Complex Exponential Fourier Series

Introduction

• Transition from trigonometric Fourier series to complex exponential Fourier series.
• Importance of complex exponential Fourier series.
• More questions on this type will be discussed.

Fourier Series Expansion

• Periodic Signal (X(t)): Focus on periodic signals for complex exponential Fourier series.
• Summation Formula:
  • Summation from n = -∞ to ∞ of C_n * e^(jnω₀*t).
  • C_n is the complex exponential Fourier coefficient.

Fourier Coefficient (C_n)

• C_n Formula:
  • C_n = (1/T₀) * ∫ (X(t) * e^(-jnω₀*t)) dt over T₀.
• First Step: Calculate the Fourier coefficient C_n.
• Substitution: Use C_n to find the Fourier series expansion for X(t).*

Conversion Between Series

• Convert complex exponential Fourier series to trigonometric Fourier series and vice versa.
• Relationship between coefficients A_n, B_n (trigonometric) and C_n (complex exponential).

Properties of C_n

• Reversal Operation:
  • Replace n with -n: C_-n = (1/T₀) * ∫ (X(t) * e^(jnω₀*t)) dt.
• Conjugate Operation:
  • Reverse imaginary part's sign: C_-n* = (1/T₀) * ∫ (conjugate(X(t)) * e^(-jnω₀*t)) dt.
• Conjugate Symmetry:
  • C_n = C_-n* implies X(t) = X(t), meaning X(t) is real._

Magnitude and Angle of C_n

• C_n expressed as:
  • |C_n| * e^(j*∠C_n).
• Reversal:
  • C_-n = |C_-n| * e^(j∠C_-n).
• Conjugate Symmetry in Magnitude and Angle:
  • |C_n| = |C_-n| (even nature).
  • ∠C_n = -∠C_-n (odd nature).*_

Conclusion

• Real signal X(t):
  • C_n is conjugate symmetric.
  • Magnitude of C_n is even.
  • Angle (phase) of C_n is odd.

Example

• Given X(t) is periodic with C_n = n² * e^(j*n³).
  • Magnitude (n²) is even.
  • Angle (n³) is odd.
  • C_n is conjugate symmetric, implying X(t) is real.

Next Steps

• Solve more questions on exponential Fourier series in future presentations.
• Questions or doubts can be discussed in the comments section.

[Applause] [Music]