Representation of Audio Signals 443● more fl exible processing possible and easy programmability;
● low cost potential; and
● processing can be independent of real time.15.3 Elementary Logical Processes ..............................................................................
We have described an outline of a binary counting scale and shown how we may
implement a count using it but some physical method of performing this is needed. We
can represent two states, a 1 and an 0 state, by using switches since their contacts will
be either open or closed and there is no half-way state. Relay contacts also share this
property but there are many advantages in representing the 1 and 0 states by the polarity
or existence of a voltage or a current, not least of which is the facility of handling such
signals at very high speed in integrated circuitry. Manipulation of the 1 and 0 signals is
referred to as logic and, in practice, is usually implemented by simple logic circuits called
gates. Digital integrated circuits comprise collections of various gates, which can number
from a single gate (as in the eight input NAND gate exemplifi ed by the 74LS30 part
number) to many millions (as can be found in some microprocessors).
All logic operations can be implemented by the appropriate combination of just three
operations:
● the AND gate, circuit symbol & , arithmetic symbol ‘. ’ ;
● the OR gate, circuit symbol |, arithmetic symbol ‘ ’ ; a n d
● the inverter or NOT gate, circuit and arithmetic symbol ‘ ’.From this primitive trio we can derive the NAND, NOR, and EXOR (exclusive-OR gate).
Gates are characterized by the relationship of their output to combinations of their inputs
( Figure 15.3 ). Note how the NAND (literally negated AND gate) performs the same logical
function as OR gate fed with inverted signals and, similarly, note the equivalent duality in
the NOR function. This particular set of dualities is known as De Morgan’s theorem.
Practical logic systems are formed by grouping many gates together and naturally there
are formal tools available to help with the design, the most simple and common of
which is known as Boolean algebra. This is an algebra that allows logic problems to be
expressed in symbolic terms. These can then be manipulated and the resulting expression
can be directly interpreted as a logic circuit diagram. Boolean expressions cope best with