With reference to Fourier Series representation of periodic signals, consider the following statements:1.Fourier Series decomposes a periodic signal into a sum of harmonically related sinusoids.2.The coefficients of the Fourier Series depend on the fundamental period of the signal.3.Fourier Series can directly represent non-periodic signals without modification.4.Both exponential and trigonometric forms of Fourier Series are equivalent representations.Which of the above statements is/are correct?

Correct Answer: C - 1, 2 and 4 only

Quick Concept Explanation

Fourier methods decompose signals into frequency components. Fourier series represents periodic signals as harmonically related sinusoids or equivalent complex exponentials; non-periodic signals are handled by Fourier transform rather than ordinary Fourier series. Key relation: Real even x(t) ⇒ X(ω) real and even.

Exam focus: Symmetry can eliminate sine/imaginary terms immediately. Common trap: Fourier series is fundamentally for periodic signals.

Statement-wise Verification

• Statement 1 - Correct.
  Fourier Series decomposes a periodic signal into a sum of harmonically related sinusoids.
  Fourier series represents periodic signals as harmonically related sinusoids or equivalent complex exponentials
• Statement 2 - Correct.
  The coefficients of the Fourier Series depend on the fundamental period of the signal.
  A real even time-domain signal has a real even Fourier transform and zero odd component.
• Statement 3 - Incorrect.
  Fourier Series can directly represent non-periodic signals without modification.
  Ordinary Fourier series is for periodic signals; Fourier transform is used for non-periodic signals
• Statement 4 - Correct.
  Both exponential and trigonometric forms of Fourier Series are equivalent representations.
  Fourier series represents periodic signals as harmonically related sinusoids or equivalent complex exponentials

Core Concept

Fourier methods decompose signals into frequency components. A real even time-domain signal has a real even Fourier transform and zero odd component. Fourier series represents periodic signals as harmonically related sinusoids or equivalent complex exponentials; non-periodic signals are handled by Fourier transform rather than ordinary Fourier series.

Formula / Key Relationship

Real even x(t) ⇒ X(ω) real and even.

Why the Other Options Are Wrong

Option A is incorrect because it omits correct statement(s) 4. Option B is incorrect because it includes incorrect statement(s) 3 and omits correct statement(s) 1, 4. Option D is incorrect because it includes incorrect statement(s) 3.

Exam Shortcut / Approach

Symmetry can eliminate sine/imaginary terms immediately.

Common Mistake

Fourier series is fundamentally for periodic signals.

Quick Revision

Trigonometric and Exponential Fourier Series is linked with Periodic Signals, Fourier Series and Signals and Systems. Symmetry can eliminate sine/imaginary terms immediately.