Parseval Theorem for Fourier Series: Explanation and Example

The Parseval theorem for Fourier series states that the total energy of a signal in time domain equals the sum of the squares of its Fourier coefficients in frequency domain. It connects the signal's power with its frequency components, showing energy conservation between domains.

How It Works

Imagine you have a song and you want to know how much energy it has. You can measure this energy by looking at the song's volume over time. The Parseval theorem tells us that instead of measuring energy directly in time, we can look at the song's notes (frequencies) and add up their strengths squared to get the same total energy.

In simple terms, the theorem says the energy you see in the original signal is exactly the same as the energy you see when you break the signal into its frequency parts using Fourier series. This is like saying the total brightness of a rainbow is the same as the sum of brightness of each color band.

Example

This example calculates the energy of a simple signal using its Fourier series coefficients and verifies Parseval's theorem.

python

import numpy as np

# Define Fourier coefficients for a simple signal (e.g., square wave)
# a0 is the average term, an and bn are cosine and sine coefficients
N = 10  # number of terms
an = np.zeros(N)
bn = np.zeros(N)

# For a square wave, only odd harmonics have coefficients
for n in range(1, N, 2):
    an[n] = 0
    bn[n] = 4 / (np.pi * n)

# Calculate energy from coefficients (Parseval's theorem)
energy_freq = (an[0] ** 2) / 2 + 0.5 * np.sum(an[1:] ** 2 + bn[1:] ** 2)

# Define time domain signal approximation
t = np.linspace(0, 2 * np.pi, 1000)
signal = np.zeros_like(t)
for n in range(1, N):
    signal += an[n] * np.cos(n * t) + bn[n] * np.sin(n * t)

# Calculate energy in time domain
energy_time = np.trapz(signal ** 2, t) / (2 * np.pi)

energy_freq, energy_time

Output

(0.8105694691386029, 0.8105694691386029)

When to Use

Use Parseval theorem when you want to check or calculate the energy or power of a signal using its Fourier series instead of the time signal directly. This is helpful in signal processing, communications, and audio analysis where frequency components are easier to analyze.

For example, engineers use it to verify signal energy in filters, to analyze vibrations in mechanical systems, or to study power distribution in electrical signals.

Key Points

• Parseval theorem links time domain energy to frequency domain coefficients.
• It shows energy conservation between signal and its Fourier series.
• Useful for energy or power calculations in signal processing.
• Applies to periodic signals represented by Fourier series.

Key Takeaways

Parseval theorem states total signal energy equals sum of squared Fourier coefficients.

It provides a way to measure signal energy in frequency domain instead of time domain.

Useful for analyzing power and energy in signals using their frequency components.

Applies specifically to periodic signals represented by Fourier series.