The important terms we are discussing in this section are
- Logic Expression - a mathematical formula consisting of logical operators and variables.
- Logic Operator - a function that gives a well defined output according to switching algebra.
- Logic Variable - a symbol representing the two possible switching algebra values of 0 and 1.
- Logic Literal - the values 0 and 1 or a logic variable or it's complement.
The analysis means a digital circuit is given and we are asked to determine its input-output relationship (its purpose, operation, what it does). One studies the circuit and then states the input-output relationship of the circuit in text or on a truth table or on an operation table or on an operation diagram. The synthesis means an input-output relationship is given and we are asked to design the digital circuit. The input-output relationship is a very crucial component of digital circuit study. Complex digital circuits are blocks are analyzed/designed individually and finally the whole circuit is analyzed/designed.
Combinational circuit analysis starts with a schematic and answers the following questions:
What is the truth table(s) for the circuit output function(s) ?
What is the logic expression(s) for the circuit output function(s) ?
Two types of analyses are possible literal as well as symbolic 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 and evaluation continues until gate outputs are circuit outputs. The literal analysis only gives us the truth table.
Symbolic analysis also starts with the circuit diagram like literal analysis. But instead of assigning values, gate output expressions are determined. Intermediate expressions are combined in following gates to form complex expressions. Symbolic analysis is more work but gives us complete information of both the truth table and logic expression.
Now we will consider an example for the analysis of combinational logic circuit shown in Figure 1.
Figure 1
Analyzing this circuit, it can be seen that
- Output of Gate G1 = AB
- Output of Gate G2 = CD
- Output of Gate G3 = AB + CD
From this we could then construct a truth table (Table 1) to calculate the output of the circuit. The truth table is constructed by considering the output of each gate in turn and then building up towards the complete output.
Table 1. Truth Table
Alternatively, the output of the circuit can be evaluated by substituting values directly into the logic equation.
For example, when A = 1, B = 1, C = 1, D = 0
then Y = AB + CD = 1 . 1 + 1. 0 = 1 + 0 = 1
This can then be repeated for all other input combinations.
The analysis is followed by synthesis i.e. we will consider how to design and implement a logic circuit to enable it to perform the desired specified operation. In this instance, we start with the equation and determine circuit to implement. For example consider the logic function
X = AB + CDE
This is composed of two terms, AB and CDE . The first term is formed by ANDing A and B and the second term is formed by ANDing together C , D and E . These two terms are then ORed together. This can then be implemented using the AND and OR gates, as shown in Figure 2. Generally, as the number of levels are increased, the overall delay is increased due to the contribution of propagation delays at each gate.
Figure 2. Implementation of X=AB+ CDE
Analysis also can be categorized into the functional analysis (determine what is computed) and the timing analysis (determine how long it takes to compute it). The logic expression is manipulated using Boolean (or switching) algebra and optimized to minimize the number of gates needed, or to use specific type of gates.
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