The causality of a linear shift-invariant system means that the output y(n) of this system only depends on the inputs at this moment and before this moment, x(n), x(n- 1), x(n-2), ..., and is different from the inputs x(n+ 1), X (n) after this time. For example, the output of the system depends not only on x(n), x(n- 1), x(n-2), …, but also on x(n+ 1), x(n+2), x(n+3), that is, the output of the system is not only taken from current and past inputs. From the convolution formula (4-6- 1), we can see that the necessary and sufficient conditions of causal system are

Geophysical digital signal analysis and processing technology

Many important networks, such as ideal low-pass filters, are non-causal physically infeasible systems. But in digital signal processing, it is often non-real-time, even if it is real-time processing, it allows a great delay. At this point, for a certain output y(n), a large number of future inputs x(n+ 1), x(n+2), ... have been recorded in memory and can be called, so these non-causal systems can be very close to being realized. That is to say, the causal system with large time delay can be used to approximate the non-causal system, which is one of the characteristics that digital system is superior to analog system. Therefore, the digital system can obtain more ideal characteristics than the analog system.

A stable system means that as long as the input is bounded, the output must be bounded. Otherwise, it is an unstable system. The necessary and sufficient condition for a stable system is that its unit impulse response is absolutely integrable (that is, it is absolutely summable in a discrete system).

Geophysical digital signal analysis and processing technology

This can be proved as follows:

Look at the necessary conditions first. If h(n) does not conform to equation (4-7-2)

Geophysical digital signal analysis and processing technology

Then when the bounded signal of the following form is input,

Geophysical digital signal analysis and processing technology

The value of the output y(n) when n=0 will be

Geophysical digital signal analysis and processing technology

That is to say, y(0) will be unbounded. Therefore, Equation (4-7-2) is a necessary condition for system stability.

Look at the sufficient conditions. If equation (4-7-2) is satisfied, then any bounded input

Geophysical digital signal analysis and processing technology

Its output is always bounded, so the system must be stable. A system that satisfies both stability and causality conditions is called a stable causal system. The stable causal system is the most important system, and the unit impulse response of this system is both unilateral and bounded, that is,

Geophysical digital signal analysis and processing technology

This stable causal system is both realizable and stable, so this system is the goal of general digital system design.