The continuous-time Fourier series is a powerful tool for analyzing periodic signals. It breaks down complex waveforms into simpler sinusoidal components, revealing their frequency content. This technique is crucial for understanding signal behavior in various engineering applications.

By representing signals as sums of harmonically related sinusoids, Fourier series enables efficient filtering, system analysis, and signal processing. It forms the foundation for more advanced transforms and provides insights into signal properties in both time and frequency domains.

Definition of Fourier series

• Fourier series is a mathematical tool used to represent periodic signals as a sum of sinusoidal components with different frequencies, amplitudes, and phases
• It allows for the decomposition of complex periodic signals into simpler sinusoidal components, facilitating analysis and processing in the frequency domain

Representation of periodic signals

• Periodic signals repeat themselves at regular intervals, with a period $T$
• Fourier series represents periodic signals as an infinite sum of sinusoidal components, each with a specific frequency, amplitude, and phase
• The frequencies of the sinusoidal components are integer multiples of the fundamental frequency $\omega_0 = \frac{2\pi}{T}$

Synthesis equation

• The synthesis equation is used to reconstruct the periodic signal $x(t)$ from its Fourier series coefficients
• The general form of the synthesis equation is: $x(t) = \sum_{n=-\infty}^{\infty} c_n e^{jn\omega_0 t}$ where $c_n$ are the complex Fourier series coefficients and $\omega_0$ is the fundamental frequency
• The synthesis equation allows for the generation of the original periodic signal by summing the sinusoidal components with their respective coefficients

Analysis equation

• The analysis equation is used to determine the Fourier series coefficients $c_n$ from the periodic signal $x(t)$
• The general form of the analysis equation is: $c_n = \frac{1}{T} \int_{0}^{T} x(t) e^{-jn\omega_0 t} dt$
• The analysis equation involves calculating the inner product between the periodic signal and the complex exponential basis functions $e^{-jn\omega_0 t}$
• The resulting coefficients $c_n$ represent the contribution of each sinusoidal component to the overall periodic signal

Properties of Fourier series

• Fourier series possesses several important properties that facilitate the analysis and manipulation of periodic signals in the frequency domain
• These properties provide insights into the behavior of signals and simplify calculations when working with Fourier series

Linearity

• The Fourier series is a linear operation, meaning that if $x_1(t)$ and $x_2(t)$ have Fourier series coefficients $c_{1n}$ and $c_{2n}$ respectively, then: $a x_1(t) + b x_2(t) \leftrightarrow a c_{1n} + b c_{2n}$ where $a$ and $b$ are constants
• Linearity allows for the superposition of signals and the scaling of Fourier series coefficients

Time shifting

• If $x(t)$ has Fourier series coefficients $c_n$, then the time-shifted signal $x(t-t_0)$ has Fourier series coefficients: $c_n e^{-jn\omega_0 t_0}$
• Time shifting in the time domain corresponds to a phase shift in the frequency domain

Time scaling

• If $x(t)$ has Fourier series coefficients $c_n$, then the time-scaled signal $x(at)$ has Fourier series coefficients: $\frac{1}{|a|} c_n$ where $a$ is a non-zero constant
• Time scaling in the time domain results in a scaling of the fundamental frequency and the Fourier series coefficients

Time reversal

• If $x(t)$ has Fourier series coefficients $c_n$, then the time-reversed signal $x(-t)$ has Fourier series coefficients: $c_{-n}$
• Time reversal in the time domain corresponds to a conjugate reversal of the Fourier series coefficients

Conjugate symmetry

• If $x(t)$ is a real-valued periodic signal, then its Fourier series coefficients exhibit conjugate symmetry: $c_{-n} = c_n^$ where $^$ denotes the complex conjugate
• Conjugate symmetry implies that the coefficients for negative frequencies are the complex conjugates of the coefficients for positive frequencies

Parseval's theorem

• Parseval's theorem relates the energy of a periodic signal in the time domain to the energy of its Fourier series coefficients in the frequency domain
• The theorem states that: $\frac{1}{T} \int_{0}^{T} |x(t)|^2 dt = \sum_{n=-\infty}^{\infty} |c_n|^2$
• Parseval's theorem provides a way to calculate the signal energy using either the time-domain representation or the Fourier series coefficients

Fourier series coefficients

• Fourier series coefficients are the complex numbers that determine the amplitudes and phases of the sinusoidal components in the Fourier series representation
• The coefficients provide information about the frequency content and the relative contribution of each sinusoidal component to the periodic signal

Calculation of coefficients

• The Fourier series coefficients $c_n$ can be calculated using the analysis equation: $c_n = \frac{1}{T} \int_{0}^{T} x(t) e^{-jn\omega_0 t} dt$
• The calculation involves evaluating the inner product between the periodic signal $x(t)$ and the complex exponential basis functions $e^{-jn\omega_0 t}$
• The resulting coefficients $c_n$ are complex numbers that represent the amplitude and phase of each sinusoidal component

Trigonometric vs exponential form

• Fourier series can be expressed in two equivalent forms: trigonometric form and exponential form
• The trigonometric form represents the periodic signal as a sum of cosine and sine functions: $x(t) = a_0 + \sum_{n=1}^{\infty} (a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t))$ where $a_0$, $a_n$, and $b_n$ are the trigonometric Fourier series coefficients
• The exponential form represents the periodic signal as a sum of complex exponentials: $x(t) = \sum_{n=-\infty}^{\infty} c_n e^{jn\omega_0 t}$ where $c_n$ are the complex Fourier series coefficients

Relationship between coefficients

• The trigonometric and exponential Fourier series coefficients are related as follows: $a_0 = c_0$ $a_n = \frac{1}{2}(c_n + c_{-n})$ $b_n = \frac{1}{2j}(c_n - c_{-n})$
• The trigonometric coefficients can be obtained from the complex coefficients and vice versa

Symmetry of coefficients

• For real-valued periodic signals, the Fourier series coefficients exhibit symmetry properties:
  • Conjugate symmetry: $c_{-n} = c_n^$
  • Even symmetry for cosine coefficients: $a_{-n} = a_n$
  • Odd symmetry for sine coefficients: $b_{-n} = -b_n$
• These symmetry properties can be exploited to simplify calculations and reduce the number of coefficients needed to represent the signal

Convergence of Fourier series

• Convergence of Fourier series refers to the conditions under which the Fourier series representation of a periodic signal converges to the original signal
• Understanding convergence is important to ensure the accuracy and validity of the Fourier series approximation

Dirichlet conditions

• The Dirichlet conditions are sufficient conditions for the convergence of a Fourier series
• The conditions state that a periodic signal $x(t)$ will have a convergent Fourier series if:
  • $x(t)$ is absolutely integrable over one period
  • $x(t)$ has a finite number of discontinuities in one period
  • $x(t)$ has a finite number of maxima and minima in one period
• If a periodic signal satisfies the Dirichlet conditions, its Fourier series will converge to the signal at all points of continuity

Gibbs phenomenon

• The Gibbs phenomenon occurs when a Fourier series approximation of a discontinuous periodic signal exhibits oscillations near the discontinuities
• The oscillations result from the truncation of the infinite Fourier series to a finite number of terms
• The magnitude of the oscillations does not decrease as more terms are added to the series, but the width of the oscillations becomes narrower
• The Gibbs phenomenon highlights the limitations of Fourier series in representing signals with discontinuities

Uniform vs pointwise convergence

• Uniform convergence means that the Fourier series approximation converges uniformly to the original signal over the entire period
• Pointwise convergence means that the Fourier series approximation converges to the original signal at each individual point, but the convergence may not be uniform
• For continuous periodic signals, the Fourier series converges uniformly
• For discontinuous periodic signals, the Fourier series may converge pointwise but not uniformly due to the Gibbs phenomenon

Fourier series for common signals

• Fourier series can be applied to various common periodic signals to obtain their frequency-domain representations
• Analyzing the Fourier series of these signals provides insights into their frequency content and helps in understanding their behavior

Square wave

• A square wave is a periodic signal that alternates between two constant values
• The Fourier series coefficients of a square wave with amplitude $A$ and period $T$ are: $c_n = \frac{A}{n\pi}(1 - e^{-jn\pi})$ for odd $n$, and $c_n = 0$ for even $n$
• The Fourier series of a square wave consists of only odd harmonics with amplitudes decreasing as $\frac{1}{n}$

Sawtooth wave

• A sawtooth wave is a periodic signal that linearly increases and then abruptly drops to its initial value
• The Fourier series coefficients of a sawtooth wave with amplitude $A$ and period $T$ are: $c_n = \frac{A}{jn\pi}$
• The Fourier series of a sawtooth wave contains both even and odd harmonics with amplitudes decreasing as $\frac{1}{n}$

Triangular wave

• A triangular wave is a periodic signal that linearly increases and then linearly decreases, forming a triangular shape
• The Fourier series coefficients of a triangular wave with amplitude $A$ and period $T$ are: $c_n = \frac{4A}{n^2\pi^2}(1 - (-1)^n)$ for odd $n$, and $c_n = 0$ for even $n$
• The Fourier series of a triangular wave consists of only odd harmonics with amplitudes decreasing as $\frac{1}{n^2}$

Full-wave rectified sine

• A full-wave rectified sine wave is obtained by taking the absolute value of a sine wave
• The Fourier series coefficients of a full-wave rectified sine wave with amplitude $A$ and period $T$ are: $c_0 = \frac{A}{\pi}$, $c_n = \frac{2A}{n\pi}$ for even $n$, and $c_n = 0$ for odd $n$
• The Fourier series of a full-wave rectified sine wave contains a DC component and even harmonics with amplitudes decreasing as $\frac{1}{n}$

Applications of Fourier series

• Fourier series has numerous applications in various fields, including signal processing, communications, and control systems
• The frequency-domain representation provided by Fourier series enables efficient analysis, filtering, and manipulation of periodic signals

Filtering in frequency domain

• Fourier series allows for the design and implementation of frequency-selective filters
• By modifying the Fourier series coefficients, specific frequency components can be attenuated or emphasized
• Low-pass, high-pass, and band-pass filters can be realized by appropriately shaping the Fourier series coefficients

Approximation of signals

• Fourier series can be used to approximate complex periodic signals by truncating the series to a finite number of terms
• The approximation improves as more terms are included in the series
• Signal compression and data reduction can be achieved by representing signals with a limited number of Fourier series coefficients

Analysis of LTI systems

• Fourier series is a powerful tool for analyzing the behavior of linear time-invariant (LTI) systems
• The response of an LTI system to a periodic input can be determined by applying the system's frequency response to the Fourier series coefficients of the input
• This approach simplifies the analysis and design of LTI systems in the frequency domain

Solving differential equations

• Fourier series can be employed to solve certain types of differential equations with periodic boundary conditions
• By expressing the solution as a Fourier series and substituting it into the differential equation, the problem can be reduced to solving for the Fourier series coefficients
• This technique is particularly useful in solving partial differential equations arising in heat transfer, vibrations, and electromagnetics

Relationship with other transforms

• Fourier series is closely related to other integral transforms commonly used in signal processing and system analysis
• Understanding the connections between Fourier series and these transforms provides a broader perspective on signal representation and manipulation

Fourier series vs Fourier transform

• The Fourier transform is a generalization of the Fourier series for non-periodic signals
• While the Fourier series represents periodic signals as a sum of discrete frequency components, the Fourier transform represents non-periodic signals as a continuous spectrum of frequencies
• The Fourier transform can be viewed as the limit of the Fourier series as the period of the signal approaches infinity

Discrete-time Fourier series

• The discrete-time Fourier series (DTFS) is the counterpart of the continuous-time Fourier series for discrete-time periodic signals
• The DTFS represents a discrete-time periodic signal as a sum of complex exponentials with discrete frequencies
• The properties and techniques associated with the continuous-time Fourier series can be adapted to the discrete-time case

Fourier series vs Laplace transform

• The Laplace transform is a generalization of the Fourier series and the Fourier transform for signals that may not be periodic or have finite energy
• The Laplace transform introduces a complex frequency variable, allowing for the analysis of signals with exponential behavior and systems with initial conditions
• The Fourier series can be obtained from the Laplace transform by evaluating it on the imaginary axis ($s = j\omega$) and considering periodic signals