Electronics and Communication- Digital and Systems

Question

Presume that while observing a periodic phenomena, you obtain complex value Fourier series

coefficients. The number of significant coefficients are 18 and the insignificant coefficients are 28.

Both (significant as well as insignificant) components decreases with an increase in the harmonic

order. Using Parseval’s theorem estimate the power that resides in significant coefficients,

insignificant coefficients as well as in the acquired periodic signal. Hint: You can refer Google

search to obtain periodic data associated with any arbitary scenario (cite the source), which can be

truncated to size it according to 18 and 28 or using the random number generation, you can

synthesize the data.

Answer 1

Since both kinds of coefficients decrease in significance as their harmonic order k increases, we can denote the 18 significant coefficients as the set {c(k)}_k=0¹⁷ and the 28 insignificant coefficients as the set {c(k)}_k=18⁴⁵.

By Parseval’s theorem, the power that resides in significant coefficients is S = ∑_k=0¹⁷|c(k)|² > 0, in insignificant coefficients is P – S = ∑_k=18⁴⁵|c(k)|² < P and in the complete signal is P = ∑_k=0⁴⁵|c(k)|² > S.