Harmonic distortion in digital sinusoidal generators An ideal sinusoidal generator produ...

Harmonic distortion in digital sinusoidal generators An ideal sinusoidal generator produces the signal

which is periodic with fundamental period N if J_o = k₀/N and k_o, N are relatively prime numbers. The spectrum of such a "pure" sinusoid consist of two lines at k = k_o and k = N — k_o (we limit ourselves in the fundamental interval . ln practice, the approximations made in computing the samples of a sinusoid of relative frequency f_o result in a certain amount of power falling into other frequencies. This spurious power results in distortion, which is referred to as harmonic distortion.

Harmonic distortion is usually measured in terms of the total harmonic distortion (THD), which is defined as the ratio

(a) Show that

where,

(b) By using the Taylor approximation

compute one period of x(n) for f₀ = 1/96, 1/32, 1/256 by increasing the number of terms in the Taylor expansion from 2 to 8.

(c) Compute the THD and plot the power density spectrum for each sinusoid in part (b) as well as for the sinusoids obtained using the computer cosine function. Comment on the results.