SECTION 8:
DIGITAL FUNDAMENTALS
MAE 2055 – Mechetronics I
2
Analog vs. Digital
In this section of notes, we’ll be discussing
digital electronic circuits. First, we’ll introduce
the concepts of analog and digital signals, and
discuss how analog signals can be converted to
digital signals.
K. Webb
MAE 2055 – Mechetronics I
Analog vs. Digital
3


Analog signals are continuous in time and
amplitude
All physical phenomena – pressure, temperature,
velocity, strain, position, etc. – are analog in nature
 They

can take on any value at any time
Digital signals are discrete in time and amplitude
 They
can only assume a finite number of discrete
values at discrete instants in time
 Digital signals are representations of analog signals
that are easily stored and processed electronically
K. Webb
MAE 2055 – Mechetronics I
Analog vs. Digital – an example
4

Temperature is an analog quantity – at any instant in time
it can assume any value
www.faqs.com

A mercury thermometer is an 
analog measure of temperature
– the mercury can be at any
height at any time
K. Webb
www.mdhb.com
A digital thermometer samples
the actual temperature at
discrete instants in time and
represents it with a finite
number of possible values
MAE 2055 – Mechetronics I
Digital Measurement System
5
Analog
Analog
Input
Signal
Signal
Conditioning/
Amplification
Digital
Analog
to Digital
Converter
(ADC)
Digital
Data/Display
Processing
Digital
Data
Output/
Display
The analog input signal is converted to a
digital signal in an analog-to-digital
converter (ADC or A/D)
K. Webb
MAE 2055 – Mechetronics I
Analog-to-Digital Conversion
6



Continuous-time and -amplitude analog signals are
converted to digital signals in A/D converters
Analog signal is sampled in time, generating a series of
discrete-time samples
Discrete-time samples are quantized – amplitudes are
mapped to a finite number of discrete amplitude
values



A continuous range of input values maps to a single
quantization level
Resulting digital signal is discrete in both time and
amplitude
Digital signal is easily processed, stored, and displayed
K. Webb
MAE 2055 – Mechetronics I
A/D Conversion – sampling
7


The first step in
converting from an
analog signal to a
digital signal is
sampling
Sampled signal is a
discrete-time
signal, but is still
continuous in
amplitude
K. Webb
MAE 2055 – Mechetronics I
A/D Conversion – quantization
8

Next step is quantization




Sampled signal becomes
a digital signal
The digital signal is
discrete in both time and
amplitude
Amplitude values of the
digital signal are
expressed as codes
# of A/D codes = 2N



K. Webb
N = # of bits
10 bit A/D has 1024
distinct quantization
levels
Digital signal stored as
binary values
MAE 2055 – Mechetronics I
Sample Rate
9

Sample Rate
 The
rate at which the measurement instrument
samples and digitizes the analog input signal
 Measured in Hz or in samples per second (Sa/sec)
 Inverse of the sampling period fs = 1/Ts
 Places an upper limit on the frequency (bandwidth) of
the input signal
 Can only accurately measure input signals whose
frequency is no more than half the sample rate
 Otherwise
K. Webb
aliasing will occur
f in 
fs
2
MAE 2055 – Mechetronics I
Sample Rate – aliasing
10

Aliasing is a phenomena that results in a higher frequency
signal appearing as a lower frequency signal

Aliasing occurs due to failure to adhere to the Nyquist Criterion
Nyquist says that the sampling frequency must be at least twice
the maximum signal frequency

f s  2 f in

The Nyquist rate or Nyquist frequency is the minimum sampling
rate for which no aliasing will occur
f nyquist  2 f in
K. Webb
MAE 2055 – Mechetronics I
Aliasing – f = 10 Hz, fs = 8 Hz
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
10 Hz signal sampled at 8 Hz

Nyquist criterion violated
Aliased signal appears at 2 Hz

K. Webb
MAE 2055 – Mechetronics I
Aliasing – f = 10 Hz, fs = 9 Hz
12

10 Hz signal sampled at 9 Hz

Nyquist criterion violated
Aliased signal appears at 1 Hz

K. Webb
MAE 2055 – Mechetronics I
Aliasing – f = 10 Hz, fs = 10 Hz
13

10 Hz signal sampled at 10 Hz

Nyquist criterion violated
Aliased signal appears at DC (0 Hz)

K. Webb
MAE 2055 – Mechetronics I
Aliasing – f = 10 Hz, fs = 20 Hz
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


10 Hz signal sampled at 20 Hz
Nyquist criterion satisfied
Measured signal could appear, in theory, at DC – in practice, this
would rarely happen
K. Webb
MAE 2055 – Mechetronics I
Aliasing – f = 10 Hz, fs = 50 Hz
15



10 Hz signal sampled at 50 Hz
Nyquist criterion satisfied
Frequency of sampled signal is the same as the analog signal
K. Webb
MAE 2055 – Mechetronics I
Digital Advantages
16

Noise immunity
Analog signals are susceptible to corruption by noise and
interference – information is conveyed the precise
instantaneous voltage level of the signal
 Digital representations of analog signals encode
information as series of bits

Bits are either 1 or 0, high or low,
 Add a little noise to a 1 or a 0, and it will still be interpreted as a
1 or a 0


Signal processing
Filtering, modulation, etc. can all be done much more
accurately, efficiently, and inexpensively with digital circuits
 Miniscule transistors on an IC replace large, inaccurate,
expensive capacitors and inductors

K. Webb
MAE 2055 – Mechetronics I
17
Number Systems
When analog signals are converted to digital signals, the
values of the samples must be converted to a numeric
form appropriate for storage and processing by digital
electronic circuits. This form is that of binary numbers.
K. Webb
MAE 2055 – Mechetronics I
Number Systems
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




After an analog signal has been sampled, the values of
those samples must be stored in memory to be processed,
displayed, played, etc.
We may think of these samples as taking on integer values
in the range of, for example, 0 – 1023 or 0 – 256 or
0 – 65535, etc.
But, how are these values represented for storage and
processing?
The answer is: in binary form
Representing the samples of an analog signal is only one
example of the use of binary numbers

K. Webb
They are used any time numbers are processed by a digital
electronic circuit
MAE 2055 – Mechetronics I
Binary
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




In digital electronics, all values are binary in nature
Binary, as its name implies, means that values can
assume one of only two possible values: 0 or 1
In electronic circuits, binary values are typically
represented by voltages: either high or low (e.g.,
0 = 0V, 1 = 5V)
In digital circuits all signals are either 1 or 0, either
high voltage or low voltage, either true or false
So, how do we represent numbers such as the
values of a sampled signal? – as binary numbers
K. Webb
MAE 2055 – Mechetronics I
Binary Number System
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


The binary number system is a base-two number
system
Binary numbers are strings of 1’s and 0’s
Each digit, called a bit, represents an integer power of
2, increasing to the left
Binary Number
11001011
1x27 + 1x26 + 0x25 + 0x24 + 1x23 + 0x22 + 1x21 + 1x20 =
128 + 64 + 0 + 0 + 8 + 0 + 2 + 1 = 203
K. Webb
MAE 2055 – Mechetronics I
Decimal Number System
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


Binary (base 2) number system is not so different from the decimal
(base 10) number system we’re used to
Each digit in a decimal number represents the coefficient of an
integer power of 10, increasing to the left
Can define number systems with any arbitrary base we want (e.g.,
2, 10, 3, 6, 16, etc.)
Decimal Number
249176
2x105 + 4x104 + 9x103 + 1x102 + 7x101 + 6x100 =
200000 + 40000 + 9000 + 100 + 70 + 6 = 249176
K. Webb
MAE 2055 – Mechetronics I
Bits, Bytes, & Nibbles
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
Each binary digit is called a bit
 Each
bit has a value of 1 or 0, so it can be represented
with a signal in a digital circuit – either a high (0) or a
low (1) voltage

A collection of 8 bits is called a byte
A
byte can represent a decimal value from 0 to 255
 Eight separate voltage signals in a digital circuit are
required to represent a byte

4 bits is half of a byte, so it is called a nibble
A
K. Webb
nibble can assume 16 values between 0 and 15
MAE 2055 – Mechetronics I
Hexadecimal Number System
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
Binary numbers require many bits to represent a
number in many applications (e.g., computers are now
moving toward 64-bit processors)



Long, cumbersome, and difficult to read in binary form
Number of bits is typically a multiple of 8 – an integer
number of bytes
The hexadecimal number system (hex) is base 16

Each digit can assume 16 values from 0 – F


0123456789ABCDEF
A hex digit is used to represent the four bits of a nibble
K. Webb
MAE 2055 – Mechetronics I
Hexadecimal Number System
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
Hex numbers often denoted with a “0x” prefix or a “h” suffix

The decimal number 500 is represented as 1F4 in hex, and may be
denoted as: 0x1F4, or 0x01F4, or 1F4h, or 01F4h, etc.
Binary Number
Nibble 3
Nibble 2
Nibble 1
Nibble 0
1100 1011 1111 0100
Nibble 3: 1x23 + 1x22 + 0x21 + 0x20 = 12 = C
Nibble 2:
1x23
+
0x22
+
1x21
+
1x20
= 11 = B
Nibble 1: 1x23 + 1x22 + 1x21 + 1x20 = 15 = F
Nibble 0: 0x23 + 1x22 + 0x21 + 0x20 = 12 = 4
K. Webb
Hexadecimal Number
0xCBF4
MAE 2055 – Mechetronics I
Binary-Coded Decimal – BCD
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
Each digit of a decimal number is represented by
four binary bits
Decimal Digit
BCD
0
0000
1
0001
2
0010
3
0011
4
0100
5
0101
6
0110
7
0111
8
1000
9
1001
K. Webb
Examples:
Decimal
BCD
37
0011 0111
249
0010 0100 1001
618
0110 0001 1000
The codes 1010, 1011, 1100, 1101, 1110 and
1111 are unused in the BCD system
A “BCD” suffix is often used to denote a BCD
number, e.g., 0111 0100BCD
MAE 2055 – Mechetronics I
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Digital Logic
Generating, storing, and processing numbers is
only one function of digital circuits. They are
also used to perform logic operations.
K. Webb
MAE 2055 – Mechetronics I
Digital Logic
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
We’ve seen that digital circuits and systems can be
used to acquire, store, and process data in the
form of numbers
 These
numbers may represent the sampled values of
an analog signal, or they may just be numbers, e.g.,
numbers used in computations

Digital circuits are also used to perform logical
operations on logic variables
 Logic
variable represented by a single bit
 Each bit is represented by the high or low voltage of a
single electrical signal
K. Webb
MAE 2055 – Mechetronics I
Digital logic
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
Say you want to design a circuit to control the heat in your home.
You want the circuit to turn the heat on under the following
conditions:


During the day if the temperature falls below the daytime set point
During the night if the temperature falls below the nighttime set point
Three logic variable inputs to the circuit:
A block diagram of your system:
Day
DayLow
NightLow
Digital
Logic
Circuit
•Day
Heat
= 1 if it’s the daytime
= 0 it it’s night
•DayLow = 1 if T < the day set point
= 0 if T > the day set point
•NightLow = 1 if T < the night set point
= 0 if T > the night set point
And one logic variable Output:
•Heat
K. Webb
= 1 to turn the heat on
= 0 if the heat should be off
MAE 2055 – Mechetronics I
Digital logic – truth table
29





The inputs are single-bit logic
variables (electrical signals with
high or low voltage levels) which,
in this case, come from
temperature sensors and a clock
The output is a single logic bit
that tell the heating system when
to turn on
We can construct a table showing
all possible combinations of input
states, along with the state of the
output corresponding to each
input combination
This is referred to as a truth table
Variable names have been
abbreviated for simplicity
K. Webb
Truth Table
Heat is
turned on
D
DL
NL
H
0
0
0
0
0
0
1
1
0
1
0
0
0
1
1
1
1
0
0
0
1
0
1
0
1
1
0
1
1
1
1
1
MAE 2055 – Mechetronics I
Digital logic – truth table
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



The truth table describes the
behavior of the digital circuit
for all possible combinations
of input variables
It specifies when the circuit
should turn the heat on based
on the state of the inputs
We need a way to implement
the logic described by the
truth table with an electronic
circuit
This is done by building a
circuit consisting of
components called logic gates
K. Webb
Truth Table
Heat is
turned on
D
DL
NL
H
0
0
0
0
0
0
1
1
0
1
0
0
0
1
1
1
1
0
0
0
1
0
1
0
1
1
0
1
1
1
1
1
MAE 2055 – Mechetronics I
31
Logic Gates
We’ve just seen how a truth table can be used
to define the behavior of a digital circuit. Now
we’ll look at the building blocks used to realize
the circuit: logic gates.
K. Webb
MAE 2055 – Mechetronics I
Logic Gates
32
 Say you want to design a circuit to control the heat in your home.
Logic
Gate
Schematic Symbol
Truth Table
You want the circuit to turn the heat on under the following
conditions:
A A B B C =C A∙B
AND
 During the day if the temperature falls below the daytime
0 0 0 0 set
00 point
Output C is high if both
 During the night if the temperature falls below the 0nighttime
0 1 1 00set point
input A and input B are
high (written A∙B)
OR
Output C is high if
either input A or input
B are high (written A+B)
INV
Output B is high if input
A is low, and low if input
A is high (written Ā)
K. Webb
11 00
11 11
AA
00
00
11
11
00
11
B B C =CA+B
0 0 00
1 1 01
0 0 01
1 1 11
A A B =B Ā
0 0 11
1 1 00
MAE 2055 – Mechetronics I
Logic Gates
33
 Say you want to design a circuit to control the heat in your home.
Logic
Gate
Schematic Symbol
Truth Table
You want the circuit to turn the heat on under the following
conditions:
A B
C =A∙B
NAND
 During the day if the temperature falls below the daytime set point
0
1set point

During
if the temperature falls below the0nighttime
Output
C isthe
lownight
if both
0 1
1
input A and input B
1 0
1
are high (written A∙B)
NOR
Output C is low if either
input A or input B are
high (written A+B)
K. Webb
1
1
0
A
0
0
1
1
B
0
1
0
1
C = A+B
1
0
0
0
MAE 2055 – Mechetronics I
Logic Gates
34
 Say you want to design a circuit to control the heat in your home.
Logic
Gate
Schematic Symbol
Truth Table
You want the circuit to turn the heat on under the following
conditions:
A B C=AB
XOR
(Exclusive
Gate)
 During
theOR
day
if the temperature falls below the daytime
set point
0
0 point

During
if the temperature falls below the0nighttime
set
Output
C isthe
highnight
if only
0 1
1
one input is high, not
1 0
1
both (written A  B )
XNOR
Output C is high if both
or neither input is high
(written A  B )
K. Webb
1
1
0
A
0
0
1
1
B
0
1
0
1
C = AB
1
0
0
1
MAE 2055 – Mechetronics I
Boolean Algebra
35


We use Boolean algebra to represent and manipulate logic
expressions
Boolean addition is equivalent to a logical OR
operation: C = A + B


Boolean multiplication is equivalent to a logical AND
operation: C = A∙B


C is 1 only if all terms in the product are 1
An inversion is written with an overbar: B = Ā


C is 1 (or high or true) if any term in the sum is 1
C is 1 if A is 0, and vice versa
Boolean expressions can include multiple variables and
multiple operations, for example:

K. Webb
F = (A∙B∙C)+(Ā∙D∙Ē)+(B∙C∙D)
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Boolean Algebra - rules
36

Familiar rules from algebra apply
 Commutative
Law:
A+B = B+A
A∙B = B∙A
 Associative
Law:
A+(B+C) = (A+B)+C
A∙(B∙C) = (A∙B)∙C
 Distributive
Law:
A∙(B+C) = A∙B + A∙C
K. Webb
MAE 2055 – Mechetronics I
Boolean Algebra –rules & identities
37






K. Webb
A+0=A
A+1=1
A∙0=0
A∙1=A
A+A=A
A+Ā=1





A∙A=A
A∙Ā=0
A + A∙B = A
A + Ā∙B = A + B
(A + B)∙(A + C) = A + BC
MAE 2055 – Mechetronics I
DeMorgan’s Laws
38

A logic expression can be transformed to an equivalent
logic expression if:
1.
2.
3.
4.
All variables in the expression are replaced with their inverses
All AND operations are replaced with OR operations
All OR operations are replaced with AND operations
The entire expression is inverted
A B  C  A  B  C
K. Webb
A B C  A B C
MAE 2055 – Mechetronics I
NAND & NOR Gates
39

The basic logic functions – AND, OR, and NOT
(inverter) – can all be implemented with only
NAND gates or only NOR gates
NAND
NOR
NOT
AND
OR
K. Webb
MAE 2055 – Mechetronics I
40
Logic Circuit Synthesis
We will now discuss how to use a truth table,
describing the desired circuit behavior, to a
physical circuit that realizes that behavior.
K. Webb
MAE 2055 – Mechetronics I
Boolean Algebra – logic circuit synthesis
41

Based on how we want a circuit to behave we can
generate a truth table
 e.g.,
we want the circuit to turn the heat on (output is
high) under certain conditions


From the truth table we can generate a Boolean
logic expression
Different forms of logic expressions describing the
circuit behavior defined by the truth table are
possible – two such forms are:
 Sum-of-products
form (SOP)
 Product-of-sums form (POS)
K. Webb
MAE 2055 – Mechetronics I
Sum-of-Products
42

Let’s generate a SOP Boolean logic expression to describe the behavior of
the circuit, as defined by the truth table
 Look at each row of the table for which H = 1
Truth Table
D
DL
NL
H
0
0
0
0
0
0
1
1
0
1
0
0
0
1
1
1
1
0
0
0
1
0
1
0
1
1
0
1
1
1
1
1
K. Webb

Using all input variables, write a logic product
that equals 1 for each of these rows
 Row 2:
D∙DL∙NL
These products
 Row 4:
D∙DL∙NL
are called
 Row 7:
D∙DL∙NL
minterms
 Row 8:
D∙DL∙NL

Create the SOP expression for the output
variable by summing the minterms
H = (D∙DL∙NL)+(D∙DL∙NL)+(D∙DL∙NL)+(D∙DL∙NL)
MAE 2055 – Mechetronics I
Sum-of-Products
43

Now we can use the SOP expression we’ve generated from the truth table
to synthesize a logic circuit using logic gates

Inverters provide
inverted values of each
input

Three-input AND gates
generate each minterm

Output, H, is generated
by combining all
minterms in a four-input
OR gate
K. Webb
MAE 2055 – Mechetronics I
Product-of-Sums
44

Let’s generate a POS Boolean logic expression to describe the behavior of
the circuit, as defined by the truth table
 Look at each row of the table for which H = 0
Truth Table
D
DL
NL
H
0
0
0
0
0
0
1
1
0
1
0
0
0
1
1
1
1
0
0
0
1
0
1
0
1
1
0
1
1
1
1
1
K. Webb

Using all input variables, write a logic sum that
equals 0 for each of these rows
 Row 1:
D+DL+NL
These products
 Row 3:
D+DL+NL
are called
 Row 5:
D+DL+NL
maxterms
 Row 6:
D+DL+NL

Create the POS expression for the output
variable by taking the product of the maxterms
H = (D+DL+NL)∙(D+DL+NL)∙(D+DL+NL)∙(D+DL+NL)
MAE 2055 – Mechetronics I
Product-of-Sums
45

Now we can use the POS expression we’ve generated from the truth table
to synthesize a logic circuit using logic gates

Inverters provide
inverted values of each
input

Three-input OR gates
generate each maxterm

Output, H, is generated
by combining all
maxterms in a fourinput AND gate
K. Webb
MAE 2055 – Mechetronics I
Logic Circuit Minimization
46


We just generated two logic circuit, a SOP circuit
and a POS circuit, that use different gates to
perform the same function
SOP and POS circuit generated following this
methodology do not necessarily realize the desired
logic function using the minimum number of gates
 The
SOP (POS) circuit in this example uses three
inverters, four 3-input AND (OR) gates, and one
4-input OR (AND) gate
 In fact it could be realized with one inverter, two 2input AND gates, and one 2-input OR gate
K. Webb
MAE 2055 – Mechetronics I
Logic Circuit Minimization
47

To see how we can reduce the number of gates, we’ll apply some of the
rules and identities of Boolean algebra to our original SOP expression:
H = (D∙DL∙NL)+(D∙DL∙NL)+(D∙DL∙NL)+(D∙DL∙NL)

Factor out a D∙NL from the first two minterms and a D∙DL from the second:
H = D∙NL∙(DL + DL) + D∙DL∙(NL + NL)

Next, apply the Boolean identity A + Ā = 1:
H = (D∙NL∙1) + (D∙DL∙1)

And the identity A∙1 = A, yielding a greatly simplified expression:
H = (D∙NL) + (D∙DL)
K. Webb
MAE 2055 – Mechetronics I
Minimum SOP Form
48

By applying a few rules and identities of boolean algebra, we’ve greatly
simplified the SOP logic expression defining our circuit, and we can now
implement the circuit with significantly fewer gates

In practice, many techniques exist for performing logic minimization

On such technique is using a tool known as Karnaugh maps, though this is beyond
the scope of this class
H = (D∙NL) + (D∙DL)
Heat is on if:
(It’s daytime AND the temp
is below the day set point)
OR
(it’s nighttime AND the
temp is below the night set
point)
K. Webb
MAE 2055 – Mechetronics I