Chapter 1 Digital Concepts
Digital Waveform:
A digital waveform: is a train of pulses represented as voltage levels changing back
and forth between HIGH and LOW levels or states
A pulse can be positive or negative:
- A positive pulse is one that starts from LOW level then goes to HIGH level and
then goes back to LOW level
- A negative pulse happens when the voltage goes from HIGH level to LOW level
and then back to HIGH level
Basically a waveform is a series of positive and negative pulses
Any pulse has two edges:
- The first edge is called the leading edge and it occurs at time
- The second edge is called the trailing edge and it occurs at time
If an edge (whether leading or trailing) goes from LOW to HIGH then we call it a
Rising Edge
If an edge (whether leading or trailing) goes from HIGH to LOW then we call it a
Falling Edge
For a positive pulse the leading edge is a Rising edge and the trailing edge is a Falling
edge
Non-ideal Pulses:
- The pulses we have seen above are ideal pulses because the rising and falling
edges are assumed to change in zero time (instantaneously)
- In reality, these transition require some time to take place and never occur
instantaneously (although for most digital work we can assume ideal pulses)
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Chapter 1 Digital Concepts
-The amplitude : is the height of the pulse from the baseline
-Rise Time : is the time required for a pulse to go from LOW level to HIGH level
-Fall Time : is the time required for a pulse to go from HIGH level to LOW level
-It is common to measure the rise time from 10% of pulse amplitude to 90% of the
pulse amplitude
- And the fall time is measured from 90% of pulse amplitude to 10% of the pulse
amplitude
- Pulse Width : is a measure of the duration of the pulse and is often defined as
the time interval between 50% points on the rising and falling edges
Digital waveforms can be either:
- Non-periodic
- Periodic
Non-periodic waveform:
- Is one that doesn’t repeat itself at fixed intervals
- It is composed of pulses of randomly differing pulse widths and/or time differing
time intervals between the pulses
Periodic waveform:
- Is one that repeat itself at fixed intervals
- The fixed interval is called Period (T)
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Chapter 1 Digital Concepts
The Frequency (f): is the rate at the waveform repeat itself. Hence it is the number
of full pulses in a second and is measured in hertz (Hz).
- It is calculated as the reciprocal of the period T
Duty Cycle:
- In a periodic digital waveform; the duty cycle is the ration of the pulse width
( )to the period T
- It can be expressed as percentage :
( )
Digital Waveforms and Binary Information:
In digital systems binary information appear as digital waveforms that represent
sequences of bits
When a waveform is HIGH a binary 1 is present and when it is LOW a binary 0 is
present
Within the waveform, each bit in the sequence occupies a defined time interval in
the waveform called a Bit Time
The Clock:
- In digital systems all waveforms are synchronized with a basic digital waveform
called the clock
- It is a periodic waveform where the interval between pulses (period) equals the
time for one bit
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Chapter 1 Digital Concepts
- Usually changes in the level of each waveform synchronized by the clock take
place at the leading edge of each pulse in the clock (some systems are designed so
that these changes occur at the trailing edge of clock pulses )
- During each bit time of the clock, the waveform is either HIGH or LOW
- These HIGHs and LOWs represent sequences of data bits
- The clock waveform itself doesn’t carry any data
Timing Diagrams:
- Is a graph that accurately displays the relationship of two or more waveforms with
respect to each other on a time basis
- By looking at timing diagram you can determine:
o the state (LOW or HIGH) of all the waveforms at any specified point of time
o the exact time that a specific waveform changes state relative to other
waveforms
Data: is group of bits that conveys some type of information
Binary data that is represented as digital waveforms must be transferred between
one circuit to the another within a digital system or from on system to another in
order to perform the required task
In computer systems binary data are transferred in two ways:
- Serial
- Parallel
Serial data transfer:
- Bits are transferred from one point to another one bit at a time along a serial line
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Chapter 1 Digital Concepts
Parallel data transfer:
- All bits in a group are sent out on a separate lines at the same time
- There is one line for each bit
Logic Gates
=========================
The Inverter:
The Inverter: performs operation called inversion or complementation:
- Changes one logic level to the opposite level
- In term of bits; it changes 1 to 0 and 0 to 1
Inverter Symbol:
Inverter Truth Table:
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Chapter 1 Digital Concepts
- In digital circuits a truth table is a table that shows the output of a given logic
component for every possible input value
- The truth table of the inverter would be:
Inverter Operation:
- when the input is LOW the output is HIGH
- when the input is HIGH the output is LOW
Inverter Timing Diagram:
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Example 3.1:
By using timing diagram show the output of an inverter based on the provided input:
Answer:
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Chapter 1 Digital Concepts
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Logic Expression of an Inverter:
- In Boolean algebra a variable is expressed by a letter
- A variable can take the value of either 1 or 0
- A complement of a variable is designated by a bar over the letter
- If the input variable of an inverter (NOT gate) is called A and the output variable is
called X, then operation of the inverter would be expressed in term of Boolean
algebra as follow:
X=
An Application:
- The following figure demonstrate a circuit that would generate the 1’s
complement of 8-bit binary number:
Bubble:
- Bubble (negation indicator) is a small circle that is inserted at the input or the
output of any logic gate or component in order to indicate inversion or
complementation
- When appearing on an input it means that 0 is the active or asserted input state.
In such case the input is called Active-LOW input
- When appearing on output it means that 0 is the active or asserted output state.
In such case the output is called Active-LOW input
- The absence of bubble on either input or output means that 1 is the active or
asserted state. And in this case the input or output is called Active-HIGH
The AND Gate:
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Chapter 1 Digital Concepts
AND gate performs what is known as logical multiplication
AND gate can have two or more inputs and a single output
AND Gate Symbol:
Operation:
- AND gate output is HIGH only if all the inputs are HIGH. If any of its inputs is LOW
then the output is LOW
- X is HIGH if A and B are HIGH.
- X is LOW if A or B are LOW
- AND gate can be used in situation where we need to determine whether certain
conditions are simultaneously true
Truth Table:
- The total number of possible combinations of binary inputs to a gate is:
Where: N: the number of possible input combinations
n: the number of input variables
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Example 3.2:
The following example demonstrates the output of 2-input AND gate with the following
input values:
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Chapter 1 Digital Concepts
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Example 3.3:
The following example demonstrates the output of 3-input AND gate with the following
input values:
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Logic Expression:
- If the input variables of AND gate are : A ,B and the output variable X, then
operation of the AND gate would be expressed logically as:
X = AB
- AND function represent what is known as Boolean multiplication:
0.0=0
0.1=0
1.0=0
1.1=1
Application:
- The following diagram represent a simple design for seat built alarm system
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Chapter 1 Digital Concepts
The OR Gate:
OR gate performs what is known as logical addition
OR gate can have two or more inputs and a single output
OR Gate Symbol:
Operation:
- OR gate output is HIGH when any of inputs is HIGH. The output is LOW when all of
the inputs are LOW
- X is HIGH if A is HIGH or B is HIGH.
- X is LOW if A and B are LOW
- OR gate can be used in situation where we need to determine whether any of
certain conditions is true
Truth Table:
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Example 3.4:
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Chapter 1 Digital Concepts
The following example demonstrates the output of 2-input OR gate with the following input
values:
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Example 3.5:
The following example demonstrates the output of 3-input OR gate with the following input
values:
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Logic Expression:
- OR function is represented mathematically by +
- If the input variables of OR gate are : A ,B and the output variable X, then
operation of the OR gate would be expressed logically as:
X=A+B
- OR function represent what is known as Boolean addition:
0+0=0
0+1=1
1+0=1
1+1=1
Application:
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Chapter 1 Digital Concepts
- The following diagram represent a simple design for open door/2windows alarm
system
The NAND Gate:
NAND gate is simply AND gate with inverted output
Symbol:
Operation:
- NAND gate output is LOW only when all inputs are HIGH. The output is HIGH when
any of the inputs is LOW
- X is HIGH if A is LOW or B is LOW or both.
- X is LOW if A and B are HIGH
- In NAND gate the LOW level (0) is the active or asserted output level as indicated
by the bubble on the output
- So NAND is simply an AND gate with Active-LOW output
Truth Table:
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Chapter 1 Digital Concepts
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Example 3.6:
The following example demonstrates the output of NAND gate with the following input
values:
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Example 3.7:
The following example demonstrates the output of 3-input NAND gate with the following
input values:
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Logic Expression:
- If the input variables of NAND gate are : A ,B and the output variable X, then
operation of the NAND gate would be expressed logically as:
-
NAND operation is equivalent to Negative-OR operation:
- Notice that NAND gate output is HIGH when any of its inputs is LOW
- So NAND can be used to detect whenever any of its inputs is LOW
- So it performs the same operation of OR gate with active-LOW inputs
- An OR gate with all inputs are active-LOW is called Negative-OR gate
- NAND gate is equivalent to Negative-OR gate
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Chapter 1 Digital Concepts
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Example 3.8:
The following example demonstrates the output of 4-input Negative-OR gate with the
following input values:
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Application:
- A manufacturing plant uses two tanks to store certain liquid chemical that is
required in manufacturing process. A system required to indicate that both tanks
have at least 25% of their capacity filled with the liquid, where a green light must
indicate this condition. The following design demonstrate such a system:
- If we wanted the same system to indicate with a red light that at least one of the
tanks level has dropped below 25% then the system would be:
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Chapter 1 Digital Concepts
The NOR Gate:
NOR gate is simply OR gate with inverted output
Symbol:
Operation:
- NOR gate output is LOW when any of its inputs is HIGH. The output is HIGH only
when all the inputs are LOW
- X is HIGH if A is LOW and B is LOW.
- X is LOW if A is HIGH or B is HIGH or both are HIGH.
- In NOR gate the LOW level (0) is the active or asserted output level as indicated by
the bubble on the output
- So NOR is simply a OR gate with Active-LOW output
Truth Table:
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Example 3.9:
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Chapter 1 Digital Concepts
The following example demonstrates the output of 2-input NOR gate with the following
input values:
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Logic Expression:
- If the input variables of NOR gate are : A ,B and the output variable X, then
operation of the NAND gate would be expressed logically as:
-
NOR operation is equivalent to Negative-AND operation:
- Notice that NOR gate output is HIGH when all inputs are LOW
- So NOR can be used as AND gate that require all its inputs to be LOW
- So it performs the same operation of AND gate with active-LOW inputs
- An AND gate with all inputs are active-LOW is called Negative-AND gate
- NOR gate is equivalent to Negative-AND gate
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Example 3.10:
The following example demonstrates the output of 4-input Negative-AND gate with the
following input values:
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Chapter 1 Digital Concepts
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The XOR Gate:
XOR (Exclusive-OR) gate can have only two inputs and one output
Symbol:
Operation:
- XOR output is HIGH only when the two inputs are at opposite logic levels
- X is HIGH when A is HIGH and B is LOW, or when A is LOW and B is HIGH
- X is LOW when A and B are both HIGH or when A and B are both LOW
Truth Table:
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Example 3.11:
The following example demonstrates the output of XOR gate with the following input
values:
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Chapter 1 Digital Concepts
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Logic Expression:
- If the input variables of NOR gate are : A ,B and the output variable X, then
operation of the NAND gate would be expressed logically as:
X=AꚚB
The XNOR Gate:
XNOR (Exclusive-NOR) gate has only two inputs and one output
XNOR output is opposite of that of XOR gate (hence the bubble on the output)
XNOR can be thought of as active-LOW XOR gate
Symbol:
Operation:
- XNOR output is HIGH when the two input levels are the same. The output is LOW
when the levels of the two inputs are different
- X is HIGH when A and B are both HIGH or both LOW
- X is LOW when A is HIGH and B is LOW or when A is LOW and B is HIGH
Truth Table:
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Chapter 1 Digital Concepts
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Example 3.12
The following example demonstrates the output of XOR and XNOR gates with the following
input values:
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