Express the functionf(t)={4cos(9t),0,7≤t<8otherwise𝑓(𝑡)={4cos⁡(9𝑡),7≤𝑡<80,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒using Heaviside functions.

The Heaviside function, also known as the step function, is defined as:

H(t) = {0 for t < 0, 1 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 function using Heaviside functions as follows:

f(t) = 4cos(9t)[H(t-7) - H(t-8)]

Here, H(t-7) is a Heaviside function that equals 1 for t ≥ 7 and 0 for t < 7. Similarly, H(t-8) is a Heaviside function that equals 1 for t ≥ 8 and 0 for t < 8.

The difference [H(t-7) - H(t-8)] equals 1 for 7 ≤ t < 8 and 0 otherwise, which matches the domain of the original function f(t). Therefore, the function f(t) = 4cos(9t)[H(t-7) - H(t-8)] is equivalent to the original function.