Ijeas0303039

International Journal of Engineering and Applied Sciences (IJEAS) ISSN: 2394-3661, Volume-3, Issue-3, March 2016

About the Recognition and Reconstruction Two Unknown Functions From Known Their Tandem Vladimir V. Arabadzhi 

on the carrier z  L . We find the single function U ( z ) and

Abstract— Paper is devoted to the area of processing of physical signals and data or artificial perception. The incoming data are presented in the form of defined sum of two unknown functions, spaced on finite carrier. Practical examples of the functions searched for can be presented: (a) by very closely spaced two atomic spectrums of radiation; (b) by the output signal of some telescope, scanning two point-like sources with angle distance between each other much less, than the angle width of main petal of telescope's directivity pattern. So the presented result can be classified as some case of the known problem of superresolution. It is shown, that two functions of tandem (mutually overlapping each other) can be reconstructed separately by using only three it is correct chosen numbers. The class of functions (and their tandems), which are overlapping each other and allows single reconstruction of both searched functions and also conditions of the correctness of the decision are formulated. The method of separation of two functions had been tested numerically on the variety of typical practical examples. The stability of suggested numerical procedure is confirmed in the presence of uncorrelated errors of measurements and calculations at each point of tandem.

single function V (z ) in Figure 1-a .

Then we do important assumption: W (z ) is presented by the tandem (all variables and numbers, used below, are assumed dimensionless) of two smooth functions U ( z ) and V (z ) , which are overlapping each other at any point of the same carrier z  L . No any point z  z * on above carrier where function U ( z*)  0 at V ( z*)  0 or V ( z*)  0 at U ( z*)  0 .

Itself determination W (z ) as "two functions" nothing do not signify. In realities possible to name such combination "one function", or suppose, that W (z ) consists of tens of functions.

To avoid the uncertainty, we must say the important thing: function U ( z ) has a centre of symmetry zU , and function V (z ) has a centre of symmetry zV . In addition the searched

functions U , V are assumed mutually linearly independent, and we must find single solution for tandem U , V on chosen class of functions (see below in Section IV). Such problems can appear in investigations of closely spaced spectrums of atomic radiation or in resolutions of astronomic sources by radio antennas [2]-[4]. Below we suppose negligible weak noise (see Figure 1-b)

Index Terms—About four key words or phrases in alphabetical order, separated by commas.

   AW , AU , AV ,

(1)

I. INTRODUCTION Usually the problem reconstruction of signals is considered in the presence of noise and interpreted as extraction of signal U ( z ) covered by noise, so the measured signal can be of the form W ( z )  U ( z )   ( z ) , where  ( z ) - stationary noise with mean-square value 2

where symbols AW >0, AU >0, AV >0

mean the

characteristic magnitudes 1/ 2

AW  [ L L W 2 ( z )dz /(2 L )]

1/ 2

    ( z ) 

1/ 2

AU  [ L L U 2 ( z )dz /(2 L )]

correlation interval

,

,

1/ 2

AV  [ L L V 2 ( z )dz /(2 L )]

   L [1].

Below we will consider some untraditional statement of the problem.

of values W (z ) , U ( z ) , V (z ) , following (1), W ( z)  U ( z)  V ( z) .

II. STATEMENT OF THE PROBLEM Usually the problem reconstruction of signals is considered in the presence of noise and interpreted as extraction of signal Statement of the problem

(2)

Both component U ( z ) , V (z ) of tandem is assumed function, symmetrical relating to some point (symmetry axis) zU , zV correspondingly. Figure 1-c illustrates the cases of generalized symmetry of the following form: value of function U (  z ) (or V (  z ) ) are predetermined by the value of function U ( z ) (or V (z ) ) by coefficient of symmetry (coefficient of reflection) as was described in the assumption ˆ in the section III).

H

We have only two unknown smooth and mutually overlapping functions U ( z ) , V (z ) , which are forming exactly predetermined smooth source function W ( z )  U ( z )  V ( z )

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About the Recognition and Reconstruction Two Unknown Functions From Known Their Tandem

(a)

AU

W ( z)

 (z )



~ U ( z )  U ( z )  W ( z )

AW

W V

z

U ( z)

ˆ  Jˆ  K ˆ =1, ˆ  H So at Aˆ  Bˆ  Сˆ  Dˆ  Eˆ  Fˆ  G ~ we will get result , V ( z )  V ( z )  W ( z ) max

(c)

V ( z)

U

L L q q a q a

(b)

q AV

z 0

~

the searched solutions which must be maximally closely to U ( z ) and V (z ) correspondingly.

z

zU zV

~

, where U ( z ) and V ( z ) represents

max

2a

Figure 1 IV. TANDEM RESTORATION PROCEDURE Now we consider the special computing procedure, which represents the base of all further investigations. Figure 2-a represents smooth function spiral process of tandem restoration. For the schematic model of pervious concrete problem we try to find a single solution for both functions of tandem.

ˆ  1 if yes, A ˆ  0 if no). a  am θ  θk ( A

Bˆ . Center point θ  ( zV  zU ) / 2 between two centers of

( Bˆ  1 if

ˆ  1 if (we assume this hypothesis is true С

ˆ  0 if no). yes, С ˆ . Symmetry center of function V (z ) is spaced in the D

7

point z  θ k  am (we assume this hypothesis is true Dˆ  1 if

11

ˆ  0 if no). yes, D Eˆ . Points zU and zV are spaced sufficiently far from

(b)

symmetry present V ,U  1 , however one can use more symmetry

U (  z )  U ( z )U ( z )

(coefficients of symmetry, coefficient of reflection,…) must provide one-to-one correspondence of functions U ( z ) and U (  z ) , V (z ) and V (  z ) , not breaking their smoothness.

~ U (  L)  W ( z )

max

max

~ , V (  L)  W ( z ) max ,

V ( z)

12

zV

zU

2

4

2

1

3

8

(c) 5

6

6

8

7

U ( z ) , V (z ) correspondingly. Table 1 shows the generation of   spiral; sequences of interpolation nodes x  { xi } , p  { p i } ,  q  {qi } ( i  1, 2,3,...,13 ), when starting from point z  θ to

,

V (  z )  V ( z )V ( z ) (Figure 1-c). Functions U ( z ) , V (z )

~ Jˆ . V (  L)  W ( z )

8

symmetry V

The appearance sequence of interpolation nodes (see below) resembles the linear structure of image light sources, caused by one light point-like source spaced in the point z  θ between parallel plane mirrors with coordinates z  θ a .

Spiral begins (see Fig 2-a) in the point θ and goes to the left.

Vertical transitions of spiral line are caused by equation (2), horizontal transitions of spiral line are caused by concrete type of symmetry U ( z ) , V ( z ) of searched functions

V

of

4

Figure 2

~ ˆ  1 , if yes, H ˆ  0 , if no). Simplest cases of (and V ( z ) ) ( H

form

1

symmetry V

7

Fˆ . the value of function U ( z ) in the start-point z  θ U (θ )  μ (we will check this hypothesis step by step Fˆ  1 if yes, Fˆ  0 if no). ~ ˆ . Function  G U ( z )  U ( z ) of symmetry of function ~ ˆ  1 if yes, G ˆ  0 , if no).

U ( z ) (and U ( z ) ) ( G ~ ˆ . Function  ( z )   ( z ) of symmetry of function V (z ) H

general

5

3

4

5

borders  L . In other words, we assume relation θ  a  L .

V

1

9

U

2

symmetry V 3

U (z)

symmetry U

6

V=W-U

point z  θ k  am

V=W-U

yes, Bˆ  0 if no). Сˆ . Symmetry center of function U ( z ) is spaced in the

q

symmetry U

10

V=W-U

symmetry is assumed placed in the point θ  θ k

(a)

U=W-V

of functions U ( z ) and V (z ) correspondingly is assumed

U=W-V

ˆ . Distance 2 a between centers z and z of symmetry A U V

U=W-V

III. ASSUMPTIONS (HYPOTHESIZES)

the left (see Figure 2-a,b).

(3) Table I Spiral sequence of interpolation

~ , U (  L)  W ( z ) at  z  L . max

x1  θ x2  θ  2a~

In other words, functions are assumed smoothly going closely to zero together with their smooth first derivatives (see Figure 1-a). ˆ . Functions W ( z ) ,U ( z ) ,V ( z ) are assumed smooth at K

x 4  θ  6a~

 z  L . Their characteristic spatial scales are of the same

x5  θ  8a~

order with L / 2 .

x6  θ  10 a~

x3  θ  4a~

69

~ p1  μ ~ p 2  p1U ( a~ ) ~ q3  q 2 / V (3a~ ) ~ p 4  p3U (5a~ ) ~ q5  q 4 / V ( 7 a~ ) ~ p6  p5 U (9 a~ )

q1  W ( x1 )  p1 q2  W ( x2 )  p2 p3  W ( x3 )  q3

q 4  W ( x4 )  p 4 p5  W ( x5 )  q5 q 6  W ( x6 )  p 6

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International Journal of Engineering and Applied Sciences (IJEAS) ISSN: 2394-3661, Volume-3, Issue-3, March 2016 ~ q7  q6 / V (11a~ ) ~ p8  q7 U (13a~ ) ~ q9  q8 / V (15 a~ ) ~ p10  p9 U (17 a~ ) ~ q11  q10 / V (19 a~ )

x 7  θ  12 a~ x8  θ  14 a~

x9  θ  16 a~ x10  θ  18 a~

x11  θ  20 a~

p 7  W ( x7 )  q 7

by variation ( i  1, 2,..., N a ; j  1,2,..., N μ ; k  1, 2,..., N θ ) and

q8  W ( x8 )  p8

selection of correct combinations of three numbers

p 9  W ( x9 )  q 9

ai , μ j , θ k

.

~

~

patterns U ( z )  U ( z; ai ;  j ; q k ) ,

Sorting

~ ~ V ( z )  V ( z; ai ;  j ; q k ) , we will search for the tandem ~ ~ U ( z ), V ( z ) satisfying all above conditions (assumptions, hypothesizes) Aˆ  Kˆ .

q10  W ( x10 )  p10

p11  W ( x11 )  q11 

Table II shows the generation of spiral sequences y  { y j } , 



  { j } ,   { j } ( j  1,2,3,...,13 ) of interpolation nodes,

V. MARKERS OF ERRORS ~

when starting from point z  θ to the right (see Figure 2-c).

sort all N a Nμ Nθ  1 patterns (and corresponding all

Table II Spiral sequence of interpolation nodes y1  θ  1  q1 1  p1 ~

~

When searching for the single U ( z ) and single V ( z ) we

ai , μ j , θ k ).

combinations

Any real combination of

y 2  θ  2a~

 2   1 / V ( a~ )

 2  W ( y2 )   2

key-numbers ai , μ j , θ k includes some small or very small

y3  θ  4 a~

~  3   2 U (3a~ ) ~  4   3 / V (5a~ ) ~  5   4 U (7 a~ ) ~  6   5 / V (9a~ ) ~  7   6 U (11a~ ) ~  8   7 / V (13a~ ) ~  9   8 U (15 a~ ) ~  10   9 / V (17 a~ ) ~ 11  10 U (19 a~ )

 3  W ( y3 )   3

errors. We can’t measure immediately errors of tandem’s restoration because both functions U ( z ) , V (z ) are assumed unknown priori.

However, we have found as empirical fact - an errors in key numbers are always accompanied: (1) by the appearance of the fluctuations function on high spatial frequency ~  / a~ and its harmonics (the breach of the hypothesis Kˆ to smoothness of U ( z ) , V (z ) , see Section III.); (2) by growing

y 4  θ  6a~ y5  θ  8a~ y 6  θ  10 a~ y 7  θ  12 a~ y8  θ  14 a~ y9  θ  16 a~ y10  θ  18 a~

y11  θ  20 a~

 4  W ( y4 )   4  5  W ( y5 )   5  6  W ( y6 )   6  7  W ( y7 )   7

 8  W ( y8 )   8

of the modules function U (  L ) , V (  L) on edges  L of

 9  W ( y9 )   9

carrier z  L (the breach of the hypothesis Jˆ , see Section

10  W ( y10 )   10

III). Thereby, we have built the quantitative criterion ~; ~ ε  ε[ a~; μ θ ] of the choice correct key numbers:

 11  W ( y11 )  11

~; ~ ε  ε[ a~; μ θ]  ε U  ε V ,

Table III Renumbering interpolation nodes from spiral structure into linear structure spiral

1

xi ; p( xi ) ; q( xi )

12 10

8

6

4

3

2

spiral linear

11 2

1

7

9

11 13

2

y j ;  ( y j ) ;  ( y j ) 13

zn ; U ( zn ) ; V ( zn )

5

9 6

4 3

5

5

7 8 7

3

1

4

6

8

Using Table III and cubic-spline interpolation [5]-[7], we obtain 26 nodes of interpolation and following continuous functions

  Where z  { z n } , U  {u n } ,

2



( 2 L)



2

2



( 2 L)

1/ 2

1/ 2

1

~ ~  U (  L)  U (  L) ,

(7)

1

~ ~  V (  L)  V (  L) ,

(8)

L ~ IUC   L U ( z ) cos(z / a~) dz ,

11 13 15 17 19 21 23 25

  ~ U ( z)  interp[cspline(z,u ), z,u,z ] ,     ~ V ( z)  interp[cspline(z, ), z,,z ] .

2

εV  IVC  IVS

10 12 14

10 12 14 16 18 20 22 24 26 9



εU  IUC  IUS

(6)

(4) (5)



  { n } ( n  1, 2,3,... ,26). 

Spatial period of sequence z  { z n } is 2 a . Note, that

Sorting

calculation procedures represented in Tables 1,2,3 guaranteed (at zero noise    0 ) the obvious equations

(9)

L ~ IVC   L V ( z ) cos(z / a~)dz ,

(10)

L ~ IUS   L U ( z ) sin(z / a~) dz ,

(11)

L ~ IVS   L V ( z ) sin(z / a~)dz .

(12)

all

values a i , μ

j

, θk

( i  1, 2,..., N a ;

j  1,2,..., N μ ; k  1, 2,..., N θ ), we search for the minimum of

the calculated value ε . But value (and its minimum) ε not ~ ~ error of restoration of searched tandem U ( z ) , V ( z ) . The

~ U ( z )  V ( z )  W ( z ) in the nodes z  z n (at any a~ , μ~ , θ !) , ~ which does not any guarantees for equations U ( z )  U ( z )  0 , ~ V ( z)  V ( z)  0 .

Now our aim is to find a single solution, V (z ) for tandem

~

minimum of the value ε[a~; μ~; θ ] , reached by choosing of three ~ correct key-numbers a~  a i  a , μ~  μ j  μ , θ  θ k  θ ,

give us the tandem, most closing to the correct true functions

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About the Recognition and Reconstruction Two Unknown Functions From Known Their Tandem (13)

e [a ; μ; θ]

3

(see below).

2

~ ~ ~ μ , U ( z ) , V ( z ) to the key-numbers errors a  a~  a , μ  μ ~ θ  θ  θ we can use only after tandem reconstruction (post factum, using true model functions U ( z ) , V (z ) and on the ~ base of key-numbers hypothesizes a~ , μ~ , θ and sorting of

~

(15)



with coefficients of symmetry V ( z )  1 , U ( z )  1

0.4



 0.4

q

0.4

θ

~

(a)

~ U

a( t )

0

~ V

1  50

3

(b)

2

W

1 0

1

0

50

 50

z

t~ ~ 5 U  U , V  V  10 W

0

50

z

at right choices t three values max

a  0, μ  0, θ  0

(16) Figure 4 ~

One can note, that with any errors U ( z )  U ( z ) ,

(17)

~ V ( z )  V ( z ) of restoration, the sum of tandem U ( z )  V ( z ) is saved equal W (z ) , following the calculation procedure

described in Section IV (excluding, of course, the interpolation errors). ~ ~ Results U ( z ), V ( z ) of restoration of searched functions

(18)

~ U ( z ), V ( z ) (a) and errors of restoration U ( z )  U ( z ) , ~ V ( z )  V ( z ) (b), admitting only one nonzero error a  0 ,

(19)

(  (U , a )   (V , a )  0.4 , see (14), (15), are represented in Fig. 6.

2



~

Sea rched s olutio ns U (z),V (z )



3 2

X U( z)

(a)

~ U

0.4

U ( z)

1

~ V

1  100  50

0

z 50

100

z

(20)

error a  0.5

(21)

;

~

(b)

V V

0.2

0 V ( z)  0.2

X V( z) 0

In this section (and next section V) we consider [2], [3] variants of Lorentz’s functions. Now consider combination of symmetric and antisymmetrical functions

V ( z )  2.2 sin0.1( z  2) / 1  0.005( z  2)

2

F (t )

VI. SYMMETRIC & ANTISYMMETRIC LORENTZ’S FUNCTIONS IN TANDEM



0



L (t ) 1

Further we will consider several examples of tandem’s components with different types of symmetry, with different relations of amplitudes, with different velocity of the decline to edge of the carrier.

U ( z )  1.8 / 1  0.005( z  2) 2

a

3

~

~ Sea rched solutions U( z), V(z )

(14)

  ~  (V , a )  Lim  Max V  V / a  , a 0 z L     ~  (U , μ)  Lim  Max U  U / μ  , μ 0 z L     ~  (V , μ)  Lim  Max V  V / μ  , μ 0 z L     ~  (U , θ)  Lim  Max U  U / θ  , θ 0 z L     ~  (V , θ)  Lim Max V  V / θ  . θ 0 z L  

0  0 .4

0.5

Results U ( z ), V ( z ) of restoration of searched functions U ( z ), V ( z ) (a) from their tandem W (z ) (b), using exact values a , μ, θ , are represented in Fig. 4.

~

 (U , a ) 

 /a, 

1

Figure 3

many patterns) given by minimums of errors markers ~; ~ ε[a~; μ θ ] ).

 ~ Lim Max U  U a 0 z L 

 0.5 0

e [a; μ; θ ]

2

1

1 0

~) , ~) ,  (V , μ Sensitivities  (U , a~ ) ,  (V , a~ ) ,  (U , μ ~ ~  (U , θ ) ,  (V , θ ) ((14)-(19)) of restored functions

e [a; μ ; θ]

2

Source function W (z)

max

Errors o f recon structed functio ns

~ ~ 5 U  U , V  V  10 W

 0.4

~

U U

 100  50

z 0

50

100

z

correct μ  0 , θ  0

Figure 5 ~

~

Results U ( z ), V ( z ) of restoration of searched functions

(22)

~ U ( z ), V ( z ) (a) and errors of restoration U ( z )  U ( z ) , ~ V ( z )  V ( z ) (b), admitting only one nonzero error μ  0

and true key numbers (21) a  2 , θ  0 , μ  U ( 0) .

Errors markers ε[a ; μ; θ] , ε[a; μ; θ] , ε[a; μ; θ] are pointing

( (U , μ)  (V , μ)  1.25 , see (16), (17)), are represented in Fig. 6.

~

to the values a~ , μ~ , θ closest with the parameters a , μ , θ of correct restoration. They are represented in Fig. 3.

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F (t ) 0

g (t )

~ V

1  50

z

0

50

t

0

~ U U

1

 50

error   0.2 ; correct a  a

z

0

(a)

L (t ) F (t )

 0.5

1

50

t

, θ 0

1 2

~ U  50

z 0

50

b ( t)

1

 50

0

z

50

t error θ  0.4

~ (b)

~ V V

0.5

F (t )

0

~ U U

1

 50

0

50

z

  U ( z )  1.8 sin0.15( z  2) /1  0.005( z  2)  V ( z )  2.2 sin0.1( z  2 / 1  0.005( z  2)2

6

10

~

~

Searched solut ions U (z ),V (z )

e [a; μ ; θ]

0 1

5 0

0.5



 0.5

0.2

b ( t)

0

 0.2

~

 0.4

2  100

z

0

2

100

;

1

(a)

~ V

L( t ) F( t )

e [a; μ; θ ]

0 1

(b)

~ U U

~ V V

 100

z

0

100

t

correct μ  0 , θ  0

2

0.4

a (t )

0.2

b (t )

0

 0.2

~

U

 0.4

 50

0

50

z

t

(b)

~ V V

~ U U  50

0

50

z

t

error   0.2 ; correct a  0 , θ  0 Figure 11

2

a

0.4

a( t )

U

4

1

0

~

0.5

~

Results U ( z ), V ( z ) of restoration of searched functions

θ

~ U ( z ), V ( z ) (a) and errors of restoration U ( z )  U ( z ) , ~ V ( z )  V ( z ) (b), admitting only one nonzero error θ  0

Figure 8 ~

(a)

(  (U , μ)   (V , μ)  4.1 , see (17), (18)), are represented in Fig. 11.

and their coefficients of symmetry (26) U ( z )  1 V ( z )  1 with coefficients of symmetry (27) a  2 , θ  0 , μ  U ( 0) .

Errors markers ε[a ; μ; θ] , ε[a; μ; θ] , ε[a; μ; θ] are pointing to ~ the values a~ , μ~ , θ closest with the parameters a , μ , θ of correct restoration Figure 8.

 0.5 0 0.5

 0, μ  0, θ  0

~ U ( z ), V ( z ) (a) and errors of restoration U ( z )  U ( z ) , ~ V ( z )  V ( z ) (b), admitting only one nonzero error μ  0 ,

(24) (25)

2

0  0.5

z

m ax a

Figure 10 ~ ~ Results U ( z ), V ( z ) of restoration of searched functions

For the following model functions

0

~

V

1

error a  0.8

; correct a  a , μ t 0

VII. ANTISYMMETRIC & ANTISYMMETRIC LORENTZ’S FUNCTIONS IN TANDEM

e [a ; μ; θ]

2

t

Figure 7

2

50

~

L (t )

 0.5

~ V

1

0

(  (U , a )   (V , a )  0.5 , see (15), (16)), are represented in Fig. 10.

~

1

g (t )

F (t ) 0

z  50

~ U ( z ), V ( z ) (a) and errors of restoration U ( z )  U ( z ) , ~ V ( z )  V ( z ) (b), admitting only one nonzero error a  0

Searched solutions U (z ),V (z )

Errors of reconstructed functions

~ ~

Searched solutions U( z),V (z )

L (t )

~

2

Results U ( z ), V ( z ) of restoration of searched functions

(  (U , θ)   (V , θ)  0.5 , see (18), (19)), are represented in Fig. 7.

U

W

1

at right choices three values

~ 5 t ~ U  U , V  V  10 W

~

2

(b) 1

Figure 9

~ ~ Results U ( z ), V ( z ) of restoration of searched functions ~ U ( z ), V ( z ) (a) and errors of restoration U ( z)  U ( z) , ~ V ( z )  V ( z ) (b), admitting only one nonzero error θ  0 ,

(a)

2

W (z ) 0

0

Figure 6

3

Source function W(z)

b ( t)

~ V V

0.5

~ V

Errors of recon structed functions

U

(b)

2

Errors of reconstructed functions

~

2 1

L (t )

1

~

(a)

~ Searched solutions U(z ),V (z )

3

Errors of reconstructed fun ctions

~ ~

Searched solutions U (z ),V (z)

International Journal of Engineering and Applied Sciences (IJEAS) ISSN: 2394-3661, Volume-3, Issue-3, March 2016

( (U , θ)  (V , θ)  2.0 , see (19), (20),are represented in Fig. 12.

~

Results U ( z ), V ( z ) of restoration of searched functions U ( z ), V ( z ) (a) from their tandem W (z ) (b), using exact values a , μ, θ , are represented in Fig. 10.

72

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0 1

0

b (t )

~

U

2

 50

0

z

50

t

 0.2

~ U U

 0.4

 50

error θ  0.3 ; correct a  0 Figure 12

1.2

0

z

50

, μ0

F( z)

e [a; μ; θ]

6

0.02

a

0

 0.005 0 0.005



0.5

~ V

~ U

0

0.1

z

~ ~z 5 U  U , V  V  10 W

~ ~

Searched solutions U (z ),V (z )

~

U

1

a ( z)

0.5

0  0.2  0.1

0

0.1

0.2

(b)

~ V V

0

b ( z)

~

 0.5

~ U U

1  0.2  0.1

0

0.1

0.2

z

a 0,θ 0

~

~ U ( z ), V ( z ) (a) and errors of restoration U ( z )  U ( z ) , ~ V ( z )  V ( z ) (b), admitting only one nonzero error θ  0 ,

0.005

0

θ

F (z )

1.5 1

(a)

1

~ V

b ( z) 1

0  0.1

0

0.1

z

1

0.2

(b)

~ V V

1

a ( z)

~

U

0.5

z

(b)

0.5 0

 0.5

~ U U

1

 0.1 A

error θ  0 .2 ; correct a  0

,μ  0

0 z

0.1

z

0.2 B

Figure 17

W

0.5

IX. TWO GAUSSIAN FUNCTIONS

0 0

0.1

z

0.2

In this section we will test the restoration procedure (section IV) on the case of two Gaussian functions (32) U ( z )  2.2 exp[ 0.005( z  2) 2 ]

at right choices tthree values m ax

z B

(  (U , θ)   (V , θ)  0.25 , see (19), (20)) are represented in Fig. 17.

0  0.005

 0.2  0.1

0.2

0.2

z

Results U ( z ), V ( z ) of restoration of searched functions

e [a; μ; θ ]

W( t )

0  0.2  0.1

(a)

~ V

0.5

L (z )

Source function W(z)

~ ~ Searched s olutions U (z ),V (z ) F (z )

0

Figure 16

~

1

L (z )

1

z

Results U ( z ), V ( z ) of restoration of searched (true) functions U ( z ), V ( z ) (a) from their tandem W (z ) (b), using exact key-numbers a , μ, θ are represented in Figure 14. (a)

 0.2 A

~

error μ  0. 3 ; correct

Figure 13 ~

1

error a  0 .001 ; correct μ  0 , θ  0

~ ~

0 .01

z B

F (z )

2

0

0.2

z

L (z )

Searched solutions U (z ),V (z )

0

0

(  (U , μ)   (V , μ)  0.6 see (17), (18)) are represented in Fig. 16.

0 .1

0 .5

 0.2

~ V V

~ U ( z ), V ( z ) (a) and errors of restoration U ( z )  U ( z ) , ~ V ( z )  V ( z ) (b), admitting only one nonzero error μ  0 ,

4

1

 0.5

1

A

0

b ( z)

0

~

(6)-(12) ε[a ; μ; θ] , ε[a; μ; θ] , ε[a; μ; θ] are pointing to the ~ values a~ , μ~ , θ closest with the parameters a , μ , θ of correct restoration Figure 13 and errors a  a~  a , μ  μ~  μ , ~ θ  θ  θ are going to zero if we are sorting patterns at . In the case considered the smallness of angle parameter a means antenna’s resolution (or superresolution).

0 .2

a ( z)

~

U

(b)

~ U U

0.5

Results U ( z ), V ( z ) of restoration of searched functions

V  1 , U  1 , θ  0 . Errors markers

e [ a ; μ; θ]

1

1

Figure 15

For simplicity we consider one-dimensional aperture antenna with characteristic directivity pattern f ( z )  sinc( 35 z ) (all variables are assumed dimensionless). The last means amplitude response of antenna to some source, spaced in antenna’s far zone, z is angle coordinate. We spaced two point–like sources in the directions with coordinates z  a  0.005 and z  a  0.005 . Angle distance 2 a between two sources is less, than the width of directivity pattern main petal. Output of antenna [4] are the powers U ( z)  sinс35( z  a)2 , V ( z)  0.64sinс35( z  a)2 , created by sources of the antenna’s output (amplitude of source in the direction z  a  0.005 is 1, other source has

1 .5

~ V

0.5

 0.2

VIII. SEPARATION OF TWO VERY CLOSELY SPACED RADIO-ASTRONOMIC POINT-LIKE SOURCES

amplitude 0.8),

(a)

1

L( z)

t

Errors of reconstructed functions

F (t )

0.2

Errors of reconstructed functions

a (t )

(  (U , a )   (V , a )  8.1 see (15), (16)) are represented in Fig. 15.

(b)

~ V V

Errors of reconstructed functions

1

L (t )

0.4

~ ~

(a)

~ V ( z )  V ( z ) (b), admitting only one nonzero error a  0 ,

Searched solutions U(z ),V (z)

2

Errors of reconstructed functions

~ ~

Searched solutions U( z),V (z )

About the Recognition and Reconstruction Two Unknown Functions From Known Their Tandem

a  a, μ  0 , θ  0

Figure 14

V ( z )  1.8 exp[0.005( z  2) 2 ] with coefficients of symmetry U ( z )  1 , V ( z )  1 .

~ ~ Results U ( z ), V ( z ) of restoration of searched functions ~ U ( z ), V ( z ) (a) and errors of restoration U ( z )  U ( z ) ,

73

(33) (34)

www.ijeas.org

0.8

e [a ; μ; θ]

e [a; μ ; θ]

10

0.4

2

0

1

a

~ ~

2

0  0 .4

0



F (t )

e [a; μ; θ ]

~ V

1

0

0  50

0

z

50

(b)

~ U U

3

0

W( t ) 2

 0.2

1

~ V V

 0.4  50

z

0

50

correct a  0t

, θ0

0

 50

Figure 21

Figure 18 ~

0

z

0.1

1

W

W( t )

0  0.2  0.1

(b)

1.5

0.5

~ ~z 5 U  U , V  V  10 W

0  0.2  0.1

0.2

0

0.1

z

L( t )

0.2

at right choicest three values

a  0 , μ  0,

m ax

(a)

~ U

2

a( t )

V

0

 50

0

errort

~

Results U ( z ), V ( z ) of restoration of searched functions

b( t)

~

F( t ) 1

θ0

Figure 19 ~

3

Errors of reconstructed functions

~ U

(  (V , θ)   (U , θ)  1.1 ( see (19), (20)), are represented in Figure 22.

~

0.5

~ U ( z ), V ( z ) (a) and errors of restoration U ( z )  U ( z ) , ~ V ( z )  V ( z ) (b), admitting only one nonzero error θ  0

~(z ) Searched solutions U (z ),V

~ V

L (z ) F (z )

Source function W(z)

~ ~

Searched solutions U(z ),V (z )

(a)

1

~

Results U ( z ), V ( z ) of restoration of searched functions

~ ~ Results U ( z ), V ( z ) of restoration of searched functions U ( z ), V ( z ) (a) from their tandem W (z ) (b), using exact values a , μ, θ , are represented in Fig. 19.

z

50

θ  0.2

~ U ( z ), V ( z ) (a) and errors of restoration U ( z )  U ( z ) , ~ V ( z )  V ( z ) (b), admitting only one nonzero error a  0 ,

1

(b)

~ V V

0.5 0

 0.5 1

~ U U  50

0

correct μ

 0t ,

50

z

a0

Figure 22

(  (U , a )   (V , a )  19 , see (15), (16)) are represented in Fig. 20.

X. IDENTICAL AND DIFFERENT SEARCHED FUNCTIONS. ASYMMETRIC SOURCE FUNCTION.

(a)

~

U

~

~

L (t ) F (t )

V

1

0  100

0

z

t

100

error a  0.5

Errors of solutions

~

Searched solutions U( z),V (z )

Figure 23-a represents the different error markers 2

e  e (a; μ; θ) , corresponding to searched Lorentz’s functions

(b)

0.4

a( t )

~ 0.2 V  V

4

2

2

U ( z )  1.2[1  0.005( z  2) ]

, V ( z )  2.2[1  0.005( z  2) ] 3 with different amplitudes (1.2 and 2.2 continuous line) and W( t )2 U ( z )  2.2[1  0.005( z  2) 2 ]

0 b ( t)  0.2

~ U U

 0.4

,

2

V ( z )  2.2[1  0.005( z  2) ]

1 ( U  V  1 ) with identical amplitudes (2.2 and 2.2, dotted 0 line). We see, that case of different amplitudes (and

 100

z

0 t

) ) gives us more sharp shape of error asymmetric  100 0 W (z100 ~ ~ ~ markers ε[a t ; μ; θ ] and more easy finding of key-numbers a , μ , θ , than the case of identical amplitudes (and symmetric W (z ) ). More great difference (Figure 23-b) in the behavior of

100

,

correct μ  0 θ  0 Figure 20

~

~

error markers ε[a~; μ~; θ ] we see in the cases of Gaussian functions with different amplitudes (1.2 and 2.2, continuous 2 line) U ( z )  1.2exp[-(( z  2)/6) 2 ] , V ( z )  2.2exp[-(( z  2)/6) ] and the pair of Gaussian functions with identical amplitudes

~

Results U ( z ), V ( z ) of restoration of searched functions ~ U ( z ), V ( z ) (a) and errors of restoration U ( z )  U ( z ) , ~ V ( z )  V ( z ) (b), admitting only one nonzero error μ  0 ,

2

2

U ( z )  2.2exp[-(( z  2)/6) ] , V ( z )  2.2exp[-(( z  2)/6) ] (2.2

(  (V , μ)   (U , μ)  1.2 , see (17), (18)) are represented in Figure 21.

and 2.2, dotted line). It is easy to see, that error marker ~; ~ ε[a~; μ θ ] is quite non-sensitive on interval 0  a  2 .

74

4

0.2

b ( t)

0.4

θ

0.4

a( t )

error μ t 0.2

0  0 .4

0.4

(a)

~

U

L (t )

1

5

0 1

Searched solutions U (z ),V (z )

And true combination of key-numbers . (35) a  2 , θ  0 , μ  U ( 0) Errors markers ε[a ; μ; θ] , ε[a; μ; θ] , ε[a; μ; θ] are pointing to ~ the values a~ , μ~ , θ closest with the parameters a , μ , θ of correct restoration are represented in Figure 18

Errors of reconstructed functions

International Journal of Engineering and Applied Sciences (IJEAS) ISSN: 2394-3661, Volume-3, Issue-3, March 2016

www.ijeas.org

About the Recognition and Reconstruction Two Unknown Functions From Known Their Tandem 6

60

(a)

~ ; μ; θ) ε( a

(b)

~ ; μ; θ) ε( a

a~ 0

by the pair (39) of Gaussian functions U ( z  a ) and U ( z  a ) with shift a in the interval (38).

a~

4 0 Figure 23

2

In other words, Gaussian function WG ( z ) can be represented

XI. CONCLUSION

4

2

Main conclusion: to separate both similar continuous smooth symmetric function of given their tandem (doing it by single way) we need only three true key-numbers, known before or founded by sorting patterns. Represented numerical results have an empirical nature and seemed sufficiently unexpected.

This is why we used many graphs in this paper.

Now we consider symmetric source function W (z ) as only one Gaussian function



W ( z )  WG ( z )  4 exp  ( z / 40)

(Figure 24-a) or only one Lorentz’s function



W ( z )  WL ( z )  4 / 1  ( z / 40 )

2

2



(36)



(37) (Figure 24-b). Let’s initiate the search (above section IV) for tandem of symmetric ( U  V  1 ) functions on the same

EFERENCES [1] [2]

carrier z  L , observing behavior of the marker of the error

[3]

~; ~ ε[a~; μ θ ] . In this case amplitudes of pairs of searched

symmetric functions can be of only identical amplitudes. In the case of Gaussian source function WG ( z ) the calculation

[4] [5]

procedure of section IV generates single pair “ U , V ” of smooth symmetric “own functions”, attached to concrete single “own number” of the “continuum spectrum of own numbers”

[6] [7]

3  a  10 .

(38)

On the other hand, Lorentz’s source function generates only ~ one pair “ U , V ”, but error marker ε[a~; μ~; θ ] does not show the key-number a sufficiently clear.

(a) 8

~ ; μ; θ) ε( a unceasing spectrum of pairs

4

<10 0

4

5

a~ 10

H. Poor, An Introduction to Signal Detection and Estimation. New York: Springer-Verlag, 1985.

Springer Handbook of Atomic, Molecular, and Optical Physics (Editors: Drake, Gordon W. F.) Springer, 2005 Oxford Master Series in Atomic, Optical and Laser Physics (Book 7), 346 pages, Oxford University Press; 1 edition (February 10, 2005) English.

Merrill Skolnik, Radar Handbook, Third Edition McGraw-Hill Education, 2008 – pages: 1328.

I.J. Schoenberg, A. Whitney, “Sur la positivite des determinants de translations de functions de frequence de Polya avec une application au probleme d`interpolation par les functions “spline,” Comptes Rendus. 1949. – Vol. 228. - P. 1996-1998.

I. J. Schoenberg, “On equidistant cubic spline interpolation” Bull.

Amer. Math. Soc. Vol. 77, N. 6 (1971), 1039-1044.

I. J. Schoenberg, On Cubic Spline Interpolation at Equidistant Nodes.

Mathematics Research Center, University of Wisconsin, 1971 – pages : 59.

Arabadzhi, Vladimir V. senior researcher, Ph.D.

(b) 10

ε( a~ ; μ; θ)

> 0 .2 0

a~

http://www.ipfran.ru/staff/Arabadzhi1.html http://www.ipfran.ru/english/staff/Arabadzhi_e.html

10

Education: Gorky State University, 1978, profession-radiophysics (01.04.03), Ph.D.–1994, topic “Investigation methods active cancellation low frequency waves”.

Professional interests: (а) searching for solutions of boundary value wave problems, which admit us to design absorbing coating of the thickness mach smaller than absorbed wavelength and without information on the direction and frequency band of incident waves. This aim is achieved by use of very small and very quick controlled elements of coating. These solutions are parametric and can’t be reduced to the combination of electric chains with parameters constant in time. (b) design of floating devises, which convert boat oscillations into a periodical sequence of one polarity stream pulses (“wave engine”).

Professional carrier: NIRFI (1978-1981),, IAP(RAS) (beginning from 1981). Engineer-senior researcher.

Figure 24 Graph of marker ε[a~; μ; θ] in Figure 24 means, for instance, that calculation procedure (described in Section IV) gives true solution U ( z ) of the equation U ( z  a )  U ( z  a )  WG ( z )

(39)

for each value a from interval (38). Taking into account the condition U ( z  a)  U ( z  a) ,

(41)

(guaranteed by Section IV), we can see, that 2U ( z  a )  WG ( z ) or 2U ( z  a )  WG ( z ) .

(42)

75

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