In a Fourier series context, when can the series be truncated without substantial loss of accuracy?

• A. The signal cannot be approximated by a finite sum.
• B. The higher harmonics must be exactly zero.
• C. The series can be truncated because higher harmonics are negligible.
• D. The fundamental frequency changes.

Answer: (C)

In a Fourier series, you can truncate the series when most of the signal’s energy sits in the lower harmonics and the higher harmonics contribute very little. If the coefficients of the higher terms are small enough, dropping them changes the reconstructed signal only a tiny amount, so the loss in accuracy is negligible. The error from truncation is essentially the tail energy of the spectrum, which can be quantified by summing the squares of the omitted coefficients. Smooth, slowly varying signals tend to have quickly decaying harmonic amplitudes, making truncation effective. This isn’t about the higher harmonics being exactly zero or about changing the fundamental; it’s about the practical reality that those higher terms may be negligible for the desired accuracy.