# Signals and systems/Continuous-time signals

## Continuous-time signals

A continuous-time real (or complex) signal is any real-valued (or complex-valued) function which is defined for all time t in an interval, most commonly an infinite interval.

A continuous signal or a continuous-time signal is a varying quantity (a signal) that is expressed as a function of a real-valued domain, usually time. The function of time need not be continuous.

The signal is defined over a domain, which may or may not be finite, and there is a functional mapping from the domain to the value of the signal. The continuity of the time variable, in connection with the law of density of real numbers, means that the signal value can be found at any arbitrary point in time.

A typical example of an infinite duration signal is:

A finite duration counterpart of the above signal could be:

...and f(t) = 0 otherwise.

The value of a finite (or infinite) duration signal may or may not be finite. For example,

...and f(t) = 0 otherwise,

is a finite duration signal but it takes an infinite value for .

In many disciplines, the convention is that a continuous signal must always have a finite value, which makes more sense in the case of physical signals.

For some purposes, infinite singularities are acceptable as long as the signal is integrable over any finite interval (for example, the t − 1 signal is not integrable, but t − 2 is).

Any analogue signal is continuous by nature.

An analog or analogue signal is any time continuous signal for which the time varying feature (variable) of the signal is a representation of some other time varying quantity, i.e analogous to another time varying signal. It differs from a digital signal in that small fluctuations in the signal are meaningful. Analog is usually thought of in an electrical context; however, mechanical, pneumatic, hydraulic, and other systems may also convey analog signals.

Essentially an analogue signal can be thought of as a simulation or duplication of one continuous time varying quantity in another, possibly different, time varying quantity. It is then a mapping of one time varying quantity to anther, often with the intent of recording or transmitting information about the former within the medium of the latter.

An analog signal uses some property of the medium to convey the signal's information. For example, an aneroid barometer uses rotary position as the signal to convey pressure information. Electrically, the property most commonly used is voltage followed closely by frequency, current, and charge.

Any information may be conveyed by an analog signal; often such a signal is a measured response to changes in physical phenomena, such as sound, light, temperature, position, or pressure, and is achieved using a transducer.

For example, in sound recording, fluctuations in air pressure (that is to say, sound) strike the diaphragm of a microphone which causes corresponding fluctuations in a voltage or the current in an electric circuit. The voltage or the current is said to be an "analog" of the sound.

Data is more easily corrupted in analogue form due to noise but may also be of higher density and processed more quickly. A 3 hour Domestic VHS cassette could hold for example 16GB of data [1].

Any measured analog signal must theoretically have noise and a finite slew rate. Therefore, both analog and digital systems are subject to limitations in resolution and bandwidth. In practice, as analog systems become more complex, effects such as non-linearity and noise ultimately degrade analog resolution to such extent that the performance of digital systems may surpass it. In analog systems, it is difficult to detect when such degradation occurs. However, in digital systems, degradation can not only be detected but corrected as well.

The primary disadvantage of analog signaling is that any system has noise – i.e., random variation. As the signal is copied and re-copied, or transmitted over long distances, these random variations become dominant. Electrically, these losses can be diminished by shielding, good connections, and several cable types such as coaxial or twisted pair.

The effects of noise make signal loss and distortion impossible to recover, since amplifying the signal to recover attenuated parts of the signal amplifies the noise as well. Even if the resolution of an analog signal is higher than a comparable digital signal, in many cases, the difference is overshadowed by the noise in the signal.

## Modulation

Another method of conveying an analog signal is to use modulation. In this, some base signal (e.g., a sinusoidal carrier wave) has one of its properties modulated: amplitude modulation involves altering the amplitude of a sinusoidal voltage waveform by the source information, frequency modulation changes the frequency. Other techniques, such as changing the phase of the base signal also work.

Analog circuits do not involve quantisation of information into digital format. The concept being measured over the circuit, whether sound, light, pressure, temperature, or an exceeded limit, remains from end to end.

Clocks with hands are called analog; those that display digits are called digital. However, many analog clocks are actually digital since the hands do not move in a smooth continuous motion, but in small steps every second or half a second, or every minute.

See digital for a discussion of digital vs. analog.

## Sources

Some of an earlier version of this article was originally taken from Federal Standard 1037C in support of MIL-STD-188.