M-TRANSFORM ALGORITHM

A new orthogonal transform, the Malakooti transform (M-TRANFORM), analogous to the Hadamard transform,

has been developed to represent the time series signals with a set of coefficients called the M coefficients. These

set of coefficients contain useful information about the spectral characteristics of the underlying time series and can

be used for data transmission and compression. Many time series signal are highly redundant; speech, image, and

other periodic signals fall into this category. The M-transform representation enables one to represent the desired

signal with fewer coefficients, resulting in a saving of transmission bandwidth and memory.

This transform, like the Hadamard transform, has a complete orthonormal set and has an important role in

signal and image processing applications. It has been shown [1] that the time series signals obtained from a two-

dimensional shape can be represented with a few coefficients for pattern recognition and shape classification.

Similarly, speech signals are represented by a set of coefficients for spectral estimation and word recognition. In all

these cases, the right singular vectors of the correlation matrix are used as an orthogonal basis for the solution

space. For this reason and many others, unitary transforms or an orthonormal basis, in particular a complete

orthonormal basis, should receive more attention than other transforms which have no unitary property.

Complete Orthonormal Set

A set of linearly independent vectors 𝑣1,, 𝑣2, … 𝑣𝑛 is said to be orthonormal if it is self-reciprocal, i.e., if the

vectors are all mutually orthogonal and have unit norm as

𝑣𝑖∗ 𝑣𝑗 = {

1 𝑖 = 𝑗0 𝑖 ≠ 𝑗.

(EQ-1)

If time series signals and are represented by a linear combination of a set of orthonormal vectors

𝑋 = ∑ αi𝑛𝑖=1 vi (EQ-2)

and 𝑌 = ∑ βi vi

𝑛

𝑖=1 (EQ-3)

Then their inner product < 𝑋, 𝑌 >, is easy to find. The inner product of X and Y is obtained as

< 𝑋, 𝑌 > = < ∑ 𝛼𝑖 𝑣𝑖 , ∑ 𝛽𝑗 𝑣𝑗 >𝑛𝑗=1

𝑛𝑖=1 (EQ-4)

= ∑ ∑ 𝛼𝑖 𝛽𝑗∗ 𝑢𝑖

∗ 𝑢𝑗

𝑛

𝑗=1

𝑛

𝑖=1

= ∑ 𝛼𝑖

𝑛

𝑖=1

𝛽𝑖∗

= < 𝛼, 𝛽 >.

An orthonormal set is said to be complete if any additional non-zero orthonormal vector is superfluous. If a

signal is approximated by a linear combination of the first m vectors of complete orthonormal set with dimension n,

then the norm of the error can be reduced by choosing m sufficiently large. In the next section, a method for

generating the complete orthonormal sets of vectors, m-transform vectors, with the Eigen analysis of the spanned

space is presented.

Generation of M-transform Matrix

Assume that the order-1 M-transform matrix, 𝑀𝑜 , is equal to one,

𝑀𝑜 = 1, (EQ-5)

and the order-2 M-transform matrix, 𝑀1 , is formed according to

𝑀1 = [𝑎𝑀𝑜 𝑎𝑏𝑀𝑜

−𝑎𝑏𝑀𝑜 𝑎𝑀𝑜] (EQ-6)

𝑀1 = [𝑎 𝑎𝑏

−𝑎𝑏 𝑎],

Where a and b are constant parameters.

The matrix M is a 2 x 2 anti-symmetric unitary matrix

𝑀1𝑇 𝑀1 = 𝑀1𝑀1

𝑇 = 𝑐 𝐼, (EQ-7)

Where the matrix I is a 2 x 2 identity matrix and constant parameters c is equal to the determinant of 𝑀1. Thus,

c = 𝑎2 (1 + 𝑏2) (EQ-8)

and 𝑀1 inverse is given as

𝑀1−1 =

𝑀1𝑇

C (EQ-9)

Similarly, the order-3 M-transform matrix, 𝑀2, can be obtained according to

𝑀2 = [𝑎𝑀1 𝑎𝑏𝑀1

−𝑎𝑏𝑀1 𝑎𝑀1] (EQ-10)

The matrix 𝑀2 is a 4 x 4 anti-symmetric unitary matrix

𝑀2𝑇𝑀2 = 𝑀2𝑀2

𝑇 = 𝐶2 𝐼, (EQ-11)

where the matrix I is an 4 x 4 identity matrix, C is given in (EQ-8), and the inverse of is calculated according to

𝑀2−1 =

𝑀2𝑇

𝐶2 , (EQ-12)

Without loss of generality, the 2𝑘 x 2𝑘 M-transform matrix, 𝑀𝑘 can be obtained from

𝑀𝑘 = [

𝑎𝑀𝑘−1 𝑎𝑏𝑀𝑘−1−𝑎𝑏𝑀𝑘−1 𝑎𝑀𝑘−1

], (EQ-13)

and 𝑀𝑘 inverse is given according to

𝑀𝑘−1 =

𝑀𝑘𝑇

𝐶𝑘 , (EQ-14)

Using the Kronecker product notation

𝐴 ⨂ 𝐵 = [

𝑎11 𝐵

𝑎21 𝐵

𝑎12 𝐵

𝑎22 𝐵⋯

𝑎1𝑛 𝐵

𝑎2𝑛 𝐵

⋮ ⋮𝑎𝑛1 𝐵 𝑎𝑛2 𝐵 ⋯ 𝑎𝑛𝑛 𝐵

], (EQ-15)

the M-transform matrices can written according to

𝑀1 = 𝑀1 ⨂ 𝑀𝑜 = [𝑎𝑀𝑜 𝑎𝑏𝑀𝑜

−𝑎𝑏𝑀𝑜 𝑎𝑀𝑜] , (EQ-16)

and 𝑀2 = 𝑀1 ⨂ 𝑀1 (EQ-17)

= 𝑀1 ⨂ (𝑀1 ⨂ 𝑀𝑜)

=(𝑀1 ⨂ 𝑀1) ⨂ 𝑀𝑜

= 𝑀1(2)

⨂ 𝑀𝑜

= 𝑀1(1)

⨂ 𝑀1,

where 𝑀1(2)

is the Kronecker power 2 of 𝑀1 and the symbol ⨂ denotes the Kronecker product. Similarly,

𝑀3 = 𝑀1 ⨂ 𝑀2 (EQ-18)

= 𝑀1 ⨂ 𝑀1(2)

⨂ 𝑀𝑜

= 𝑀1(3)

⨂ 𝑀𝑜

= 𝑀1(2)

⨂ 𝑀1

.

.

𝑀𝑘 = 𝑀1 ⨂ 𝑀𝑘−1 (EQ-19)

= 𝑀1⨂ 𝑀1(𝑘−1)

⨂ 𝑀𝑜

= 𝑀1(𝑘)

⨂ 𝑀𝑜

= 𝑀1(𝑘−1)

⨂ 𝑀1,

It has been shown that, (EQ-64) - (EQ-67), that the eigenvalues of a 4 x 4 matrix D, 𝜆𝑖

D = 𝐴 ⨂ 𝐵, (EQ-20)

can be calculated from the product of the eigenvalues of B, 𝜇𝑖, and the eigenvalues of A, 𝛾𝑖, according to

𝜆1 = 𝜇1 𝛾1 , (EQ-21)

𝜆2 = 𝜇2 𝛾1, (EQ-22)

𝜆3 = 𝛾1𝜇2, (EQ-23)

𝜆4 = 𝛾2𝜇2, (EQ-24)

Thus, the eigenvalues of the M-transform matrices can be obtained from a recursive algorithm proposed in the

following section.

Eigenvalues-Eigenvectors of M-transform Matrices

Assume that constant parameters a and b are given as a=1 and b=2.

Thus,

𝑀1 = [1 2

−2 1], (EQ-25)

and

𝑀2 = [

1−2−2 4

2 1−4−2

2 −4 1

−2

4 2 2 1

], (EQ-26)

The eigenvalues, 𝜆𝑖(1)

, and eigenvectors, 𝑋𝑖(1)

, of 𝑀1 are,

𝜆1(1)

= 1 + 𝑗2 , (EQ-27)

𝜆2(1)

= 1 − 𝑗2, (EQ-28)

𝑋1(1)

= { 𝑗0.7071

−0.7071, (EQ-29)

𝑋2(1)

= {0.7071

−𝑗0.7071, (EQ-30)

where the eigenvalues of are complex conjugates of each other. Using the Kronecker product relationship

between 𝑀1 and 𝑀2, Equation (EQ-20) , the eigenvalues of 𝑀2, 𝜆𝑖(2)

, are calculated according to

𝝀𝟏(𝟐)

= 𝝀𝟏(𝟏)

𝝀𝟏(𝟏)

, (EQ-31)

= (𝟏 + 𝒋𝟐) (𝟏 + 𝒋𝟐)

= -3+j4

𝝀𝟐(𝟐)

= 𝝀𝟐(𝟏)

𝝀𝟏(𝟏)

, (EQ-32)

= (1 − 𝑗2) (1 + 𝑗2)

= 5

𝝀𝟑(𝟐)

= 𝝀𝟏(𝟏)

𝝀𝟐(𝟏)

, (EQ-33)

= (1 + 𝑗2) (1 − 𝑗2)

= 5

= 𝝀𝟐∗(𝟐)

𝝀𝟒(𝟐)

= 𝝀𝟐(𝟏)

𝝀𝟐(𝟏)

, (EQ-34)

= (1 − 𝑗2) (1 − 𝑗2)

= −3 −j4

= 𝝀𝟏∗(𝟐)

.

The matrix 𝑀2 has two complex conjugate eigenvalues. Using the complex conjugate property half of the

eigenvalues of 𝑀2 can be obtained without any calculation.

In general eigenvalues of the 2𝐿 ∗ 2𝐿 transform, 𝑀𝐿, are calculated recursively form the proposed algorithm as

follows

1. Calculate the eigenvalues of 𝑀1

𝜆1(1)

= 𝑎 + 𝑗𝑎𝑏 (EQ-35)

𝜆2(1)

= 𝑎 − 𝑗𝑎𝑏 (EQ-36)

2. for K=2 to L do

N=2𝑘 (EQ-37)

for i=1 to N/2 do;

𝜆𝑖(𝑘)

= 𝜆𝑖(𝑘−1)

𝜆1(1)

(EQ-38)

𝜆𝑁−𝑖+1(𝑘)

= 𝜆𝑖∗(𝑘)

(EQ-39)

End do

End do

The eigenvectors of the L-th order M-transform are obtained from a new procedure based on the eigenvectors

of the lower order M-transform. The proposed eigenvector algorithm calculates half of the eigenvectors of the 𝑀𝐿

matrix from a simple procedure. This method, which requires few operations, is incomparable with a direct method

where the dimension of 𝑀𝐿 is high. To show the effectiveness of the proposed eigenvector algorithm, the

eigenvectors of the 𝑀2 matrix are calculated using the Eigen characterization of the 𝑀1 matrix.

The characteristic equation of the 𝑀1 matrix is given as

𝑓(𝜆) = 𝜆2- 2a𝜆+ 𝑎2(1 + 𝑏2) (EQ-40)

or

𝜆2 = 2𝑎𝜆 − 𝑎2(1 + 𝑏2) (EQ-41)

Using the Cayley-Hamilton theorem gives

𝑀12 = 2𝑎𝑀1 − 𝑎2 (1 + 𝑏2)𝐼

= 𝑡𝑟[𝑀1]𝑀1 − 𝑑𝑒𝑡[𝑀1]𝐼. (EQ-42)

Assume that 𝜆 is the eigenvalue of 𝑀2 corresponding to the eigenvector, X,

X = [Xu

Xl] (EQ-43)

Thus, the eigenvectors of the 𝑀2 matrix are related by the following relationships

𝑀2X = 𝜆X (EQ-44)

(𝑀2 − 𝜆𝐼 )𝑋 = 0. (EQ-45)

Substituting for 𝑀2 and x into (EQ-45) gives

[00

] = [ 𝑎𝑀1 − 𝜆𝐼 𝑎𝑏𝑀1

−𝑎𝑏𝑀1 𝑎𝑀1 − 𝜆𝐼] (EQ-46)

or

(𝑎𝑀1 − 𝜆𝐼) 𝑥𝑢 + 𝑎𝑏𝑀1 𝑥𝑙 = 0 . (EQ-47)

and

−𝑎𝑏𝑀1 𝑥𝑢 + (𝑎𝑀1 − 𝜆𝐼)𝑥𝑙 = 0. (EQ-48)

Since ab≠ 0, 𝑥𝑙 can be obtained from (EQ-48) as

𝑥𝑙 = − 1

𝑎𝑏 𝑀1

−1 (𝑎𝑀1 − 𝜆𝐼)𝑥𝑢 . (EQ-49)

Substituting for 𝑥𝑙 into (EQ-48) gives

−𝑎𝑏𝑀1𝑥𝑢 − (𝑎𝑀1 − 𝜆𝐼) 1

𝑎𝑏𝑀1

−1 (𝑎𝑀1 − 𝜆𝐼) 𝑥𝑢 = 0 (EQ-50)

or

1

𝑎𝑏 [2𝑎𝜆𝑀1 − 𝑎2(1 + 𝑏2) 𝑀1

2 − 𝜆2𝐼] 𝑀1−1 𝑥𝑢 = 0. (EQ-51)

Similarly, 𝑥𝑢 can be obtained from (EQ-48)

𝑥𝑢 = 1

𝑎𝑏𝑀1

−1 (𝑎𝑀1 − 𝜆𝐼)𝑥𝑙. (EQ-52)

Substituting for 𝑋𝑢 into Equation (EQ-48) gives

(𝑎𝑀1 − 𝜆𝐼) [1

𝑎𝑏 𝑀1

−1 (𝑎𝑀1 − 𝜆𝐼) 𝑋𝑙] + 𝑎𝑏𝑀1 𝑋𝑙 = 0 (EQ-53)

Or

1

𝑎𝑏 [𝑎2 (1 + 𝑏2)𝑀1

2 − 2𝑎𝜆𝑀1 + 𝜆2𝐼]𝑀1−1 𝑋𝑙 = 0 (EQ-54)

Substituting for 𝑀12 from (EQ-42) into (EQ-53) gives

2𝑎[𝑎2 (1 + 𝑏2) − 𝜆] [𝑀1 −1

2𝑎 (𝑎2 (1 + 𝑏2) + 𝜆)𝐼] 𝑀1

−1 𝑋𝑙 = 0 (EQ-55)

Similarly, Substituting for from (EQ-42) into (EQ-50) gives

2𝑎[𝑎2 (1 + 𝑏2) − 𝜆] [𝑀1 −1

2𝑎 (𝑎2 (1 + 𝑏2) + 𝜆)𝐼] 𝑀1

−1 𝑋𝑢 = 0 (EQ-56)

Two eigenvalues of 𝑀2 are calculated from (EQ-57) and (EQ-58) according to

𝜆2(2)

= 𝑎2(1 + 𝑏2)

= 𝑑𝑒𝑡[𝑀1] (EQ-57)

and

𝜆3(2)

= 𝑎2(1 + 𝑏2)

= 𝑑𝑒𝑡 [𝑀1] (EQ-58)

or

𝜆2(2)

= 𝜆3(2)

= 𝑎2(1 + 𝑏2)

= (𝑎 + 𝑗𝑎𝑏) (𝑎 − 𝑗𝑎𝑏) (EQ-59)

where (𝑎 + 𝑗𝑎𝑏)and (𝑎 − 𝑗𝑎𝑏) are the eigenvalues of the 𝑀1 matrix. Thus,

𝜆2(2)

= 𝜆3(2)

(EQ-60)

𝜆1(1)

= 𝜆2(1)

= 𝑑𝑒𝑡 [𝑀1]. (EQ-61)

The remaining two eigenvalues of 𝑀2 are calculated from the following relationship,

𝑀1 − 1

2𝑎 (𝑎2(1 + 𝑏2) + 𝜆)𝐼 = 0 (EQ-62)

and

2𝑎𝑀1 − 𝑎2 (1 + 𝑏2)𝐼 − 𝜆𝐼 = 0 (EQ-63)

Equation (EQ-63) indicates that the remaining eigenvalues of 𝑀2 are related to the eigenvalues of 𝑀1 according to

𝜆1(2)

= 2𝑎𝜆1(1)

− 𝑎2 (1 + 𝑏2) (EQ-64)

= 𝑡𝑟 [𝑀1] 𝜆1(1)

− 𝑑𝑒𝑡 [𝑀1]

= 2𝑎 𝜆1(1)

− 𝜆1(1)

𝜆2(1)

= 𝜆1(1)

[2𝑎 − 𝜆2(1)

].

Substituting for 𝜆2(1)

from (EQ-36) into (EQ-64) gives

𝜆1(2)

= 𝜆1(1)

[2𝑎 − (𝑎 − 𝑗𝑎𝑏)]

= 𝜆1(1)

(𝑎 + 𝑗𝑎𝑏)

= 𝜆1(1)

𝜆1(1)

(EQ-65)

Similarly,

𝜆4(2)

= 2𝑎 𝜆2(1)

− 𝑎2 (1 + 𝑏2) (EQ-66)

= 𝑡𝑟 [𝑀1] 𝜆2(1)

− 𝑑𝑒𝑡 [𝑀1]

= 2𝑎 𝜆2(1)

− 𝜆1(1)

𝜆2(1)

= 𝜆2(1)

[2𝑎 − 𝜆1(1)

]

= 𝜆2(1)

[2𝑎 − (𝑎 + 𝑗𝑎𝑏)]

= 𝜆2(1)

(𝑎 − 𝑗𝑏).

Thus,

𝜆4(2)

= 𝜆2(1)

𝜆2(1)

. (EQ-67)

Assume that 𝑋4(2)

is an eigenvector of 𝑀2 , where

𝑋4(2)

= [

1

𝑎𝑏(𝑎𝑀1 − 𝜆4

(2)𝐼)𝑋2

(1)

𝑀1𝑋2(1)

]. (EQ-68)

𝑋2(1)

is the eigenvector of 𝑀1 , and 𝜆4(2)

, and 𝑋4(2)

are the eigenvalues and eigenvector of 𝑀2 , respectively. Using

Equation (EQ-45), the eigenvalues- eigenvector of 𝑀2 can be written according to

[𝑀2 − 𝜆4(2)

𝐼]𝑋4(2)

= 0

= [(𝑎𝑀1 − 𝜆4

(2)𝐼)

1

𝑎𝑏(𝑎𝑀1 − 𝜆4

(2)𝐼)𝑋2

(1)+ 𝑎𝑏𝑀1

2𝑋2(1)

−𝑎𝑏𝑀1 (1

𝑎𝑏) (𝑎𝑀1 − 𝜆4

(2)𝐼)𝑋2

(1)+ (𝑎𝑀1 − 𝜆4

(2)𝐼)𝑀1𝑋2

(1)]

= [1

𝑎𝑏(𝑎2(1 + 𝑏2)𝑀1

2 − 2𝑎𝜆4(2)

𝑀1 + 𝜆42(2)

𝐼)𝑋2(1)

0] (EQ-69)

Substituting for 𝑀12 from (EQ-42) into (EQ-69) gives

[𝑀2 − 𝜆4(2)

𝐼]𝑋4(2)

= [1

𝑎𝑏(𝑎2(1 + 𝑏2) − 𝜆4

(2))[2𝑎𝑀1 − (𝑎2(1 + 𝑏2) + 𝜆4

(2))𝐼]𝑋2

(1)]

= [1

𝑎𝑏(𝑑𝑒𝑡𝑀1 − 𝜆4

(2))[𝑡𝑟(𝑀1)𝑀1 − (𝑑𝑒𝑡(𝑀1) + 𝜆4

(2))𝐼]𝑋2

(1)] (EQ-70)

Substituting for 𝜆4(2)

from (EQ-67) into (EQ-70) gives

[𝑀2 − 𝜆4(2)

𝐼]𝑋4(2)

= [1

𝑎𝑏[2𝑑𝑒𝑡(𝑀1) − 𝜆2

(1)𝑡𝑟(𝑀1)]𝑡𝑟 (𝑀1)[𝑀1 − 𝜆2

(1)𝐼]𝑋2

(1)]

= [00

] (EQ-71)

Thus, 𝑋4(2)

, is an eigenvector of 𝑀2 corresponding to 𝜆4(2)

.

Similarly, 𝑋1(2)

is an eigenvector of 𝑀2 corresponding to 𝜆1(2)

, where

𝑋1(2)

= [1

𝑎𝑏(𝑎𝑀1 − 𝜆1

(2)𝐼)𝑋1

(1)

𝑀1

]. (EQ-72)

Since 𝑀1 is nonsingular, 𝑋1(2)

and 𝑋4(2)

are linearly independent. The other two eigenvector of 𝑀2 , 𝑋2(2)

and 𝑋3(2)

are

selected so that

𝑇 = [𝑋1(2)

, 𝑋2(2)

, 𝑋3(2)

, 𝑋4(2)

] (EQ-73)

Are linearly independent, and

Λ = 𝑇−1 𝑀2 𝑇 (EQ-74)

is a diagonal matrix.

This analysis clearly shows that half of the eigenvectors of 𝑀𝑙 can be obtained from a straight-forward

procedure and the other half can be selected so that T = 𝑋1(𝑘)

, 𝑋2(𝑘)

, … , 𝑋𝑛(𝑘)

is a linearly independent set. The

proposed M-transform, whose eigenvalues are calculated from a simple recursive algorithm and half of its

eigenvectors are calculated from a few simple operations, can be used as an orthogonal basis to represent many

signal and image processing applications. Moreover, the number of distinct eigenvalues of 𝑀𝐿 is L+1 as opposed to

an L-th order Hadamard transform, 𝐻𝐿 , which only has two distinct eigenvalues[2]. The eigenvalues of the 𝑀𝐿

transform can be used as feature parameters if the elements of the 𝑀𝐿 matrix are the autocorrelation lags of the

observation and by proper selection of the a and b constants.

References:

[1] Mohammad V Malakooti, Keith Teague, “CARMA Model method of two-dimensional shape

classification: An Eigen system approach vs. the LP Norm, ICASSP, Vol. 12, 1987.

[2] Clark R. Givens, “Some observations on eigenvectors of Hadamard Matrices of order 2n”

, Linear Algebra and its

Applications, Vol. 56, Jan. 1984, pp245-250.