Fourier Series Representation of Periodic Signals

Why it matters

Fourier Series is crucial for analyzing and synthesizing periodic signals in electronics and communication systems. It allows engineers to break down complex periodic signals into simpler sinusoidal components, making it easier to design and troubleshoot circuits and communication systems.

Key ideas

• Periodic Signals: Signals that repeat after a fixed interval, known as the period.
• Fourier Series: A way to represent a periodic signal as a sum of sinusoidal functions (sines and cosines).
• Harmonics: Sinusoidal components of a signal that are integer multiples of the fundamental frequency.
• Coefficients: The amplitudes of the sinusoidal components in the Fourier Series, which determine the contribution of each harmonic.
• Convergence: The Fourier Series converges to the original signal under certain conditions, such as Dirichlet's conditions.

At a jump discontinuity, under the usual convergence conditions the series converges to the midpoint of the one-sided limits. In the example it gives zero at the jump points. The sum below starts at n = 1.

Formulas

• x(t) = a_0 + Σ (a_n * cos(nω_0t) + b_n * sin(nω_0t))

  • x(t): Periodic signal
  • a_0: DC component
  • a_n, b_n: Fourier coefficients
  • ω_0: Fundamental angular frequency (rad/s)
  • t: Time (s)

• a_0 = (1/T) ∫ x(t) dt over one period

  • T: Period of the signal (s)

• 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

Worked example

Given: A periodic signal x(t) with period T = 2s is defined as x(t) = 1 for 0 ≤ t < 1 and x(t) = -1 for 1 ≤ t < 2.

1. Calculate a_0:

   • Formula: a_0 = (1/T) ∫ x(t) dt
   • Calculation: a_0 = (1/2) [∫ from 0 to 1 (1) dt + ∫ from 1 to 2 (-1) dt]
   • Result: a_0 = 0

2. Calculate a_n:

   • Formula: a_n = (2/T) ∫ x(t) * cos(nω_0t) dt
   • Calculation: a_n = (1) [∫ from 0 to 1 (1 * cos(nπt)) dt + ∫ from 1 to 2 (-1 * cos(nπt)) dt]
   • Result: a_n = 0 for all n

3. Calculate b_n:

   • Formula: b_n = (2/T) ∫ x(t) * sin(nω_0t) dt
   • Calculation: b_n = (1) [∫ from 0 to 1 (1 * sin(nπt)) dt + ∫ from 1 to 2 (-1 * sin(nπt)) dt]
   • Result: b_n = (2/nπ) [1 - cos(nπ)]

Final Answer: The Fourier Series representation is x(t) = Σ (b_n * sin(nω_0t)) where b_n = (2/nπ) [1 - cos(nπ)].

Common mistakes

• Forgetting to divide by the period T when calculating coefficients.
• Incorrectly setting up the integral limits for different segments of the signal.
• Ignoring the convergence conditions, leading to incorrect series representation.

For GATE EC

Questions often involve finding the Fourier coefficients for given periodic signals or analyzing the convergence of the Fourier Series. Practice setting up integrals correctly and understanding the physical significance of harmonics and coefficients.

Quick check

1. What is the fundamental frequency of a signal with period T = 4s?
2. How do you calculate the DC component of a periodic signal?
3. What is the significance of the b_n coefficients in the Fourier Series?

Answers: 1. f0 = 0.25 Hz, equivalently ω0 = π/2 rad/s, 2. a_0 = (1/T) ∫ x(t) dt, 3. They represent the amplitude of the sine components.