Question 6.11: Total area under a function using the CTFT Find the total ar......
Total area under a function using the CTFT
Find the total area under the function x(t) = 10sinc((t + 4)/7).
Step-by-Step
Ordinarily we would try to directly integrate the function over all t.
Area = \int_{-\infty }^{\infty } x(t)dt = \int_{-\infty }^{\infty } 10sinc(\frac{t+4}{7})dt = \int_{-\infty }^{\infty }10 \frac{sin(\pi (t+4)/7)}{\pi (t+4)/7}dtThis integral is a sine integral (first mentioned in Example 6.6) defined by
Si(z) = \int_{0}^{z}{\frac{sin(t)}{t}}dt.
The sine integral can be found tabulated in mathematical tables books. However, evaluation of the sine integral is not necessary to solve this problem. We can use
X(0) = \int_{-∞}^{∞} x(t)dt.
First we find the CTFT of x(t), which is X( f ) = 70rect(7 f )e^{ j8πf} . Then Area = X(0) = 70.
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