Control with binary code. William Sandqvist

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1 Control with binry code

2 Dec Bin He Oct DA

3 E 1.1c Deciml to Binäry binry weights: ? 2

4 E 1.1c Deciml to Binäry binry weights: ? ( )

5 E. 1.2 Binry to Deciml binry weights: ? 10

6 E. 1.2 Binry to Deciml binry weights: ? ( )

7 E 1.3c Binry/Octl/Hedeciml ? 16? 8

8 E 1.3c Binry/Octl/Hedeciml ? 16?

9 E 1.3c Binry/Octl/Hedeciml ? 16?

10 Venn-digrm in common with y together with y in common with outside y

11 E. 3.2 De Morgns theorem with Venn digrm Prove De Morgns theorem with the use of Venn Digrm.

12 E. 3.2 De Morgn

13 E. 3.2 De Morgn

14 E. 3.2 De Morgn

15 E. 3.2 De Morgn

16 E. 3.2 De Morgn Now proved!

17 (E. 5.1) How to open the code-lock? (minterm) Which buttons should be simultneously pressed in order to light up the lmp? ( open up the lock)

18 (E. 5.1) How to open the code-lock? (minterm) Which buttons should be simultneously pressed in order to light up the lmp? ( open up the lock) Answer: 4,d nd 2,h but you must simultneously void pressing b c e f g i nd k!

19 (E. 5.1) How to open the code-lock? (minterm) Which buttons should be simultneously pressed in order to light up the lmp? ( open up the lock) Answer: 4,d nd 2,h but you must simultneously void pressing b c e f g i nd k!

20 (E. 5.1) How to open the code-lock? (minterm) Which buttons should be simultneously pressed in order to light up the lmp? ( open up the lock) Answer: 4,d nd 2,h but you must simultneously void pressing b c e f g i nd k! A product-term with ll vribles is clled minterm.

21 E 3.3 Venn Digrm ) Drw Venn Digrm for thre vribles nd mrk ll truth tble minterms in the digrm. b) Minimize this function with the help of the Venn Digrm f

22 E. 3.3 Truth Tble Venn digrm

23 E. 3.3b simplified epression Orginl epression.

24 E. 3.3b simplified epression Orginl epression. Simplified!

25 Boole s lgebr rules Logicl ddition "", OR, nd logicl multipliction " ", AND, brodly follows the usul norml lgebric distributive, commuttive nd ssocitive lws (with one eception).

26 Theorems Rules

27 E. 4.1(, b, c, h) Boolen lgebr

28 E. 4.1 f c d d { fctor d} d ( c 1) d

29 E. 4.1b b b b b b c b b c b f ) ( 0 ) (

30 E. 4.1c f b b c

31 E. 4.1c f b b c ( ) b b c b b b c b ( b b) c

32 E. 4.1h f ( b )

33 E. 4.1h f ( b) { demorgn} b b

34 E. 4.4 De Morgn

35 E. 4.4 bc c bc b b c bc bc bc bc bc bc bc c b c b c b c b c b c b bc c b c b c b bc bc c b c b ) ( ) ( ) ( ) ( ) ( ) )( )( ( ) )( )( ( ) )( )( ( Duplicte!

36 Logic gtes

37 (E. 4.5) Gtes Enter the nme nd output 1/0 for the following si gte types when the input signls re s shown in the figure.

38 (E. 4.5) Gtes Enter the nme nd output 1/0 for the following si gte types when the input signls re s shown in the figure. AND 0

39 (E. 4.5) Gtes Enter the nme nd output 1/0 for the following si gte types when the input signls re s shown in the figure. AND 0 OR 1

40 (E. 4.5) Gtes Enter the nme nd output 1/0 for the following si gte types when the input signls re s shown in the figure. AND 0 OR 1 XOR 0

41 (E. 4.5) Gtes Enter the nme nd output 1/0 for the following si gte types when the input signls re s shown in the figure. AND NAND 0 0 OR 1 XOR 0

42 (E. 4.5) Gtes Enter the nme nd output 1/0 for the following si gte types when the input signls re s shown in the figure. AND NAND 0 0 OR NOR 1 1 XOR 0

43 (E. 4.5) Gtes Enter the nme nd output 1/0 for the following si gte types when the input signls re s shown in the figure. AND NAND 0 0 OR NOR 1 1 XOR XNOR 0 1

44 E. 4.7 Timing digrm nd Truth Tble

45 E. 4.7

46 E. 4.7

47 E. 4.7

48 E. 4.7

49 E From tet to Boolen equtions A combintoricl circuit with si input signls 5, 4, 3, 2, 1 nd three output signls u 2, u 1, u 0, is described in this wy: u 0 1 if nd only if either both 0 nd 2 re 0 or 4 nd 5 re different u 1 1 if nd only if 0 nd 1 re equl nd 5 is the inverse of 2 u 2 0 if nd only if 0 is 1 nd some of 1 5 is 0

50 ÖH 4.12 XOR u 0 1 if nd only if either both 0 nd 2 re 0 or 4 nd 5 re different AND XOR NOT u ( 5)

51 ÖH 4.12 u 1 1 if nd only if 0 nd 1 re equl nd 5 is the inverse of 2 XNOR AND XOR ) ( ) ( ) ( u

52 ÖH 4.12 u 2 0 if nd only if 0 is 1 nd some of 1 5 is 0 NOT AND OR NOT ) ( ) ( ) ( u u

53 Logic circuits of SoP-form All the logicl functions cn be relized by using gte types AND nd OR combined in two steps. We ssume here tht the input vribles re lso vilble in inverted form, if not then you of course inverters too. AND-OR logic, SoP-form One cn relize the gte circuit direct from the truth tble. Ech "1" in the tble is minterm. The function is the sum of these minterms. One sys tht the function is epressed in the SoP form (Sum of Products). However, there my eist simpler circuit with fewer gtes tht do the sme job.

54 E. 5.2 SoP nd PoS norml form A locic function hs this Truth Tble: Write the function on SoP norml form: Write the function on PoS norml form:

55 E. 5.2 SoP-form A logicl function hs the following truth tble. Specify the function of SoP-norml form (sum of products).

56 E. 5.2 SoP-form A logicl function hs the following truth tble. Specify the function of SoP-norml form (sum of products).

57 E. 5.2 SoP-form A logicl function hs the following truth tble. Specify the function of SoP-norml form (sum of products). f b c b c b c b c

58 E. 5.2 SoP-form A logicl function hs the following truth tble. Specify the function of SoP-norml form (sum of products). f b c b c b c b c

59 Logik circuits of PoS-form Alterntively, one cn focus on the truth tble 0s. If gte circuit reproduces the function 0's correct then of course the 1's re right to! OR-AND logic, PoS form Thus, if the function is to be "0" for prticulr vrible combintion (, b) for emple (0.0) one is forming the sum ( b). This sum could only be "0" for the combintion (0.0). Such sum is clled mterm. The function is epressed s product of ll such mtermer. Ech mterm contributes with 0 from the truth-tble. The function is sid to be epressed in the PoS form (Product of Sums).

60 E. 5.2 PoS-form A logicl function hs the following truth tble. Specify the function of PoS-norml form (product of sums).

61 E. 5.2 PoS-form A logicl function hs the following truth tble. Specify the function of PoS-norml form (product of sums).

62 ) ( ) ( ) ( ) ( c b c b c b c b f E. 5.2 PoS-form A logicl function hs the following truth tble. Specify the function of PoS-norml form (product of sums).

63 ) ( ) ( ) ( ) ( c b c b c b c b f E. 5.2 PoS-form A logicl function hs the following truth tble. Specify the function of PoS-norml form (product of sums).

64 nd Π SoP nd PoS-forms re usully simplifies to list of the included mtermerm s / mintermerm s seril number: f(,b) m(1,2) f(,b) ΠM(0,3)

65 E. 5.3 SoP nd PoS -form A minimized function is given on SoP form (Sum of Products). Specify this function with minterms on SoP norml form, nd with mterms on PoS (Product of Sums) norml form.

66 E. 5.3 ) )( ( (0,7) ),, ( 6) 5, 4, 3, 2, (1, 011,100,101,110) 010, (001, ),, ( ) ( ) ( ) ( ),, ( z y z y M z y f m m z y f yz yz yz yz yz yz z y y yz z z y z yz y z y f

67 complete logic NAND-NAND OR AND nd NOT could be produces with NAND gtes. For logic functions on the SoP form, you cn chnge the AND-OR circuit to NAND-NAND "stright off". The cost, the number of gtes, will be the sme!

68 complete logic NOR-NOR OR AND nd NOT cn lso be produced with NOR gtes. For logic functions on the PoS form, you cn replce the OR- AND circuit to NOR-NOR "stright off". The cost, the number of gtes, will be the sme!

69 E. 5.5 NAND-gtes Chnge to NAND gtes!

70 E. 5.5 Chnge to NAND gtes Algebriclly: b c &? & b c b c b c

71 (E. 4.11) Europen nd Americn Symbols Try out yourself

72 (E. 4.11) Europen nd Americn Symbols Try out yourself

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