Fourier Transform: Phase and Magnitude Sign Flip

1. Fourier Transform Basics

For a signal x(t), its Fourier transform X(f) can be written in polar form:

X(f) = |X(f)| e^{j φ(f)}

Where:

• |X(f)| is the magnitude spectrum
• φ(f) = arg(X(f)) is the phase spectrum

2. Changing the Sign of the Magnitude

If we flip the sign of the magnitude:

X_new(f) = -|X(f)| e^{j φ(f)}

This is equivalent to:

- |X(f)| e^{j φ(f)} = |X(f)| e^{j (φ(f) + π)}

Key Insight: Changing the sign of the magnitude adds π (180°) to the phase.

3. Deep Meaning

• Sign flips in frequency → phase shift: Each frequency rotates 180° in the complex plane.
• Signal reconstruction: Inverse Fourier transform of X(f) vs -X(f) gives x(t) vs -x(t).
• Intuition: Magnitude shows "how much" of each frequency is present, phase shows "how to align" them. Flipping the sign of magnitude inverts the signal in time domain.

4. Example

Suppose X(f) = 2 e^{j π/4}:

• Original phase: π/4
• Flip sign: -2 e^{j π/4} = 2 e^{j (π/4 + π)} = 2 e^{j 5π/4}

The phase plot jumps by π.

5. Summary Table

Operation on |X(f)|	Effect on Phase φ(f)	Effect on Time-Domain Signal
None	None	Original signal
Flip sign	Add π everywhere	Signal inverted (-x(t))