Цифр. Элек. Уральский Тех. Универ. Хмелевский
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Transcript of Цифр. Элек. Уральский Тех. Универ. Хмелевский
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2005
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681.332.973.20273-65
:(. . ....);
., . . ..()
: .., ..
-65 . . 1.: / .. , .. . 2- ., . . -
: -, 2005. 87 .
ISBN 5-321-00516-8
, -, , -, -.
230100
. 11, . 30.
"".
681.3
32.973.20273
ISBN 5-321-00516-8 -, 2005
(. .)
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1. ..... 5
1.1..................................................................................................5
1.2. ............................................................................................7
1.3. ..............................................................................8
1.4. ()....................................................................................................11
1.5. ...............................................................................................14
1.6. ..........................................................171.6.1. ........................................................................................................191.6.2. ................................................................................................20
1.7. ........................................................................................................20
...........................................................................21
2. .......................................... 22
2.1. .......................................................22
2.2.................................................................222.2.1. ....................232.2.2. ...................242.2.3.-.................................26
2.3. ......................................................26
2.4. ..........................................27
2.5..................................................................................28
2.5.1. ..................................................................................................282.5.2. ................................................................................................282.5.3. ...............................................................................................292.5.4......................................................................................................30
2.6. , .....................................312.6.1. ...............................................................................................312.6.2. ..........................................................................................322.6.3...........................................................................332.6.4. ...................35
2.6.5. ...................................362.6.6...................................................372.6.7. -
...................................................................................................372.6.8. .................................................................382.6.9..............................................39
2.7. . -.....................................................................40
2.7.1. ...................................................402.7.2. ............................................................412.7.3. () ................................................................482.7.4. .......................................................................................492.7.5. .......................................................502.7.6..........................................51
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2.8. ............................................................................522.8.1. -.......................................522.8.2. -..................562.8.3. -.................582.8.4.-........................59
2.9. ...................................................................59
2.10. -.......................60
2.11. ..................................................................61
...........................................................................62
2.....................................................................631. ......................................................................63 2. -
...........................................................................................................................63(2) ..................66
3. ...... 72
3.1. ()............................................................75
3.2. .........................................................................773.2.1. .........................................................................783.2.2. ...................................................80
...........................................................................85
3 .....................................................................85
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2.
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...SSSS...SSX 221
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2 1/2.
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. 101101(2).
( )1045=
101101
222222
12481632012345 ,
.. ( )10012345 45212021212021 =+++++ .
, . -(2), . : 1/2, 1/22, 1/23, 1/24, 1/251/2, 1/4, 1/8, 1/16. : 2-1= 0.5, 2-2= 0.25, 2-3= 0.125, 2-4= 0,0625.
, 1101101.10111.
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2.2.2.
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101
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57(10)=111001(2)22828
0
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3. -2. , -. : 2*0.75=1.5.4. 1.5>1, -, -1.5, .. 0.5: 2*0.5=1.
5. , -1 (), -., 0.375(10)= 0.011(2).S = 2 -
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00*2
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11.50.75*2
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4.02.0*2
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4.02.0*2
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2.2.3.-
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57352
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4231
24.13100010.011001 82 =
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2.4.
S = 16. -
:
...16161616...1616 221
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0
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r
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FEDCBAi ,,,,,,9,0= .
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121110
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)10()16()10()16()10()16(
===
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- .
2.5.3.
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.
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+
11001100 = 204(10)
10001 - 3
10001 - 2
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, -
. . -
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==
.sign(Y)sign(X),
sign(Y)sign(X),sign(Z)
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, , -. -, . - , .. , .. (.. , ). , - ( ),. -X Y.
: ( -
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) ( -).
.X*Y=1101(2)*1011(2)=13(10)*11(10)= 143(10).
Pi i-.1. :
110
+
P41101
1Y=
1 11010
P30000
P4+ P311010
1 110100+
P21101
P4+P3+P21000001
1 10000010+
P11101P4+P3+P2+P1() = 143(10)10001111
2. :
110
+
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1Y=
1 01101
P21101
P1+ P2100111
1 100111+
P30000 P1+P2+P3100111
1 0100111+
P41101
P1+P2+P3+P4()=143(10)10001111
2.5.4.
, , -, , , . -, .
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-
-
-
-
-10001, .. 10001(2)=17(10)
0
0
0
0
110
011
01
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1100110011001100
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, , , . ,, .
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0
28(10)=10011100(2)+28(10)=00011100(2)
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10010
10010
10010
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, , -: , . - . , - . , -
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.
.
010+1
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1, -,
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: 1101 -5(10).
=1010
001
1001
101+
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1
0
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.
a
1
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.7(10)-3(10).
1110
.
.
011
+
1
7(10)+
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=0100
011
1110
100
+
101
, .
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(-)
(-)(-)
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1
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101
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1
1
1
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+
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1
101
.
100+ 1
0
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(-)
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-
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-2k-1+2k-1-1, k , . , --128 +127 0 255 ( ). .2.1 4- 0000 1111
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.
2.1
4 -
0000 +0 0
0001 +1 1
. . . . . . . . . .
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0111 +7 7
1000 -8 8
1001 -7 9
1010 -6 10
. . . . . . . . . .
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1111 -1 15
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1010100
1010011
0101100
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1
1
1
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100
1
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0101
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..
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1
0
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, .
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2.6.4.
(), - () () . -. 2.4.
. , - () ,
, . -. -() .
.
. , ,. :
: ;
: . -
, .
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- 58(10) 0011 1010(2)
-
- - -()
1110 1001(2)
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+0011 1010 ..
() = 35(10)1 0010 0011(2)1110 1001
.26-34.
. - 26(10) 0001 1010(2)
-
-
1101 1110(2)
-34(10)1010 0010(2)
-. , 1.
+ 0001 1010
0 1111 1000(2)1101 1110
..0..0
..0
..1..1
+11
. 2.4. : ;
-
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., :
1111 1000(2) ()
, +1000 0111
() = - 8(10)1000 1000(2)1
.-5-1.
+1011
-5(10) 1101 () 1011 ()
1 1010 (2)1111
-1(10) 1001 () 1111 ()
, :
()
1010(2) ()
+1101
() = - 6(10)1110(2)
1
. , 1.
2.6.5.
, , , .. , -
, -. , ,
.
( ) ().
, .
, -, .
. ,, -4 +3. 1 -2 .
1. .
0
2+1=3
+
1(10)001(2) = 3(10)
2(10)010(2)
011(2)
001(2)
010(2)
0
-, .
, 3.
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2. .
1
1
-3-1=-4
+
=-4(10)-1(10)101(2)111(2)
-3(10)111(2)101(2)
100(2)
111(2)
101(2)
, .
, -4.3. .().
10
2+2=4
+
= ?(10)
2(10)010(2)
2(10)010(2)
100(2)
010(2)
010(2)
, -, .. , -.
4 . -4.4. .().
0
1
-3-2=-5
+
=?(10)-2(10)010(2)110(2)
-3(10)111(2)101(2)
011(2)
110(2)
101(2)
-5 3- . +3.
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(. 2.6). - (N-).
.
.1110 .
(2)
-232
22
12
0
0111 () = -8+4+2 = -2(10)
, :
, ..
010
110
+001
1
1
1
1010(2) = -(0+21+0) = -2(10)
1
.101100 .
101100(2)() = -25+23+22= -32+8+4 = -20(10)
:
, ..
10100
01100
+10011
1
1
1
110100(2)=-(2 +2 )=-20(10)
1
2.6.8.
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11. () . -
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01. . .N-2 N-3N-1
202
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N-1 2
N-22
N-3
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. . .
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00 11, , 01 10 . . 2.6.5.
2 + 1 = 3
0011
+0001
0010
2 + 2 = 4
0110
+0010
0010
-3 - 2 = -5
1011
+1110
1101
-3 - 1 = -4
1100
+1111
1101
-, , --, . -.
2.6.9.
, , - ,(1-2 ). -, 1 . -
-128 127. , -. (16 ) -32768 32767. . - (1 23 ). -8388608 8388607, .. .
. . - . ,
. -., . ,, - . , -. -.
,
. , ( , ).
-
5/20/2018 . . . .
41/88
40
2.7. .
N -. N-. - .
. , , .
:
(), ();
() (), -:
- (|X| < 1);- ().
2.7.1.
, -. , , ( ). , . . , , - , IBM-360/370 , .
-,
32- , IBM-360 (. 2.7). - 32- , -, .
(32 ), (16 ).
. . .
- (16 ), (. 2.8), , -.
, - - (16 8 ), (32
16 ). (8 ) . , , , (1,2,4 ).
0 313021
202
12
292
30
. 2.7.
. . .
. . .
-
5/20/2018 . . . .
42/88
41
(, .. ). N -, N-1 (). , ,
N-2 0
N-1 =2
-; |X|max+1 = 2
-1
1+
0 0 0 0 . . . 0
|X|max= 1 1 1 1 . . . 1
, |X|max= 2N-1-1 0 |X|2N-1- 1.
- -2N-1. (N ) . N-:
-2N-1X2N-1-1 .
. 2.2 8-, 16- 32-.
2.2
N 8 16 32
Xmax 127 32767 2147483647
Xmin -128 -32768 - 2147483648
- , , , .
--. - , , -, . -.
2.7.2.
-, -. -
15 011314
202
12
132
14
7 0156
202
12
52
6
. 2.8. -
. . .
. . .
. . .
. . .
-
5/20/2018 . . . .
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, . -, . -, (-).
:
X = Sp*q; |q||q|1/S, .. S- , . , , 0.00726
*10-3 , 0.726
*10-5 .
. , . . 2.9- .
r S-. :- r ;- Pr ;- r .
, .
..:
001*0.010=(2 *0.25)(10)=0.5(10)qp
:
N-k-2k+11
N-1 0
P q
. 2.9. -
. . .
-
5/20/2018 . . . .
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43
p=p-1; q - 1
000*0.100=(2 *0.50)(10)=0.5(10)qp
..:
011*0.011=(23*0.375)(10)=3(10)
qp
:
p=p-1; q - 1
010*0.110=(2 *0.750)(10)=3(10)qp
, ,. :
- ;- P.-
S, 2, .. S=2W. , q , W -S=2W. , , S = 2, 16.
S>2 :
- ;- , (S=16 4 ).
.(S=16):
X=1000 * 0.0000 0101 = 168*5/16
2=5*16
6(10)
: 1/161/16
0 5(10)8(10)
. q - , .. 4 , P 1.
X=0111 * 0.0101 = 16 *5/16=5*16 (10)
: 1/16
5(10)7(10)
, S , P. -q.
q , . S, 2, q..
-
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44
, , , q. -32-, (S=16)(S=2) .
S=16 . 2.10.
q 24 . N 32 -
. (64 ) (128 ). P 7 ( ). , -k (k = 6). -
P 0 2k-1 0 63, - = (-64) (+63).
12212RPP 1kkkCM =+=+= + .
, S=16 P= 0 127., -
-.
S=2 . 2.11.
N 32 . -, N = 64 . 8. , -
P= 0...255 P = -128...+127 .
, q , , ., q 24() 56 ().
.
S ,P.
(S=2): (k=7) Xmax=21271038 .
d6d5d2d1P
283124271215811170
d , 4
. 2.10. S=16
. . .
31 02230...23
P q (23 )
. 2.11. S=2
. . .
-
5/20/2018 . . . .
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45
(S=16): (k=6) Xmax=16631076 .
S . S=16, S=2 q -24 :
S=2: 24 , ;
S=16: (N = 32) , .. q , . , . 8-:
S=1611110101
S=2111110.1010
0.0001
S=16 , d1 . . S=2 . S=16 .
(). . -ANSI/IEEE 754-1985 Standard for Binary Floating-Point Arithmetic.
: (single 32 ) (double 64 ). IEEE 754 , (79 ).
, , -
, , . : 23- f, 8--
e, s (. . 2.12).
. 2.3 .
2.3
0 < e< 255(-1)s 2e-127 1.f()
e= 0; f0 (, )(-1)s 2-126 0.f()
e= 0; f= 0 () (-1)s 0.0 ()
e= 255; f= 0 () INF ()
e= 255; f0 (, ) NaN (Not-a-Number)
( ) , . 23--24 .
31 02230
e[30:23]s f[22:0]
23
2.12.
. . . . . .
-
5/20/2018 . . . .
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: 53- f, 11-
e, s. 32- , . 2.13. x86 , , , SPARC.
. 2.4 .
2.4
0 < e< 2047 (-1)s 2e-1023 1.f()
e= 0; f0 (-1)s 2-1022 0.f( )e= 0; f= 0 (-1)s 0.0 ()
s= 0; e= 2047; f= 0 +INF ()
s= 1; e= 2047; f= 0 -INF ()
e= 2047; f0 NaN (Not-a-Number)
( ) ,
. 53--54 .(SPARC )
: 112- f,15-e, s. 32- , . 2.14. SPARC .
. 2.5
SPARC.
f[31:0]
031
63 325162
e[52:62]s
52
f[51:32]
. 2.13.
. . .
. . .. . .
127 96111126
e[126:112]s f[111:96]
112
f[95:64]95 64
f[63:32]
63 32
f[31:0]
31 0
. 2.14. (SPARC )
. . . . . .
. . .
. . .
. . .
-
5/20/2018 . . . .
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2.5
0 < e< 32767 (-1)s 2e-16383 1.f()
e= 0; f0 (-1)s 2-16382 0.f()
e= 0; f= 0 (-1)s 0.0 ()
s= 0; e= 32767; f= 0 +INF ()
s= 1; e= 32767; f= 0 -INF ()
e= 32767; f0 NaN (Not-a-Number)
(86 ) 4- : 63- f,
j, 15-e, s.86-
8-. UNIX System V Application Binary Interface Intel 386Processor Supplement (Intel ABI) , -32-, 16
, . 2.15.
. 2.6 86.
2.6
j = 0; 0 < e< 32767
j = 1; 0 < e< 32767 (-1)s 2e-16383 1.f()
j = 0; e= 0; f0 (-1)s 2-16382 0.f()
j = 1; e= 0(-1)s 2-16382 0.f ( )
j = 0; e= 0; f= 0 (-1)s 0.0 ()j = 1; s= 0; e= 32767; f= 0 +INF ()
j = 1; s= 1; e= 32767; f= 0 -INF ()
j = 1; e= 32767; f0 quiet signaling NaN
IBM S/360 -
- (HF) ESA/390 ( S/390) BFP, IEEE 754. -
3 , 87 .BFP ESA/390 , 1998 .
12 FR (FR 16). -
f[31:0]
31 0
32
f[62:32]j
6263
64
e[78:64]s
78798095
. 2.15. (86 )
. . .
. . .
. . .
. . .
-
5/20/2018 . . . .
49/88
48
, -. 8 , , 4 -HFP BFP.
HFP-26 , -
BFP-. , , - .
NEC SX-4 (1995 ) -32-, 64-. IEEE 754 (32-, 64-). , SX-4 - 128- -, PVP-Cray IBM. SX-4 , .
2.7.3. () Z=XY , |X||Y|.
:
( )zP
zyPxP
y
xxPy
P
yxP
x SqS
qqSSqSqYXZ =
===
.
:
, , -S- ,
(Px-Py), .. ; () ,
();
; .
.() Z=X+Y S = 2.
- qx
- Z
- Y
21*0.5=1 (10)
2.+
1. 0.01010
PX qX
qYPY 0.10
22*0.75=3 (10)
1.00
0.11
0.11
Y 001
X 010
3. 1.00010
- Z4.
0.10 = 2* 0.5= 4(10)
0.10
Z=011
011PZ qZ
q , .. , -.
-
5/20/2018 . . . .
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49
, , , - () ().
2.7.4.
YXZ = . :
Z=X*Y=qxSPx
*qySPy= qxqyS
(Px+Py)=qzSPz .
:
; ; ; , -
, :
=+=
==
.sign(Y)sign(X),
sign(Y)sign(X),sign(Z)
,Y*X*sign(Z)Y*XZ
, -. -, , . , .
.:
-;
;
. .
. . , -. () -
: "" "-".
, , -, . , . .
" " , -.
-
5/20/2018 . . . .
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50
.Z=X*Y=0.1101(2)* 0.1011(2)= 0.8125(10)* 0.6875(10)= 0.55859375(10).
X=110. 01Y=100. 11
1 0.0100 111
+
P10.1101
1 1 0.0110
1 P1+ P21.0011
11 1 0.1001
P20.1101
+
0.0000+
+P40.1101
1 0.1000 1111
P1+P2+P3+P41.0001 111
P1+P2+P3110.1001
P30.0000
- 0.1000 1111
, Z=0.1000(2)=0.5(10), -.
( ) , . , X, - . , , 0.1000(2). : -1, 1. -0.1001(2).
:
X Y 0, Z=0; PXPYPZ0, -, Z , qZ-
.
2.7.5.
, , N , -.
, -, , - . :
;
(); .
-
5/20/2018 . . . .
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1. . , .2. . . .X Y 1 . Z=X*Y. -
65 Z - X Y, X Y . :
X Y
00...00 000...00000...00
Z
.........
8 16 8
......... .........
...........................
11..11 11..11 111..111
65
. : S=16
(4 ). 4 . , , -
.
. X Y 16-. -
Z=X*Y. ,. 16-, .
Z = X*Y = (x15... x0)*(y15 ... y0) =
= (x15...x8000...0 + 000...0x7...x0)* (y15...y8000...0 + 000...0y7...y0) == 216(x15...x8) (y15...y8) + 28(x15...x8) (y7...y0) + 2
8(x7...x0) (y15...y8) ++ (x7...x0)*(y7...y0) .
, 8--. 8- -, 16,8,8,0 .
2.7.6.
-. , .
-
5/20/2018 . . . .
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52
2.8.
-. , - , -- ,
. . , - . -, 0...9 4- (0000 1001). 8421 (, -).
.-.
( 347 )(10)=001101000111(2-10)
( 3 4 7 )(10)011101000011
- 1 , .. 1 0 99.
-..
, , - -. , Z=X-Y X
-
5/20/2018 . . . .
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, .. 1 (j+1) .-:
1
, , Zj10
1 - 1 -
1 1 - -r3 r2 r1 r0
8 4 2 1
:
1.)r(r)r(rf 1323 == .Zj=Xj+Yj= 5(10)+7(10) , j .
Zj1100(2-10)
Yj0111
Xj0101+
. . 1 2(10)
7
5+
+1 j
j ,
10ZZ jj = . 1
, 16ZZ jj += . ,
Zj= Zj- 10(10) + 16(10) = Zj + 6(10).
.., Zj=00101 j+1
=12(10) Zj10010
6(10)0110
Zj1100+0110Zj1100
+
2. Zj=16,17,18 = (8+8, 8+9, 9+9)1 16(10).
10. -, 6.
.Zj=Xj+Yj= 8(10)+9(10)= 17(10),j .
Xj 1000
Yj 1001
Zj 1 0001 (2-10)
+
=17(10)Zj 1 0111
6(10) 0110
Zj 1 0001+
, , -
:-
f=1, ( 6(10))., -.
-
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54
.Z = X + Y = 927 + 382 = 1309.
Z=0001 0011 0000 1001(2-10)= 1309(10)
Z
Z
Y
X
2- 3-
, f=1
- 3-
- 2-
0111
001010000011
0010
1
1
1101
0000
0000
0000
0011
0110
0110
. . .. . .
1001
1001
1001
100110101100
1001
+
+
. .. .
- --.
:1. 6, .. --0110. -. -.2. -.3. Z
j 10,
(), -
Zj16. - - 8421 ., , , , .4. -, --6. :
0110(2)= 6(10);
16(10), .. 10(10)= 1010(2)...
.Z = X + Y = 132 + 57 = 189.
Z0001 1000 1001
Y 0000 0101 0111
X 0001 0011 0010
- Z
+
. -Z (6(10)):
-
5/20/2018 . . . .
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55
Z60111 1110 1111
0110 0110 0110
Z0001 1000 1001
- . Z
+
, +6 (XY). .
Z6 -,Z6. 10(10)= 1010(2):
Z=0001 1000 1001(2-10)=189(10)
Z 0001 1000 1001
Z60111 1110 1111
111
+ 1010 1010 1010
.Z = X + Y = -93(10)- 48(10)= -(93+48)(10)= -141(10).
X 0000 1001 0011+
Z0000 1101 1011
Y 0000 0100 1000- Z
. Z (6(10)):
Z0000 1101 1011
Z60111 0100 0001 - . Z
+0110 0110 0110
- 11
, +6 (XY). . - , ():
Z = -(0001 0100 0001)(2-10) = -141(10)
Z 0001 0100 0001
Z60111 0100 0001
1
1010+
.Z = X + Y = 99(10)+ 99(10)= 198(10).
Z 1 0011 0010Y 1001 1001
X 1001 1001
- Z
+
- 11
. :
-
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Z0001 0011 0010+
Z60111 1001 1000
0110 0110 0110
- . Z
6 , . , -
, , :
Z=0001 1001 1000(2-10)=198(10)
Z60111 1001 1000
Z 0001 1001 1000
1
1010+
2.8.2. -
X-Y
X + (-Y). -, . . -(, ) -.
:1. -(8421). () 6.:
- ;- 1.
, (mod) +1 . 6, -10, 16.2. (X) (Y) .3. , -, "+", .. . , .4. ,
"-", .. . (.. --1).5. . , 2 ( -), 10(10)1010(2). -. , , . , -
(X - Y = Z, |X||Y|), .. Z0 -Z . , Z - , :
-
5/20/2018 . . . .
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57
) ;) ;
) Z0 (.. . , ), Y. Y , () Z -, XY.
.Z=X-Y=49(10)-238(10)=-189(10).
Y 0010 0011 1000
X 0000 0100 1001
|Y| 6:
Y1101 1100 0111+
Y1101 1100 1000
0001
:X0000 0100 1001
+
Z 1110 0001 0001
Y1101 1100 1000
- Z
1 1 - , -
(.. ). :
Z 1110 0001 0001
Z 0001 1110 1110+
Z 0001 1110 1111
0001
- Z
. 5 :
Z 0001 1110 1111
|Z|= (0001 1000 1001) = 189(10)
+0000 1010 1010
0001 1000 1001
11
, .. ,
(-). :
Z= -(0001 1000 1001) = -189(10).Z=X-Y=143(10)-58(10)=85(10).
Y 0000 0101 1000
X 0001 0100 0011
|Y| 6:
Y1111 1010 0111+
Y 1111 1010 1000
0001
-
5/20/2018 . . . .
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58
:
X0001 0100 0011+
Z 0000 1110 1011
Y1111 1010 1000
- Z
- 1
, -(.. ).
. 3 :
Z 0000 1110 1011
Z= 1000 0101(2-10)=85(10)
+ 1010 1010
0000 1000 0101
11
2.8.3. -
--.
:1. .2. , , , .3. , , . 2, () -
. , "" "".4. , - - 6(.. 0110 , f=1).
.Z = X * Y = 25(10)* 13(10)= 325(10).X = 25(10)= 0010 0101(2-10); Y = 13(10)= 0001 0011(2-10).
"-
". . 1 P0=0. (Pi).
Y = 0001 0011 X = 0010 0101
0000 0000+
0010 0101
0010 0101 0000+
1(10)P0=0P1 (1-. .)
1=P0 + P1=P1-
0010 0101
0010 01010010 0101
+
+
3(10)P2 (2-. .)
-
5/20/2018 . . . .
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59
- , 3(10), . P2, -, P2:
0010 0101 2
(f=1). . .+
+0010 0101 2
0000 0110
+
2. 0111 0101P2
0101 0000P2 . , 20010 0101 2
0100 1010P'2 . , 2. .
, , ,
P2= 0111 0101.
,.. :
0000 0110 0000
0010 1100 0101
0111 0101
0010 0101 0000
2=P1+P2
P2
P1
. (f=1)
. . .+
+
0011 0010 01012=P1+P2
. .
: Z = 0011 0010 0101(2-10)= 325(10)., -
, f=1.
2.8.4.-
, -, . . , .
2.9.
, -, . :
:- , (-);
:- , PZ>Pmax;- , PZ
-
5/20/2018 . . . .
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60
- , -, , ;- .
2.10. - .
, -, .. -.
-. -28=256 . , - . -. , -
. , , ,, , , , -.
ASCII, - . ASCII 7 , .. 2
7=128 , :
= - 0011 1101 - 3D(16)
+ - 0010 1011 - 2B(16)
Z - 0101 1010 - 5A(16)
. . .
B - 0100 0010 - 42(16)
A - 0100 0001 - 41(16)
9 - 0011 1001
. . .
2 - 0011 0010
1 - 0011 0001
0 - 0011 0000
. -, .
. , -
(), 8- - (-8). 7- -7 , -7 ASCII.-8 8--, .
-. , -. , -, , -. , .
-
5/20/2018 . . . .
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61
8--16-, -. - Unicode, Unicode Consor-tium. Unicode Consortium , Unicode, -
, -.
Unicode ,, , . Unicode, Apple, HP,IBM, JustSystem, Microsoft, Oracle, SAP, Sun, Sybase, Unisys, ., , , XML, Java, ECMAScript (JavaScript), LDAP, CORBA 3.0,WML . Unicode - ISO/IEC 10646, . , -
. Unicode . -, Unicode -.
Unicode , -, - , . Unicode --, --. .
2.11.
- . , -, , . -
. , .
1. . .1. . 3- ., . . / . ; . .. .; .; : , 2000. 729 .2. . .2: . 3-., .. / . ; . .. .; .; : , 2000. 832.3. : .. . .: . .., 2001. 328 .4. : :
/ ., .; .: , 1999. 256 .5. : . .3-., . . / .. ;.: , 1991. 592 .
-
5/20/2018 . . . .
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6. : .. / .., .. .: -, 1991. 303 .7. -: . / .. ; .: , 1988. 368 .8. / . ; . .. -
. .: , 1984. 334 .
1. ?2. , -?3. -?4. , -?5. , ?
6. ?7. ?8. ?9. -.10. .11. .12. , -?13. .14. ? ?15. ?16. , .17. ?18. ?19. ? .20. ?21. -?22. ?23. ?
-
5/20/2018 . . . .
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2
:1 .2 ..
1. 1. , .2. . . 2.7 n-1 -, n . , -
2.7
anan-1
0 1 2 3 4
0 1,5,9,13,19 2,6,10,14,20 3,7,11,15,21 4,8,12,16,22 1,7,12,17,231 3,8,10,18,20 4,6,12,13,22 2,7,9,16,23 1,5,11,14,21 3,6,9,14,19
2 1,6,9,18,19 2,5,10,17,20 1,8,11,16,21 3,5,9,18,22 2,7,9,17,19
3 1,5,9,13,19 2,6,10,14,20 3,7,11,15,21 4,8,12,16,22 1,7,12,17,23
4 3,8,10,18,20 4,6,12,13,22 2,7,9,16,23 1,5,11,14,21 3,6,9,14,19
5 1,6,9,18,19 2,5,10,17,20 1,8,11,16,21 3,5,9,18,22 2,7,9,17,19
6 1,5,9,13,19 2,6,10,14,20 3,7,11,15,21 4,8,12,16,22 1,7,12,17,23
7 3,8,10,18,20 4,6,12,13,22 2,7,9,16,23 1,5,11,14,21 3,6,9,14,19
8 1,6,9,18,19 2,5,10,17,20 1,8,11,16,21 3,5,9,18,22 2,7,9,17,19
9 1,5,9,13,19 2,6,10,14,20 3,7,11,15,21 4,8,12,16,22 1,7,12,17,23
anan-1
5 6 7 8 9
0 2,8,9,15,19 3,5,10,16,20 4,6,11,13,21 1,8,11,17,22 2,5,12,18,23
1 4,7,10,13,20 1,6,12,14,23 2,7,11,15,20 3,5,11,18,21 4,8,10,15,22
2 1,6,10,14,22 3,7,12,16,19 4,7,12,15,23 2,5,10,13,23 4,6,9,17,21
3 2,8,9,15,19 3,5,10,16,20 4,6,11,13,21 1,8,11,17,22 2,5,12,18,23
4 4,7,10,13,20 1,6,12,14,23 2,7,11,15,20 3,5,11,18,21 4,8,10,15,22
5 1,6,10,14,22 3,7,12,16,19 4,7,12,15,23 2,5,10,13,23 4,6,9,17,21
6 2,8,9,15,19 3,5,10,16,20 4,6,11,13,21 1,8,11,17,22 2,5,12,18,23
7 4,7,10,13,20 1,6,12,14,23 2,7,11,15,20 3,5,11,18,21 4,8,10,15,22
8 1,6,10,14,22 3,7,12,16,19 4,7,12,15,23 2,5,10,13,23 4,6,9,17,219 2,8,9,15,19 3,5,10,16,20 4,6,11,13,21 1,8,11,17,22 2,5,12,18,23
2.
1. , , - , -, .2. , .. "-" "".3. .4. ,
.5. - . . 2.8 n-1
-
5/20/2018 . . . .
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64
, n . . 2.8 -, . 2.9.
2.8
anan-1
0 1 2 3 4 5 6 7 8 9
0 1 19 34 27 5 14 20 30 9 261 28 2 31 4 13 6 15 8 32 10
2 21 35 3 12 22 33 7 16 23 18
3 29 24 11 36 28 25 30 29 17 34
4 20 9 12 8 22 5 15 4 27 1
5 11 36 21 35 14 32 24 33 18 31
6 19 10 13 7 23 6 16 3 26 2
7 17 25 1 15 34 33 27 29 12 20
8 14 2 22 5 35 8 36 9 21 11
9 3 16 4 18 6 19 7 13 10 17
1. , 1-. . 2 .
2. , 2 : 4 , - 8 . . -. .
3., -.
2.91 () 2 () 3 (2-10)
-
X+Y X-Y X*Y X+Y X-Y X*Y X+Y X-Y X*Y
1 2 3 4 5 6 7 8 9 10 11
01XY
1533
1533
1533
15.3333.15
15.3333.15
15.3333.15
153331
153331
153331
02XY
3267
3267
3267
32.6767.32
32.6767.32
32.6767.32
326673
326673
326673
03 XY 1737 1737 1737 17.3737.17 17.3737.17 17.3737.17 173371 173371 173371
04XY
3063
3063
3063
30.6363.30
30.6363.30
30.6363.30
306633
306633
306633
05XY
1941
1941
1941
19.4141.19
19.4141.19
19.4141.19
194411
194411
194411
06XY
2859
2859
2859
28.5959.28
28.5959.28
28.5959.28
285592
285592
285592
07XY
2145
2145
2145
21.4545.21
21.4545.21
21.4545.21
214452
214452
214452
08
X
Y
26
55
26
55
26
55
26.55
55.26
26.55
55.26
26.55
55.26
265
552
265
552
265
552
09XY
2349
2349
2349
23.4949.23
23.4949.23
23.4949.23
234492
234492
234492
-
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65
. 2.91 2 3 4 5 6 7 8 9 10 11
10XY
2451
2451
2451
24.5151.24
24.5151.24
24.5151.24
245512
245512
245512
11XY
2553
2553
2553
25.5353.25
25.5353.25
25.5353.25
255532
255532
255532
12 XY
2247
2247
2247
22.4747.22
22.4747.22
22.4747.22
224472
224472
224472
13XY
2757
2757
2757
27.5757.27
27.5757.27
27.5757.27
275572
275572
275572
14XY
2043
2043
2043
20.4343.20
20.4343.20
20.4343.20
204432
204432
204432
15XY
2961
2961
2961
29.6161.29
29.6161.29
29.6161.29
296612
296612
296612
16XY
3854
3854
3854
38.5454.38
38.5454.38
38.5454.38
385543
385543
385543
17XY
3165
3165
3165
31.6565.31
31.6565.31
31.6565.31
316653
316653
316653
18XY
1635
1635
1635
16.3535.16
16.3535.16
16.3535.16
163351
163351
163351
19XY
1331
1331
1331
13.3131.13
13.3131.13
13.3131.13
133331
133331
133331
20XY
1872
1872
1872
18.7272.18
18.7272.18
18.7272.18
187721
187721
187721
21XY
1548
1548
1548
15.4848.15
15.4848.15
15.4848.15
154481
154481
154481
22X
Y
41
58
41
58
41
58
41.58
58.41
41.58
58.41
41.58
58.41
415
584
415
584
415
58423
XY
2281
2281
2281
22.8181.22
22.8181.22
22.8181.22
228812
228812
228812
24XY
1974
1974
1974
19.7474.19
19.7474.19
19.7474.19
197741
197741
197741
25XY
4673
4673
4673
46.7373.46
46.7373.46
46.7373.46
467734
467734
467734
26XY
3862
3862
3862
38.6262.38
38.6262.38
38.6262.38
386623
386623
386623
27XY
1451
1451
1451
14.5151.14
14.5151.14
14.5151.14
145511
145511
145511
28 XY
2336
2336
2336
23.3636.23
23.3636.23
23.3636.23
233362
233362
233362
29XY
3471
3471
3471
34.7171.34
34.7171.34
34.7171.34
347713
347713
347713
30XY
1964
1964
1964
19.6464.19
19.6464.19
19.6464.19
196641
196641
196641
31XY
4269
4269
4269
42.6969.42
42.6969.42
42.6969.42
426694
426694
426694
32XY
3568
3568
3568
35.6868.35
35.6868.35
35.6868.35
356683
356683
356683
33XY
2175
2175
2175
21.7575.21
21.7575.21
21.7575.21
217752
217752
217752
34XY
1766
1766
1766
17.6666.17
17.6666.17
17.6666.17
176661
176661
176661
-
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66
. 2.9
35XY
3552
3552
3552
35.5252.35
35.5252.35
35.5252.35
355523
355523
355523
36XY
2883
2883
2883
28.8383.28
28.8383.28
28.8383.28
288832
288832
288832
(2)
1 () 2 () 3 (2-10)
-
X+Y X-Y X*Y X+Y X-Y X*Y X+Y X-Y X*Y
XY
1833
1833
1833
18.3333.18
18.3333.18
18.3333.18
183331
183331
183331
1. , 1-. .
2 .1. Z = X+Y = 18(10)+ 33(10)= 51(10).X = 18(10)= 0001 0010(2); Y = 33(10)= 0010 0001(2).:
()X 0001 0010Y
+0010 0001
(X+Y) 0011 0011(2) = 51(10)
: Z = 0011 0011(2) = 51(10).
2. Z = X-Y = 18(10) 33(10)= -15(10).
- X = 18(10)0001 0010(2)
-
- (-)
1101 1111(2)
Y = -33(10)1010 0001(2)
-. , 1.
+0001 0010
0 1111 0001(2)
1101 1111
., :
1111 0001(2) ()
, +1000 1110
() = - 15(10)1000 1111(2)
1
: Z = 1000 1111(2)= 15(10).
3. Z = X*Y = 18(10)*33(10)= 594(10).X = 18(10)= 0001 0010(2); Y = 33(10)= 0010 0001(2).
-
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:
000000010010
100
+
P100010010
0Y=
1 000010010
P200000000P1+ P2000010010
1 0000010010+
P300000000
P1+P2+P30000010010
1 00000010010
+ P400000000
P1+P2+P3+P400000010010
0100
00000000
0000000010010
P5
P1+P2+P3+P4+P5
1
00010010
0001001010010
00001001010010
P6
P1+P2+P3+P4+P5+P6
1
+
+
000000010010 1
2
:
0000001001010010(2)= 594(10).Z =
2. , 2 : 4 , - 8 . . -. .
X = 18.33(10); Y = 33.18(10).
, 0.33(10), :
00.480.24*2
00.240.12*2
11.120.56*2
00.560.28*2
11.280.64*2
00.640.32*2
11.320.66*2
)(00.660.33*2
=
=
=
=
=
=
=
=
, 0.33(10)= 0.01010100(2), a X = 18.33(10)= 00010010.01010100(2).
X , 8 (-):
-
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68
0000*10010.011 .
qxPx
X=
X:
Px=x+5; qx- 5
0101*0.10010011.qxpx
X=
Y, 0.18(10), :
00.080.04*2
11.040.52*2
11.520.76*2
11.760.88*2
00.880.44*2
11.440.72*2
00.720.36*2
()00.360.18*2
=
==
=
=
=
=
=
, 0.18(10)= 0.00101110(2), a Y = 33.18(10)= 00100001.00101110(2).
Y , 8 (-):
0000*100001.01 .
qyPy
Y=
Y:
Py=y+6; qy- 6
0110*0.10000101.qypy
Y=
1. Z = X+Y = 18.33(10)+ 33.18(10)= 51.51(10).
- qY
- Z
- X
2.+
1. 0.010010010110
PY qY
qXPX0.10010011
0.11001110
0.10000101
0.10000101
X 0101
Y 0110
3. 0.110011100110
: Z = 0110 * 0.11001110(2)= 2
6
* 0.8046875 = 51.5(10)
PZ qZ
2. Z = X-Y = 18.33(10)+ 33.18(10)= -14.85(10).
-
5/20/2018 . . . .
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69
: Z = (-) 0110 * 0.00111100(2)= - 2
6* 0.234375 = - 15(10)
3). Z = X*Y = 18.33(10)*33.18(10)= 608.1894(10).
0101 0.10010011 ,
qXPXX= Y= 0110 0.10000101
qYPY
( -):
qX= 0.1 0 0 1 0 0 1 1
qY= 0.1 0 0 0 0 1 0 1
10010011 P1
01001001 1 1
00000000 2+
01001001 1 1+2
10010011 P3
00100100 11 1
10110111 11 1+2+3
+
4 4
00000101 1011111 4
10010011 P4
01001100 01011111 1
10011000 1011111 1+2+3+4
+
00000000 5
01001100 01011111
+
01011011 111 1
-
5/20/2018 . . . .
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70
:
0101 - PX
0110 - PY
1011 - PX+ PY
+
:
PZ = Z -1; qZ- 1
1010 0.10011110 .qZpZ
: Z = 1010 * 0.10011000(2)= 210* 0.59375(10)= 608(10).
3., -.
1. Z=X+Y= 183(10)+ 331(10)= 514(10).X = 183(10)= 0001 1000 0011(2-10); Y = 331(10)= 0011 0011 0001(2-10).
. .
0001 1000 0011 - X
0011 0011 0001 - Y+
0101 0001 0100 - X+Y
0000 0110 0000 - 2-+
0100 1011 0100 - X+Y
: Z = 0101 0001 0100(2-10)= 514(10).
2. Z = X Y = 183(10) 331(10)= -148(10).
|Y| 6:
Y1100 1100 1111
0001
Y1100 1100 1110+
:
Z 1110 0101 0010
Y1100 1100 1111X0001 1000 0011
- Z
1 1 -
+
, -(.. ).
:
Z 1110 0101 0010
Z 0001 1010 1101
Z 0001 1010 1110
0001
- Z
+
-
5/20/2018 . . . .
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71
. 5 -:
Z 0001 1010 1110
0000 1010 1010
|Z|= (0001 0100 1000) = 148(10)
0001 0100 1000
11
+
, .. , (-).
: Z = - ( 0001 0100 1000)(2-10)= -148(10).
3. Z = X * Y = 183(10)* 331(10)= 60573(10)X = 183(10)= 0001 1000 0011(2-10); Y = 331(10)= 0011 0011 0001(2-10).
" ". . 1 P0=0. (Pi).
Y = 0011 0011 0001 X = 0001 1000 0011
0000 0000 0000+
0001 1000 0011
0001 1000 0011
0000 0001 1000 0011
0001 1000 00110001 1000 0011
0001 1000 0011
+
+
+
1(10)
3(10)
P0=0
P1 (1-. .)
1=P0 + P1=P1-
P2 (2-. .)
3(10) 0001 1000 0011
0001 1000 0011
0001 1000 0011
+
+P3 (3-. .)
+
, 3
(10), -
. -P2 ( P2 = 3 ), , P2:
0000 0110 0000
0011 0000 0110
0001 1000 0011
0001 1000 0011
P'2 .
()
, 2
2
2
+
+
2, 0101 0100 10010001 1000 0011
0011 0110 0110+
P2 .
P2
, 2
2
1
-
5/20/2018 . . . .
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72
, () , ,
P2= 3= 0101 0100 1001(2-10).
, -, .. .
0000 0000 0000 0110 0000
0000 0101 0110 0001 0011
0000 0101 0100 1001
0000 0000 0001 1000 0011
2=P1+P2
P2
P1
()
1+ 2
+
+
0000 0101 0110 0111 00112=P1+P2 1+2
0000 0110 0110 0000 0000
0101 1001 1111 0111 0011
3=P1+P2+P3 (f=1)1+ 2+ 3+
0110 0000 0101 0111 00113=P1+P2+P3 1+2+3
0101 0100 1001P3
..
1
: Z = 0110 0000 0101 0111 0011(2-10)= 60573(10).
3.
, , , - ,
, ( )., , -, - - ( )., -, , - . , -- , - . , -.
(, , -), . , , -.
:- ;- ;- ;- ;
- ;- ;- +1 -1 ().
-
5/20/2018 . . . .
74/88
73
.:
- ;- ;- . , , -
() (). . ,
... ,
. , , ,
, .., .. -. , , -, , ,
. -, .
. 3.1.,
, .-
, : R20, () 1 .. - , ,, .
n -, n S
1,...,S
n, -
. ,.. -, . .
Pm
Z
.
X
P1
.
S1
(+, -, /, * ..)
. 3.1.
Sn
;
;
-
5/20/2018 . . . .
75/88
74
S1,...,Sn ( ), , P1,...,Pm, .
, -, -
-,.-
. , , (, , ), , . -:1. - , .. "". , ' ', , , .
. .
, , , - , -. -.
, , - 70- 4-5 , -. .
, , () -. :
- -;- -;- ();- ();- ();- ().
,
, -, . -, .2. , , - , -.3. , - . , . 3.2. - , -
(.. ), - - , .
-
5/20/2018 . . . .
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75
4. , - , -.
5. -
:- ;- ;- (. . 3.2) , - , , -;- , -, , , , , -.
3.1. ()
"" , (, , ) . , , , () . , - , .
, , -:
- ;
- () ;- ;- ;- ();- : , ( ), () ...
, -, , -.
. (. 3.3) -
, , . , -
. 3.2.
-
-
5/20/2018 . . . .
77/88
76
, -.
, -, . - , ,
, -.
, . - , -.
(), -.
, , -, , . , -.
. -, , - (). , , , -(). -. ,
. -. (- ). -, (). , , , -, (-).
. 3.4.
X
S
Z
P
X ; S ; Z ; P
. 3.3.
-
5/20/2018 . . . .
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77
, -. -.
3.2.
, (. 3.5) , -(+, -, /, * ..). -, -, (P1,...,Pm) .
, ,P.
. -
. () - .
. 3.4.
P
S
. 3.5.
; S (); -; ;P ()
-
5/20/2018 . . . .
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78
, -. , , .
, - , , , , .
. - , - .
.
3.2.1.
, (. 3.6), . . , -, -. ,
(- , , , -..) .
-, , -, . . -.
-().
, , ,
, , , .
. .
-
; ;
;
. 3.6.
-
5/20/2018 . . . .
80/88
79
, , -, , -, . ( -) .
. -.
- (). , -. - , . , ,
.
. 3.7.
, . -
""
""
&
&
&
1
1
1
1
1
1
1
1
1
. 3.7.
1
2
r
k
1
k
1
k
1
n
1
n
1
n
1
2
k
1
2
n
11
1'
2
2'
n
n'
-
5/20/2018 . . . .
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80
, . "". "" "". , .
(, ).
, . . , , . , , -, -, .
. -. , .
-
, - , , .
XILINX, ALTERA, LATTICE, AT&T, INTEL. , , .., EPLD FPGA.
EPLD -).
FPGA -).
- -EPLD FPGA -.
3.2.2.
, 1951., - . . 3.8.
.
. - (). (). , , , , .
, -. , .. . ( ). , (-) .
-
5/20/2018 . . . .
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81
3.2.2.1. -.
(. 3.9, ).i+1-
i. --.
() . 3.9, . i+1i. -
--2 .
-(. 3.9, ). ,
. i+2, -
i+1, i+1.
S1 Sn
P
; ; ; ; ; ;
; ; ; ; ;
P ; ()
. 3.8.
-
5/20/2018 . . . .
83/88
82
-
. , ,
, (. 3.10, ).
, - 1. --.
- , - () . , -
(. 3.10, ).(, = 1 ).
. (), . -, :
- := + 1;- , -. , := , -, := + 1.
MKi
MKi+1
MKi+2
t
B P
B P
B P
MKi
MKi+1
MKi+3
t
B P
B P
B P
1
2
3
MKi
MKi+1
MKi+2
t
B P
B P
B P
; ; 1 3 ; 2
. 3.9. : ; ;-
-
5/20/2018 . . . .
84/88
83
3.2.2.3.
, -, .. -.
1. (. 3.11, ). -, .
, -, .. , -.
(-) , , .2. (. 3.11, ). . - , , ,.
a
a
. 3.10. :- ;- ;-
a
1 L
L1
1 i nk
1 2
21
1 m
1 n
1 2 3 8
1
4
1 n
1 n
1 n
; ;
. 3.11. : ; ; ;
-
5/20/2018 . . . .
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84
-.3. (. 3.11, ). , .
- .
. 1, 2, 3 .
, .4. (. 3.11, ). . , .
. 3.11 1, 2.
00 ;01 ;10 ;11 .
264 .-
, , ..
(, . .).
3.2.2.4.
. , .
(. 3.12, ) .
(. 3.12, ) , -. -.
1
n
1
1 n
& & & &&
&
;
. 3.12. : ;
-
5/20/2018 . . . .
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85
( ), , . (), -() .
1. .2. ?3. -?4. ? ?5. ?6. .7. .8.
.9. .10. ?11. ?12. ?13. ?14. .15. -.16. .17. ?18. .19. . .20. -.21. ?
3
1. , .2. -. .3.1 n-1 , n . , .
-
5/20/2018 . . . .
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86
3.1
anan-1
0 1 2 3 4 5 6 7 8 9
01,5,9,13,17
3,6,10,14,18
4,7,11,16,20
2,8,12,15,19
3,7,9,13,17
1,6,10,13,21
4,7,9,13,20
2,5,9,15,18
1,6,10,14,18
4,7,11,14,17
1
2,8,12,
16,17
4,8,11,
13,21
1,5,9,
16,20
3,8,10,
15,18
4,7,11,
13,18
2,5,10,
16,21
3,6,10,
13,17
1,7,12,
16,21
4,7,9,
16,19
2,8,12,
14,18
23,6,10,16,20
1,8,12,16,18
4,7,11,15,19
2,6,9,15,17
1,5,9,14,18
3,6,10,15,19
4,8,12,14,19
2,8,12,13,20
3,5,10,14,21
1,5,9,15,19
32,6,9,15,21
3,7,11,13,20
1,7,11,14,18
1,5,9,13,17
3,6,10,14,18
4,7,11,16,20
2,8,12,15,19
3,7,9,13,17
1,6,10,13,21
4,7,9,13,20
42,5,9,15,18
1,6,10,14,18
4,7,11,14,17
2,8,12,16,17
4,8,11,13,21
1,5,9,16,20
3,8,10,15,18
4,7,11,13,18
2,5,10,16,21
3,6,10,13,17
51,7,12,16,21
4,7,9,16,19
2,8,12,14,18
3,6,10,16,20
1,8,12,16,18
4,7,11,15,19
2,6,9,15,17
1,5,9,14,18
3,6,10,15,19
4,8,12,14,19
62,8,12,
13,20
3,5,10,
14,21
1,5,9,
15,19
2,6,9,
15,21
3,7,11,
13,20
1,7,11,
14,18
1,5,9,
13,17
3,6,10,
14,18
4,7,11,
16,20
2,8,12,
15,197
3,7,9,13,17
1,6,10,13,21
4,7,9,13,20
2,5,9,15,18
1,6,10,14,18
4,7,11,14,17
2,8,12,16,17
4,8,11,13,21
1,5,9,16,20
3,8,10,15,18
84,7,11,13,18
2,5,10,16,21
3,6,10,13,17
1,7,12,16,21
4,7,9,16,19
2,8,12,14,18
3,6,10,16,20
1,8,12,16,18
4,7,11,15,19
2,6,9,15,17
91,5,9,14,18
3,6,10,15,19
4,8,12,14,19
2,8,12,13,20
3,5,10,14,21
1,5,9,15,19
2,6,9,15,21
3,7,11,13,20
1,7,11,14,18
1,5,9,13,17
-
5/20/2018 . . . .
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..
20.06.2005 60 84 1/16 . . . 4,99.-. . 6,7 150
--620002, , . , 19
-
620002, , . , 19
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