Discrete Integer Fourier Transform in Real Space

Discrete Integer Fourier Transform in Real Space: Elliptic Fourier Transform Artyom M. Grigoryan Merughan M. Grigoryan

Department of Electrical and Computer Engineering The University of Texas at San Antonio One UTSA Circle, San Antonio,TX 78249-0669, USA Tel: (210) 458-7518 Fax: (210) 458-5589 [email protected] [email protected]

Outline Introduction  Discrete Fourier transform (DFT) in the real space  N-block T-transform generated discrete transform (T-GDT)  Roots of the identity matrix  N-block elliptic DFT  Properties of the elliptic DFT  Comparison with the DFT  Examples  Summary 

Art and Merughan, SPIE 2009

2

Introduction 





The traditional N-point DFT is defined as the decomposition of the signal by N roots of the unit, which are located on the unit circle. Multiplication of a complex number by a twiddle factor can be considered in matrix form; as the Givens transformation. The definition of the DFT in the real space can be generalized by using a two-point transform T different from the rotations. It is assumed that the matrix of T defines a one-parametric group with period N. We introduce a concept of the T-generated N-block discrete transformation, or N-block T-GDT. For the N-block T-GDT the inner product is defined with respect to which rows/columns of the matrix are orthogonal. T is parameterized; selection of parameters can be done among the integer numbers, which leads to integer-valued metric.


3

DFT in Real Space  Wk

The N-point DFT: decomposition of the signal by N roots of the unit WNk

e

j

2 k N

ck

jsk

cos(2 k / N )

j sin( 2 k / N )

Multiplication in matrix form: x Tkx

W kx

( x1 , x2 ) ck sk

sk ck

(ck

jsk )( x1

cos sin

x1 x2

N-point DFT of vector

sin cos

k k

f

jx2 ) k k

x1 x2

( f 0 , f1 , f 2 ,..., f N 1 )

N 1

W np f n , p

Fp

0 : ( N 1).

n 0

Matrix of the N-point DFT

[ FN ]

1 1 1 1

1 W1 W2 ...

1 WN

1 W2 W4 ... 1

WN

... 1 ... W N 1 ... W N 2 ... ... 2

...

W1


4

Transform: CN to R2N 

Vector representation f

f

(r0 , i0 , r1 , i1 , r2 , i2 ,..., rN 1 , iN 1 )

with the vector component fn

( f 2n , f 2n 1 )

(rn , in )

(Re f n , Im f n ) .

N-point DFT as 2N-point in R2N Fp

Rp Ip

N 1

rn , p in

N 1

T

np

T np

fn

n 0

n 0

0 : ( N 1),

has the following matrix: [ FN b ]

I

I

I

I I I

T1 T2 ...

T2 T4 ...

I TN

T k1

k2

T k1T k2 ,

1

TN

...

I

... T N 1 ... T N 2 , ... ... 2

k1 , k 2 ,

...

(T 0

T1

TN

I ).


5

Exm: 6-point DFT in R12 Six-point DFT with the matrix



X

1 0

1

0 ...

1

0

1

0

0

1

0

1 ...

0

1

0

1

1 0

c1

s1 ...

c4

s4

c5

s5

0

s1

c1 ...

s4

c4

s5

c5

1

... ...

... ... ...

... ...

1 0

c4

s4 ...

c4

s4

... ... , det( X ) c2 s 2

0

1

s4

c4 ...

s4

c4

s2

c2

1 0

c5

s5 ...

c2

s2

c1

s1

0

s5

c5 ...

s2

c2

s1

c1

1

66.

sk for Since c6 k ck and s6 k k=1,2 and c3 1, s3 0,

Xˆf

R4

1

1

1

1

1

1 0 1 0 1

c1 s1 c1 s1 1

c1 s1 c1 s1 1

1 0 1 0 1

c1 s1 c1 s1 1

R2 , I 4

I2

and

1 f0 c1 f1

R0 R1

f2 f3 f4 f5

I1 R2 I2 R3

s1 c1 s1 1

R5

R1 , I 5

[ Two multiplications by s1 , (c1

I1 . 0.5) ]


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The inner product 

The DFT is linear in the R2N and the inner product is preserved for extended DFT: 2N 1

( f , g)

2N 1

fn gn n 0

( Xf , Xg )

[ F ]k [G ]k k 0

where [ F ]k and [G]k denote components of the extended DFTs of the vectors f and g, respectively.


7

Integer representation 

The definition of the DFT in the real space R2N can be generalized by 2-D transforms different from the rotations. The 2N-point transform of the vector f

( f 0 , f1 , f 2 , f 3 ,..., f 2 N 2 , f 2 N 1 )

is defined by Fp

F2 p F2 p 1

N 1

N 1

T n 0

np

fn

T n 0

np

f 2n f 2n

, 1

p 0 : ( N 1).

T is a matrix 2 2, det T=1, and it defines a one-parametric group with period N. F We call the transformation X : f the T-generated N-block discrete transformation, or the N-block T-GFT. Case T=W: N-block W-GFT (N-block DFT) when ( f 2 n , f 2 n 1 ) (rn , in ). Art and Merughan, SPIE 2009

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N=6 case (space R12) Consider the 6-block T-GDT: T I

X

X (T )

1

1

1

0

I

I

I

T1 T 2

I

T2 T4 3

I

T

I I

T4 T2 T5 T4

1, T 6

det T

, I

I

I,

I

T3 T4 T5 T2 T4

I

I

3

T

I

T

3

, det( X ) 66.

I T4 T2 T 3 T 2 T1

The inverse matrix: X

1 6

1

I

I

I

I

T

1

T

2

I

T

2

T

4

I

T

3

I I

T T

4 5

I

I T T

T

4

3

I T

2

I

f

I

I

4

T

5

I

T5 T4

T

2

T

4

I

T4 T2

T

3

I

T3

T T

2

I I

T2 T4 T1 T 2

I T T 3 T

Cyclic shift:

I

T

3

X (T ) 4

I

I 4 2

1

X (T 1 ) 4

I

I

I

T3 T2 I T3

I T1

T4 T2 I

T3

I T2 T4 T3 T4 T5

62 I .

 f

( f 0 , f1 , f 2 , f 3 , f 4 , f 5 ) ( f 5 , f 0 , f1 , f 2 , f 3 , f 4 )  ( Xf ) p ( Xf ) p T p ( Xf ) p , p 0 : 5,  f , f R12 . Art and Merughan, SPIE 2009

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N=6: Inner product ( f , g)

( Xf , Xg )

We define the inner product as ( f , g)A

f Ag

where A≠I is a matrix 12 12. For that we first define a matrix R T RT and, then, A I 6 R.  We obtain solutions R

a

R ( a , b)

Example:

R

| x | 2 ( x, x ) R

a b a

x02

|f|

( f , f )A

1 1

x12

.

x0 x1 ,

11

f n 0

.

1 0

R(1,0)

11 2

b

2 n

f 2 n f 2 n 1. n 0 Art and Merughan, SPIE 2009

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“Inner product” in 

12 R

The metric can be defined as g |2 ( f

d( f , g) | f

g, f

g)A

however ( f , g ) A ( g , f ) A , f To obtain the property ( f , g)A

(g, f ) A

b

g

0

R12 .

a / 2.

Example: R

1 R(2, 1) 3

( f , g)A

( Xf , Xg ) A

1 3

1 3

2

1

1

2

5

[ f 2n

f 2n 1 ]

n 0

1

5

6 3

p 0

g 2n 2 g 2n 1

2

1

1

[ F2 p F2 p 1 ]

f,g

, det R 1.

2 1

G2 p 2 G2 p 1 1

R12 . Art and Merughan, SPIE 2009

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Roots of the unit matrix ? Integer matrix such that T I . 2 / 5, c cos( ) 0.3090, For 5

C C( )

c

c 1

0.3090

0.6910

c 1

c

1.3090

0.3090

cU V c

1 1

0

1

1 1

1

0

and C 5 I . • 5-block C-GDT is defined by det(C) 1 I X (C )

I

I

I

I

I C1 C 2 C 3 C 4 I C 2 C 4 C 1 C 3 , X 4 (C ) I C 3 C1 C 4 C 2

I C4 C3 C2 I I 1 1 X 1 (C ) X (C 1 ) I 5 5 I

25 I , det X (C ) 55.

C1 I I I I C 4 C 3 C 2 C1 C 3 C1 C 4 C 2 . C 2 C 4 C1 C 3

I C1 C 2 C 3 C 4 Art and Merughan, SPIE 2009

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Elliptic DFT 

x

Consider the orbit of a point x with respect to the group of motion {C k ; k 0 : 4}, y1

Cx

y2

Cy1

y3

Cy2

y4

Cy3

x

Fig 1: (a) Location of all points y k and (b) scheme of the movement of the point (1,0).

The points move along the perimeter of the ellipse, x

2

y2 1, 2 b

b

1 c 1.3764 1 c


13

General Elliptic DFT 2 / N, Given N 1 and angle the following matrix is defined 

C

C( )

cos cos

C C( ) R

R( )

cos

1

1

cos

cos

I sin

0

tan / 2

cot / 2

0

cos

U V.

R, , det R 1.

Such a matrix can be also defined as C C( ) cos CN ( )

I,

I sin N

R.

1.

• Definition: Given N 1 and angle the matrix C

C ( ) cos

I sin

R( )

is called the generalized elliptic matrix (GEM). Art and Merughan, SPIE 2009

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Example: N=32 Consider the sinusoidal signal xr (t )

cos(2 0t ),

x

0

2 / N,t

tn

[0,2 ].

( xr (0),0, xr (1),0,...,0, xr ( N 1),0)

y

X (C) x

yr

( y (0), y (2), y (4), y (6),..., y (2 N

yi

( y (1), y (3), y (5), y (7),..., y (2 N 3), y (2 N 1)) 2

6

4), y (2 N

2))

0

0

Fig 2: (a,c) Sinusoidal signals and (b,d) the 32-block DEFTs of the signals. Art and Merughan, SPIE 2009

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Imaginary part of EDFT The N-block EDFT recognizes the carrying frequency at the frequencypoints p=6 and 28=32-p.

Fig 2: Imaginary parts of the (a) 32-block EDFT and (b) 32-block DFT of the signal cos(6 0t ).

DFT does not have maximums on the carrying frequency-points 6,26. The imaginary part of the DFT by amplitude is smaller than the EDFT. Art and Merughan, SPIE 2009

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Properties of GEM 

GEM defines a one-parametric group with period N C ( 1 )C (

) C (

),

, 2. C N ( ) C ( N ) C (2 ) I .

x

y1

Cx

y2

2

Cy1

y3

Cy2

1

2

...

yN

1

Cy N

1

2

x

Cy N

1

/ 30

/ 20

/ 15

/ 10

Fig 4: The scheme of the movement of the point (1,0) when (a) N=10 and (b) N=15.

x

2

y2 b2

1, b cot

2

.

The ellipse does not depend on the angle

.


17

General case Matrix R satisfies the condition 2

R ( )

I.

• Given N 1 and matrix R such that I , we call the matrix

R2

C C ( ) cos

I sin

R

the generalized elliptic matrix (GEM). The matrix R has the form R

R(a, b, c)

when a 2 bc Example: R

a

b

c

a

1.

R( 1,2, 1)

1 2 1 1

.


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N-block GEFT: C

C ( ) - generated matrix I

X (C )

I

I

I C1 C2 I C2 C4 ... ... ... I C N 1 C 2N N

I

2

C3 C6 ... C 3N 3

...

I

... C N 1 ... C 2 N 2 ... ... ... C 1

2 / N.

parameterized by and . The following property holds for the N-block GEFT as for the DFT: R( x) N k R( x) k , I ( x ) N k I ( x) k k 1 : ( N / 2 1),

where R(x) and I(x) denote the “real” and “imaginary” parts of the N-block GEFT of a real input x , respectively. Art and Merughan, SPIE 2009

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Example: 5-block GEFT 

For N=5, the 5-block GEFT has the following matrix: I

X (C )

X (C ( 5

C1

C3

2 )) 5

0.3090

0.3090

2.9271

0.3090

,

0.8090

0.1910

1.8090

0.8090

C5

I

1 0 0 1

1 X (C 1 ) 5

I

I

I

I

C1 C 2

I I

C 2 C 4 C1 C 3 , C 3 C1 C 4 C 2

I

C4

C3 C2

C1 0.1910

1.8090

0.8090

0.3090 0.3090

, C4

,

C3 C4

0.8090

C2

2.9271 0.3090

,

,

det X (C) 3125.

I X 1 (C )

I

I 1 I 5 I I

I

I

I

I

C 4 C 3 C 2 C1 C 3 C1 C 4 C 2 . C 2 C 4 C1 C 3 C1 C 2

C3 C4


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Example: 512-block GEFT 2 / 512 and /6 Consider and a “complex” signal of length 512

C

/6

( )

0.9999

0.0033

0.0458

0.9999

.

Fig 5: (a) Signal of length 512, (b) the real and (c) imaginary parts of the 512-block GEFT of the signal.


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Comparison 

Images of matrices GEFT and DFT.

Fig 6: The image of the matrix of the N-block GEFT when (a) N=15, (b) N=30, and (c) N=52.

Fig 7: The image of the matrix of the N-block DFT when (a) N=15, (b) N=30, and (c) N=52.


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Summary By constructing matrices C ( ) which are roots of the identity matrix, the N-block EFT has been defined whose block-matrices are composed by powers of C( ). The set of matrices {C ( n ); n 0 : ( N 1)}, where n 2 n / N , compose a oneparametric group of order N. The matrix C ( ) is the matrix of rotation around the ellipse y

a x b

2

1 2 x 2 b

r 2.

The ellipse turns into the circle of radius r when a=0 and b=1. Then C is the Givens rotation and the elliptic DFT is the traditional DFT. Art and Merughan, SPIE 2009

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References 1.

A.M. Grigoryan and V.S. Bhamidipati, “Method of flow graph simplification for the 16-point discrete Fourier transform,” IEEE Trans. on Signal Processing, vol. 53, no. 1, pp. 384-389, January 2005.

2.

A.M. Grigoryan and M.M. Grigoryan, Brief Notes in Advanced DSP: Fourier Analysis with MATLAB®, Taylor & Francis Group / CRC Press, January 2009 (scheduled publication).


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Example: 12-point Block Transform Consider the integer matrix 3x3 1 H

0

0 3

1 2

H4

H 4bl

2 2 , H 1

I and I

1

H

I

I

I

I

I I

H1 H2

H2 I

H3 H2

I

H3

H2

H1

5

4

2

6 3

5 2

2 , det( H ) 1

H2

H3

1,

0.

1 0 0

1

0

0

1

0

0

1

0

0

0 1 0 0 0 1

0 0

1 0

0 1

0 0

1 0

0 1

0 0

1 0

0 1

1 0 0 0 1 0

1 0

0 1

2 2

5 6

4 5

0 0

5 6

4 5

2 2

0 1 0 0 1 0 0

3 5 6 0 5 6 3

2 4 5 0 4 5 2

1 0 0 1 2 2 1

0 1 0 0 5 6 0

0 0 1 0 4 5 0

1 0 0 1 0 0 1

3 5 6 0 1 0 3

2 4 5 0 0 1 2

1 0 0 1 2 2 1

0 0 1 0 0 1 0

1 0 0 1 0 0 1

This is not Fourier-like transform: H2≠I. Art and Merughan, SPIE 2009

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Example: 15-point Block Transform Consider the complex matrix 3x3 H

- 0.1483 - 1.3080i 0.3416

0.0566 - 0.5002i - 0.4714

0.5834 - 0.3570i - 0.8131 ,

1.1971 1.3080i

0.8862 0.5002i - 0.6893 0.3570i det( H ) 1.

H5

H 5 bl

H 5 bl1

I and I

H

H2

H3 H4

I I I I

I H1 H2 H3

I H2 H4 H1

I H3 H1 H4

I H4 H3 H2

I

H4

H3

H2

H1

I I I I

I H H H

I H H H

I

H

1 2 3 4

H

H 5 bl H 5 bl1

2 4 1 3

I H H H H

3 1 4 2

I H H H H

0.

4 3 2 1

5I .

This is a Fourier-like transform. Art and Merughan, SPIE 2009

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