Overview
This lecture reviews the origins and properties of Fourier series, focusing on the analogy between vectors and signals, orthogonality, basis sets, completeness, and relationships between trigonometric and exponential representations.
Vector and Signal Analogy
- Any vector in a defined vector space can be represented as a linear combination of orthogonal basis vectors.
- The analogy is extended to signals: a periodic signal can be represented as a linear combination of known, orthogonal basis signals.
Orthogonality in Signal Space
- Signals x₁(t) and x₂(t), both with period T₀, are orthogonal if their inner product (integral over a period) is zero.
- For real signals: ∫ x₁(t)x₂(t) dt over one period equals 0.
- For complex signals: ∫ x₁(t)x₂*(t) dt (where * denotes conjugate) must equal 0.
Fourier Series Representations
- Trigonometric Fourier series expresses signals as sums of sine and cosine terms.
- Exponential Fourier series expresses signals as sums of complex exponentials e^(j2πf₀nt).
- These complex exponentials are orthogonal for different integer values of n and span all possible periodic signals (completeness).
Fourier Coefficient Calculation and Properties
- Any periodic signal g(t) of period T₀ can be written as the sum Σ cₙ e^(j2πf₀nt), n from -∞ to ∞.
- Coefficient cₙ = (1/T₀) ∫ g(t) e^(-j2πf₀nt) dt over one period.
- For real signals, c₋ₙ equals the complex conjugate of cₙ.
Symmetry in Fourier Representations
- The amplitude spectrum |cₙ| is even-symmetric; |c₋ₙ| = |cₙ|.
- The phase spectrum θₙ is odd-symmetric; θ₋ₙ = -θₙ.
Time and Frequency Domain Interpretation
- A periodic signal can be decomposed into sinusoids of different frequencies—frequency domain representation.
- The exponential form includes both positive and negative frequencies, explaining symmetry in amplitude and phase plots.
- Trigonometric form only contains positive frequencies (no negative frequencies).
Mapping Between Trigonometric and Exponential Series
- cₙ can be written as |cₙ|e^(jθₙ); c₋ₙ as |cₙ|e^(–jθₙ).
- Pairing cₙ and c₋ₙ terms in the exponential series yields a cosine term with amplitude 2|cₙ| and phase θₙ.
- There is a direct relationship: R = 2|cₙ|, Φ = –θₙ, and Φ = tan⁻¹(Bₙ/Aₙ), R = √(Aₙ² + Bₙ²).
Key Terms & Definitions
- Orthogonality — Two signals are orthogonal if their inner product over a period is zero.
- Fourier Coefficient (cₙ) — The weight for each basis function in the Fourier series.
- Completeness — The ability of the basis set to represent any signal in the space.
- Amplitude Spectrum — The magnitudes of Fourier coefficients plotted against frequency.
- Phase Spectrum — The phases of Fourier coefficients plotted against frequency.
- Frequency Domain — Representation of a signal by its frequency components instead of time.
Action Items / Next Steps
- Next class: Discussion on the concept of negative frequency.
- Review the relationship between trigonometric and exponential Fourier coefficients.
- Practice computing Fourier coefficients and plotting amplitude and phase spectra.