SOP POS.pdf
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Transcript of SOP POS.pdf
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SOP vs. POS
A sum of products (SOP) expressioncontains:
Only OR (sum) operations at theoutermost level
Each term that is summed must be
a product of literals If any input to OR gate is true then
f is true
Aproduct of sums (POS) expressioncontains:
Only AND (product) operations atthe outermost level
Each term must be a sum of
literals If any input to the AND gate is
false then f is false
f(x,y,z) = y (x + y + z) (x + z)f(x,y,z) = y + xyz + xz
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Minterms vs. Maxterms
A minterm is a special product ofliterals, in which each input variableappears exactly once. Each minterm is true for exactly
one combination of inputs:
A maxterm is a sum of literals, inwhich each input variable appearsexactly once.
Each maxterm is false for exactly onecombination of inputs:
Minterm Is true when Shorthand
xyz x=0, y=0, z=0 m0xyz x=0, y=0, z=1 m1xyz x=0, y=1, z=0 m2xyz x=0, y=1, z=1 m3xyz x=1, y=0, z=0 m4xyz x=1, y=0, z=1 m5xyz x=1, y=1, z=0 m6
xyz x=1, y=1, z=1 m7
Maxterm Is false when Shorthand
x + y + z x=0, y=0, z=0 M0x + y + z x=0, y=0, z=1 M1x + y + z x=0, y=1, z=0 M2x + y + z x=0, y=1, z=1 M3x + y + z x=1, y=0, z=0 M4x + y + z x=1, y=0, z=1 M5x + y + z x=1, y=1, z=0 M6
x + y + z x=1, y=1, z=1 M7
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Billy's Burger
Billy will eat a burger with mustard or a burgerwithout mustard, but will not eat mustardwithout a burger. (trivially Billy cannot eat
without either)
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Billy's Burger Truth Table
Billy will eat a burger with mustard or a burgerwithout mustard, but will not eat mustardwithout a burger. (trivially Billy cannot eat
without either)
b m e
0 0 0
0 1 01 0 1
1 1 1
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Billy's Burger - m/M
Billy will eat a burger with mustard or a burgerwithout mustard, but will not eat mustardwithout a burger. (trivially Billy cannot eat
without either)
b m e m/M
0 0 0 M0
0 1 0 M11 0 1 m3
1 1 1 m4
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Billy's Burger SOP vs. POS
E = bm'+bm
True if a burger and nomustard
E = (1*1)+(1*0)E = 1 + 0 = 1
True if a burger andmustard
E = (1*0)+(1*1)
E = 0 + 1 = 1
Otherwise false
E = (b+m).(b+m')
False if no burger andno mustard
E = (0+0)(0+1)E = (0)(1) = 0
False if no burger andmustard
E = (0+1)(0+0)
E = (1)(0) = 0
Otherwise true
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Minimal Sum of Products
Minimal sum of products (MSP) form:
There are a minimal number of product terms in theexpression.
Each term has a minimal number of literals. Circuit-wise, this leads to a minimal two-level
implementation.
What is the MSP for Billy's burger?
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Minimal Sum of Products
Minimal sum of products (MSP) form:
There are a minimal number of product terms in theexpression.
Each term has a minimal number of literals.
Circuit-wise, this leads to a minimal two-levelimplementation.
What is the MSP for Billy's burger?
Logically we can see that Billy will eat a burger with orwithout ketchup and we can use Boolean Algebra to easilyprove it
E = bm'+bm = b(m'+m) = b(1) = b
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Billy's Burger Getting Complicated
This is just two variables. What if we want tooptimize a function to make Billy a burger, buthave more toppings?
Ketchup Pickles
Lettuce
Tomato
etc.
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Karnaugh Map
The Karnaugh map (K-map for short), Maurice Karnaugh's 1953refinement of Edward Veitch's 1952 Veitch diagram, is a method tosimplify Boolean algebra expressions.
takes advantage of humans' pattern-recognition capability
In a Karnaugh map the boolean variables are transferred andordered according to the principles ofGray code
Once the table is generated and the output possibilities aretranscribed, the data is arranged into the largest possible groupscontaining 2^n cells (n=0,1,2,3...) and the MSP is generated
through the axiom laws of boolean algebra.
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Billy's Toppings
We know Billy will eat a burger plain or with justmustard so we will concentrate on four toppings
Mustard(m), Ketchup(k), Special Sauce(s), and
Pickles(p)
k m p s e k m p s e
0 0 0 0 1 1 0 0 0
0 0 0 1 1 0 0 1
0 0 1 0 1 0 1 00 0 1 1 1 0 1 1
0 1 0 0 1 1 1 0 0
0 1 0 1 1 1 0 1
0 1 1 0 1 1 1 0
0 1 1 1 1 1 1 1
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Billy's Toppings
We know Billy will eat a burger plain or with justmustard so we will concentrate on four toppings
Mustard(m), Ketchup(k), Special Sauce(s), and
Pickles(p)
k m p s e k m p s e
0 0 0 0 1 1 0 0 0 0
0 0 0 1 0 1 0 0 1 0
0 0 1 0 0 1 0 1 0 10 0 1 1 0 1 0 1 1 1
0 1 0 0 1 1 1 0 0 0
0 1 0 1 1 1 1 0 1 0
0 1 1 0 0 1 1 1 0 1
0 1 1 1 1 1 1 1 1 1
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Billy's Topping's - m/M
It is easier to find the minterms and maxtermsof e(k,m,p,s) before transferring it to theKarnaugh Map
k m p s e m/M k m p s e m/M
0 0 0 0 1 m0 1 0 0 0 0 M8
0 0 0 1 0 M1 1 0 0 1 0 M9
0 0 1 0 0 M2 1 0 1 0 1 m10
0 0 1 1 0 M3 1 0 1 1 1 m11
0 1 0 0 1 m4 1 1 0 0 0 M12
0 1 0 1 1 m5 1 1 0 1 0 M13
0 1 1 0 0 M6 1 1 1 0 1 m14
0 1 1 1 1 m7 1 1 1 1 1 m15
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K-map
Using the minterms and Maxterms we caneasily fill out the K-map
P
m0 m1 m3 m2
m4 m5 m7 m6M
Km12 m13 m15 m14
m8 m9 m11 m10
S
P
M0 M1 M3 M2
M4
M5
M7
M6 M
KM12 M13 M15 M14
M8 M9 M11 M10
S
P
M
K
S
P
M
K
S
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K-map
Using the minterms or Maxterms we can easilyfill out the K-map
P
m0 m1 m3 m2
m4 m5 m7 m6M
Km12 m13 m15 m14
m8 m9 m11 m10
S
P
M0 M1 M3 M2
M4 M5 M7 M6M
KM12 M13 M15 M14
M8 M9 M11 M10
S
P
1
1 1 1M
K1 1
1 1
S
P
0 0 0
0M
K0 0
0 0
S
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K-Map Group and Reduce
Now we can group terms and reduce them tofind the MSP
P
1 0 0 0
1 1 1 0M
K0 0 1 1
0 0 1 1
S
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Billy's Toppings - MSP
An answer would be
But that answer isn't unique
How can we tell if something is the MSP?
Covered on Monday!
e k , m , p , s =k ' p ' s 'k ' ms ' pk
e k , m , p , s =k ' p ' s 'k ' mp 'mps pk