S1 =
[
1 1
1 −1
]
, St =
⊗t
S1, t ≥ 2, (4)
is an extremal case (jacket width 2t−1) of such jacket matrices
(“all jacket”).
Example 3.3: The matrix Ct of the CBT of length 2
t
C1 =
[
1 −i
1 i
]
,
C2 =
[
S1 S1
C1 −C1
]
,
Cm =
[
Cm−1 Cm−1
C1 ⊗ Sm−2 −C1 ⊗ Sm−2
]
,m ≥ 3.
(5)
is permutation equivalent to a jacket matrix, for t ≥ 2.
Proof. For t ≥ 2, Ct is normalised and the first two rows of
Ct consist of 2
t−1 copies of S1. By induction the (2
t−1+1)st
column of Ct is [1 −1]
, where 1 has length 2t−1. Rotating
the second row to the bottom and the (2t−1 + 1)st column to
the right of Ct produces a jacket matrix. 
Example 3.4: The matrix F2n = [ ω
jk ]0≤j,k≤2n−1, where
ω = e−πi/n, n ≥ 1, of the DFT of length 2n is permutation
equivalent to a jacket matrix.
Proof. [7, Theorem 1, Definition 1] Represent the indices in
mixed radix notation j = j1n + j0 = (j1, j0), with index set
G = Z2 × Zn. The permutation (j1, j0) → (j1, (1− j1)j0 +
(n−1−j0)j1) leaves the first n indices unchanged and reverses
the order of the last n indices. Under this permutation on rows
and columns, F2n is equivalent to Kn(ω) =
[ω{(1−j1)j0+(n−1−j0)j1+j1n}{(1−k1)k0+(n−1−k0)k1+k1n}],
(6)
the matrix of Lee’s complex RJT. 
If K,K ′ are jacket matrices indexed by G,G′ of orders
2n, 2n′ respectively, with entries from R∗, then the tensor
product K ⊗ K ′ is a jacket matrix indexed by G × G′ of
order 4nn′, with entries from R∗, since the border condition
is easily seen to be satisfied. In fact, a tensor product of jacket
matrices is a jacket matrix of width ≥ 2.
Lemma 3.5: If Ki is a width mi jacket matrix with entries
from R∗, for i = 1, 2, then K1 ⊗ K2 is a jacket matrix of
width at least 2m1m2.
Proof. Let K1 have order 2n and K2 have order 2n
′. Permute
Ki to K˜i, so K1⊗K2 is permutation equivalent to K˜1⊗ K˜2.
Let i ∈ {1, . . . ,m1, 2n − m1 + 1, . . . , 2n} be an index of
an all–(±1)s row in K˜1. The corresponding i
th block row in
K˜1⊗K˜2 consists of 2n copies of ±1K˜2, so each row indexed
{2, . . . ,m2, 2n
′ − m2 + 1, . . . 2n
′} of each copy consists of
all–(±1)s, contributing 2m2−1 all–(±1) rows to the i
th block
row of K˜1 ⊗ K˜2. If i = 1 the top row is all 1s and if i > 1
the top row consists of n 1s and n −1s. Those in the top m1
block rows of K˜1 ⊗ K˜2 may be permuted to occupy the top
2m1m2 rows and those in the bottom m1 block rows to the
bottom 2m1m2 rows, and similarly for columns. 
IV. TENSOR DECOMPOSITION
If a jacket matrix may be decomposed as a tensor product
of two smaller jacket matrices, the decomposition may be
repeated until no further tensor product decomposition is
possible.
Definition 4.1: A jacket matrix of length 2n is a primary
jacket matrix Kn if it is minimal with respect to tensor
product, that is, there are no jacket matrices K, K ′ such that
Kn is permutation equivalent to K ⊗K
′.
Examples of primary jacket matrices for n = 1, . . . , 4 are
K1 = S1,
K2(r) =


1 1 1 1
1 −r r −1
1 r −r −1
1 −1 −1 1

 , r = ±1 ∈ R∗ (7)
K3(α) =


1 1 1 1 1 1
1 α α2 α5 α4 −1
1 α2 α4 α4 α2 1
1 α5 α4 α α2 −1
1 α4 α2 α2 α4 1
1 −1 1 −1 1 −1


, (8)
where α is a primitive 6th root of unity in an integral domain
R, and
K4(i) =


1 1 1 1 1 1 1 1
1 i −i 1 −1 i −i −1
1 −i −1 i i −1 −i 1
1 1 i i −i −i −1 −1
1 −1 i −i i −i 1 −1
1 i −1 −i −i −1 i 1
1 −i −i −1 1 i i −1
1 −1 1 −1 −1 1 −1 1


. (9)
The matrix K1 is the unique 2 × 2 jacket matrix, and
K2(1) = K1⊗K1 = S2, so is not primary. The matrix K2(r)
for r = ±1 ∈ R∗ is a “centre-weighted Hadamard transform”
(CWHT) matrix [7] and for r = i ∈ C∗, is K2(i). The matrix
K3(α) with α = e
iπ/3 is K3(e
iπ/3) and with α the fourth
power of a primitive root in GF (25) is an “extended” complex
RJT matrix [7, Example 4]. The matrix K4(i) is described for
the first time here: it is equivalent to the back-circulant matrix
derived from a quadriphase perfect sequence [1, Example 2]
of length 8; such sequences are rare.
Corollary 4.2: A jacket matrix is permutation equivalent
to a tensor product of one or more primary jacket matrices.
Conversely, any tensor product of primary jacket matrices is
a jacket matrix.
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V. CONSTRUCTION OF PRIMARY JACKET MATRICES
By Lemma 3.5 any jacket matrix of width 1 is primary. The
tensor product of two jacket matrices is a jacket matrix, but in
fact, to construct a jacket matrix it is enough that one factor
is a jacket matrix and the other a normalised GBH matrix.
Theorem 5.1: Let B be a normalised GBH matrix and K
a jacket matrix, both with entries in R∗. Then B ⊗ K is
permutation equivalent to a jacket matrix (B ⊗K)†.
Proof. Let B have order m and K have order 2n. The first
2n rows of B ⊗K consist of m blocks K,K, . . . ,K, so the
1st row is all 1’s and the 2nth row is all ±1s, and similarly
for columns. Cyclically permute row 2n to row 2mn and row
2n+i to row 2n+i−1, i = 1, . . . , 2n(m−1), and similarly for
columns. This shifts row 2n to the bottom of the matrix and
column 2n to the right of the matrix, leaving the order of the
other rows and columns otherwise unchanged. This permuted
matrix (B ⊗K)† is of the form (3). 
This result explains the generation of some primary jacket
matrices and is fundamental to the construction of Generalised
Hadamard Transforms.
Example 5.2: Let β = 1 ∈ R∗ satisfy β2 + β + 1 = 0, so
β3 = 1. Let
B3 =

 1 1 11 β β2
1 β2 β

 ,
so B3 is a normalised GBH(3, 3). Then
(B3 ⊗K1)
† =


1 1 1 1 1 1
1 β β β2 β2 1
1 β −β β2 −β2 −1
1 β2 β2 β β 1
1 β2 −β2 β −β −1
1 1 −1 1 −1 −1


.
This jacket matrix relates to the DFT matrix of (8) as fol-
lows. A second permutation (2543) cycling central rows and
columns gives the jacket matrix

1 1 1 1 1 1
1 −β β2 −β2 β −1
1 β2 β β β2 1
1 −β2 β −β β2 −1
1 β β2 β2 β 1
1 −1 1 −1 1 −1


.
When α is a primitive 6th root of unity in R∗ with α2 = β
and α5 = γ, then this matrix equals K3(γ).
The following matrix is a new 12×12 primary jacket matrix,
since it has width 1.
Example 5.3: Let r = ±1 ∈ R∗.
Let K6(β, r) = (B3 ⊗K2(r))
† =


1 1 1 1 1 1 1 1 1 1 1 1
1 −r r 1 −r r −1 1 −r r −1 −1
1 r −r 1 r −r −1 1 r −r −1 −1
1 1 1 β β β β β2 β2 β2 β2 1
1 −r r β −rβ rβ −β β2 −rβ2 rβ2 −β2 −1
1 r −r β rβ −rβ −β β2 rβ2 −rβ2 −β2 −1
1 −1 −1 β −β −β β β2 −β2 −β2 β2 1
1 1 1 β2 β2 β2 β2 β β β β 1
1 −r r β2 −rβ2 rβ2 −β2 β −rβ rβ −β −1
1 r −r β2 rβ2 −rβ2 −β2 β rβ −rβ −β −1
1 −1 −1 β2 −β2 −β2 β2 β −β −β β 1
1 −1 −1 1 −1 −1 1 1 −1 −1 1 1


Those GHT matrices which are jacket matrices have addi-
tional structure, by virtue of their tensor product decomposi-
tion into primary jacket matrices and their jacket form, which
may particularly suit them to specific applications.
Consider the set of jacket matrices with entries in in an
integral domain R
{K = (⊗K1)⊗K2(r)
 ⊗Kn(α)
δ;  ≥ 0, , δ ∈ {0, 1}},
(10)
where r = ±1 ∈ R∗, α is a primitive 2nth root of unity
and where by M0 we mean the 1 × 1 identity matrix. When
 ≥ 1,  = 0, δ = 0, this is the WHT. When  = 0,  = 0, δ = 1
and R = C this is equivalent to the 2n-point DFT. When
 = 1, δ = 0 and R = R, this is the CWHT. When  = 0, n =
2, α = i ∈ C∗, δ = 1, or when  = 1, r = i ∈ C∗, δ = 0,
this is the complex RJT, and when  = 0, δ = 1, this is the
extended complex RJT.
This uniform classification may make it possible to recog-
nise common fast algorithms.
Finally, the examples and constructions of primary jacket
matrices above are all instances of a class of matrices called
cocyclic, investigated by the author and colleagues over the
past decade [4], [5].
VI. CONCLUSION
To summarise: a generalisation of Butson’s Hadamard ma-
trices determines a Generalised Hadamard Transform (GHT).
The GHT includes the Fourier and Generalised Transform
families (in particular the WHT and DFT) and the centre-
weighted Walsh-Hadamard, Complex Reverse Jacket and ex-
tended Complex Reverse Jacket Transforms.
In the jacket case, GHT matrices can be permuted into
tensor products of primary jacket matrices. New primary
jacket matrices may be constructed as tensor products of a
Generalised Butson Hadamard matrix which is not a primary
jacket matrix, and a primary jacket matrix. New examples in
orders 8 and 12 have been given.
Application of the GHT to image processing, error-control
coding and decoding and sequence design are obvious direc-
tions for future research.
REFERENCES
[1] K. T. Arasu and W. de Launey, Two-dimensional perfect quaternary
arrays, IEEE Trans. Inform. Theory 47(4) (2001) 1482–1493.
[2] A. T. Butson, Generalised Hadamard matrices, Proc. Amer. Math. Soc.
13 (1962) 894–898.
[3] D. F. Elliott and K. R. Rao, Fast Transforms: Algorithms, Analyses,
Applications, Academic Press, New York, 1982.
[4] K. J. Horadam, An introduction to cocyclic generalised Hadamard matri-
ces, Discrete Appl. Math. 102 (2000) 115–131.
[5] K. J. Horadam and P. Udaya, Cocyclic Hadamard codes, IEEE Trans.
Inform. Theory 46(4) (2000) 1545–1550.
[6] D. K. Maslen and D. N. Rockmore, Generalised FFTs - a survey of some
recent results, DIMACS Ser. Discr. Math. Theoret. Comp. Sci. 28 (1997)
183–237.
[7] Moon Ho Lee, B. Sunder Rajan and J. Y. Park, A generalized reverse
jacket transform, IEEE Trans. Circuits Syst. II, 48(7) (2001) 684–690.
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