# Control Systems/Digital and Analog

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## 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:

### 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:

### 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:
* 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.
* 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.
* 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:

## Analog

By definition:

### Analog

    A signal is considered analog if it is defined for all points in time and if it can take any real magnitude value within its range.
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

If we have a given motor, we can show that the output of the motor (rotation in units of radians per second, for instance) is a function of the voltage that is input to the motor. We can show the relationship as such:
    ${\dot {\theta }}(v)=f(v)$
Where ${\dot {\theta }}$ is the output in terms of rad/sec, and $f(v)$ is the motor's conversion function between the input voltage ($v$) and the output. For any value of $v$ we can calculate out specifically what the rotational speed of the motor should be.

### Example: Analog Clock

Consider a standard analog clock, which represents the passage of time though the angular position of the clock hands. We can denote the angular position of the hands of the clock with the system of equations:
    $\phi _{h}=f_{h}(t)$
    $\phi _{m}=f_{m}(t)$
    $\phi _{s}=f_{s}(t)$
Where $\phi _{h}$ is the angular position of the hour hand, $\phi _{m}$ is the angular position of the minute hand, and $\phi _{s}$ is the angular position of the second hand. The positions of all the different hands of the clock are dependent on functions of time.
Different positions on a clock face correspond directly to different times of the day.

## Digital

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

### Digital

    A signal or system is considered digital if it is both discrete-time and quantized.
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 | Digital Waveform |

### Example: Digital Clock

As a common example, let's consider a digital clock: The digital clock represents time with binary electrical data signals of 1 and 0. The 1's are usually represented by a positive voltage, and a 0 is generally represented by zero voltage. Counting in binary, we can show that any given time can be represented by a base-2 numbering system:
    MinuteBinary Representation 11 101010 3011110 59111011
But what happens if we want to display a fraction of a minute, or a fraction of a second? A typical digital clock has a certain amount of precision, and it cannot express fractional values smaller than that precision.

## 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

Most modern automobiles today have integrated computer systems that monitor certain aspects of the car, and actually help to control the performance of the car. The speed of the car, and the rotational speed of the transmission are analog values, but a sampler converts them into digital values so the car computer can monitor them. The digital computer will then output control signals to other parts of the car, to alter analog systems such as the engine timing, the suspension, the brakes, and other parts. Because the car has both digital and analog components, it is a hybrid system.

## 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:
* Discrete time (sampled)
* Discrete magnitude (quantized)
* 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:
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