1.4. Properties of discrete-time systems – 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 pulses:

The output of an invariant continuous linear system can therefore be expressed in the following form:

The output y(k) thus corresponds to the convolution product between the input x(k) and the impulse response h(k):

We see that the convolution relation has its own legitimacy; that is, it is not obtained by a discretization of the convolution relation obtained in continuous systems. Using the example of a continuous system, we need only two hypotheses to establish this relation: those of invariance and linearity.

1.4.3. Causality

The impulse response filter h(k) is causal when the output y(k) remains null as long as the input x(k) is null. This corresponds to the philosophical principle of causality, which states that all precedent causes have consequences. An invariant linear system is causal only if its output for every k instant (that is, y(k)), depends solely on the present and past (x(k), x(k−1),… and so on).

Given the relation in equation (1.34), its impulse response satisfies the following condition:

An impulse response filter h(k) is termed anti-causal when the impulse response filter h(-k) is causal; that is, it becomes causal after inversion in the sense of time. The output of rank k then depends only on the inputs that are superior, or equal to k.

1.4.4. Interconnections of discrete-time systems

Discrete-time systems can be interconnected either in cascade (series) or in parallel to obtain new systems. These are represented, respectively, in Figures 1.10 and 1.11.

Figure 1.10. Interconnection in series

For interconnection in series, the impulse response of the resulting system h(k) is represented by h(k) = h1 (k)* h2(k). Thus, subject to the associativity of the law *, we have:

Figure 1.11. Interconnection in parallel

For a interconnection in parallel, the impulse response of the system h(k) is written as h(k) = h1 (k) + h2(k).

So we have: