Parseval's theorem is a fundamental concept in signal processing and harmonic analysis. It asserts that for a periodic function, the average power of the signal over one period equals the sum of the squared magnitudes of all its complex Fourier coefficients. This theorem, named after Marc-Antoine Parseval, provides a powerful tool for analyzing the energy distribution in signals.

Interestingly, Parseval's theorem also holds for the trigonometric form of the Fourier series, which expresses a function in terms of sine and cosine functions. Here, the Fourier coefficients can be related to the trigonometric series coefficients, allowing the theorem to be applied in this alternate form.

To validate Parseval's theorem, we start by considering a function x(t) with a complex Fourier series representation:

Where C_n are the complex Fourier coefficients, and ω₀ is the fundamental angular frequency. The theorem states:

Where T is the period of the function. Substituting the Fourier series into the left-hand side and solving confirms the equality, thus proving the theorem.

Parseval's theorem is crucial in practical applications, particularly in audio processing. It allows for comparing the energy contained in an original sound wave to that in its compressed version. This comparison is essential in ensuring that the compression process does not significantly degrade the quality of the audio signal by losing too much energy.

From an engineering perspective, Parseval's theorem offers valuable insights. For instance, if the function in question represents an electrical signal such as current or voltage, then the square of this function represents the instantaneous power dissipated in a 1-ohm resistor. Consequently, the theorem links the energy dissipated in the resistor over one period to the Fourier series representation of the signal. This relationship is expressed in two different forms: one using the trigonometric Fourier series and the other using the amplitude-phase form of the Fourier series. Thus, Parseval's theorem not only serves as a powerful analytical tool but also bridges theoretical concepts with practical engineering applications.