. , s(t), - S(f). s(t) F ( Dt = 1/F) s(t) Dt(t) = d(t-kDt):
sDt(t) = s(t)×Dt(t) = s(t) d(t-kDt) = s(kDt)d(t-kDt). (4)
Dt(t) Û (1/T) d(f-nF) = FF(f), (5)
- sDt(t):
SF(f) = S(f) * F×F(f). (6)
, :
SF(f) = F×S(f) * d(f-nF) = F S(f-nF). (7)
, F, F×S(f) s(t) -fN fN, fN = 1/2Dt = F/2. fN ( wN = p/Dt) . SF(f) .
, ( ) Dt = 1, -0.5 £ f £ 0.5, , , -p £ w £ p. s1(t) = exp(-a|t|) s2(t) = exp(-bt2) ( ) . 5.2.1 5.2.2.
. 5.2.1. . . 5.2.2. .
, , ( ), , fmax (fmax £ fN = F/2). , :
F = 1/Dt ³ 2fmax, (8)
, S2(w) . 5.2.2.
, fmax . . (8) , . . 5.2.2 , s1(t) , , s1(t) s2(t) ( ).
(8) . 5.2.3. .
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1. . , , ( , ).
2. . , , , , , ( ). .
. 5.2.3. .
3. . , ( ). () (aliasing). "" ( ), . 5.2.2 S1(w), . ( ) , .
-. (7) , . , , . (6) F(f), 1 [-F/2,F/2] , , F×S(f) :
F×S(f) = F×[S(f) * F(f)]×F(f). (9)
. (9):
F[S(f) * F(f)] Û sDt(t), F(f) Û F×sinc(pFt).
F×s(t) = sDt(t) * F×sinc(pFt).
s(t) = sinc(pFt) * s(kDt)d(t-kDt),
sDt(t) = s(kDt)d(t-kDt) , Dt, , s(t) kDt. h(t)= sinc(pFt)= sin(pFt)/pFt . ,
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d(t-kDt) * sinc(pFt) = sinc[pF(t-kDt)],
, :
s(t) = s(kDt) sinc[pF(t-kDt)] = s(kDt) sinc[p(t/Dt-k)]. (10)
-. , s(t) , s(kDt), k = 0,1,2,..., . . (sampling teorem).
.., 1908-2005. , . 1931 . 1931 . 1941 . . 1933 . , . (1941-1945 .) . 1948 . 1953 . . 1953 . 1954 - . , . , . , , " " I .
, (10) (2) v(t, kDt)= sinc(pF(t-kDt))= sinc(p(t/Dt k)), s(t). :
v(t,nDt) v(t,mDt) dt = .
(10) , , . 5.2.4. sinc[pF(t-kDt)] t= kDt, s(kDt), sinc[pF(t-(kj)Dt))], j= 1,2, . t= nDt k :
sinc[pF(nDt-kDt)] º Dt(t).
, , , . t , t, .
. 5.2.4. .
. 5.2.5. . |
, , , . (. . 5.2.5). . 5.2.4 , , , , , , , . (10) .
. 5.2.6. . |
(10) , :
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s(t) = s(kDt) sinc[pF(mt-kDt)].
:
s(nDtnew) = s(kDt) sinc[pF(nDtnew-kDt)].
. 5.2.7.
. 5.2.7. -.
. 5.2.8 , .
, . () , , 0-2fN. , (8), ( , fN).
S1 (8). 2 , 2 0.5fN , . S1a S1b, ( 0-2fN, fN ), 0-2fN . , . 0-2fN , S1 . , ( . 5.2.2 S1), , S1 ( ).
, , s2. , , , .
. 5.2.8. .
, , , , . , .. . , 2-4 , .
, . , . , , . - . ( ) , . . 5.2.8 - S2(f) s2(t), sd(kDt) ↔ S2(f). 0-fN SM(f) sm(mDm) 1.5fN- fN, 2 S2(f), 0.5fN 1.5fN. () .
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. , . r :
s(kDt) = (1/r) s(t) dt. (11)
:
sDt(t) = (1/r)[s(t) * r(t)]Dt(t). (12)
:
SF(f) = [S(f)×sinc(pfr)] * F×F(f). (13)
, S(f) S(f)×sinc(pfr), . -, s(t) s'(t) = s(t) * r(t)/r, h(t) = r(t)/r, .. "" r.
r=lDt, l£1, F=2afmax, a³1. : H(f) = sin[(pl/2a)(f/fmax)]. 3%, : sinc(pl/2a)³0,97. a=1 , l l£0.27, .. 27 % .
, (11) r . r, , (12) r/2, wr/2 ( (5.2.10) exp(-jpfr)).
. , , . Df = 1/Df. ( ) :
Df £ 1/T. (14)
, 0 , -/2 /2.
-.
( ) , 0- 0 fmax. , , :
Dt = 1/2fmax, Nt = T/Dt. (15)
Nt . , , .. fmax £ 1/2Dt.
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1/Dt = 2fN. :
Df = 1/T = 1/(DtNt), fN = 1/2Dt, (16)
Nf = 2fN/Df = Nt. (17)
- (), , , 2fN ( -fN fN). . 1/Df = T, :
Dt = 1/2fN, T = 1/Df, (18)
Nt = T/Dt = Nf. (19)
s(kDt) Û S(nDf), S(nDf) Û s(kDt), Nf = Nt . (15-19) . N , .
. 5.2.9. |
( ), , .. , , . , , . , -. . 5.2.9.
s(kDt) 5.2.9 ( ) s1(t) . 5.2.1, Dt = 1. , , (. 5.2.2). s(kDt) Þ S(nDf) S(nDf) 5 s(kDt), .. Nf = 5Nt. S(nDf) Þ z(kDt), (18-19), 5 (Dt = 0.2). . 5.2.9 ( z(kDt)). s(kDt) - Dt = 0.2. , (8), , .. .
. (8) , , , , . , , . , .
, . , . , , . , W , . :
|s(t)|2 dt = k |s(t)|2 dt, (20)
|S(w)|2 dw = k |S(w)|2 dw, (20')
k- () , , , 0,9 0,99.
, s(t) [0, ] . s(t) T(t):
sT(t) = s(t) T(t).
ST(f) sT(f) :
ST(f) = S(f) * ×sinc(pfT). (21)
ST(f) , sinc(pfT). , sT(t) , .. . , ST(f) [-W,W]:
S'T(f) = ST(f)×2W(f),
, , ST(f) :
|ST(f)| 1/W, f Ï (-W,W). (22)
s(t) s'(t), S'T(f), .. sT(f):
eT(t) = sT(t) s'T(t).
, :
s2 = e2T(t) 1/W, s 1/ . (23)
. 5.2.10. |
, , sT(t)s'T(t). . 5.2.10. , .
, , . (23) . W sT(t) , F = 2W N=TF=2WT.
.
, , , . , , , .. sinc(pfT) (21) . - . H(f) - H(f) = r(f) h(t) Û H(f) . hT(t) = h(t)T(t) ( ): HT(f) = r(f) * T(f) Þ r(f)×× sinc(pfT), - 9% .
, , T(t) Û ×sinc(pfT), , . , , . , , ( ).
. ST(f) = S(kDf) sT(t) (Df = 1/T). S(f) s(t), , T Þ ¥. :
sT(t) = s(t) * T(t).
:
ST(f) = (1/T)×S(f)×1/T(f) = S(kDf),
ST(f) = (1/T) S(f)d(f-k/T). (24)
, , .
, s(t) sT(t) T(t):
s(t) = sT(t)×T(t).
:
S(f) = T×ST(f) * T(f) = S(kDf)×sinc[pT(f-k/T)], (25)
.. ( - ).