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13.3 Logic Systems
Digital signals are actually analog voltage (or current) levels that vary continuously as they change. Digital simulation assumes that digital signals may only take on a set of logic values (or logic states —here we will consider the two terms equivalent) from a logic system . A logic system must be chosen carefully. Too many values will make the simulation complicated and slow. With too few values the simulation may not accurately reflect the hardware performance.
A twovalue logic system (or twostate logic system) has a logic value '0' corresponding to a logic level 'zero' and a logic value '1' corresponding to a logic level 'one'. However, when the power to a system is initially turned on, we do not immediately know whether the logic value of a flipflop output is '1' or '0' (it will be one or the other, but we do not know which). To model this situation we introduce a logic value 'X' , with an unknown logic level, or unknown . An unknown can propagate through a circuit. For example, if the inputs to a twoinput NAND gate are logic values '1' and 'X' , the output is logic value 'X' or unknown. Next, in order to model a threestate bus, we need a highimpedance state . A highimpedance state may have a logic level of 'zero' or 'one', but it is not being driven—we say it is floating. This will occur if none of the gates connected to a threestate bus is driving the bus. A fourvalue logic system is shown in Table 13.2 .
13.3.1 Signal Resolution
What happens if multiple drivers try to drive different logic values onto a bus? Table 13.3 shows a signalresolution function for a fourvalue logic system that will predict the result.
TABLE 13.3 A resolution function R {A, B} that predicts the result of two drivers simultaneously attempting to drive signals with values A and B onto a bus. 

A resolution function, R {A, B}, must be commutative and associative . That is,
R {A, B} = R {B, A} and R {R {A, B}, C} = R {A, R {B, C}}.(13.4)
Equation 13.4 ensures that, if we have three (or more) signals to resolve, it does not matter in which order we resolve them. Suppose we have four drivers on a bus driving values '0' , '1' , 'X' , and 'Z' . If we use Table 13.3 three times to resolve these signals, the answer is always 'X' whatever order we use.
13.3.2 Logic Strength
In CMOS logic we use n channel transistors to produce a logic level 'zero' (with a forcing strength) and we use p channel transistors to force a logic level 'one'. An n channel transistor provides a weak logic level 'one'. This is a new logic value, a resistive 'one' , which has a logic level of 'one', but with resistive strength . Similarly, a p channel transistor produces a resistive 'zero' . A resistive strength is not as strong as a forcing strength. At a highimpedance node there is nothing to keep the node at any logic level. We say that the logic strength is high impedance . A highimpedance strength is the weakest strength and we can treat it as either a very highresistance connection to a power supply or no connection at all.
With the introduction of logic strength, a logic value may now have two properties: level and strength. Suppose we were to measure a voltage at a node N with a digital voltmeter (with a very high input impedance). Suppose the measured voltage at node N was 4.98 V (and the measured positive supply, V _{ DD} = 5.00 V). We can say that node N is a logic level 'one', but we do not know the logic strength. Now suppose you connect one end of a 1 k W resistor to node N , the other to GND, and the voltage at N changes to 4.95 V. Now we can say that whatever is driving node N has a strong forcing strength. In fact, we know that whatever is driving N is capable of supplying a current of at least 4.95 V / 1 k W ⊕ 5 mA. Depending on the logicvalue system we are using, we can assign a logic value to N . If we allow all possible combinations of logic level with logic strength, we end up with a matrix of logic values and logic states. Table 13.4 shows the 12 states that result with three logic levels (zero, one, unknown) and four logic strengths (strong, weak, highimpedance, and unknown). In this logic system, node N has logic value S1 —a logic level of 'one' with a logic strength of 'strong'.
The Verilog logic system has three logic levels that are called '1' , '0' , and 'x' ; and the eight logic strengths shown in Table 13.5 . The designer does not normally see the logic values that result—only the three logic levels.
The IEEE Std 11641993 logic system defines a variable type, std_ulogic , with the nine logic values shown in Table 13.6 . When we wish to simulate logic cells using this logic system, we must define the primitivegate operations. We also need to define the process of VHDL signal resolution using VHDL signalresolution functions . For example, the function in the IEEE Std_Logic_1164 package that defines the and operation is as follows 1 :
function "and"(l,r : std_ulogic_vector) return std_ulogic_vector is
alias lv : std_ulogic_vector (1 to l'LENGTH ) is l;
alias rv : std_ulogic_vector (1 to r'LENGTH ) is r;
variable result : std_ulogic_vector (1 to l'LENGTH );
constant and_table : stdlogic_table := (


( 'U', 'U', '0', 'U', 'U', 'U', '0', 'U', 'U' ),   U 
( 'U', 'X', '0', 'X', 'X', 'X', '0', 'X', 'X' ),   X 
( '0', '0', '0', '0', '0', '0', '0', 'U', '0' ),   0 
( 'U', 'X', '0', '1', 'X', 'X', '0', '1', 'X' ),   1 
( 'U', 'X', '0', 'X', 'X', 'X', '0', 'X', 'X' ),   Z 
( 'U', 'X', '0', 'X', 'X', 'X', '0', 'X', 'X' ),   W 
( '0', '0', '0', '0', '0', '0', '0', '0', '0' ),   L 
( 'U', 'X', '0', '1', 'X', 'X', '0', '1', 'X' ),   H 
( 'U', 'X', '0', 'X', 'X', 'X', '0', 'X', 'X' ),    );
if (l'LENGTH /= r'LENGTH) then assert false report
"arguments of overloaded 'and' operator are not of the same
result(i) := and_table ( lv(i), rv(i) );
If a = 'X' and b = '0' , then (a and b) is '0' no matter whether a is, in fact, '0' or '1' .
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