Is life a non-causal stochastic system

Basic system properties

An essential prerequisite for the feasibility of a system is the requirement that the values ​​of the output signals at a given point in time only depend on values ​​of the input signals at this point in time or at an earlier point in time, but not on future values. This behavior is defined for time-continuous systems as the property of the causality of a system, which is also essential in the discrete case. If a system description is available via a difference equation, the causality can be assessed directly.

(4.36)

Is the coefficient c0 ¹ 0, the assumption c0 = 1 are hit. If this is not the case, the equation is replaced by c0 divided. With this assumption, the difference equation can be solved for y [k], and it results

(4.37)

Since all indices m and n are greater than or equal to zero, a system that can be described by a linear difference equation of the form from equation (4.37) is a causal system. If in the difference equation (4.36) the coefficient c0 = 0, cannot be solved for y [k]. Solving for y [k - 1] results in the equation

(4.38)

The output value y [k − 1] is therefore dependent on the future input value u [k]. The system is therefore for c0 = 0 not causal.

Due to the causality requirement, some system considerations of digital signal processing are complex. Therefore, for reasons of simplicity, non-causal systems are also considered below. However, these systems can often be made causal and thus realizable through a shift.

Example: causality of the moving average

When describing time-discrete systems, the moving average is presented. He has the difference equation

(4.39)

Because the output signal y [k] only depends on current and past input values, the system is causal. However, it has a time delay that is still shown when calculating frequency responses. A system with the difference equation

(4.40)

is no longer causal, but - as will be shown later - has no time lag. Often all measured values ​​are recorded first in data processing and then evaluated. In this case, future values ​​are also available. The causality of systems is not absolutely necessary in this case. Figure 4.9 shows the step response of a causal and a non-causal system for calculating a moving average.

Figure 4.9: Step response of a causal and a non-causal system for calculating a moving average

It can be clearly seen that the non-causal system reacts before the jump in the input signal has taken place, while the causal signal reacts only after the actual stimulation.