Lecture Notes: Fourier Series Expansion

Introduction to Fourier Series

• New chapter: Fourier Series Expansion.
• Introduced by Joseph Fourier, a French mathematician and physicist.
• Key topics:
  • What is Fourier Series?
  • Uses of Fourier Series.
  • Types of Fourier Series expansions: Fourier series and Fourier transform.

Key Concepts

Definition of Fourier Series

• Used for periodic signals.
• Expands periodic signals in terms of harmonics (sinusoidal and orthogonal).

Types of Periodic Signals

1. Continuous Time Periodic Signals
2. Discrete Time Periodic Signals

• Corresponding Fourier Series Expansions:
  • Continuous Time Fourier Series
  • Discrete Time Fourier Series

Focus of the Chapter

• This chapter will discuss Continuous Time Fourier Series Expansion.
• Discrete Time signals will be covered later.

Analysis and Applications

• Fourier Series is for analysis of periodic signals.
• Fourier Transform is used for non-periodic signals.
• Non-periodic signals are more common in real-life applications.
• Non-periodic signals have two types:
  1. Continuous Time Non-Periodic Signals
  2. Discrete Time Non-Periodic Signals

Transition from Fourier Series to Fourier Transform

• Fourier Series is limited to periodic signals.
• Fourier Transform is necessary for analyzing non-periodic signals.

Differences Between Transforms

1. Laplace Transform
   • Used for designing systems.
   • Helps in obtaining transfer functions and checking system stability.
2. Z Transform
   • Used for discrete time systems.

Periodic Signals

• Periodic signals repeat a particular structure from -∞ to +∞.
• Time period (T): Interval after which the signal repeats.
• Condition for periodicity:
  • If Xt is a periodic signal, then Xt + nT = Xt.

Existence of Fourier Series

• Existence depends on Dirichlet Conditions (to be discussed in the next lecture).
• Even if a signal is periodic, it must satisfy these conditions for Fourier series to exist.

Definition of Frequency and Harmonics

• Frequency: Number of cycles per second.
• Harmonics: Integral multiples of the fundamental frequency.
• Harmonics can be even or odd, and their dominance can vary depending on the signal.

Importance of Harmonics in Analysis

• Harmonics allow analysis of the signal by expressing it in terms of fundamental and other frequency components.
• Example:
  • A periodic signal expressed as something like 2 sin(ωt) + sin(2ωt) + 7 sin(3ωt).

Types of Fourier Series Expansions

1. Trigonometric Fourier Series Expansion
2. Complex Exponential (Exponential) Fourier Series Expansion
3. Polar or Harmonic Fourier Series Expansion

• Focus will be on the first two types.

Conclusion

• This lecture was an introduction to Fourier Series Expansion and its significance in periodic signal analysis.
• Encouraged students to review periodic signals if not familiar.
• Next lecture will cover Dirichlet Conditions.

• For any doubts, students are encouraged to ask in the comment section.
• Upcoming lectures will include tips and tricks for competitive examination preparation.