Filtering System

The questions below are due on Thursday May 01, 2025; 02:00:00 PM.

You are not logged in.
Please Log In for full access to the web site.
Note that this link will take you to an external site (https://shimmer.mit.edu) to authenticate, and then you will be redirected back to this page.

Consider a system whose input x[n] and output y[n] are related as shown in the box labeled S below

where H(\Omega) represents a linear, time-invariant system. H(\Omega) is a periodic function of \Omega with period 2\pi, and one period of H(\Omega) is described by the following:

H(\Omega) = \cases{ 1& if $0\le|\Omega|\le{\pi \over 3}$\cr 0& if ${\pi\over3}\lt|\Omega|\lt \pi$\cr}

Part a. Let Y_1(\Omega) represent the DTFT of the output y_1[n] that results when the DTFT of the input x_1[n] has the following form:

Note that X_1(\Omega) is periodic in \Omega with period 2\pi.

Sketch |Y_1(\Omega)| on the axes below. Label the important points of your sketch.

Part b. Is the system S linear? Briefly explain.

Part c. Is the system S time-invariant? Briefly explain.

Part d. Can system S be regarded as a lowpass filter or as a highpass filter or as a bandpass filter? If so, describe which and specify the cutoff frequency or frequencies.

Please upload a single pdf file that contains your answers to all parts of this problem:

No file selected