Overview
This lecture reviews Fourier series representation of periodic signals, explains how signal energy can be computed both in the time and frequency domains, and introduces Parseval's theorem for relating these approaches.
Fourier Series Representation
- Any periodic signal can be represented as a sum of sinusoidal or exponential basis functions (Fourier series).
- The exponential Fourier series is expressed as ( w(t) = \sum_{n=-\infty}^{\infty} D_n e^{j n 2\pi f_0 t} ), where ( f_0 = 1/T_0 ).
- Fourier series coefficients ( D_n ) provide both amplitude (magnitude) and phase information for each frequency component.
- For real signals with real ( D_n ), the phase spectrum is zero everywhere.
- The complete signal can be reconstructed by summing up all basis components with the correct amplitude and phase._
Spectral Components and Signal Construction
- Each frequency component is plotted using its amplitude from the amplitude spectrum and phase from the phase spectrum.
- For real signals, positive and negative frequency components are combined to form cosine functions.
- Components with zero amplitude make no contribution to the reconstructed signal.
Signal Energy in Time and Frequency Domain
- For real signals, energy is computed as ( \int_a^b |g(t)|^2 dt ).
- For complex signals, energy is ( \int_a^b |g(t)|^2 dt = \int_a^b g(t)g^*(t)dt ).
- The energy of a signal can be interpreted as the squared length (norm) analogous to vectors.*
Orthogonality and Energy Addition
- If two signals are orthogonal, their cross-product integral over a period is zero.
- The energy of a linear combination of orthogonal signals is the sum of the energies of each component.
- All cross-terms vanish due to orthogonality, simplifying energy computation.
Parseval's Theorem
- Parseval's theorem states that the total energy of a periodic signal over one period equals the sum of the squared magnitudes of its Fourier coefficients.
- Mathematically: Total energy = ( \sum_{n=-\infty}^{\infty} |D_n|^2 ).
- For real signals: ( |D_0|^2 + 2 \sum_{n=1}^{\infty} |D_n|^2 ).
- This mirrors the Pythagorean theorem for vectors: measurement (energy) is the sum of orthogonal component measurements.
Key Terms & Definitions
- Fourier Series — Representation of a periodic signal as a sum of sinusoidal or exponential components.
- Coefficient (( D_n )) — Complex number representing amplitude and phase of each frequency component.
- Orthogonal Signals — Signals whose cross-product integral over a period is zero.
- Parseval's Theorem — The total energy of a signal equals the sum of the squares of its Fourier coefficients.
Action Items / Next Steps
- Review how to compute Fourier coefficients for various signals.
- Prepare for the next lecture on representing and analyzing non-periodic (energy) signals.