Lecture Notes: Introduction to Fourier Transform

Overview

• Introduction to the Fourier Transform chapter.
• Focus on properties and implementation.
• Importance of watching related lectures for understanding.

What is Fourier Transform?

• A mathematical tool used for frequency analysis of signals.
• Analyzes signals in the frequency domain.
• Known as the frequency domain representation of the original signal.

Frequency Analysis of Signals

• Studies signal variations with changes in frequency.

Comparison with Laplace Transform

• Fourier Transform: Used for frequency domain analysis of signals.
• Laplace Transform: Used for analysis of systems and circuits.

Existence of Fourier Transform

• Can exist for:
  • Energy signals
  • Power signals
  • Impulse related signals (an exception)
• Impulse related signals:
  • Absolutely integrable, unlike other neither energy nor power signals.
• Energy signals are absolutely integrable.
• Power signals require using properties to obtain Fourier Transform.

Existence of Laplace Transform

• Exists for:
  • Energy signals
  • Power signals
  • Neither energy nor power signals (limited extent)

Fourier Series Expansion vs. Fourier Transform

• Fourier Series: Only for periodic signals.
• Fourier Transform: Can be used for aperiodic signals.

Representation of Fourier Transform

• Signal representation:
  • Signal ( x(t) )
  • Fourier Transform: ( X(j\Omega) ) or ( X(f) )
  • Units: ( X(j\Omega) ) in radians per second, ( X(f) ) in Hertz.
• Note on notation: ( X(j\Omega) ) may be simplified to ( X(\Omega) ).

Fourier Transform as a Complex Number

• ( X(j\Omega) ) is a complex number with magnitude and angle.
• Conversion between ( X(j\Omega) ) and ( X(f) ):
  • Replace ( \Omega ) by ( 2\pi f ).

Fourier Transform Formulas

• Transform from Time to Frequency Domain:
  • ( X(j\Omega) = \int_{-\infty}^{\infty} x(t) e^{-j\Omega t} , dt )
• Inverse Transform from Frequency to Time Domain:
  • ( x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(j\Omega) e^{j\Omega t} , d\Omega )
• Known as Fourier Transform Pair.
• Valid only for absolutely integrable signals.

Conversion of Laplace Transform to Fourier Transform

• Replace ( s ) by ( j\Omega ) in Laplace Transform to obtain Fourier Transform.
• Valid only for absolutely integrable signals.

Conclusion

• Introduction to Fourier Transform and related concepts.
• Open for questions or comments in subsequent lectures.