2.1.1. Linear System

A system refers to any entity that produces a unique output in response to a legitimate input. In wireless communications, communication channels, base stations, mobile stations, receivers, transmitters, frequency synthesizers, and even filters all are physical systems with different complexity. Mathematically, a system can be described in

(2.1.1)

where x( t) is the input (or excitation), y( t) is the output (or response), t is an independent variable usually representing time, and T is the operation performed by the system. Then the system is also viewed as a transformation (or mapping) of x( t) into y( t).

A system is linear if and only if the principle of superposition holds i.e., its output can be expressed as a linear combination of responses to individual inputs:

(2.1.2)

where ? ₁and ? ₂are arbitrary scalars. A system that does not satisfy the superposition relationship (2.1.2) is classified as nonlinear.

There is a class of linear systems called linear time-invariant ( LTI) systems that play particularly important role in communication system theory and design. A system is referred to as time-invariant if a time shift, ?, in the input signal x( t- ?) causes the same time shift in the output signal y( t). This is expressed as

(2.1.3)

where L is LTI system operator. LTI systems provide accurate models for a large...