Lecture Digital logic design - Lecture 12: More about combinational analysis and design procedures

The main contents of the chapter consist of the following: Logic circuit analysis, logic circuit analysis, verification - circuit analysis, symbolic analysis, literal analysis, analysis versus design, digital design overview, design procedure (mano), combinational logic design,...

Lecture 12 More about Combinational Analysis and Design Procedures Logic Circuit Analysis Analysis: Determining the behavior of a system given its description The description of the system is often provided in the form of a circuit diagram o o  Logic Circuit Analysis For two-level circuits, the analysis process is simple The Boolean expression representing the circuit can often be written by inspection For multilevel circuits, the analysis process is much more complicated Cannot write a Boolean expression by inspection Must follow a procedure to implement the analysis Logic Circuit Analysis Identify inputs and outputs Track circuit behavior from input to output Determine Boolean expression for output(s) Determine Truth Table Examine circuit timing, power dissipation, etc Logic Circuit Analysis x 1  P 1  x 2  P 9  x 5  x 3  P 4  f  P 7  P 2  P 3  x 4  P 6  P 8  P 10 P 5  Veri ficat ion ° Circuit analysis involves figuring out what some circuit does Circ uit • Every circuit computes some function, which can be described with Ana Boolean expressions or truth tables lysi • So, the goal is to find an expression or truth table for the circuit s ° The first thing to is to figure out what the inputs and outputs of the overall circuit are Inputs: x, y,z Output: f Sy mb olicstart with the circuit diagram ° We Ana • We determine gate output expressions lysi • Intermediate expressions are combined in following gates to form s complex expressions - It might help to some algebraic simplification along the way • We repeat until we have the output function and expression ° Symbolic analysis gives both the truth table and logic expression Literal Analysis ° Literal analysis is process of manually assigning a set of values to the inputs, tracing the results, and recording the output values • For ‘n’ inputs there are 2n possible input combinations • From input values, gate outputs are evaluated to form next set of gate inputs • Evaluation continues until gate outputs are circuit outputs ° Literal analysis only gives us the truth table ° Once you know the number of inputs and outputs, list all the possible input combinations in your truth table • A circuit with n inputs should have a truth table with 2n rows x 0 0 1 1 y 0 1 0 1 z 1 1 f Lite ral Ana ° You lysi can simulate the circuit by hand to find the output for s each possible combination of inputs x y z 0 0 1 1 0 1 0 1 1 1 f Lite ral Ana the same thing for all the other input combinations ° Doing lysi the complete truth table yields s ° This is simple, but tedious x y z f 0 0 1 1 0 1 0 1 1 1 1 0 1 10 Multilevel Logic Circuits Two techniques that can be used to realize multilevel logic circuits: Factoring Functional Decomposition 52 Factoring Example: Realize a logic circuit that has a maximum fan-in of two for the following Boolean expression F(A G) = ACF' + ADEF' + BCG + BDEG 53 Factoring Example: Design the minimum-cost logic circuit that implements the following Boolean expressions F1(A,B,C,D) = m(1,2,3,7,11,15) F2(A,B,C,D) = M(0,1,2,3,4,8,12) 54 Functional Decomposition Example: Design a minimum-cost logic circuit to implement the following Boolean expression F(A,B,C,D) = A'BC + AB'C + ABD + A'B'D 55 NAND and NOR Circuits As with two-level circuits, multilevel circuits can be realized using NAND or NOR gates only 56 NAND and NOR Circuits Example: Realizing a NAND Circuit 57 NAND and NOR Circuits x 1  x 2  x 3  f  x 4  x 5  x 6  x 7  58 NAND and NOR Circuits x 1  x 2  x 3  x 4  x 5  x 6  f  x  7  59 NAND and NOR Circuits x 1  x 2  x 3  x 4  f  x 5  x 6  x 7  60 NAND and NOR Circuits Example: Realizing a NOR Circuit 61 NAND and NOR Circuits x 1  x 2  x 3  f  x 4  x 5  x 6  x 7  62 NAND and NOR Circuits x 1  x 2  x 3  f  x 4  x 5  x 6  x 7  63 NAND and NOR Circuits x 1  x 2  x 3  f  x 4  x 5  x 6  x 7  64 Co mbi nati ona Step l Step Capture the Log ic function Des Step Convert to ign Pro equations ces s Step Implement as a gatebased circuit Description Create a truth table or equations, whichever is most natural for the given problem, to describe the desired behavior of the combinational logic This step is only necessary if you captured the function using a truth table instead of equations Create an equation for each output by ORing all the miniterms for that output Simplify the equations if desired For each output, create a circuit corresponding to the output’s equation (Sharing gates among multiple outputs is OK optionally.) 65 Summary ° Analysis and Design Procedures (Combinational) ° Important concept – analyze digital circuits • Given a circuit - Create a truth table - Create a minimized circuit ° Approaches • Boolean expression approach • Truth table approach ° Both results can then be minimized using K-maps (Leads to minimized hardware) ° Need to formulate circuits from problem descriptions 1.Determine number of inputs and outputs 2.Determine truth table format 3.Determine K-map 4.Determine minimal SOP o There may be multiple outputs per design o Solve each output separately 66 ... no feedback or storage elements) 27 Digital Design Overview ° Design digital circuit from specification ° Digital inputs and outputs known • Need to determine logic that can transform data ° Start... inputs and outputs Derive truth table Obtain simplified Boolean functions Draw logic diagram and verify correctness A B C RS 0 0 0 1 0 S=A+B+C R = ABC 1 1 0 1 1 1 0 1 1 1 29 Combinational logic design. .. write the SumOf-Products logic expression 22 Exa mpl Example e #2: Analyze the logic circuit shown below to Circ determine the logic expression for the output F2 uit Using the logic expression,