Fourier Transform Lecture Notes

Introduction to Fourier Transform

• Definition: A mathematical tool used for frequency analysis of signals.
• Purpose: Analyze signal variations while changing the frequency.
• Frequency Domain Representation: Fourier transform provides a representation of the original signal in the frequency domain.

Comparison with Laplace Transform

• Laplace Transform: Another mathematical tool used for the analysis of systems and circuits.

Existence of Fourier Transform

• Conditions for Existence:
  • Energy Signals: Exist as they are absolutely integrable.
  • Power Signals: Exist but require properties for their Fourier Transform calculation.
  • Impulse Related Signals: Also exist; considered an exception because they are absolutely integrable despite being neither energy nor power signals.
  • Neither Energy nor Power Signals: Do not have a Fourier transform.

Important Notes

• Absolutely Integrable Signals: Can be represented in Fourier Transform.
• Existence of Laplace Transform:
  • Exists for energy signals, power signals, and to some extent, neither energy nor power signals.

Fourier Series vs. Fourier Transform

• Fourier Series: Applicable only to periodic signals.
• Fourier Transform: Can analyze both periodic and non-periodic signals.

Representation of Fourier Transform

• Notation:
  • Fourier Transform of signal XT: X(jΩ) or X(F).
  • Units: X(jΩ) in radians/second; X(F) in Hertz.
• Complex Nature: Fourier Transform provides complex numbers which include magnitude and angle.
• Conversion: To find X(F), first compute X(jΩ) and then replace Ω with 2πF.

Formulas

1. Fourier Transform:
   [ X(jΩ) = \int_{-\infty}^{\infty} x(t)e^{-jΩt} dt ]
   • Converts time domain signal x(t) to frequency domain X(jΩ).
2. Inverse Fourier Transform:
   [ x(t) = \frac{1}{2π} \int_{-\infty}^{\infty} X(jΩ)e^{jΩt} dΩ ]
   • Converts frequency domain signal X(jΩ) back to time domain x(t).
   • Together, they form the Fourier Transform Pair.
   • Important Note: Both formulas are valid only for absolutely integrable signals.

Conversion from Laplace to Fourier Transform

• Conversion Rule: To convert Laplace Transform to Fourier Transform, replace s with jΩ.
• Validity: This conversion is only applicable to absolutely integrable signals.

Conclusion

• This lecture covered the fundamentals of the Fourier Transform and its properties.
• Future presentations will expand on these concepts and provide practical applications.