Signal (information theory)

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In information theory, a signal is the sequence of states of a communications channel that encodes a message. In a communications system, a transmitter encodes a message into a signal, which is carried to a receiver by the communications channel. For example, the words "Mary had a little lamb" might be the message spoken into a telephone. The telephone transmitter converts the sounds into an electrical voltage signal. The signal is transmitted to the receiving telephone by wires; and at the receiver it is reconverted into sounds.

Information theory studies both continuous signals, commonly called analog signals; and discrete signals (Shannon 2005, 3), or quantized signals, of which the most common today are digital signals . The information carried by a signal may be measured either on a per second basis, or per transmitted symbol; that is, either in continuous or discrete time (ibid, 19).

In information theory, the message is generated by a stochastic process, and the transmitted signal derives its statistical properties from the message. Conversely, usage of signal in reference to a process that generates a transmitted sequence of states in a communications channel implies that this process is stochastic. When it is not stochastic, misunderstandings can be created. Oldberg. (2005) reports that misunderstandings of this type plague the field of Defect Detection Testing.

Analog and digital signals

The two main types of signals are analog and digital. In short, the difference between them is that digital signals are discrete and quantized, as defined below, while analog signals possess neither property.

Discretization

Main article: Discrete signal

One of the fundamental distinctions between different types of signals is between continuous and discrete time. In the mathematical abstraction, the domain of a continuous-time (CT) signal is the set of real numbers (or some interval thereof), whereas the domain of a discrete-time signal is the set of integers (or some interval). What these integers represent depends on the nature of the signal.

DT signals often arise via of CT signals. For instance, sensors output data continuously, but since a continuous stream is impossible to record, a discrete signal is used as an approximation. Computers and other digital devices are restricted to discrete time.

Quantization

Main article: Quantization (signal processing)

If a signal is to be represented as a sequence of numbers, it is impossible to maintain arbitrarily high precision - each number in the sequence must have a finite number of digits. As a result, the values of such a signal are restricted to belong to a finite set; in other words, it is quantized.

Examples of signals

  • Motion. One can conceive of a signal representing the motion of a particle - say, a mote of dust, through some suitable space. The domain of a motion signal is one-dimensional (time), and the range is generally three-dimensional.
  • Sound. Since a sound is a vibration of a medium (such as air), a sound signal associates a pressure value to every value of time. In the real world, sound signals are analog.
  • Compact discs (CDs). CDs contain discrete signals representing sound, recorded at 44,100 samples per second. Each sample contains data for a left and right channel (since CDs are recorded in stereo).
  • Pictures. A picture assigns a color value to each of a set of points. Since the points lie on a plane, the domain is two-dimensional. If the picture is a physical object, such as a painting, it's a continuous signal. If the picture a digital image, it's a discrete signal. It's often convenient to represent color as the sum of the intensities of three primary colors, so that the signal is vector-valued with dimension three.
  • Videos. A video is a series of images. A point in a video is identified by its position (two-dimensional) and by the time at which it occurs, so a video signal has a three-dimensional domain.

Frequency analysis

Main article: Frequency domain

It is remarkably useful to analyze the frequency spectrum of a signal. This technique is applicable to all signals, both continuous and discrete. For instance, if a signal is passed through an LTI system, the frequency spectrum of the resulting output signal is the product of the frequency spectrum of the original input signal and the frequency response of the system.

Entropy

Another important propery of a signal (actually, of a statistically defined class of signals) is its entropy or information content.

See also

Works cited

Oldberg, T., 2005, "An Ethical Problem in the Statistics of Defect Detection Test Reliability," ndt.net, http://www.ndt.net/article/v10n05/oldberg/oldberg.htm .

Shannon, C. E., 2005 [1948], "A Mathematical Theory of Communication," (corrected reprint), accessed Dec. 15, 2005. Orig. 1948, Bell System Technical Journal, vol. 27, pp. 379-423, 623-656.