: (), .
1.
-, -, ( , , . .) , .
() (. 1, ) (. 1, ) (. 1, ), (, , , ) . - (-1), .
. 1
( ), ( ) . . , .
(. 2) ( k- kF kF+F kF-F, F - , F - ). F .
.2
, .
() , . , , . . k=1,2,...,n-1, .
, () , , .
1.1.
n , n - 1 .
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, , M=2n - 1.
, , Ui ti . (n-1) [1]:
k=1, 2,..., n-1. (1)
, , .
(1) . , , , /ti , , 2. (1) . , U1. (1) Ui . .
:
;
;
.
1.2.
, Uo, Ui, ti.
n , , n - 1 , .. n . , - .
, n - , , , .
, , M=2n + 1.
Ui ti .
t, i- ti= it. .
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[1]:
(2)
, , , .. n, , . . n+1.
2.
2.1.
. ( ) 2.3.
2.1.1. ( ):
n = 3 - ;
T = 1 - ( );
U1 = 1 - ;
t1 = T/4 - ;
t2 = T/8 - ;
t3 = T/16 - .
MathCAD . , :
u1 = U1, u2 = 6, u3 = -4.5.
( ) MathCAD Given find:
Given
u1 sin(p t1/T) + u2 sin (p t2/T) + u3 sin (p t3/T) = 0,
u1 sin(p 2t1/T) + u2 sin(p 2t2/T) + u3 sin(p 2t3/T) =0,
u = find(u2, u3) - , ,
-8.886
u = - .
13.807
, :
U1 = U1 = 1, U2 = u0 = -8.886, U3 = u1 = 13.807.
2.1.2. ():
n = 2 - , ;
T = 1 - ( );
A0 =1 - () ;
t = T/8 - ;
a0 = A0, a1 = -7, a2 = 8 - .
:
Given
a0 + 2 (a1 + a2) = 0,
a0 + 2 (a1 cos(2 p 1 t/T) + a2 cos(2 p 2 t/T)) =0,
a = find (a1, a2).
-0.707
a = - .
0.207
:
A0 = a0 = 1, A1 = a0 = -0.707, A2 = a1 = 0.207.
2.2. MathCAD
.
2.2.1. .
k ( 2, MathCAD fft(x) - . , K:=512 k:=0.. K-1.
MathCAD : dt1:= T1/K - ,
t1k:= dt1×k - MathCAD.
. :
s1 k:= if (t1 k £ 0, 0, if (t 1k £ t1, U1, 0)) - ;
s2 k:= if (t1 k £ (t1-t2)/2, 0, if (t1 k £ (t1-t2)/2 +t2, U2, 0)) - ;
s3 k:= if (t1 k £ (t1-t3)/2, 0, if (t1k £ (t1-t3)/2 + t3, U3, 0)) - ;
s k:= s1 k + s2 k +s3 k - .
2.2.2. .
MathCAD : dt2:= T2/K - ,
t2k:= dt2×k - MathCAD.
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:
S0 k:= if (t2 k £ n × t, 0, if (t2 k £ (n+1) × t, A0, 0)),
n - , , t , - t .
:
S j1 k:= if (t2 k £ (n- j)× t, 0, if (t2 k £ (n+1 - j)× t, A j, 0)) - c ,
S j2 k:= if(t2 k £ (n+j)× t, 0, if (t2 k £ (n+1+ j)× t, A j, 0)) - ,
j = 1, 2,..., n - ,
A j - ,
S j k:= S j1 k + S j2 k - j- ,
S k:= S0 k + S1 k + S2 k +... - .
: , , : 1, dt1, t1k, T2, dt2, t2k.
2.3.
2.3.1.
t | T, | U1 0, | ||
0, 1 0, 1, 2 0, 1 0, 1, 2 0, 1 0, 1 1, 2 1, 2, 3 2, 3 1, 2, 3 | 16:1 20:1 8:1 16:1 10:1 12:1 (2:4:16):1 (2:4:8:32):1 (2:4:16):1 (2:4:8:16):1 (3:10):1 (4:6):1 | 2.5 |
1...6 , 7...12 - .
2.3.2.
. 2.3.1 | . 2.3.1 | ||
1, 7 2, 11 3, 9 4, 12 5, 8 6, 10 | 1, 10 2, 12 3, 7 4, 11 5, 8 6, 9 |
3.
3.1. MathCAD , , , . , . 2.1, 2.2.
3.2. , MathCAD fft(x), x, 2:
3.2.1. n:= 0.. K/2;
3.2.2. :
srektr1:= fft(s)
:
spektr2:= fft (S);
3.2.3. :
mod1 n:= | spektr1 n | mod2 n:= | spektr2 n |;
3.2.4. :
f n:= n / T.
3.3. .
3.4. , .
:
s1:= fft(s1), s2:= fft(s2), s3:= fft(s3).
:
S0:= fft(S0), S1:= fft(S1), S2:= fft(S2).
3.5. , , :
sum1 n:= s1 n + s2 n + s3 n,
sum2 n:= S0 n + S1 n + S2 n,
1 n:= | sum1 n |,
2 n:= | sum2 n |.
3.6. .
4.
4.1. .
4.2. .
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4.3. MathCAD.
4.4. .
4.5. .
4.6. .
4.7. , .
5.
5.1. ?
5.2. , .
5.3. , - ?
5.4. :
;
?
5.5. , n :
;
?
5.6. , , n ?
5.7. ?
. ., . ., . . / . . . . .: , 1996. . 76 - 87.
充...1
1.
腅..1
2.
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