For a real-valued even signal in time domain, consider the following statements:1.The signal satisfies the condition x(t) = x(−t) for all time.2.Its Fourier Transform is purely real-valued without any imaginary component.3.The odd component of such a signal is identically zero.4.The imaginary part of its Fourier Transform is an even function.Which of the above statements is/are correct?

Correct Answer: A - 1, 2 and 3 only

Quick Concept Explanation

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. 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.
  The signal satisfies the condition x(t) = x(−t) for all time.
  A real even time-domain signal has a real even Fourier transform and zero odd component.
• Statement 2 - Correct.
  Its Fourier Transform is purely real-valued without any imaginary component.
  A real even time-domain signal has a real even Fourier transform and zero odd component.
• Statement 3 - Correct.
  The odd component of such a signal is identically zero.
  A real even time-domain signal has a real even Fourier transform and zero odd component.
• Statement 4 - Incorrect.
  The imaginary part of its Fourier Transform is an even function.
  For a real even signal the Fourier transform is purely real, so its imaginary part is identically zero

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 B is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 1, 3. Option C is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 2, 3. Option D is incorrect because it includes incorrect statement(s) 4.

Exam Shortcut / Approach

Symmetry can eliminate sine/imaginary terms immediately.

Common Mistake

Fourier series is fundamentally for periodic signals.

Quick Revision

Fourier Transform of Real Even Signals is linked with Signal Symmetry, Fourier Transform and Signals and Systems. Symmetry can eliminate sine/imaginary terms immediately.