2 Properties of causal functions
Even though a function may be causal we shall still often use the step function to emphasize its causality and write . The following properties are easily verified.
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The sum of casusal functions is causal:
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The product of causal functions is causal:
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The derivative of a causal function is causal:
- The definite integral of a causal function is a constant.
Calculating the definite integral of a causal function needs care.
Consider where . There are 3 cases to consider and which are described in Figure 10:
Figure 10
- If then and so
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If
then
since, in the first integral and so whereas, in the second integral and so .
- If then since and so
Task!
If then find and
Find the derivative first:
Now obtain another integral representing :
You should obtain since
This follows because in the range to the step function and so that part of the integral is zero. In the other part of the integral .
Now complete the integration:
You should obtain 8.9817 (to 4 d.p.) since
Exercises
- Find the derivative with respect to of .
- Find the area under the curve between and .
- Find the area under the curve between and .