Fourier Series Representation of Periodic Signals

Why it matters

Understanding the Fourier Series Representation of Periodic Signals is crucial for analyzing and designing systems that process periodic signals, such as communication systems and signal processing applications. It allows engineers to break down complex periodic signals into simpler sinusoidal components, making it easier to analyze their frequency content.

Key ideas

• Periodic Signals: Signals that repeat after a fixed interval, known as the period.
• Fourier Series: A mathematical tool used to express a periodic signal as a sum of sinusoidal functions (sines and cosines) with different frequencies and amplitudes.
• Harmonics: Sinusoidal components of a signal that are integer multiples of the fundamental frequency.
• Convergence: Under standard Dirichlet-type conditions, it converges to the signal at continuity points and to the average of the left and right limits at a jump.
• Applications: Used in signal processing, communications, and control systems to analyze and synthesize signals.

Formulas

• Fourier Series Representation: x(t) = a_0 + Σ (a_n * cos(nω_0t) + b_n * sin(nω_0t)) where:

  • x(t) is the periodic signal.
  • a_0 is the average value of the signal over one period.
  • a_n and b_n are the Fourier coefficients.
  • ω_0 is the fundamental angular frequency, ω_0 = 2π/T.
  • T is the period of the signal.

• Fourier Coefficients: a_0 = (1/T) ∫ x(t) dt over one period a_n = (2/T) ∫ x(t) * cos(nω_0t) dt over one period b_n = (2/T) ∫ x(t) * sin(nω_0t) dt over one period

The sum runs over n = 1,2,… . This convention uses a₀ as the mean; some texts write a₀/2 and consequently define a₀ differently.

Worked example

Given a periodic signal x(t) = 3 + 2cos(2πt) + sin(4πt), find the Fourier coefficients.

1. Identify the period T. Here, T = 1 second.
2. Calculate a_0: a_0 = (1/T) ∫ x(t) dt = 3
3. Calculate a_n and b_n for n = 1, 2:
   • For n = 1, a_1 = 2, b_1 = 0
   • For n = 2, a_2 = 0, b_2 = 1
4. The Fourier series is x(t) = 3 + 2cos(2πt) + sin(4πt)

Answer: The Fourier coefficients are a_0 = 3, a_1 = 2, b_1 = 0, a_2 = 0, b_2 = 1.

Common mistakes

• Confusing the period T with the fundamental frequency f_0.
• Incorrectly calculating the integrals for Fourier coefficients.
• Forgetting to multiply by the factor (2/T) for a_n and b_n.

For GATE EE

Questions often involve finding Fourier coefficients for given periodic signals or reconstructing signals from given coefficients. Practice calculating integrals and understanding the physical meaning of each term in the Fourier series.

Quick check

1. What is the fundamental frequency of a signal with period T = 2 seconds?
2. How do you calculate a_0 for a periodic signal?
3. What is the role of harmonics in a Fourier series?

Answers: 1. f_0 = 0.5 Hz; 2. a_0 = (1/T) ∫ x(t) dt over one period; 3. Harmonics represent higher frequency components of the signal.