Problem #1

For the set-up shown below...

Assume that the input signal is x(t) = 5cos(2Π2000t) + 4cos(2Π2050t) and assume that the sampling rate of the ADC is Fs = 20,000 Hz.

(a) Determine the signal x[n] and express it as x[n] = 5cos(Ω₁n) + 4 cos(Ω₂n) by determining the values of Ω₁ and Ω₂.

(b) Use MATLAB to simulate the scenario where you collect 64 samples (starting at n=0), then do the following:

• Don't zero-pad
• compute the DFT using MATLAB's fft command call the result X,
• use the command fftshift to re-order the points and put them back in X,
• plot the magnitude of the resulting DFT versus Ω.
 - You'll need to determine the proper values of Ω and put them in a variable called omega
 - Then plot using plot(omega, abs(X),'-o')
• Zoom-in on the plot to see the range near Ω1 and Ω2.
• Provide the plot and your code

(c) Repeat (b) except now zero-pad to a total of 2048 samples (so tack on 2048 - 64 zeros),

(d) Repeat (b) except now collect 1024 samples but don't zero-pad

(e) Repeat (b) except now collect 1024 samples and zero-pad to a total of 8192 samples

(f) Give reasons for what causes the results (b) - (d) to be not desired.

Problem #2

You wish to use the DFT to detect two complex sinusoids based on 1000 collected samples (noise is assumed to be negligibly small). You know that one of them is always in the frequency range θ∈[Π/8, Π/2] and that the other is always in the frequency range θ∈[-Π/8, -Π/2]. The amplitude of one of them is always 1 and the amplitude of the other is always in the range [10^-2.5, 1]. Specify an appropriate window to use for this and state your reasons. Provide matlab code and results that verify that your solution works.