метода цос
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y(0) = h(0)x(0) + h(1)x(–1) + h(2)x(–2) + … = h(0)x(0) = 1·2 = 2 |
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y(1) = h(0)x(1) + h(1)x(0) + h(2)x(–1) + … = h(0)x(1) + h(1)x(0) = 1·1 + 2·2 = 5 |
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y(2) = h(0)x(2) + h(1)x(1) + h(2)x(0) + h(3)x(–1) + … = |
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= h(0)x(2) + h(1)x(1) + h(2)x(0) = 1·0 + 2·1 + 2·2 = 6 |
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y(3) = h(0)x(3) + h(1)x(2) + h(2)x(1) + h(3)x(0) + h(4)x(–1) + … = h(0)x(3) + |
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+h(1)x(2) + h(2)x(1) + h(3)x(0) = 1·0 + 2·0 + 2·1 + 1·2 = 4 |
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4 |
y(4) = h(0)x(4) + h(1)x(3) + h(2)x(2) |
+ h(3)x(1) + h(4)x(0) + h(5)x(–1) + … = |
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= h(0)x(4) + h(1)x(3) + h(2)x(2) + h(3)x(1) + h(4)x(0) = 1·0 + 2·0 + 2·0 + 1·1 + 0·2 = 1 |
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5 |
y(5) = h(0)x(5) + h(1)x(4) + h(2)x(3) + h(3)x(2) + h(4)x(1) + h(5)x(0) + |
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+ h(6)x(–0) + … = h(0)x(5) + h(1)x(4) + h(2)x(3) + h(3)x(2) + h(4)x(1) + h(5)x(0) = |
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= 1·0 + 2·0 + 2·0 + 1·0 + 0·1 + 0·1 = 0 |
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y(6) = 0 |
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y(7) = 0 |
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0 1 2 3 4 5
B"(. 1.9. B$:)2"?
15
1.3.2. '<E0:580:. ?7<-0.01.
D;:",1(8?;= ,$9'- 81;'$%(+8"$, x(nT ) " *$:)2"$% y(nT ) –
(11+#1O$#"$ 8@1'/8&@1' – ,19$+ 10"(&8:+=(? +3F2'<=2$5 7+3(2.-
20.5 (B6)
N )1 |
M )1 |
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y(n) $ ,bi xB(n ) i)T C) |
,ak yB(n ) k)T C, |
(1.14) |
i$0 |
k $1 |
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>'$ bi , ak – )1RCC"2"$#+& -*:8#$#"? (8$S$(+8$##&$ )1#(+:#+&); x(nT ), y(nT ) – 81;'$%(+8"$ " *$:)2"? (8$S$(+8$##&$ "<" )1,-
0<$)(#&$ (">#:<&);
i, k – ;#:.$#"? ;:'$*9$) 81;'$%(+8"? " *$:)2"" (11+8$+(+8$##1; N, M – )1#(+:#+&;
xB(n ) i)T C, yB(n ) k)T C – 81;'$%(+8"$ " *$:)2"?, ;:'$*9:##&$ #: i " k 0$*"1'18 '"()*$+";:2"" (11+8$+(+8$##1.
M1RCC"2"$#+& bi " ak #:;&8:A+(? (27=+.22050 @3+35.=+350
(@3+35.=+350) !G3.
G<? #1*,"*18:##1>1 8*$,$#" *:;#1(+#1$ -*:8#$#"$ (1.14) 0*"- #",:$+ 8"'
N )1 |
M )1 |
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y(n) $ ,bi x(n ) i) ) |
,ak y(n ) k) . |
(1.15) |
i$0 |
k $1 |
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!"#$%#:? '"()*$+#:? ("(+$,:, (11+#1O$#"$ 8@1'/8&@1' )1+1- *1% 10"(&8:$+(? 8 8"'$ *:;#1(+#1>1 -*:8#$#"? (1.15), 1+8$.:$+ -(- <18"?, 60F0B.<&': +.340F7.5'<=0: 0*" #-<$8&@ #:.:<=#&@ -(<18"- ?@ (1.9) *$:)2"? #$ ,19$+ 81;#")#-+= *:#=O$ 81;'$%(+8"?; ;#:.$#"? *$:)2"" 8 ):9'&% ,1,$#+ 8*$,$#" n ;:8"(?+ +1<=)1 1+ +$)-S$>1 " 0*$'O$(+8-AS"@ ;#:.$#"% 81;'$%(+8"?, #1 #$ ;:8"(?+ 1+ $>1 01(<$- '-AS"@ ;#:.$#"% ((,. 0. 1.3).
#3F2'<=2'. 7+3(2.20. (1.15) *$O:$+(? 5.=';'5 @+95': @';- <=32'(&0 0*" 274.($H 23B34-2$H 7<4'(09H (1.9), (<$'18:+$<=#1, 1#1
#$01(*$'(+8$##1 10"(&8:$+ 341'+0=5 8&."(<$#"? *$:)2"" 01 ";- 8$(+#1,- 81;'$%(+8"A " 0:*:,$+*:, !G3.
B719.7 1.2. B$O"+= *:;#1(+#1$ -*:8#$#"$ y(n) $ x(n) ) 0,5y(n )1)
,$+1'1, 0*?,1% 01'(+:#18)" 0*" ;:':##1, 81;'$%(+8"" " #-<$8&@ #:.:<=#&@ -(<18"?@
16
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x(n) $ 0,1n . |
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7*$/-$+(? 10*$'$<"+= 5 1+(.$+18 *$:)2"". |
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'.S.01. 0*"8$'$#1 8 +:/<. 1.2. |
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7:/<"2: 1 . 2 |
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D&."(<$#"$ *$:)2"" ,$+1'1, 0*?,1% 01'(+:#18)" |
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n |
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,:E/.358-1. |
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'.<6I1D |
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0 |
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x(0) = 1 |
y(0) = x(0) |
– 0,5y(–1) = 1 – 0,5·0 = 1 |
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1 |
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x(1) |
= 0,1 |
y(1) |
= x(1) |
– 0,5y(0) = 0,1 – 0,5·1 = 0,1 – 0,5= –0,4 |
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2 |
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x(2) |
=0,01 |
y(2) |
= x(2) |
– 0,5y(1) |
= 0,01 – 0,5·(–0,4) = 0,01 + 0,2=0,21 |
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3 |
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x(3) |
= 0,001 |
y(3) |
= x(3) |
– 0,5y(2) |
= 0,001 – 0,5·0,21 = 0,001 – 0,105 = |
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= –0,104 |
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4 |
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x(4) |
= 0,0001 |
y(4) |
= x(4) |
– 0,5y(3) |
= 0,0001 – 0,5·(–0,104) = |
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= 0,0001+ 0,052 = 0,0521 |
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" |
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1.3.3. '.6?751-04. 1 0.7.6?751-04. =10.304. /1567.804. 5158.94
!"#$%#:? '"()*$+#:? ("(+$,: #:;&8:$+(? +.&7+<0(2':, $(<" @1- +? /& 1'"# "; )1RCC"2"$#+18 ak *:;#1(+#1>1 -*:8#$#"? (1.15) #$ *:8$# #-<A:
ak ( 0 @1+? /& '<? 1'#1>1 "; ;#:.$#"% k.
I'+9;&'5 *$)-*("8#1% !G3 #:;&8:A+ 01*?'1) B6 (1.15), +. $. maxE*M )1+, *N )1+F.
31><:(#1 (1.15) *$:)2"? y(n) +.&7+<0(2': !G3 8 ):9'&% ,1,$#+ 8*$,$#" n 10*$'$<?$+(?:
-+$)-S", 1+(.$+1, 81;'$%(+8"? x(n);
-0*$'&(+1*"$% 81;'$%(+8"? x(n ) i), i $ 1, 2, ..., N )1;
-0*$'&(+1*"$% *$:)2"" y(n ) k), k $ 1, 2, ..., M )1 .
4*",$*& *:;#1(+#&@ -*:8#$#"% *$)-*("8#1% !G3: |
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- |
@.+('1' 01*?'): |
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y(n) $ b0 x(n) # b1x (n )1) ) a1 y(n )1) ; |
(1.16) |
- |
(='+'1' 01*?'): |
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y(n) $ b0 x(n) # b1x(n )1) # b2 x(n ) 2) ) a1y(n )1) ) a2 y(n ) 2) .(1.17)
!"#$%#:? '"()*$+#:? ("(+$,: #:;&8:$+(? 2.+.&7+<0(2':, $(<" 8($ )1RCC"2"$#+& ak *:;#1(+#1>1 -*:8#$#"? (1.15) *:8#& #-<A:
ak $ 0 , k $ 1, 2,..., M )1 .
17
G<? #$*$)-*("8#1% !G3 *:;#1(+#&$ -*:8#$#"? (1.14)–(1.15) 0*"#",:A+ 8"' (11+8$+(+8$##1
N )1 |
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y(n) $ ,bi xB(n ) i)T C; |
(1.18) |
i$0 |
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N )1 |
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y(n) $ ,bi x(n ) i) . |
(1.19) |
i$0
I'+9;'& #$*$)-*("8#1% !G3 *:8$# (N )1) .
31><:(#1 B6 (1.19) *$:)2"? y(n) 2.+.&7+<0(2': !G3 8 ):9'&% ,1,$#+ 8*$,$#" n 10*$'$<?$+(?:
-+$)-S", 1+(.$+1, 81;'$%(+8"? x(n) ;
-0*$'&(+1*"$% 81;'$%(+8"? x(n ) i), i $ 1, 2, ..., N )1.
4*",$* B6 #$*$)-*("8#1% !G3 8+1*1>1 01*?')::
y(n) $ b0 x(n) # b1x(n )1) # b2 x(n ) 2) . (1.20)
1.3.4. #+P- 1 Q+P-5158.94
B:((,1+*", 1(1/$##1(+" 05@74-<2$H H3+3&=.+0<=0& *$)-*("8- #&@ " #$*$)-*("8#&@ !G3. 3 R+1% 2$<=A 0*"8$'$, 0*",$*& 8&."(- <$#"? FN 01 ;:':##1,- *:;#1(+#1,- -*:8#$#"A, *$O:? $>1 ,$+1'1, 0*?,1% 01'(+:#18)" 0*" #-<$8&@ #:.:<=#&@ -(<18"?@.
B719.7 1.3. D&."(<"+= ",0-<=(#-A @:*:)+$*"(+")- 2.+.&7+- <0(2': !G3 8+1*1>1 01*?'):, (11+#1O$#"$ 8@1'/8&@1' )1+1*1% 10"(&8:$+(? *:;#1(+#&, -*:8#$#"$, (1.20):
y(n) $ b0 x(n) # b1x(n )1) # b2 x(n ) 2) .
'.S.01.. 31><:(#1 10*$'$<$#"A FN – R+1 +.3&809 #: 2"C*1- 81% $'"#".#&% ",0-<=( (*"(. 1.6), 01R+1,-, 8&01<#"8 ;:,$#-
'x(n) 4 u0 |
(n); |
(1.21) |
& |
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%y(n) 4 h(n), |
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0$*$0"O$, B6 8 8"'$
h(n) $ b0u0 (n) # b1u0 (n )1) # b2u0 (n ) 2)
" 8&."(<", 1+(.$+& FN ,$+1'1, 0*?,1% 01'(+:#18)" 0*" #-<$8&@ #:.:<=#&@ -(<18"?@ ((,. 0. 1.3.2):
h(0) $ b0u0 (0) # b1u0 ()1) # b2u0 ()2) $ b0 G1 # b1 G 0 # b2 G 0 $ b0 ; h(1) $ b0u0 (1) # b1u0 (0) # b2u0 ()1) $ b0 G 0 # b1 G1 # b2 G 0 $ b1 ; h(2) $ b0u0 (2) # b1u0 (1) # b2u0 (0) $ b0 G 0 # b1 G 0 # b2 G1 $ b2 ;
18
h(3) $ b0u0 (3) # b1u0 (2) # b2u0 (1) $ b0 G 0 # b1 G 0 # b2 G 0 $ 0 ; h(n) $ 0 0*" n 1 3 .
B:(0*1(+*:#?? 01<-.$##&$ *$;-<=+:+& #: FN #$*$)-*("8#1% !G3 0*1";81<=#1>1 01*?'):, ,19#1 ('$<:+= (<$'-AS"$ 8&81'&:
-",0-<=(#:? @:*:)+$*"(+"): 2.+.&7+<0(2': !G3 ",$$+ &'2.B- 27A '<"+$<=#1(+=;
-;#:.$#"? 1+(.$+18 FN *:8#& )1RCC"2"$#+:, *:;#1(+#1>1 -*:8#$#"?
h(n) $ bi , 0*" n $ i $ 0, 1,..., N )1 .
41R+1,- #$*$)-*("8#&$ !G3 #:;&8:A+ ("(+$,:," ( )1#$.#1% ",0-<=(#1% @:*:)+$*"(+")1% (J*K-<0<=.5350).
B719.7 1.4. D&."(<"+= ",0-<=(#-A @:*:)+$*"(+")- +.&7+<0(- 2': !G3 0$*81>1 01*?'):, (11+#1O$#"$ 8@1'/8&@1' )1+1*1% 10"(&- 8:$+(? *:;#1(+#&, -*:8#$#"$,
y(n) $ b0 x(n) ) a1 y(n )1) .
'.S.01.. D&01<#"8 ;:,$#- (1.21), 0$*$0"O$, B6 8 8"'$ h(n) $ b0u0 (n) ) a1h(n )1)
" 8&."(<", 1+(.$+& FN ,$+1'1, 0*?,1% 01'(+:#18)" 0*" #-<$8&@ #:.:<=#&@ -(<18"?@:
h(0) $ b0u0 (0) ) a1h()1) $ b0 ; h(1) $ b0u0 (1) ) a1h(0) $ )a1b0 ;
h(2) $ b0u0 (2) ) a1h(1) $ )a1 ()a1b0 ) $ a12b0 ; h(3) $ b0u0 (3) ) a1h(2) $ )a1 (a12b0 ) $ )a13b0 .
"
D&."(<$#"? ,19#1 0*1'1<9:+= /$()1#$.#1 01 C1*,-<$ h(n) $ ()1)n a1nb0 , n $ 4, 5, ...
B:(0*1(+*:#?? 01<-.$##&$ *$;-<=+:+& #: FN +.&7+<0(2': !G3 0*1";81<=#1>1 01*?'):, ,19#1 ('$<:+= 8&81': ",0-<=(#:? @:*:)+$- *"(+"): +.&7+<0(2': !G3 ",$$+ ,.<&'2.B27A '<"+$<=#1(+=.
41R+1,- *$)-*("8#&$ !G3 #:;&8:A+ ("(+$,:," ( /$()1#$.#1% ",0-<=(#1% @:*:)+$*"(+")1% (L*K-<0<=.5350).
1.3.5. !58:3K1-:58> =10.304T /1567.804T 5158.9
!G3 #:;&8:$+(? 7<=':B0(':, $(<" 0*" 1>*:#".$##1, 81;'$%(+-
8""
19
max x(n) " Rx ,
n
>'$ Rx – <A/1$, ()1<= ->1'#1 /1<=O1$ 01<19"+$<=#1$ ."(<1, #$
*:8#1$ /$()1#$.#1(+", " 0*1";81<=#&@, #1 1>*:#".$##&@ #:.:<=#&@ -(<18"?@ *$:)2"? +:)9$ 1>*:#".$#:
max y(n) " Ry ,
n
>'$ Ry – <A/1$ ()1<= ->1'#1 /1<=O1$ 01<19"+$<=#1$ ."(<1, #$ *:8- #1$ /$()1#$.#1(+".
1.3.6. %I.06< ?58:3K1-:581 C: 19C?=>50:3 T<7<68.715816.
3-S$(+8-A+ '8: &+0=.+09 -(+1%."81(+" !G3. K'"# "; #"@ 01- ;81<?$+ 12$#"+= -(+1%."81(+= !G3 01 $$ ",0-<=(#1% @:*:)+$*"(+"- )$ 81 8*$,$##1% 1/<:(+", '*->1% – 01 z-";1/*:9$#"A R+1% @:*:)+$- *"(+")" 8 z-1/<:(+" ((,. 0. 1.4). D&/1* )*"+$*"? ;:8"("+ 1+ -'1/(+8: $>1 0*:)+".$()1>1 "(01<=;18:#"?.
M*"+$*"%, 01;81<?AS$% 12$#"+= -(+1%."81(+= !G3 01 $$ ",- 0-<=(#1% @:*:)+$*"(+")$, C1*,-<"*-$+(? (<$'-AS", 1/*:;1,: '<? +1>1 .+1/& !G3 /&<: -(+1%."8:, #$1/@1'",1 " '1(+:+1.#1 8&01<-
#$#"? -(<18"? :/(1<A+#1% (@1'",1(+" *?':
-
, h(n) / - . (1.21:)
n$0
G:##&% )*"+$*"% -(+1%."81(+" (8"'$+$<=(+8-$+ 1 +1,, .+1 2.-
+.&7+<0(2$. !G3 (MFN-("(+$,&) (<.1;3 7<=':B0($, 01()1<=)- "@ ",0-<=(#:? @:*:)+$*"(+"): )1#$.#:.
4*$9'$ .$, '$<:+= 8&81' 1/ -(+1%."81(+" *$)-*("8#&@ !G3, *:((,1+*", 0*1(+1% 0*",$*.
B719.7 1.5. K0*$'$<"+=, -(+1%."8: <" *$)-*("8#:? !G3, ",- 0-<=(#:? @:*:)+$*"(+"): )1+1*1% 10"(&8:$+(? '"()*$+#1% R)(01-
#$#+1% (1.6)
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(1.21:), 01<-.", *?' |
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