Can f(t) = u(t)5cos(2pi t) be represented directly as a fourier series?

yes, but only if u(t) is a periodic function. If f(t) is nonzero over a certain range of t only, it can be represented by a Fourier transform.

well the text in my notes just states that u(t) is a step function..and therefore how do i determine if its periodic .. To my understanding the unit step signal is not periodic..is that assumption right?

I believe then that this funtion can only be represented by fourier transform and laplace transform?

A Fourier series is periodic. If there is a single step function increase in u at some value of t, f(t) is not periodic. If the step function repeated each time t increases by 1, then f(t) would be periodic, and u(t) would be a kind of square wave. In that case you could use a Fouier series.

I am going to stop there. There may be more to u(t) than you are telling me.

To determine if the function f(t) = u(t)5cos(2πt) can be represented directly as a Fourier series, we need to consider its properties and the conditions for a function to be represented as a Fourier series.

A Fourier series is an expansion of a periodic function into an infinite sum of sines and cosines. For a function to be represented directly as a Fourier series, it needs to satisfy two conditions:

1. The function must be periodic: A function f(t) is periodic if there exists a nonzero T such that f(t + T) = f(t) for all values of t. In this case, the function f(t) = u(t)5cos(2πt) is indeed periodic as it is multiplied by the unit step function u(t) and, therefore, repeats itself for every value of t.

2. The function must have a finite number of discontinuities, with each discontinuity being finite or removable: A function can only be represented directly as a Fourier series if it is continuous or has a finite number of discontinuities. Discontinuities that are infinite, oscillating, or infinite in number cannot be represented directly.

In the given function f(t) = u(t)5cos(2πt), the unit step function u(t) acts as a switch that turns on the cosine function after t = 0. Since the cosine function is continuous, f(t) has no discontinuities, and thus, satisfies the second condition.

Therefore, the function f(t) = u(t)5cos(2πt) can be directly represented as a Fourier series since it is periodic and has no infinite or oscillating discontinuities.