The signals can be grouped into classes. The classes of signals which are relevant to linear systems are listed below.
A brief explanation for each class of signals is provided now. This topic will be presented in greater detail for the subject on Signals and Systems. Those undertaking the study of Circuit Analysis need to know the nature of signals and hence this page is presented.
We call a signal a continuous-time signal if it is defined for all time t. An example of a continuous-time signal is shown in Fig. 1. Here t is the independent variable. Many of the signals encountered are continuous-time signals. Let us take the case of a thermo-couple. The voltage at the terminals of a thermo-couple is a function of the temperature at the junction of the thermo-couple, and it is a continuous-time signal.
On the other hand, if a signal is defined only at discrete values of t, then it is a discrete-time signal. An example of a discrete-time signal is shown in Fig. 1. The discrete values of a discrete-time signal have fixed intervals, or in other words, they are spaced uniformly. A discrete-time signal can be obtained from a continuous-time signal by sampling it at a constant or uniform rate. The scope of this text does not extend to finding the response of an electrical circuit to a discrete-time signal.
There are differences between an analog signals and a continuous-time signal. In Fig. 1, f(t) is a continuous-time signal. Here t is the independent variable, and f(t) is defined for all values of t or a continuum of values of t. We can call f(t) as the dependent variable. If the amplitude of the dependent variable can take on any value, then we have analog signal. Now the question arises whether an analog signal should be continuous. The amplitudes of many signals, such as voltage, current, temperature, and pressure tend to be continuous, these signals are analog, continuous-time signals. We can sample an analog, continuous-time signal at a uniform rate and generate an analog, discrete-time signal. The plots in Fig. 2 show an analog, continuous-time signal and an analog, discrete-time signal.
The amplitude of an analog signal can be any value, whereas the amplitude of a digital signal can be one of a finite number of values. More often than not, a digital signal has only two values. The plots in Fig. 3 show digital signals.
To illustrate what periodic and aperiodic signals are, a few waveforms are presented below.
Fig. 4: A periodic sinusoidal signal
Fig. 5: A periodic square-wave signal
Fig. 5: An aperiodic exponential signal
A signal y(t) is said to be periodic over time T if equation (1.1) is valid. The period T is a positive, constant value and it is the smallest value time that renders the the waveform to be periodic or repetitive. The smallest period for the periodicity of the waveform is the fundamental period of y(t). The reciprocal of fundamental period is the fundamental frequency. On the other hand, a waveform that exhibits no periodicity is aperiodic. An exponential signal is aperiodic.
By definition, a periodic signal is assumed to last for ever, from
.
Otherwise equation (1.1) will not be valid for all t. Another property of a periodic function is expressed by equation (1.2).
The value of integral over a cycle period is the same for a periodic waveform, irrespective of the value of the lower limit of the integral. It is true because of the periodicity of waveform. Since a periodic waveform lasts for ever, it is an everlasting signal. An everlasting signal exists in theory, but not in real world.
A signal with finite energy is an energy signal. An exponentially decaying signal that exists only for t > 0 is an energy signal. On the other hand, a signal that has a finite and nonzero power is a power signal. The periodic signals are power signals, and since these signals are everlasting, they have infinite energy. Since the periodic signals have infinite energy, we calculate the power delivered by them. Power by definition is the energy per second. To summarize, a power signal has infinite energy and a finite power, and and an energy signal has a finite energy and zero power, where the average power is computed as energy over infinite time. It is clear that a signal cannot be both a power signal and an energy signal. It has to be either of the two. But it is possible for a signal to be neither of the two. For instance, a ramp signal is neither a power signal nor an energy signal. A ramp signal has infinite power and infinite energy. Such a signal exists in theory, but not in real world. If an everlasting exponential signal is defined as exp(at), it is neither an energy signal nor a power signal, as long as a is a positive or a negative real value. If the value of a is either zero or imaginary, then the signal is a power signal. If the value of a is zero, then the signal is a dc signal, and if the value of a is imaginary, then the signal is an alternating signal.
In network theory, the principal aim remains the analysis of the response of a circuit to a single signal or to a group of signals. Hence mathematical description of the signal itself becomes an essential requirement. It is shown that this mathematical description leads to an important classification of signals. It may be possible to predict a signal accurately as a function of time and then such a signal is called a deterministic signal. All past, present, and future values of a deterministic signal are known precisely without any ambiguity or uncertainty.
On the other hand, there are some signals which cannot be predicted accurately by a mathematical expression. Such a signal is called a random signal. A random signal can be described only in terms of probability. For example, the output of a noise generator is a random signal. In a car, the sparks across a spark plug produce noise signals which may be picked by an antenna at random. The scope of this text is restricted to response of a circuit to deterministic signals.
A continuous-time signal y(t) can be said to be an even signal, if equation (1.3) is satisfied. If equation (1.4) is satisfied, then y(t) can be said to be an odd signal. A continuous-time signal which is neither even nor odd can be shown to be the sum of an even part part and an odd part. More attention will be paid to this aspect of signals, when we take up the topic of Fourier Series.
In practice, the input signals to a circuit start at t = 0. Signals that start at t = 0 referred to as causal signals. The effect of signals applied prior to t = 0 are represented as the initial conditions in a circuit. In this text, we deal mostly with causal signals.
The operations performed on signals can be divided into two groups.
Operations on the signal
Operations on the independent variable
Amplitude Scaling
Given a signal y1(t), it can be multiplied by a scalar constant to yield y2(t), which has an amplitude equal to c times the amplitude of y1(t).
Addition
Given signals y1(t) and y2(t), the sum of these signals yields signal y3(t). Signal y3(t) is obtained by adding the amplitudes of y1(t) and y2(t) at every instant.
Differentiation and Integration
Given a continuous-time signal, its derivative and integral can be obtained as shown by equations (1.7) and (1.9). In a circuit, the inductor voltage is proportional to the derivative of its current, where the capacitor voltage is obtained by integrating its current. Equations (1.8) and (1.10) how expressions for the inductor voltage and the capacitor voltage can be obtained.
Time Scaling
An example is presented now to illustrate what time scaling is. There are two waveform presented in Fig. 6. Here the operation is performed on the independent variable t.
Equation (1.11) shows how the relationship between y(t) and x(t). In equation (1.11), a is a positive factor. If a > 1, then signal y(t) is compressed in time, as shown by the sketch in Fig. 6. On the other hand, if a < 1, then signal y(t) is expanded in time. We can obtain expression for x(2t) as shown below.
We can substitute t in equation (1.12) by 2t, as shown by equation (1.13). We get equation (1.14) from equation (1.13). It can be seen that the span of y(t) is compressed in time domain.
Time shifting or Translation in Time Domain
The plots in Fig. 7 show x(t) and x(t - 1), where x(t) is shifted in time by 1 second.
Equation (1.15) shows how the relationship between y(t) and x(t). In equation (1.15), a is a positive factor. Given x(t) as shown by equation (1.16), is obtained, as displayed by equations (1.17) and (1.18). When a is a positive, the waveform is shifted to the right, and the waveform is shifted to the left, when a is a negative. More examples are presented in the pages to follow.
Time reversal /Folding / Reflection
The folded or the reflected signal of x(t) is obtained by folding the signal or rotating the signal about the vertical axis. The folded waveform y(t) looks like the reflection of x(t).
Equation (1.19) shows how the relationship between y(t) and x(t). Given x(t) as shown by equation (1.20), equations (1.21) and (1.22) show how an expression for y(t) can be obtained.
The pages to follow present more information and some worked examples. The next page is on singularity functions.