Finding Fourier Series Coefficients

Question

I'm just beginning to learn about Fourier series and I'm trying to figure out how to find the Fourier series coefficients for $$x(t) = e^{j100\pi t}$$

I know that $$x(t) = \sum_{-\infty}^{\infty} a_{k} e^{jk(2\pi/T)t}$$

How do I go about finding the coefficient(s)? I know that I can get T = 1/50, but beyond that I don't even know where to begin. I think I'm supposed to be able to do this just by looking at it without having to solve the integral equation for $a_k$, but I don't know what I'm supposed to be looking for.

Answer 1

The answer is right there in the question. You have expressed the signal as linear sum of scaled complex exponentials.

Now the question is fourier series coefficients of a complex exponential$$e^{j100\pi t}$$

So you can see from the summation of x(t) that you have mentioned in the question that there is only one fourier series coefficient $a_1$ with a fundamental frequency $\omega_0$ because there is only one complex exponential in the input.The value of $\omega_0$ is $$ 100\pi$$. So $f_0$ = $(100\pi)/(2\pi)$ = 50

The value of $a_1$ is 1.

In other words you can express $$x(t)=(1).\ e^{(100 \pi)(1)t}$$ to see that k=1 and $a_1$=1.

Answer 2

The answer depends on what period you assume / assign the signal to have (i.e. what sampling rate you use ). As long as the ground frequency is divisible by $50$ you will get exactly one non-zero Fourier coefficient, the position being dependent on the quotient between 50 and the ground frequency.

The special case Karan brings up above is the Fourier series with highest possible ground frequency still able to represent the function.