In the computer generation of musical sounds, pure tones need to be windowed to make them more interesting. Windowing mimics the way a musician would approach the generation of a certain sound. Increasing the richness of the harmonic frequencies is the result of the windowing as we will see in this problem. Consider the generation of a musical note with frequencies around f_A = 880Hz. Assume our “musician” while playing this note uses three strokes corresponding to a window w₁(t) = r(t) − r(t − T1) − r(t − T2) + r(t − T₀), so that the resulting sound would be the multiplication, or windowing, of a pure sinusoid cos(2π f_At) by a periodic signal w(t) with w₁(t) a period that repeats every T₀ = 5T where T is the period of the sinusoid. Let T₁ = T₀/4, and T₂ = 3T₀/4.

(a) Analytically determine the Fourier series of the window w(t) and plot its line spectrum using MATLAB. Indicate how you would choose the number of harmonics needed to obtain a good approximation to w(t).

(b) Use the modulation or the convolution properties of the Fourier series to obtain the coefficients of the product s(t) = cos(2π f_At) w(t). Use MATLAB to plot the line spectrum of this periodic signal and again determine how many harmonic frequencies you would need to obtain a good approximation to s(t).

(c) The line spectrum of the pure tone p(t) = cos(2π f_At) only displays one harmonic, the one corresponding to the f_A = 880Hz frequency, how many more harmonics does s(t) have? To listen to the richness in harmonics use the function sound to play the sinusoid p(t) and s(t) (use F_s =2 × 880Hz to play both).

(d) Consider a combination of notes in a certain scale, for instance let

p(t) = sin(2π × 440 t) + sin(2π × 550 t) + sin(2π × 660 t).

Use the same windowing w(t), and let s(t) = p(t) w(t). Use to plot p(t) and s(t) and to compute and plot their corresponding line spectra. Use sound to play p(nT_s) and s(nT_s) using F_s = 1000.