"" (signal, . signum ) . :
, - , , ;
, - - (, , , ..) , - ;
, , , ..
. , , - , , , , , .
(data) (information). , , .
, , , . ( . datum ), , , - , , , [2]. , - (). "" , (, , .) - (, , , , .). , - , , , . (, , , , ), - , . , .
, , , . - U(t).
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, .
. 2.1. |
- , () , - ,
() , , .
: , [2]. (, ..) (, ), , , .
2.2 [2], [5]
, , - (. 2.2).
. 2.2. .
, , , .
. , , , , - , , , , .. [5]
. - , , .. , , , ..
, , [5].
, . . , .
, , . , , . .
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, . , , , .
. () , . , .. .
, () . , . : ( ). , ( ), () [5].
2.3 [2],[5]
, , , s(t), - .. (. 2.1 2.2.)
. 2.3. .
, - , , , , . .
, . , , . .
2.4 [2],[3],[5],[6]
, - . , () s(), y(t) ..[1] ( ). , . .
2.5 [2],[3],[5],[6]
( ) f(t1, t2,tn) , [2],[6]. , , . , , - , ( ), . , , .
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f(t1, t2,tn) , [3].
() x xi: x=Sxi. ( -), sin(x)/x ( -), ( ), (-) [7].
, Â, (), .. , () {vi}, fÎÂ {vi} [5]
f=c1v1+c2v2++cnvn+ (2.5.1)
{i} - , vi - , .
f :
i=<f,vi> (2.5.2)
(1) :
f=Si(<f,vi>vi) (2.5.3)
, - , .. , f(t1, t2,tn) . , . , . , .
, , . , . - . () , - , [6].
, , , , [5]. , . , . , , ( ) [5].
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(. 2.5 2.12).
2.6 [2]
. : (.2.4).
. 2.4. .
: .
. s(t) = s(t + kT), k = 1, 2, 3,... - , - , .
. 2.5. |
( ), :
s(t) = A×sin (2pft+f) = A×sin (wt+f), (2.6.1)
s(t) = A×cos(wt+j),
, fo, wo, j, f - , : - , f - , w = 2pf - , j f- . T = 1/f = 2p/wo. j = f-p/2 . f ( t = 0).
:
s(t) = An sin (2pfnt+jn), (2.6.2)
s(t) = y(t kTp), k = 1,2,3,..., - y(t), . fp =1/Tp . (f=0) ( - ) An jn, , fp. , fp, , , . , - ( ).
. 2.6 , ( 0) . :
s(t) = Ak×cos(2×p×fk×t+jk),
: Ak = {5, 3, 4, 7} - ; fk = {0, 40, 80, 120} - ; jk = {0, -0.4, -0.6, -0.8} - ; k = 0, 1, 2, 3. 40 .
. 2.6. .
. 2.7. .
( ) . 2.7. , s(t), , .
, f = 1/. , Df = fp:
s(t) = (ak cos 2pkDft + bk sin 2pkDft), (2.6.3)
ao = (1/T) s(t) dt, ak = (2/T) s(t) cos 2pkDft dt, (2.6.4)
bk = (2/T) s(t) sin 2pkDft dt. (2.6.5)
K = kmax fmax , fmax < Kfp. fmax ¥, s(t).
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:
s(t) = Sk cos (2pkDft-jk),
Sk = , jk = argtg (bk/ak). (2.6.6)
. 2.8. ().
() . 2.8. t=0, , jk .
( , , ..), . , , , , ( ). :
, , :
(1/) s(t) dt.
- :
(1/) s(t) dt.
- :
(1/) |s(t)| dt.
- :
.
. .
. 2.9. .
. ( ), , , ( ) , . , , 2fo 3.5fo (2/3.5 ) 0.5fo, 4 7 . fo, , 2/ . , , . ( ), .
. . 2.10 , (0, ¥):
s(t) = exp(-a×t) - exp(-b×t),
a b , a = 0.15, b = 0.17.
. 2.10. .
. 2.11. .
, , , . , , , . , . 2.11, .
[0, ¥]. , (1.3) Df 0 kDf f.
s(t) = (a(f) cos 2pft + b(f) sin 2pft) df = S(f) cos(2pft-j(f)) df. (2.6.7)
a(f) = s(t) cos 2pft dt, b(f) = s(t) sin 2pft dt, (2.6.8)
S(f) = , j(f) = argtg (b(f)/a(f)). (2.6.9)
a(f), b(f) S(f) , . (2.6.8-2.6.9) , (2.6.7) .
[0, ], , , .. , (2.6.3-2.6.6).
. . 2.12.
. 2.12. .
:
s(t) = u(t) cos(2pfot+jo).
cos(2pfot+jo) , u(t) . fo, . , .
: () .
( ) :
|s(t)|2dt < ∞.
, , 2- 1- []. , , 2- , , . .
, ( ). .
2.7 [2]
, , . , . () , . :
) ;
) .
. . , . , .
3 3. [3]
, .
. 3.1. . |
(analog signal) , .. . , , , , , () . : y(t) = 4.8 exp[-(t-4)2/2.8] (. 3.1), , , y1 £ y £ y2, t1 £ t £ t2. - , . , - , , .
(discrete signal) (. 3.2) , .
. 3.2
() (samples) y(nDt), y1 £ y £ y2, Dt - ( , sample time), n = 0, 1, 2,...,N. , : f = 1/Dt, (sampling frequency). (sampling) , , nDt.
, . 3.1, . 3.2.
. 3.3.
(digital signal) . yn = Qk[y(nDt)], Qk - k, , , - . , , (discrete series) - Dt = const, .
. , ( ) .
4 [2],[3]
, , .
(discretization) (), , . ( ), s(t) Þ s(nDt), s(nDt) s(t) t = nDt, n = 0, 1, 2,..., N. , , . , , , , s(t) Þ s(tk), k=1, 2, , K, . () .
- (; Analog-to-Digital Converter, ADC) s(tn) s(n) = sn s(tn), n = 0, 1, 2,.., N, , . (quantization), (quantization error, quantization noise).
.