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9.3 DETECTION AND ELIMINATION HAZARDS 399

9.3 DETECTION AND ELIMINATION HAZARDS IN MULTILEVEL
XOR-TYPE FUNCTIONS

Conventional static hazard analysis used in two-level logic does not address the problem
of hazards in multilevel XOR-type functions. This section presents a simple but general
procedure for the detection and elimination of static hazards in these functions. It is shown
that all static hazards can be eliminated with redundant cover derived by using a method
based on lumped path delay diagrams (LPDDs), and that this method is of unrestricted
applicability. The problems associated with dynamic hazards, as they relate to static hazard
cover, are also considered.
Multilevel XOR-type functions of the type considered in this section find use in arithmetic
circuits, such as dedicated arithmetic logic units, and in error detection circuits. These
functions can be obtained from logic synthesis algorithms or from K-map extraction as was
demonstrated in Chapter 5. If steady, clean outputs from these functions are required, it is
necessary to remove any logic (hazard) noise that may be present.
Modern CMOS 1C technology has produced XOR and EQV gates whose speed and
compactness are close to those of other two-input gates (see, e.g., Figs. 3.26 and 3.27). This
has made the use of XOR and EQV gates more practical and has led to the development
of various methods of multilevel function implementation that take advantage of these
gates. These implementations can produce gate-minimum results not possible with two-
level logic. When fan-in restrictions on two-level implementations are considered, multilevel
implementations become even more attractive.
The simpler multilevel functions include the XOR/SOP and EQV/POS forms. The
XOR/SOP form connects p-terms with OR and XOR operators, while the EQV/POS form
connects s-terms with AND and EQV operators. XOP and EOS forms are special cases of
XOR/SOP and EQV/POS, respectively, and are considered to be special two-level forms
of representation. The XOP form connects p-terms only with XOR operators, and the EOS
form connects s-terms only with EQV operators. Multilevel functions more complex than
XOR/SOP and EQV/POS are classified simply as compound multilevel forms for lack of a
practical classification scheme.
As was stated in Section 9.2, a static hazard is a glitch in an otherwise steady-state
output signal and is produced by two asymmetric paths from a single input. Figure 9.7 is
a generalization of the condition that allows the static hazard to form in multilevel circuits.
The coupled variable must be an input to the initial gate in each path (Gates 1 and 3 in

Path 1

| t | Path 2
FIGURE 9.7
Alternative paths of the coupled variable to the hazard gate that is necessary for static hazard formation
in a multilevel logic circuit.