ic
Section 8.2 Logic Networks
647
a bit more about why we want to minimize. When logic networks were built from
separate gates and inverters, the cost of these elements was a considerable factor
in the design, and it was desirable to have as few elements as possible. Now, however, most networks are built using integrated circuit technology, a development
that began in the early 1960s. An integrated circuit is itself a logic network representing a certain truth function or functions, just as if some gates and inverters had been combined in the appropriate arrangement inside a package. These
integrated circuits are then combined as needed to produce the desired result.
Because the integrated circuits are extremely small and relatively inexpensive, it
might seem pointless to bother minimizing a network. However, minimization is
still important because the reliability of the final network is inversely related to the
number of connections between the integrated circuit packages.
Moreover, the designers of integrated circuits are highly interested in the
minimization problem. Integrated circuits are embedded into a substrate of silicon
or other semiconductor material. The resulting chips may be tiny, yet they can
contain the equivalent of 3 billion transistors for implementing truth functions.
The distance between two gates may be as small as 45 nanometers (about 1/1000
of the width of a human hair). Minimizing the number of components and the
amount of wiring required to realize a desired truth function makes it possible to
embed more functions in a single chip.
Programmable Logic Devices
Instead of designing a custom chip to implement particular truth functions, a pLD
(programmable logic device) can be used. A PLD is a chip that is already implanted with an array of AND gates and an array of OR gates, together with a rectangular grid of wiring channels and some inverters. Once Boolean expressions in
sum-of-products form have been determined for the truth functions, the required
components in the PLD are activated. Although this chip is not very efficient and is
practical only for smaller-scale circuit logic, on the order of hundreds of gates, the
PLD can be mass-produced, and only a small amount of time (i.e., money) is then
required to “program” it for the desired functions. A FpGa ( field-programmable
gate array) is a big brother to the PLD. The term “field-programmable” suggests
that, like a PLD, the user can configure the chip for a specific purpose. An FPGA
basically connects a number of PLDs in a reconfigurable way, and it can support
thousands of gates. Often the FPGA also contains hardwired components such as
multipliers or even processors and memory, thus producing a small reconfigurable
computer that is “programmed” in hardware rather than software.
example 17
Figure 8.13a shows a PLD for the three inputs x 1 , x 2 , and x 3 . There are four output
lines, so four functions can be programmed in this PLD. When the PLD is programmed, the horizontal line going into an AND gate will pick up certain inputs,
and the AND gate will form the product of these inputs. The vertical line going
into an OR gate will, when programmed, allow the OR gate to form the sum of
certain inputs. Figure 8.13b shows the same PLD programmed to produce the truth
functions f 1 from Example 13 (x 1 x 2 x 3 + x 1 x 2
′x 3 + x 1 x 2
′x 3
′ + x 1
′x 2
′x 3 ) and f 2 from
Practice 11 (x 1 x 2 x 3 + x 1 x 2
′x 3 + x 1 x 2
′x 3
′ + x 1
′x 2
′x 3 + x 1
′x 2
′x 3
′). The dots represent activation points.
Section 8.2 Logic Networks
647
a bit more about why we want to minimize. When logic networks were built from
separate gates and inverters, the cost of these elements was a considerable factor
in the design, and it was desirable to have as few elements as possible. Now, however, most networks are built using integrated circuit technology, a development
that began in the early 1960s. An integrated circuit is itself a logic network representing a certain truth function or functions, just as if some gates and inverters had been combined in the appropriate arrangement inside a package. These
integrated circuits are then combined as needed to produce the desired result.
Because the integrated circuits are extremely small and relatively inexpensive, it
might seem pointless to bother minimizing a network. However, minimization is
still important because the reliability of the final network is inversely related to the
number of connections between the integrated circuit packages.
Moreover, the designers of integrated circuits are highly interested in the
minimization problem. Integrated circuits are embedded into a substrate of silicon
or other semiconductor material. The resulting chips may be tiny, yet they can
contain the equivalent of 3 billion transistors for implementing truth functions.
The distance between two gates may be as small as 45 nanometers (about 1/1000
of the width of a human hair). Minimizing the number of components and the
amount of wiring required to realize a desired truth function makes it possible to
embed more functions in a single chip.
Programmable Logic Devices
Instead of designing a custom chip to implement particular truth functions, a pLD
(programmable logic device) can be used. A PLD is a chip that is already implanted with an array of AND gates and an array of OR gates, together with a rectangular grid of wiring channels and some inverters. Once Boolean expressions in
sum-of-products form have been determined for the truth functions, the required
components in the PLD are activated. Although this chip is not very efficient and is
practical only for smaller-scale circuit logic, on the order of hundreds of gates, the
PLD can be mass-produced, and only a small amount of time (i.e., money) is then
required to “program” it for the desired functions. A FpGa ( field-programmable
gate array) is a big brother to the PLD. The term “field-programmable” suggests
that, like a PLD, the user can configure the chip for a specific purpose. An FPGA
basically connects a number of PLDs in a reconfigurable way, and it can support
thousands of gates. Often the FPGA also contains hardwired components such as
multipliers or even processors and memory, thus producing a small reconfigurable
computer that is “programmed” in hardware rather than software.
example 17
Figure 8.13a shows a PLD for the three inputs x 1 , x 2 , and x 3 . There are four output
lines, so four functions can be programmed in this PLD. When the PLD is programmed, the horizontal line going into an AND gate will pick up certain inputs,
and the AND gate will form the product of these inputs. The vertical line going
into an OR gate will, when programmed, allow the OR gate to form the sum of
certain inputs. Figure 8.13b shows the same PLD programmed to produce the truth
functions f 1 from Example 13 (x 1 x 2 x 3 + x 1 x 2
′x 3 + x 1 x 2
′x 3
′ + x 1
′x 2
′x 3 ) and f 2 from
Practice 11 (x 1 x 2 x 3 + x 1 x 2
′x 3 + x 1 x 2
′x 3
′ + x 1
′x 2
′x 3 + x 1
′x 2
′x 3
′). The dots represent activation points.