Express the functionf(t)={4cos(9t),0,7≤t<8otherwise𝑓(𝑡)={4cos(9𝑡),7≤𝑡<80,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒using Heaviside functions.Note: You must use the notation u(t−c)𝑢(𝑡−𝑐) rather than uc(t)𝑢𝑐(𝑡) in order for your answer to be accepted by Bolster Academy.
Question
Express the functionf(t)={4cos(9t),0,7≤t<8otherwise𝑓(𝑡)={4cos(9𝑡),7≤𝑡<80,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒using Heaviside functions.Note: You must use the notation u(t−c)𝑢(𝑡−𝑐) rather than uc(t)𝑢𝑐(𝑡) in order for your answer to be accepted by Bolster Academy.
Solution
The Heaviside function, also known as the step function, is defined as:
u(t) = {1 for t >= 0, 0 for t < 0}
We can use this function to express the given function f(t) in terms of Heaviside functions. The given function f(t) is defined as 4cos(9t) for 7 ≤ t < 8 and 0 otherwise.
We can express this using Heaviside functions as follows:
f(t) = 4cos(9t) * [u(t - 7) - u(t - 8)]
Here, u(t - 7) is 1 for t >= 7 and 0 for t < 7, and u(t - 8) is 1 for t >= 8 and 0 for t < 8.
So, the term [u(t - 7) - u(t - 8)] is 1 for 7 ≤ t < 8 and 0 otherwise, which matches the condition for f(t) to be 4cos(9t).
Therefore, f(t) = 4cos(9t) * [u(t - 7) - u(t - 8)] is the expression of the given function using Heaviside functions.
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