# Discrete Fourier transform of sampled continuous signal

1. Aug 25, 2011

### Bromio

1. The problem statement, all variables and given/known data
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].

2. Relevant equations

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

3. 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.