SOP and POS Forms Notation

SOP Given a Table of Combinations What is the SOP form for the following 3 input / 1 output digital device? S A B f 1

Computing the SOP (2) This SOP has 4 minterms: f = S'AB' + S'AB + SA'B + SAB S A B f minterm name 1 m2 m3 m5 m7

Canonical SOP Boolean functions can use shorthand notation when in SOP form: f = S'AB' + S'AB + SA'B + SAB f(S,A,B) = (m2,m3,m5,m7) or f(S,A,B) = m(2,3,5,7)

Canonical SOP Example f(x1,x2,x3) = m(1,4,5,6) f = minterm x1 x2 x3 f 1 2 3 4 5 6 7 x1'x2'x3 + x1x2'x3' + x1x2'x3 + x1x2x3'

Product of Sums Form An alternate canonical “two-level” format “Product of sums”  POS Two levels OR level followed by AND level Again, NOT doesn’t count as a level Not a common as SOP, but can be useful in some situations Which ones?

Computing the POS Identify rows with “0” on output (f = 0) Represent the input for each 0 row as a maxterm A logical “sum” of the input bits which guarantees that term will be “0” (sum of literals) A B f 1

Canonical POS Example f(x1,x2,x3) = (M0,M2,M3,M7) = M(0,2,3,7) f = maxterm x1 x2 x3 f 1 2 3 4 5 6 7 (x1+x2+x3)(x1+x2'+x3)(x1+x2'+x3')(x1'+x2'+x3')

Example: 3 Way Light Control L(A,B,C) = m(5,6) or L(A,B,C) = M(0,1,2,3,4,7) SOP: L = POS: A B C L 1 (A B' C)+(A B C') (A+B+C)(A+B+C') (A+B'+C)(A+B'+C') (A'+B+C)(A'+B'+C')

Question: Under what conditions would POS form be better? (assuming we aren’t doing further reductions)

Inverters in Two-Level Circuits Inverters are not always required for two-level logic This is why we do not always count them among the cost of a circuit Later, we will see that many variables will be available to us in both normal and inverted form  don't need to invert them We show them only for completeness at this point

NAND/NOR Circuits

Completeness of NAND Any Boolean function can be implemented using just NAND gates. Why? Need AND, OR, and NOT NOT: 1-input NAND (or 2-input NAND with inputs tied together) AND: NAND followed by NOT OR: NAND preceded by NOTs Likewise for NOR

Using NAND as Universal Logic NOT AND OR

SOP Using NORs & POS Using NANDs NANDs are natural for SOP networks You can extend this idea to multi-level circuits as long as the levels alternate AND/OR/AND/OR ending with OR You can implement an SOP circuit using only NOR gates All gates become NORs; just add an extra “inverter” following the final NOR NORs are natural for POS networks You can extend this idea to multi-level circuits as long as the levels alternate OR/AND/OR/AND ending with AND You can implement a POS circuit using only NAND gates All gates become NANDs; just add an extra inverter following the final NAND

SOP Using NAND Networks SOP can be implemented with just NAND gates “pushing the bubbles” Every gate just becomes a NAND! x 1 2 3 4 5

2x1 MUX Using NANDs Implement f = S'A + SB with NAND gates only This one is complicated by the inverter on S!

POS Using NOR Networks POS can be implemented with just NOR gates Every gate just becomes a NOR x 1 2 3 4 5

Schematics of DeMorgan’s Laws (x ∙ y)' = x' + y' (x + y)' = x' ∙ y'

Universal Logic Families Any logic function can be designed using only: AND, OR, NOT NAND NOR These are called “universal logic families” Actual components are often designed using either NAND or NOR gates only NAND and NOR require fewer transistors to build Just having a single gate design is simpler than having 3!

AND/OR Networks  NAND/NAND Convert multi-level AND/OR net  NAND/NAND

And Again … But Be Careful conserve the polarity of the input/output signals

Some Useful Circuits

Decoders and Multiplexors Decoder: Popular combinational logic building block, in addition to logic gates Converts input binary number to one high output 2-input decoder: four possible input binary numbers So has four outputs, one for each possible input binary number Internal design AND gate for each output to detect input combination Decoder with enable e Outputs all 0 if e=0 Regular behavior if e=1 n-input decoder: 2n outputs i0 i1 d0 d1 d2 d3 1 i0 d0 d1 d2 d3 i1 i0 i1 d0 d1 d2 d3 e 1 i1’i0’ i1’i0 i1i0’ i1i0

Multiplexor (Mux) Mux: Another popular combinational building block Routes one of its N data inputs to its one output, based on binary value of select inputs 4 input mux  needs 2 select inputs to indicate which input to route through 8 input mux  3 select inputs N inputs  log2(N) selects Like a railyard switch

Mux Internal Design 2x1 mux 4x1 mux i0 (1*i0=i0) 1 i0 (0+i0=i0) s0 d × 1 2 × 1 2 × 1 i1 i0 s0 d 1 i0 (0+i0=i0) i0 i0 d d i1 i1 s0 s0 a 2x1 mux i0 4 × 1 i2 i1 i3 s1 s0 d 4x1 mux

Muxes Commonly Together -- N-bit Mux 2 × 1 a3 i0 Simplifying notation: d b3 i1 s0 4 4-bit 2 × 1 4 C a2 i0 2x1 d A I 4 b2 i1 s0 4 D C is short B I 1 f or 2 × 1 a1 i0 s0 d b1 i1 c3 s0 s0 c2 2 × 1 a0 i0 d c1 b0 i1 s0 s0 c0 Ex: Two 4-bit inputs: A (a3 a2 a1 a0) and B (b3 b2 b1 b0) 4-bit 2x1 mux (just four 2x1 muxes sharing a select line) can select between A or B

N-bit Mux Example Four possible display items Temperature (T), Average miles-per-gallon (A), Instantaneous mpg (I), and Miles remaining (M) -- each is 8-bits wide Choose which to display using two inputs x and y Use 8-bit 4x1 mux