Digital Filters Design for Signal and Image Processing

1.4. Properties of discrete-time systems

1.4.1. Invariant linear systems

The important features of a system are linearity, temporal shift invariance (or invariance in time) and stability.

A system represented by the operator T is termed linear if x₁, x₂ a₁, a₂ so we get:

A system is called time-invariant if the response to a delayed input of l samples is the delayed output of l samples; that is:

and this holds, whatever the input signal x(k) and the temporal shift l.

As well, a continuous linear system time-invariant system is always called a stationary (or homogenous) linear filter.

1.4.2. Impulse responses and convolution products

If the input of a system is the impulse unity δ(k), the output is called the impulse response of the system h(k), or:

Figure 1.9. Impulse response

A usual property of the impulse δ(k) helps us describe any discrete-time signal as the weighted sum of delayed ...