ECE and Homework Assignment #2

Transcription

1 ECE nd -002 Homework Assignment #2 1) Do the tions elow or eh eqution:. (,,) = ( XOR ) ( XOR ). (,,) = ( + ) ( + ) ( + ). (,,) = '( + ) + (' (' + ) + ) Note: XOR is the "exlusive-or" untion. A XOR B = AB' + A'B (produes 1 i either A is 1 or B is 1, ut not i oth re 1). Drw the logi iruit digrm tht diretly implements the untion (i.e., OR nd AND gtes should mth sum nd produt terms in the originl eqution). Construt the truth tle. From the truth tle, rewrite the eqution in nonil sum-o-produts (SOP) orm, nd then use shorthnd SOP nottion. From the nonil SOP, drw the iruit digrm with only NAND gtes. From the truth tle, rewrite the eqution in nonil produt-o-sums (POS) orm, nd then use shorthnd POS nottion. From the nonil POS, drw the iruit digrm with only NOR gtes. Whih iruit uses ewer gtes: the nonil SOP iruit or nonil POS iruit? (Note: Do not inlude inverters in your gte ount, i.e., ssume oth the unomplemented nd omplemented versions o eh input vrile re lredy ville.) () (,,) = ( XOR ) ( XOR ) = ( + )( + ) ' ' ' Cnonil SOP: = + Shorthnd SOP: =,, (2, 5) NAND-gte iruit: ' '

2 Cnonil POS: = ( + + )( + + )( + + )( + + )( + + )( + + ) Shorthnd POS: =,, (0, 1, 3, 4, 6, 7) NOR-gte iruit: ' ' ' ' ' ' SOP iruit hs ewer gtes (only 3 gtes, ompred to 7 or POS iruit). () (,,) = ( + ) ( + ) ( + ) Cnonil SOP: = Shorthnd SOP: =,, (2, 4, 6, 7) NAND-gte iruit: ' '

3 Cnonil POS: = ( + + )( + + )( + + )( + + ) Shorthnd POS: =,, (0, 1, 3, 5) NOR-gte iruit: ' SOP nd POS iruits hve the sme numer o gtes. () (,,) = '( + ) + (' (' + ) + ) ' ' ' ' (Hint: Derive truth tle ster y simpliying originl expression into sum-o-produt orm (,,) = '( + ) + (' (' + ) + ) = + + ( + ) + =

4 Cnonil SOP: = + + Shorthnd SOP: =,, (3, 5, 6) NAND-gte iruit: ' ' Cnonil POS: = ( + + )( + + )( + + )( + + )( + + ) Shorthnd POS: =,, (0, 1, 2, 4, 7) NOR-gte iruit: ' ' ' ' SOP iruit hs ewer gtes (only 4 gtes, ompred to 6 or POS iruit). 2) Do the ollowing or the iruit drwn elow. Derive the Boolen eqution orresponding to the iruit (do not simpliy the expression -- sum/produt terms should extly mth OR/AND gtes in the iruit digrm). Simpliy the Boolen eqution using only swithing lger theorems (no Krnugh Mps) -- SHOW ALL OF YOUR SIMPLIFICATION STEPS! Express the simpliied eqution in sum-o-produt orm. How mny (1) AND gtes nd (2) OR gtes re needed to implement your simpliied expression? Derive the sum-o-produt Boolen eqution tht produes the ext opposite (omplement) o the iruit drwn elow.

5 F = (X + Y + X Z)(X + Y + XZ)(YZ + W) Simpliy: (XX + XY + XXZ + XY + YY + XYZ + XX Z + X YZ + X ZXZ)(YZ + W) = (X + XY + XZ + Y + XYZ + X YZ)(YZ + W) = (X(1 + Y + Z) + Y(1 + XZ + X Z))(YZ + W) = (X(1) + Y(1))(YZ + W) = (X + Y)(YZ + W) = XYZ + WX + YZ + WY = YZ (X + 1) + WX + WY = YZ (1) + WX + WY = YZ + WX + WY 3 (2-input) AND gtes, 1 (3-input) OR gte F = (YZ + WX + WY) = ((YZ ) + (WX) + (WY)) = (Y + Z)(W + X )(W + Y ) = (Y + Z)(W + W Y + W X + X Y ) = (Y + Z)(W (1 + Y + X ) + X Y ) = (Y + Z)(W + X Y ) = W Y + X Y + W Z + X Y Z = W Y + W Z + X Y (1 + Z) = W Y + W Z + X Y 3) A voting iruit mkes deision sed on the mjority o votes. The logi reeives our inputs w, x, y, nd z. Together, these input vriles re the inry representtion o the numer o memers who vote yes out o 15 totl memers. For exmple, wxyz = 1000 indites tht 8 memers voted yes, wxyz = 0101 indites tht 5 memers voted yes, nd so on. The voting iruit outputs 1 i the mjority o memers vote yes out o 15 totl memers, nd outputs 0 otherwise.

6 Express the voting logi untion in truth tle. Express it in nonil SOP orm. Using oolen lger theorems, simpliy the logi untion so tht it uses minimum numer o gtes. (Hint: pply theorem 10) Solution: Output 1 i numer o yes votes is greter thn or equl to 8 ( mjority), i.e., 1000 through Truth Tle: w x y z Cnonil SOP orm: = WX Y Z + WX Y Z + WX YZ + WX YZ + WXY Z + WXY Z + WXYZ + WXYZ Simpliy: = (WX Y Z + WX Y Z) + (WX YZ + WX YZ) + (WXY Z + WXY Z) + (WXYZ + WXYZ) = WX Y (Z + Z) + WX Y(Z + Z) + WXY (Z + Z) + WXY(Z + Z) = (WX Y + WX Y) + (WXY + WXY) = WX (Y + Y) + WX(Y + Y) = (WX + WX) = W(X + X) = W 4) A prity iruit determines the prity o 3-it input x, y, nd z. Odd prity mens there is n odd numer o 1 s in the 3-it input (x,y,z), nd even prity mens there is n even numer o 1 s in the 3-it input (x,y,z). The prity iruit outputs 1 i the 3-it input (x,y,z) ontins n odd numer o 1 s. Express the prity logi untion in truth tle.

7 Express it in nonil SOP orm. Using oolen lger theorems, simpliy the logi untion so tht it uses minimum numer o gtes. Truth Tle: x y z Cnonil SOP orm: = x y z + x yz + xy z + xyz Simpliy: nnot e simpliied urther 5) Simpliy the ollowing expressions using only oolen lger (show ll your steps or redit).. (,,) = ( XOR ) ( XOR ) = ( + )( + ) = = +. (,,) = ( + ) ( + ) ( + ) = ( )( + ) = ((1 + + ) + )( + ) = ( + )( + ) = = + +. (,,) = ' (+) + [' (' + ) + ] = ' + ' + '('+) + = ' + '' + ' + = ' + ' +

8 6) Apply DeMorgn's theorem to ind the inverse untion ', showing your intermedite steps.. (,,) = ( XOR ) ( XOR ) = [ (( ) + ( )) (() + ( )) ] = (( + )( + )) + (( + )( + )) = ( + )( + ) + ( + )( + ) (Note: My lso simpliy irst, then pply DeMorgn s Theorem on simpliied.). (,,) = ( + ) ( + ) ( + ) = [ ( + )( + )( + ) ] = ( ) + ( ) + ( ) = + +. (,,) = '( + ) + (' (' + ) + ) = [ ( + ) + ( ( + ) + ) ] = [ ( ( + )) + ((( ( + )) + ( ))) ] = ( + + ( )) ((( + ( ))( + )) + ) = ( + + ) (( + )( + ) + ) (Note: My lso simpliy irst, then pply DeMorgn s Theorem on simpliied.)