The smoothness of the signal determines the frequency content of its spectrum. Consider the signals

x(t) = u(t + 0.5) − u(t − 0.5),

y(t) = (1 + cos(π t))[u(t + 0.5) − u(t − 0.5)]

(a) Plot these signals. Can you tell which one is smoother?

(b) Find X(Ω) and carefully plot its magnitude |X(Ω)|vs frequency ­Ω.

(c) Find Y(Ω)(use the Fourier transform properties) and carefully plot its magnitude |Y(Ω)|vs frequency ­.

(d) Which one of these two signals has higher frequencies? Can you now tell which of the signals is smoother? Use MATLAB to decide. Make x(t) and y(t) have unit energy. Plot 20 log₁₀|Y(Ω)|and 20 log₁₀|X(Ω)|using MATLAB and see which of the spectra shows lower frequencies. Use fourierfunction to compute Fourier transforms.