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| The mathematical apparatus of this technology is based on Fourier and Maclaurin
series and some additional conditions which I discovered. This apparatus was developed by me during my 4-th and 5-th year as a computer
science student for my own satisfaction. When simulated, a devise based on this apparatus demonstrates incredibly correct recognition
even in an extremely harsh signal-to-noise ratio conditions. Based on this evidence I am working on manufacturing a metal detector based
on this technology. However since it will take a while before I can market the device, in interest of the public and as all knowledge belongs to God I decided to
publish the mathematical apparatus of the device, but for now I will not publish the simulating/emulating programs. Note that
this document was created before I came to England and my English may be imprecise at times. This document is a copyrighted matter. |
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Let us have the signal s(t). Let us apply the discrete Fourier transformation over it.
The discrete s (t ) d signal, which is in respect to the analog signal s(t) is:
... (1)
where:
We apply the discrete Fourier transformation over (1):
As it is well known that:
Then
The following frequency is:
... (7)
That is trivial discrete Fourier transformation in forward direction.
From the other hand if ( ) sin( ) 0s nT = W n , where 2 m/ N 0 W = p , and m and N are integer
numbers, we can make expansion over that signal by Fourier transformation or by easier
way using the formula
sin( )
= - - , (comes from Euler's formula
e jx = cos(x) + j sin( x) ):
= 1 p - - p ... (8)
using backward Fourier transformation:
we can determine that:
W = - , for k = -m ... (11)
S kW = , for k ¹ ±m ... (12)
For example if m=3 and N=5
The Metal Detector works in the case like shown above - with exactly recognizable sin waves.
From the other hand we can present the sin(x) by Maclaurin's series:
= , where 0<q <1 (15) f (x)= sin(x )
... (16)
Applying (16) in (13) ((14) and (15)) we get:
Now we can say that:
(x ) A(x) R (x) l sin = +
, ... (19)
where:
( ) ...
1! 3! 5! 7! 9!
The sign '...' in (18) and (20) depends on accuracy of the computer and is represented by l.
1/(2j) 1/(2j) 1/(2j)
-k ... -3 -2 -1 0 1 2 3 4 5 6 7 8 ... k
-1/(2j) -1/(2j)
2
The metal detector works in this way: We send the sin(x) wave with exactly recognizable
frequency. We apply FFT over received returned sin(x+ph). The ph represent the kind of
the metal object, which has reflected the wave sent by the detector.
It is clear that when we apply sin(x) to the metal detector (every computer), actually we
apply A(x) instead of sin(x). The sign ‘...’ depends on ADC and calculation accuracy.
That’s why to describe the real signal processing in digital circuits we should not use
sin(x), but A(x) in (1) (as Metal Detector reaches (10)-(12) by the way: (1)-(7)).
From the other hand from (19):
A(x) (x) R (x) l = sin - ... (21)
Applying (21) in (1), and as in our case x º t , therefore sin(x)º sin(t) for discrete signal
we get:
We apply the discrete Fourier transformation over(22):
As it is well known that: e jwt d (t nT )dt e- jwnT
The following frequency is:
That is trivial discrete Fourier transformation in forward direction.
The metal detector works with exactly recognizable frequency, i.e. the metal detector
sends (and later receives) a wave = sin(W0 n), where W0=2pm/N , where m and N are
int eger 2 , i.e. m and N are integer. Therefore we can represent the sum: ( ) å¥
in (27) by the way: (8) - (12), where k = 0,±1,±2,...,±N /2 :
Or using (28), (27) become:
where ( ). ; ( 0, 1, 2,..., / 2)
(30)
is the difference between the real ( ) W k S
and the calculated ( ) W k S
. The difference
depends on the exactness of the ADC and calculations.
Now let us look the situation when there are more sources of waves. Unswitched signal is:
s(t ) ( t) ( t ) ( t ) ( pt ) = sin w1 + sin w2 + sin w3 + ... + sin w ... (31)
Switching (31) and taking in mind (21) and the preceding explanations and conditions:
Applying forward Fourier transformation over (32):
sin . sin . ... sin sin . w . w ... (34)
Including (26)
Now for simplification all applied frequencies are exactly recognizable, i.e. T m T i i = . ,
where i m and T are integer, and T is sampling period. With other words we had
( n) i sin W , where 2 m / N 1 1 W = p , 2 m / N 2 2 W = p , ..., m N p p W = 2p / ; i,( i 1,2,...,p) m = and N
are integer numbers.
We will describe (35) by the way (8)-(12) only for k ³ 0 , as k < 0 repeat k ³ 0 only with
changed sign, and doesn’t carry additional information:
... (36)
It easy to be seen that when the number of source signal (p) incre a s) e s , S ( k W)
for
(k m , (i 1,.., p)) i ¹ = , increases too instead of being zero, by the formula
for (k m ,(i 1,.., p)) i = = decrease by the same
way. This means that p ® wrong and p ® recognition ¯ .
0 w1 w2 w3
Rl(w1) Rl(w2)
Rl(w3)
This boring characteristic of digitizing is very useful for the metal detector. By it we
could determine increasing and decreasing of sources of signal with great accuracy
in large area of noise tolerance. In the metal detector this method is applied to
recognize whether under the metal detector during calibration of the device has had
a metal object or now under it has source of inverted for that metal wave.
It is realized in the file WB.m.
Literature
1. Signals and Systems, Alan V. Oppenheim, Alan S. Willsky, Ian T. Young, Prentice-
Hall, Inc., 1983
2. Theory of signals, George D. Nenov, Technique, 1990 /Bulgarian/
3. Higher Mathematics - 3, Spas Manolov, Technique, 1977 /Bulgarian/
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