Express the functionf(t)={4cos(9t),0,7โคt<8otherwise๐(๐ก)={4cosโก(9๐ก),7โค๐ก<80,๐๐กโ๐๐๐ค๐๐ ๐using Heaviside functions.Note: You must use the notation u(tโc)๐ข(๐กโ๐) rather than uc(t)๐ข๐(๐ก) in order for your answer to be accepted by Bolster Academy.
Question
Express the functionf(t)={4cos(9t),0,7โคt<8otherwise๐(๐ก)={4cosโก(9๐ก),7โค๐ก<80,๐๐กโ๐๐๐ค๐๐ ๐using Heaviside functions.Note: You must use the notation u(tโc)๐ข(๐กโ๐) rather than uc(t)๐ข๐(๐ก) 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 โค 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 โค 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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