. , , = b-a (, -, 1- ), :
s(t) = Sn exp(jnDwt), Sn = S(nDw), Dw = 2p/T, (1)
Sn :
Sn = (1/T) s(t) exp(-jnDwt) dt. (2)
exp(jnDwt) , . S(nDw) - s(t). , .. n , : Dw = 2p/ ( Df = 1/T). n = 1, w1 = 1×Dw = 2p/T ( f1 = 1/T), ( ), nw1 n>1 . S(nDw) n .
exp(jnDwt), -¥ < n < ¥ L2[a,b] - , Sn (2) s(t) . , s(t) (1) L2[a,b], Sn exp(jnDwt). (2)
exp(jwt) = cos(wt) j×sin(wt)
:
Sn = (1/T) s(t) [cos(nDwt) - j sin(nDwt)] dt = n - jBn. (3)
An ≡ A(nDw) = (1/T) s(t) cos(nDwt) dt, (4)
Bn ≡ B(nDw) = (1/T) s(t) sin(nDwt) dt. (5)
. 4 ( (1-3.3), =40) . , A(nDw) = A(-nDw), A(nDw) (4) cos(nDwt) = cos(-nDwt). B(nDw) = -B(-nDw), (5) sin(nDwt) = - sin(-nDwt).
. 4. .
(3) . , :
Sn = Rn exp(jjn), (3')
Rn2 ≡ R2(nDw) = A2(nDw)+B2(nDw),jn ≡ j(nDw) = arctg(-B(nDw)/A(nDw)).
|
|
. 5. .
R(nDw) - , ( j(nDw)) - . : R(nDw) = R(-nDw), : j(nDw) = -j(-nDw). , . 4, . 5. 2p ( -p -2p).
s(t) , B(nDw) (5) , .. s(t)sin(nDwt) . , . , s(t) (nDw) ( ) . . . 6() , . 6() . , .
. 6. .
n = 0 = 0, :
S0 ≡ Ao ≡ Ro ≡ (1/T) s(t) dt.
2.5. .
( , S0), :
s(t) = +2 (An cos(nDwt) + Bn sin(nDwt)), (6)
s(t) = +2 Rn cos(nDwt + jn). (6')
An, Bn (4-5), Rn jn - (3').
(6) s(t) ( ) , (.. 2×An, 2×Bn) , nDw. ( nDw) . . 4, , ( , , (6), ). ( ). (6'). - , . (. . 5) . 0 p, p/2.
. , () ( ).
|
|
7 N = 8 ( , n = ws/Dw), N = 16 ( ) N=40 ( ). , , . . , . . ( ) ( ), , .
. , .
, (, , ..) (a,b), . , (1-6) ( ) = b-a. , , . 8. , . . ( ) . .
. 7. ()
. 8.