How do higher harmonics behave for smooth periodic signals in a Fourier series?

How do higher harmonics behave for smooth periodic signals in a Fourier series?

• A. They grow with frequency.
• B. They decay with frequency.
• C. They vanish completely after a finite number of harmonics.
• D. They increase randomly.

Answer: (C)

For smooth periodic signals, the Fourier coefficients get smaller as you go to higher harmonics. The rate of decay depends on how smooth the signal is. If the signal has p continuous derivatives, the coefficients typically fall off like 1/n^(p+1). If the signal is infinitely differentiable, the decay is faster than any power; if it’s analytic, the decay is often exponential. This means higher harmonics contribute progressively less to the signal’s reconstruction, and the spectrum is dominated by the lower frequencies.

Importantly, they do not vanish completely after a finite number of harmonics. Unless the signal is already a finite sum of a few sinusoids (a band-limited case that happens to be smooth), there will be nonzero components at arbitrarily high frequencies, just very small ones.