CONTINUOUS-TIME AND DISCRETE-TIME SIGNALS

ANALOG AND DIGITAL SIGNALS

PERIODIC AND APERIODIC SIGNALS

ENERGY AND POWER SIGNALS

DETERMINISTIC AND RANDOM SIGNALS

ODD AND EVEN SIGNALS

CAUSAL SIGNALS

BASIC OPERATIONS ON SIGNALS

The signals can be grouped into classes. The classes of signals which are relevant to linear systems are listed below.

- Continuous-time and discrete-time signals.
- Analog and digital signals.
- Periodic and aperiodic signals.
- Energy and Power signals.
- Deterministic signals and random signals.
- Odd and even signals.
- Causal signals.

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

We call a signal a * continuous-time signal* if it is defined for all time

On the other hand, if a signal is defined only at discrete values of * t*, then it is a

There are differences between an analog signals and a continuous-time signal. In Fig. 1, * f*(

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*(

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(

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*(

In practice, the input signals to a circuit start at * t* = 0. Signals that start at

The operations performed on signals can be divided into two groups.

- Operations on the signal itself.
- Operations on the independent variable.

Operations on the signal

- Amplitude Scaling
- Addition
- Differentiation of continuous-time signal
- Integration of continuous-time signal

Operations on the independent variable

- Time scaling
- Time shifting
- Time reversal
**/**Folding**/**Reflection

**
Amplitude Scaling**

Given a signal **y**_{1}(* t*), it can be multiplied by a scalar constant to yield

**Addition**

Given signals **y**_{1}(* t*) and

**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*(

We can substitute * t* in equation (1.12) by 2

**Time shifting or Translation in Time Domain**

The plots in Fig. 7 show * x*(

Equation (1.15) shows how the relationship between * y*(

**Time reversal /Folding / Reflection **

The folded or the reflected signal of * x*(

Equation (1.19) shows how the relationship between * y*(

The pages to follow present more information and some worked examples. The next page is on singularity functions.