Perhaps the most common activity in science, medicine and engineering is the extraction of information from a signal, a procedure encompassing the field of signal analysis. In this context a signal will be a generic term describing a time varying voltage, v(t). Of course a signal can be viewed more generally as any physical quantity, such as pressure or temperature for example, which varies with time. In these cases a transducer is introduced which converts the physical variable to a voltage.
Since signals are time varying, they are by their very nature transient. An important aspect is the capturing or recording of the signal, or an associated attribute, to allow time for detailed analysis.
The act of signal processing invariably alters the nature of the signal. This may in fact be desirable. What is really important is that the desired information contained in the signal be preserved. In any event the alteration is mathematically represented by a response function.
The representation of a signal is worthy of consideration. In order to analyze any physical quantity a mathematical representation is introduced. This representation is not unique and its arbitrariness is often obvious. Consider the case of a 20 km south west wind. In a geographical coordinate system with y-axis oriented from south to north and x-axis from west to east, the wind velocity is represented by a vector with approximate components of 14.1 km along each axis, i.e., v = (14.1,14.1). In a coordinate system with the x-axis oriented along the wind direction the velocity vector becomes v = (20,0). Despite the different mathematical representations the physical quantity is still a 20 km wind out of the south-west. The relationship between different representations of the same physical object is a coordinate transformation, in this example a rotation. An analogous but more subtle situation arises with the representation of a signal. Clearly the most obvious and natural representation is as a function of time. This is for example the way it is observed on an oscilloscope. It is referred to as the time domain representation. An alternative and very useful representation of the signal is as a function of frequency. It is important to reiterate that in a sense this is simply an alternative coordinate system in which the same physical entity, the signal, is described. It is very worth noting that frequency and time are physically conjugate variables, frequency being rescaled to energy through Planck's constant.
If the value of a signal at a particular time is sufficient to uniquely determine the value at a later time, the signal is deterministic. In contrast, many signals are random or stochastic so that it is impossible to predict future values based upon the present, weather conditions being the most familiar example. An important random signal is the natural background signal present in all materials, referred to as noise. Noise is unavoidably superimposed upon all deterministic signals so in principle the latter are a philosophical idealization which can never really exist in the strictest sense. In practice it may be that the ratio of effects due to the signal is enormous compared to the contribution from the noise so the signal is deterministic from the practical point of view. One can always think of the situation as a superposition between two components, the desired signal and an unwanted interfering signal, the latter having noise as its fundamental limit. A major task is the recovery of the signal information in the presence of significant noise contribution, referred to as improving the signal-to-noise ratio. In many situations the desired signal is itself stochastic. Thus techniques developed to analyze random signals are generally important.