Download presentation

1
SOP and POS Forms Notation

2
**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

3
**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

4
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)

5
**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'

6
**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?

7
**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

8
**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')

9
**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')

10
**Question: Under what conditions would POS form be better?**

(assuming we aren’t doing further reductions)

11
**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

12
NAND/NOR Circuits

13
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

14
**Using NAND as Universal Logic**

NOT AND OR

15
**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

16
**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

17
**2x1 MUX Using NANDs Implement f = S'A + SB with NAND gates only**

This one is complicated by the inverter on S!

18
**POS Using NOR Networks POS can be implemented with just NOR gates**

Every gate just becomes a NOR x 1 2 3 4 5

19
**Schematics of DeMorgan’s Laws**

(x ∙ y)' = x' + y' (x + y)' = x' ∙ y'

20
**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!

21
**AND/OR Networks NAND/NAND**

Convert multi-level AND/OR net NAND/NAND

22
**And Again … But Be Careful**

conserve the polarity of the input/output signals

23
Some Useful Circuits

24
**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

25
**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

26
**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

27
**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

28
**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

Similar presentations

OK

Logic Gates How Boolean logic is implemented. Transistors used as switches to implement Boolean logic: ANDOR Logic with Transistors.

Logic Gates How Boolean logic is implemented. Transistors used as switches to implement Boolean logic: ANDOR Logic with Transistors.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on disaster management in india Ppt on hindu religion pictures Ppt on total parenteral nutrition solutions Ppt on application of trigonometry in different fields Ppt on blood stain pattern analysis definition Display ppt on tv Ppt on places in our neighbourhood Ppt on protein energy malnutrition Ppt on fema act 1999 Ppt on tools of data collection