[1.12 SUMMARY](#page-8-0)

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1.12 SUMMARY

A signal is a set of data or information. A system processes input signals to modify them or extract additional information from them to produce output signals (response). A system may be made up of physical components (hardware realization), or it may be an algorithm that computes an output signal from an input signal (software realization).

A convenient measure of the size of a signal is its energy, if it is finite. If the signal energy is infinite, the appropriate measure is its power, if it exists. The signal power is the time average of its energy (averaged over the entire time interval from −∞ to ∞). For periodic signals, the time averaging need be performed over only one period in view of the periodic repetition of the signal. Signal power is also equal to the mean squared value of the signal (averaged over the entire time interval from t = −∞ to ∞).

Signals can be classified in several ways.

• 1. A continuous-time signal is specified for a continuum of values of the independent variable (such as time t). A discrete-time signal is specified only at a finite or a countable set of time instants.
• 2. An analog signal is a signal whose amplitude can take on any value over a continuum. On the other hand, a signal whose amplitudes can take on only a finite number of values is a digital signal. The terms discrete-time and continuous-time qualify the nature of a signal along the time axis (horizontal axis). The terms analog and digital, on the other hand, qualify the nature of the signal amplitude (vertical axis).
• 3. A periodic signal x(t) is defined by the fact that x(t) = x(t +T0) for some T0. The smallest positive value of T⁰ for which this relationship is satisfied is called the fundamental period. A periodic signal remains unchanged when shifted by an integer multiple of its period. A periodic signal x(t) can be generated by a periodic extension of any contiguous segment of x(t) of duration T0. Finally, a periodic signal, by definition, must exist over the entire time interval −∞ < t < ∞. A signal is aperiodic if it is not periodic.
• 4. An everlasting signal starts at t = −∞ and continues forever to t = ∞. Hence, periodic signals are everlasting signals. A causal signal is a signal that is zero for t < 0.
• 5. A signal with finite energy is an energy signal. Similarly a signal with a finite and nonzero power (mean-square value) is a power signal. A signal can be either an energy signal or a power signal, but not both. However, there are signals that are neither energy nor power signals.
• 6. A signal whose physical description is known completely in a mathematical or graphical form is a deterministic signal. A random signal is known only in terms of its probabilistic description such as mean value or mean-square value, rather than by its mathematical or graphical form.

A signal x(t) delayed by T seconds (right-shifted) can be expressed as x(t − T); on the other hand, x(t) advanced by T (left-shifted) is x(t + T). A signal x(t) time-compressed by a factor a(a > 1) is expressed as x(at); on the other hand, the same signal time-expanded by factor a(a > 1) is x(t/a). The signal x(t) when time-reversed can be expressed as x(−t).

The unit step function u(t) is very useful in representing causal signals and signals with different mathematical descriptions over different intervals.

In the classical (Dirac) definition, the unit impulse function δ(t) is characterized by unit area and is concentrated at a single instant t = 0. The impulse function has a sampling (or sifting) property, which states that the area under the product of a function with a unit impulse is equal to the value of that function at the instant at which the impulse is located (assuming the function to be continuous at the impulse location). In the modern approach, the impulse function is viewed as a generalized function and is defined by the sampling property.

The exponential function est, where s is complex, encompasses a large class of signals that includes a constant, a monotonic exponential, a sinusoid, and an exponentially varying sinusoid.

A real signal that is symmetrical about the vertical axis (t = 0) is an even function of time, and a real signal that is antisymmetrical about the vertical axis is an odd function of time. The product of an even function and an odd function is an odd function. However, the product of an even function and an even function or an odd function and an odd function is an even function. The area under an odd function from t = −a to a is always zero regardless of the value of a. On the other hand, the area under an even function from t = −a to a is two times the area under the same function from t = 0 to a (or from t = −a to 0). Every signal can be expressed as a sum of odd and even functions of time.

A system processes input signals to produce output signals (response). The input is the cause, and the output is its effect. In general, the output is affected by two causes: the internal conditions of the system (such as the initial conditions) and the external input.

Systems can be classified in several ways.

• 1. Linear systems are characterized by the linearity property, which implies superposition; if several causes (such as various inputs and initial conditions) are acting on a linear system, the total output (response) is the sum of the responses from each cause, assuming that all the remaining causes are absent. A system is nonlinear if superposition does not hold.
• 2. In time-invariant systems, system parameters do not change with time. The parameters of time-varying-parameter systems change with time.
• 3. For memoryless (or instantaneous) systems, the system response at any instant t depends only on the value of the input at t. For systems with memory (also known as dynamic

systems), the system response at any instant t depends not only on the present value of the input, but also on the past values of the input (values before t).

• 4. In contrast, if a system response at t also depends on the future values of the input (values of input beyond t), the system is noncausal. In causal systems, the response does not depend on the future values of the input. Because of the dependence of the response on the future values of input, the effect (response) of noncausal systems occurs before the cause. When the independent variable is time (temporal systems), the noncausal systems are prophetic systems, and therefore, unrealizable, although close approximation is possible with some time delay in the response. Noncausal systems with independent variables other than time (e.g., space) are realizable.
• 5. Systems whose inputs and outputs are continuous-time signals are continuous-time systems; systems whose inputs and outputs are discrete-time signals are discrete-time systems. If a continuous-time signal is sampled, the resulting signal is a discrete-time signal. We can process a continuous-time signal by processing the samples of the signal with a discrete-time system.
• 6. Systems whose inputs and outputs are analog signals are analog systems; those whose inputs and outputs are digital signals are digital systems.
• 7. If we can obtain the input x(t) back from the output y(t) of a system S by some operation, the system S is said to be invertible. Otherwise the system is noninvertible.
• 8. A system is stable if bounded input produces bounded output. This defines external stability because it can be ascertained from measurements at the external terminals of the system. External stability is also known as the stability in the BIBO (bounded-input/bounded-output) sense. Internal stability, discussed later in Ch. 2, is measured in terms of the internal behavior of the system.

The system model derived from a knowledge of the internal structure of the system is its internal description. In contrast, an external description is a representation of a system as seen from its input and output terminals; it can be obtained by applying a known input and measuring the resulting output. In the majority of practical systems, an external description of a system so obtained is equivalent to its internal description. At times, however, the external description fails to describe the system adequately. Such is the case with the so-called uncontrollable or unobservable systems.

A system may also be described in terms of certain set of key variables called state variables. In this description, an Nth-order system can be characterized by a set of N simultaneous first-order differential equations in N state variables. State equations of a system represent an internal description of that system.