1. Boolean algebra and Logic gates
2. Combinational logic blocks(Adders, Multiplexers, Encoders, de-coder)
3. Sequential logic blocks(Latches, Flip-Flops, Registers, Counters)
• Binary information is represented in digital computers by physical quantities called SIGNALS.
• There are two states that represent a binary variable i.e. 1 or 0.
• The manipulation of binary information is done by logic circuits called GATES.
• The I/P-O/P relationship of the binary variables for each gate can be represented by TRUTH TABLES.
A weak signal may be boosted by means of two
A Boolean Algebra Is An Algebra Consisting Of A Set Of Elements (>=2) , Together With 3 Operations (AND, OR And NOT).
Boolean Algebra is used to analyze and simplify the digital (logic) circuits. It uses only the binary numbers i.e. 0 and 1. It is also called as Binary Algebra or logical Algebra.
Boolean algebra was invented by George Boole in 1854.
Basic Laws Of Boolean Algebra Are Depicted In The Next Slide.
This Table Lists The Most Basic Identities Of Boolean Algebra. The First 8 Show The Basic
QUESTION
Simplification of the Boolean expression (A + B)'(C + D + E)' + (A + B)' yields which of the following results?
A + B A'B'
C + D + E A'B'C'D'E’
A'B' is CORRECT. By applying the property x + xy = x, with x = (A + B)'= A'B' and y = (C + D + E)'= C'D'E', we get the result.
They Are The One’s In Which There Are NO Feedback Paths From Output To Input And There Is NO Memory.
We Will Be Discussing The
Following Combinational Circuits:
• Adder Half-adder
Half adder is a combinational logic circuit with two input and two output. The half adder circuit is designed to add two single bit binary number A and B. It is the basic
building block for addition of two single bit numbers. This circuit has two outputs carry and sum.
Full adder is developed to overcome the drawback of Half Adder circuit. It can add two one-bit numbers A and B, and carry c. The full adder is a three input and two output
combinational circuit.
Multiplexer is a special type of combinational circuit. There are n-data inputs, one output and m select inputs with 2m = n. It is a digital circuit which selects one of the n data
inputs and routes it to the output. The selection of one of the n inputs is done by the selected inputs. Depending on the digital code applied at the selected inputs, one out of n data sources is selected and transmitted to the single
output Y. E is called the strobe or enable input. It is
generally an active low terminal, that means it will perform the required operation when it is low.
A demultiplexer performs the reverse operation of a
multiplexer i.e. it receives one input and distributes it over several outputs. It has only one input, n outputs, m select input. At a time only one output line is selected by the
select lines and the input is transmitted to the selected output line.
Encoder is a combinational circuit which is designed to perform the inverse operation of the decoder. An encoder has n number of input lines and m number of output lines. An encoder produces an m bit binary code corresponding to the digital input number. The encoder accepts an n input digital word and converts it into an m bit another digital word.
A decoder is a combinational circuit. It has n input and to a maximum m = 2n outputs. Decoder is identical to a
demultiplexer without any data input. It performs operations which are exactly opposite to those of an encoder.
The combinational circuit does not use any memory. Hence the previous state of input does not have any effect on the present state of the circuit. But sequential circuit has
memory so output can vary based on input. This type of circuits uses previous output, input, clock and a memory element.
Flip flop is a sequential circuit which generally samples its inputs and changes its outputs only at particular instants of time and not continuously.
Characteristics
- 2 stable states
- Memory capability
- Operation is specified by a Characteristic Table
Triggering: This means making a circuit active.
Level Triggering: In level triggering the circuit will become active when the gating or clock pulse is on a particular level. This level is decided by the designer. We can have a negative level triggering in which the circuit is active when the clock signal is low or a positive level triggering in which the circuit is active when the clock signal is high.
Edge Triggering: In edge triggering the circuit becomes active at negative or positive edge of the clock signal. For example if the circuit is positive edge triggered, it will take input at exactly the time in
which the clock signal goes from low to high.
A latch is a device with exactly two stable states. These states are high-output and low-output. A latch has a
feedback path, so information can be retained by the
device. Therefore latches can be memory devices, and can store one bit of data for as long as the device is powered. As the name suggests, latches are used to "latch onto" information and hold in place. Latches are very similar to flip-flops, but are not synchronous devices, and do not operate on clock edges as flip-flops do.
An SR latch (Set/Reset) is an asynchronous device: it
works independently of control signals and relies only on the state of the S and R inputs. In the image we can see that an SR latch can be created with two NOR gates that have a cross-feedback loop. SR latches can also be made
from NAND gates, but the inputs are swapped and negated. In this case, it is sometimes called an SR latch.
S-R FLIP FLOP
In some situations it may be desirable to dictate when the latch can and cannot latch. The gated SR latch is a simple extension of the SR latch which provides an Enable line
which must be driven high before data can be latched. Even though a control line is now required, the SR latch is not
synchronous, because the inputs can change the output even in the middle of an enable pulse.
Master slave JK FF is a cascade of two S-R FF with
feedback from the output of second to input of first. Master is a positive level triggered. But due to the presence of the inverter in the clock line, the slave will respond to the
negative level. Hence when the clock = 1 (positive level)
the master is active and the slave is inactive. Whereas when clock = 0 (low level) the slave is active and master is
inactive
Delay Flip Flop or D Flip Flop is the simple gated S-R latch with a NAND inverter connected between S and R inputs. It has only one input. The input data is appearing at the
output after some time. Due to this data delay between i/p and o/p, it is called delay flip flop. S and R will be the
complements of each other due to NAND inverter. Hence S = R = 0 or S = R = 1,these input condition will never
appear. This problem is avoid by SR = 00 and SR = 1 conditions
The D latch (D for "data") or transparent latch is a simple extension of the gated SR latch that removes the possibility of invalid input states.
The D latch outputs the D input whenever the Enable line is high
Toggle flip flop is basically a JK flip flop with J and K
terminals permanently connected together. It has only input denoted by T is shown in the Symbol Diagram. The symbol for positive edge triggered T flip flop is shown in the Block Diagram
In a computer, a register is one of the small set of data holding places that are part of a computer processor. A register may hold a computer instruction, a storage
address, or any kind of data (such as a bit sequence or
individual characters). Some instructions specify registers as part of the instruction. For example, an instruction may specify that the contents of two defined registers be added together and then placed in a specified register.
REGISTER
Shift Registers
D Q C
D Q C
D Q C
In digital logic and computing, a counter is a device which stores (and sometimes displays) the number of times a
particular event or process has occurred, often in relationship to a clock signal.
COUNTER
J K Q
J K Q
J K Q
J K Q
Clock
Counter Enable
A0 A1 A2 A3
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BOOK- computer architecture and organization by M.Morris Mano
