##### 2.1 INTRODUCTION

Linear systems are those that satisfy both homogeneity and additivity.

**(i) Homogeneity:** Let *x* be the input to a linear system and *y* the corresponding output, as shown in Fig. 2.1. If the input is doubled (2*x*), then the output is also doubled (2*y*). In general, a system is said to exhibit homogeneity if, for the input *nx* to the system, the corresponding output is *ny* (where *n* is an integer). Thus, a linear system enables us to predict the output.

**FIGURE 2.1** A linear system

**(ii) Additivity:** For two input signals *x*_{1} and *x*_{2} applied to a linear system, let *y*_{1} and *y*_{2} be the corresponding output signals. Further, if (*x*_{1} + *x*_{2}) is the input to the linear system and (*y*_{1} + *y*_{2}) the corresponding output, it means that the measured response will just be the sum of its responses to each of the inputs presented separately. This property is called additivity. Homogeneity and additivity, taken together, comprise the principle of superposition.

**(iii) Shift invariance:** Let an input *x* be applied to a linear system at time *t*_{1}. If the same input is applied at a different time instant *t*_{2}, the two outputs should be the same except for the corresponding shift in time. A linear system that exhibits this property is called a shift-invariant linear system. All linear systems are not necessarily shift invariant.

A circuit employing linear circuit components, namely, *R, L* and *C* can be termed a linear circuit. When a sinusoidal signal is applied to either *RC* or *RL* circuits, the shape of the signal is preserved at the output, with a change in only the amplitude and the phase. However, when a non-sinusoidal signal is transmitted through a linear network, the form of the output signal is altered. The process by which the shape of a non-sinusoidal signal passed through a linear network is altered is called linear waveshaping. We study the response of high-pass *RC* and *RL* circuits to different types of inputs in the following sections.