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# Signal (information theory)

A signal is an abstract element of information, or (more commonly) a flow of information (in one or more dimensions).

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## Definition

A signal is any physical phenomenon that can be modeled as a function from time or position to some scalar- or vector-valued domain that can be real or complex valued.

### Continuous versus discrete

The distinction made between a continous and discrete signal is only with respect to the time. A continuous signal is a continuous function of time while a discrete signal is a discrete function of time.

### Analog versus digital

Unlike the previous comparison of continuous vs. discrete, this distinction considers all dimensions of a signal. A signal that is continuous in all dimensions is an analog signal while a digital signal is discrete in all dimensions. A digital signal is realized from an analog signal when the signal is sampled in time and quantized or sampled in the other dimensions.

For example, CD quality music is 16-bit (the quantization) and is sampled at 44.1 kilohertz. A digital image is sampled in the two spatial dimensions (giving the image a fixed width and height (e.g., 800x600 pixels)) and quantized in intensity (i.e. the brightness of the image at that point). It might be worth noting that even a photograph is a discrete signal because the film used contains photo-sensitive grains.

## Sampling

With the increasing use of computers the usage and need of digital signal processing has increased. In order to use an analog signal on a computer it must be digitized with an analog to digital converter (ADC). In order to properly sample an analog signal the Nyquist-Shannon sampling theorem must be satisfied. In short, the sampling frequency must be greater than twice the bandwidth of the signal (provided it is filtered appropriately). A digital to analog converter (DAC) is used to convert the digital signal back to analog. The use of a digital computer is a key ingredient into digital control systems.

## Abstract definition

In more abstract terms, a signal could be considered a n-tuple: $D:\, \left(d_1, d_2, \cdots, d_n \right)$.

### Examples of signals

#### Compact discs

For compact discs (CDs), there are 88,200 samples per second for two channels, which corresponds to 44,100 samples per second per channel. Digital stereo audio could then be represented as a 3-tuple $D:\, \left(value, channel, time \right)$ where value is the amplitude; channel is either the left or right channel; and "time" is an integer index from 1 to the number of samples in the music. All three indices of CD 3-tuple are discretized and it is, therefore, a discrete signal.

#### Digital images

For digital cameras, the pixel count represents the number of spatial samples contained within the image. For example, Canon's digital camera, the 1DsII , has 16,700,000 pixels (or 16.7 megapixels or 16.7 MP) per image. A color digital image could then be described with a 5-tuple $D:\, \left(R, G, B, x, y \right)$ where R, G and B are the red, green and blue values, respectively, and x and y are integer location indices with range as the width and height of the image. All five indices of a digital color image are discretized and it is, therefore, a discrete signal.

#### Digital video

Digital video would be a 6-tuple $D:\, \left(R, G, B, x, y, time \right)$ that is the 5-tuple for an image but an added element of time. All six indices of a digital video are discretized and it is, therefore, a discrete signal.

A Radar-like plot (assumed to be instantaneous instead of sweeping) could be considered a 4-tuple $D:\, \left(velocity, distance, angle, elevation \right)$ where velocity is the velocity of the object; distance is how far away the object is; angle is the rotational angle; and elevation is the elevation the object is at.

A "digital video" of radar-like plot would be a 5-tuple $D:\, \left(velocity, distance, angle, elevation, time\right)$.

#### Theoretical sampler

In a theoretical sampler, a continuous signal is multiplied by a Dirac comb. This sampled signal is discretized in time but not in value (all other values in the continuous signal are zero) but the value is still continuous. Only when this signal is quantized does it become a digital audio signal where all three indices are discretized.

### Dimensionality

In signal processing, signals are usually labeled with a dimension. The dimensionality can be observed from the n-tuple representation by counting the number of indices that represent time or space. The dimensionality of the above examples (and some others) are:

• Monaural audio is 1-dimensional (the channel index can be dropped)
• Stereo audio is 2-dimensional
• Dolby's 5.1 audio is 2-dimensional (with six channels)
• A digital image is 2-dimensional
• A digital video is 3-dimensional
• A digital video with Dolby 5.1 audio would be 4-dimensional (2 for video, 1 for audio and a common time index)
• A radar plot is 3-dimensional

## Analog signals

While analog signals exist on paper, they do not exist in reality. This is the result of Planck time, Planck length, and Planck energy units. In other words, all reality-based signals are digital signals but with extremely small quantization levels. In treating a signal as an analog signal it is for mathematical purposes or for simplicity's sake by considering the extremely small quantizations as negligible. This realization that analog signals are only theoretical is one not usually made and so assuming such quantizations as negligible is not one to quibble over except in theoretical arguments.

A second view: The original author (above) raises a very interesting point, but this author feels that his/her depiction of quantization effects at the Planck and Heisenberg level is a bit of a misrepresentation. Microscopic, Nano-scopic, and Pico-scopic signals (such as the sound waves resulting from the orderly transmission of a vibration through the atmosphere or other fluid type substance, be it liquid, gas or other) are indeed only idealized as mathematically continuous functions in the sense of the real numbers. But, neither are they digital signals in the modern sense of sampled data with characteristic sampling rates, bit depths and associated engineering baggage necessary for the adequate perception, conceptualization and processing of such a form of a real world signal.

The important point to note here is that of data granularity. Digitized signals have obvious granularity, and the first author attempts to draw a parallel with the granularity described by quantum physics. One important distinction to consider is the uncertainty inherent in quantum mechanics which we have from Heisenberg, as well as the non-linear dynamics and other mechanisms of chaotic and complex systems theories, which we have from statistical mechanics (thermodynamics and its cousins). Computer digital signals attempt as a general rule to be as precise as possible, ideally with zero variation and no ambiguity in any of the measured features of the data. While this is proveably akin to a dream at some resolutions, at higher resolutions, the effects can be ignored, as the first author states. In between, however, lies the realm of noise effects. Brownian motion is a famous example of this.

If the granularity of the reality which one measures is finer than the granularity of one's instruments, then one can map the system to the continuous real numbers, and its higher dimensional forms ( 2D, 3D spaces and higher). But keep the point in mind that these mathematical systems are indeed mere approximations to the real system (even if one has a good theoretical framework to describe the system!), and one should always be aware of the underlying dogma of one's assumptions, most especially the context in which these assumptions are reasonably valid.

## Frequency spectrum

See frequency domain and frequency spectrum.

## Other notes

Another important propery of a signal (actually, of a statistically defined class of signals) is its entropy or information contents, measured in bits (or bits per second, or bits per square millimeter, etc.).