# Control Systems/Digital and Analog

## Digital and Analog

There is a significant distinction between an **analog system** and a
**digital system**, in the same way that there is a significant
difference between analog and digital data. This book is going to
consider both analog and digital topics, so it is worth taking some time
to discuss the differences, and to display the different notations that
will be used with each.

### Continuous Time

A signal is called **continuous-time** if it is defined at every time t.

A system is a continuous-time system if it takes a continuous-time input
signal, and outputs a continuous-time output signal. Here is an example
of an analog waveform:

{{-}}

![](Analog_Waveform.svg "Analog_Waveform.svg"){width="400"}

### Discrete Time

A signal is called **discrete-time** if it is only defined for
particular points in time. A discrete-time system takes discrete-time
input signals, and produces discrete-time output signals. The following
image shows the difference between an analog waveform and the sampled
discrete time equivalent: {{-}}

![](Sampled_Waveform.svg "Sampled_Waveform.svg"){width="400"}

### Quantized

A signal is called **Quantized** if it can only be certain values, and
cannot be other values. This concept is best illustrated with examples:

1.  Students with a strong background in physics will recognize this
    concept as being the root word in \"Quantum Mechanics\". In quantum
    mechanics, it is known that energy comes only in discrete packets.
    An electron bound to an atom, for example, may occupy one of several
    discrete energy levels, but not intermediate levels.
2.  Another common example is population statistics. For instance, a
    common statistic is that a household in a particular country may
    have an average of \"3.5 children\", or some other fractional
    number. Actual households may have 3 children, or they may have 4
    children, but no household has 3.5 children.
3.  People with a computer science background will recognize that
    integer variables are quantized because they can only hold certain
    integer values, not fractions or decimal points.

The last example concerning computers is the most relevant, because
quantized systems are frequently computer-based. Systems that are
implemented with computer software and hardware will typically be
quantized.

Here is an example waveform of a quantized signal. Notice how the
magnitude of the wave can only take certain values, and that creates a
step-like appearance. This image is discrete in magnitude, but is
continuous in time:

![](Quantized_Waveform.svg "Quantized_Waveform.svg"){width="400"}

## Analog

By definition:

An analog system is a system that represents data using a direct
conversion from one form to another. In other words, an analog system is
a system that is continuous in both time and magnitude.

### Example: Motor

### Example: Analog Clock

## Digital

Digital data is represented by discrete number values. By definition:

Digital data always have a certain granularity, and therefore there will
almost always be an error associated with using such data, especially if
we want to account for all real numbers. The tradeoff, of course, to
using a digital system is that our powerful computers with our powerful,
Moore\'s law microprocessor units, can be instructed to operate on
digital data only. This benefit more than makes up for the shortcomings
of a digital representation system.

Discrete systems will be denoted inside square brackets, as is a common
notation in texts that deal with discrete values. For instance, we can
denote a discrete data set of ascending numbers, starting at 1, with the
following notation:

:   x\[n\] = \[1 2 3 4 5 6 \...\]

**n**, or other letters from the central area of the alphabet (m, i, j,
k, l, for instance) are commonly used to denote discrete time values.
Analog, or \"non-discrete\" values are denoted in regular expression
syntax, using parenthesis. Here is an example of an analog waveform and
the digital equivalent. Notice that the digital waveform is discrete in
both time and magnitude:

+----------------------------------+----------------------------------+
| ![](Analog_Waveform.svg "An      | ![](Digital_Waveform.svg "Dig    |
| alog_Waveform.svg"){width="400"} | ital_Waveform.svg"){width="400"} |
+----------------------------------+----------------------------------+
| ```{=html}                       | ```{=html}                       |
| <center>                         | <center>                         |
| ```                              | ```                              |
| **Analog Waveform**              | **Digital Waveform**             |
|                                  |                                  |
| ```{=html}                       | ```{=html}                       |
| </center>                        | </center>                        |
| ```                              | ```                              |
+----------------------------------+----------------------------------+

### Example: Digital Clock


:   {\| class=\"wikitable\"

!Minute !! Binary Representation \|- \|1 \|\| 1 \|- \|10 \|\| 1010 \|-
\|30 \|\| 11110 \|- \|59 \|\| 111011 \|}

## Hybrid Systems

**Hybrid Systems** are systems that have both analog and digital
components. Devices called **samplers** are used to convert analog
signals into digital signals, and Devices called **reconstructors** are
used to convert digital signals into analog signals. Because of the use
of samplers, hybrid systems are frequently called **sampled-data
systems**.

### Example: Automobile Computer

## Continuous and Discrete

A system is considered **continuous-time** if the signal exists for all
time. Frequently, the terms \"analog\" and \"continuous\" will be used
interchangeably, although they are not strictly the same.

Discrete systems can come in three flavors:

1.  Discrete time (sampled)
2.  Discrete magnitude (quantized)
3.  Discrete time and magnitude (digital)

**Discrete magnitude** systems are systems where the signal value can
only have certain values.**Discrete time** systems are systems where
signals are only available (or valid) at particular times. Computer
systems are discrete in the sense of (3), in that data is only read at
specific discrete time intervals, and the data can have only a limited
number of discrete values.

A discrete-time system has a **sampling time** value associated with it,
such that each discrete value occurs at multiples of the given sampling
time. We will denote the sampling time of a system as T. We can equate
the square-brackets notation of a system with the continuous definition
of the system as follows:

$$x[n] = x(nT)$$

Notice that the two notations show the same thing, but the first one is
typically easier to write, *and* it shows that the system in question is
a discrete system. This book will use the square brackets to denote
discrete systems by the sample number n, and parenthesis to denote
continuous time functions.

## Sampling and Reconstruction

The process of converting analog information into digital data is called
\"Sampling\". The process of converting digital data into an analog
signal is called \"Reconstruction\". We will talk about both processes
in a later chapter. For more information on the topic than is available
in this book, see the Analog and Digital
Conversion wikibook. Here is
an example of a reconstructed waveform. Notice that the reconstructed
waveform here is quantized because it is constructed from a digital
signal:

![](Reconstructed_Waveform.svg "Reconstructed_Waveform.svg"){width="400"}

