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.
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
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
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
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
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.
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