Discrete Fourier transform of sampled continuous signal

[Bromio]

Homework Statement

Let a system that converts a continuos-time signal to a discrete-time signal. The input x(t) is periodic with period of 0.1 second. The Fourier series coefficients of x(t) are $$X_k = \displaystyle\left(\frac{1}{2}\right)^{|k|}$$. The ideal lowpass filter $H(\omega)$ is equal to 0 for $\left|\omega\right| > 205\pi$. The sampling period is T = 0.005 seconds.

Determine the Fourier series coefficients of x[n].

Homework Equations

$X\left(\Omega\right) = X_s\left(\omega\right), \omega = \Omega/Ts$

The Attempt at a Solution

The Fourier transform of $X\left(\omega\right) = 2\pi\displaystyle\sum_{k=-\infty}^{\infty} X_k\delta(\omega-20\pi k)$.

The output of the filter is $X_c\left(\omega\right) = 2\pi\displaystyle\sum_{k=-10}^{10} X_k\delta(\omega-20\pi k)$ and the last impulse has $\omega = 200\pi$.

When $X_c(\omega)$ is multiplied by $P(\omega)$, I obtain $X_s(\omega) = \displaystyle\frac{1}{T_s}\sum_{k=-\infty}^{\infty} X_c\left(\omega-\omega_s k\right)$

With the expression written in 2., I've the Fourier transform of x[n].

So, I think that $X_k = \displaystyle\frac{1}{T_s}\left(\frac{1}{2}\right)^{|k|},\;|k| = 0, 1, 2,...10$.

Is this correct?

Thank you.

 

[KataKoniK]

Based on the information provided, it seems like you have the right approach to solving this problem. Your calculation for X_c(\omega) and X_s(\omega) looks correct, and your final expression for X_k also seems reasonable.

However, I would suggest double-checking your calculations and making sure that you have correctly accounted for all the factors involved in the Fourier series and transform equations. Also, it might be helpful to plot the resulting X_k values to visually confirm that they match the given Fourier series coefficients for x(t).

Additionally, it would be beneficial to provide more context or explanation for your solution so that others can better understand your approach and reasoning. Overall, your solution seems to be on the right track, but it would be great to see more details and explanations to fully confirm its correctness.