Week6 Quiz Question2 Solution:
Suppose $ X(e^{j\omega}), X_e(e^{j\omega}), Y(e^{j\omega}) $ is the DTFT of $ x(n), x_e(n), y(n) $.
Then $ X(e^{j\omega}) $ has a period of $ 2\pi $.
After the upsampling. All of the original information of x(n) will be contained in the interval $ [\frac{-\pi}{L}, \frac{\pi}{L}] $.
New aliases occur in the interval of $ [-\pi,\frac{-\pi}{L}],[\frac{\pi}{L},\pi] $ which need to be filtered out.
Thus, the cut-off frequency of the LP filter is $ \frac{\pi}{L} $.
