ECE  171  Winter  2011  

Homework  4  Solutions      

1. [30]  Consider  the  following  two  functions:    

! !,!,!,! =   0,2,3,4,6,7,10,11    ! !,!,!,! = (0,4,8,9,10,11,12,13)    

 a) Show   K-­‐maps   and   minimal   equations   for   each   of   the   functions.     How  many   total   gates   are  

required  (not  including  inverters)?    You  can  assume  AND  and  OR  gates  with  any  fan-­‐in  you  like.    

       

   

   

 ! !,!,!,! =  ! ∙ ! + ! ∙ ! + ! ∙ !  ! !,!,!,! = ! ∙ ! + ! ∙ ! + ! ∙ !  

 Requires  8  gates.  

 b) Now  consider  that  the  functions  are  to  be  implemented  as  part  of  the  same  circuit.      Show  K-­‐

maps   that   reduce   the   total   number   of   gates   needed.       How  many   total   gates   are   required?        Hint:  consider  product  term  sharing.  

             

   ! !,!,!,! =  ! ∙ ! ∙ ! + ! ∙ ! + ! ∙ ! ∙ !    ! !,!,!,! = ! ∙ ! ∙ ! + ! ∙ ! + ! ∙ ! ∙ !    

 Requires 6 gates.

     

5 points for correctly filled in and circled K-maps, 5 points for correct equations, 5 points for correct number of gates.

5 points for correctly filled in and circled K-maps, 5 points for correct equations, 5 points for correct number of gates.

2. [15]  Simplify  the  Boolean  function  ! !, !,!, ! =   1,3,7,11,15)+   !" 0,2,5 .    

There  are  two  possible  (equivalent)  solutions:            

   

3.  [10]  What  function  is  implemented  by  the  schematic  below?    

   

!2 = ! ∙ ! ∙ ! + ! ∙ ! ∙ ! + ! ∙ !        

5 points for correctly filled in and K-map; 5 points for correctly circled K-map; 5 points for correct minimal equation

 4. [10]  What  function  is  implemented  by  the  schematic  below?      Give  your  answer  in  SOP  form  using  

only  AND,  OR,  and  NOT.    

   

! !,!,!,! =! ∙ ! ∙ ! + ! ∙ !        

5. [10]  Draw  a  schematic  to  implement  F A,B,C,D = A ∙ B ∙ C+ A ∙ B ∙ D+ B ∙ C ∙ D  using  a  SOP  NAND/NAND  implementation  (that  is  using  only  NAND  gates  and  inverters).    You  do  not  need  to  minimize  the  function.  

 

     

6. [15]  Obtain  a  minimized  equation  for  the  function  F X,Y, Z = 0,1,3 + md(5,6,7)  then  draw  a  schematic  to  implement  the  minimized  function  in  SOP  form  using  AND,  OR  and  inverters.      Draw  a  schematic  implementing  the  function  using  only  NOR  and  inverters.  

 

 

 

5 points each for correct K-map (with circling), AND/OR circuit, NOR/NOR circuit.

7. [10]  Can  a  74AC08  output  drive  a  74AS138  input?      Why  or  why  not?      Be  specific.      Assume  both  devices  operate  at  a  supply  voltage  Vcc  =  4.5V.  

   

Yes.    The  Voh  for  the  74AC08  is  at  least  3.7V  under  the  worst  conditions,  still  greater  than  the  Vih  of  the  74AS138.    Its  Vol  is  at  most  0.44V,  less  than  the  74AS138’s  Vil  of  0.7V.  

 

10 points: 5 points for “Yes”, 5 points for correct explanation specifically referring to Voh/Vih