Summary of "Introduction to Signal Processing: Fourier Series (Lecture 11)"

This lecture introduces the concept of Fourier Series as a fundamental tool in Signal Processing, particularly in the context of Linear Time-Invariant (LTI) systems. The main focus is on representing signals as sums of sinusoidal components (sines and cosines) or equivalently Complex Exponentials, and understanding why these representations are powerful for analyzing and processing signals.

Main Ideas and Concepts

Fourier Series and Signal Representation:
- Fourier Series expresses any signal as a sum of sinusoids (cosines and sines) or Complex Exponentials.
- These sinusoids form a basis set (coordinate system) for signals, allowing infinite-dimensional representation.
- The harmonics (cosine/sine components at multiples of a fundamental frequency) build up the signal.
- Fourier Series is a coordinate transformation of signals into frequency domain components.
Historical Context:
- Joseph Fourier developed this method while studying heat conduction.
- His ideas were initially controversial (e.g., LaGrange opposed the concept).
- Later theorems proved that under mild conditions, any function can be represented as a sum of sines and cosines.
Fourier Series and LTI Systems:
- Complex Exponentials \( e^{s t} \) are eigenfunctions of LTI systems.
- Inputting an exponential into an LTI system results in the same exponential scaled by a complex number (eigenvalue).
- This property simplifies analyzing the system's response to any signal because signals can be decomposed into exponentials.
- The system's behavior is characterized by eigenvalues \( H(s) \), which scale each exponential component.
Mathematical Formulation:
- Continuous-time signal \( x(t) \) can be represented as: \[ x(t) = \sum_{k=-\infty}^{\infty} a_k e^{i k \Omega_0 t} \] where \( \Omega_0 = \frac{2\pi}{T} \) is the fundamental frequency for period \( T \).
- Orthogonality of basis functions \( e^{i k \Omega_0 t} \) ensures coefficients \( a_k \) can be uniquely calculated.
- Coefficients \( a_k \) are computed by: \[ a_n = \frac{1}{T} \int_0^T x(t) e^{-i n \Omega_0 t} dt \]
Discrete-Time Systems:
- Similar principles apply with discrete signals \( x[n] \) and discrete exponentials \( Z^n \).
- The system output is the input exponential scaled by eigenvalue \( H(Z) \).
- Convolution in time domain corresponds to multiplication by eigenvalues in frequency domain.
Superposition Principle:
- Because LTI systems are linear, the response to a sum of exponentials is the sum of responses to each exponential.
- This allows analyzing complex signals by decomposing them into simpler exponential components.
Periodicity and Orthogonality:
- Fourier Series assumes signals are periodic with period \( T \).
- Basis functions \( e^{i k \Omega_0 t} \) share this period.
- Orthogonality of these basis functions is key to calculating Fourier Coefficients.
Real Signals and Complex Conjugates:
- Real-valued signals require Fourier Coefficients to satisfy \( a_{-k} = a_k^* \) (complex conjugate symmetry).
- This ensures the imaginary parts cancel out, yielding a real signal composed of sines and cosines.
Visualization and Examples:
- Adding successive harmonics (modes) builds increasingly complex signals.
- Even discontinuous or non-sinusoidal signals can be represented as sums of sinusoids.
- The lecture hints at future demonstrations with computational tools to visualize Fourier Series approximations.

Methodology / Instructions for Fourier Series Representation

Identify the Period \( T \) of the signal.
Determine the fundamental frequency: \[ \Omega_0 = \frac{2\pi}{T} \]
Express the signal as a sum of Complex Exponentials: \[ x(t) = \sum_{k=-\infty}^{\infty} a_k e^{i k \Omega_0 t} \]
Use orthogonality to calculate coefficients \( a_k \):
- Multiply both sides by \( e^{-i n \Omega_0 t} \).
- Integrate over one period \( [0, T] \):
\[ a_n = \frac{1}{T} \int_0^T x(t) e^{-i n \Omega_0 t} dt \]