Question 1. Consider the following scenario: You have a system intended to receive a sinusoidal RF signal whose frequency is unknown but is known to be one of the following 9 frequencies (all in GHz):

f₁ = 1.01 f₂ = 1.02 f₃ = 1.03 f₄ = 1.04 f₅ = 1.05

f₆ = 1.06 f₇ = 1.07 f₈ = 1.08 f₉ = 1.09

Note: spacing is 10 MHz

In other words, you are receiving the signal x(t) = Acos(2Πf_kt + Φ) with f_k taking the value of one of the 9 values above - but you don't know which one. Each time you receive the signal its frequency could be any one of those 9 values. For simplicity here assume that A = 1 and Φ = 0 .

Assume that some sampling system operates on this signal and converts it into its DT equivalent lowpass (ELP) signal that has been sampled such that -Π corresponds to the original 1.0 GHz and +Π corresponds to the original 1.1 GHz.

Make a sketch of the resulting DT ELP's spectrum for each of the nine cases.

Tips:

1. If you do this "sketch" using something like Powerpoint it will be easy to use copy and paste to get the other 8 cases once you "sketch" one of them.)

2. You can actually put all 9 spectra on a single plot and color code them!

3. You'll want to show "sketches" of the intermediate results that lead up to your final sketch.

Question 2. Imagine that the real-valued x(t) consists of two bandpass signals as shown below (only the positive frequencies are shown). Find the conditions on Fs, fa and fb that give the spectra shown at the output of the system shown below. (Note: I have used the symbol Ω to indicate DT frequency - as I have done in many of the note sets. The textbook does not use that symbol).

Note: You can assume all things can be done "ideally". So for example... the two spectra are perfectly bandlimited to B Hz width and are exactly 0 outside of their band (the first signal's band is [ f_a , f_a + B] and the second signal's band is [ f_b, f_b + B] ).