2.1 Trigonometric Fourier Series

Trigonometric Fourier Series are the building blocks of signal processing. They break down complex periodic signals into simple sine and cosine waves, making it easier to analyze and manipulate them. This powerful tool helps us understand the frequency components of signals.

By representing signals as sums of sinusoids, we can extract valuable information about their characteristics. This approach is crucial for various applications, from audio processing to image compression, and forms the foundation for more advanced signal analysis techniques.

Representing periodic signals with Fourier series

Periodic signals and their properties

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• A periodic signal is a function that repeats itself at regular intervals, with a period T
  • f(t) = f(t + T) for all values of t
  • Examples of periodic signals include sine waves, square waves, and sawtooth waves
• The fundamental frequency (ω0) of a periodic signal is the reciprocal of its period (T)
  • ω0 = 2π/T
  • The fundamental frequency represents the lowest frequency component in the Fourier series

Trigonometric Fourier series representation

• Trigonometric Fourier series represent periodic signals as a sum of sinusoidal functions with different frequencies, amplitudes, and phases
  • The sinusoidal functions are integer multiples of the fundamental frequency (harmonics)
  • The amplitudes and phases of the sinusoidal functions are determined by the Fourier coefficients
• The general form of a trigonometric Fourier series for a periodic signal f(t) with period T is:
  • f(t) = a0 + Σ(n=1 to ∞) [an * cos(nω0t) + bn * sin(nω0t)]
  • a0 is the constant term (DC component) representing the average value of the signal
  • an and bn are the Fourier coefficients for the cosine and sine terms, respectively
• The Fourier series representation is unique for a given periodic signal
  • There is only one set of Fourier coefficients that accurately describes the signal
  • The uniqueness property allows for the analysis and synthesis of periodic signals using Fourier series

Determining Fourier coefficients

Formulas for calculating Fourier coefficients

• Fourier coefficients (a0, an, and bn) are calculated using specific formulas that involve integrating the product of the periodic signal and the corresponding basis functions over one period
  • The basis functions are 1 for a0, cos(nω0t) for an, and sin(nω0t) for bn
  • The limits of integration are from 0 to T, representing one period of the signal
• The constant term a0 represents the average value (DC component) of the periodic signal and is calculated using the formula:
  • a0 = (1/T) * ∫(0 to T) f(t) dt
• The cosine coefficients an represent the amplitudes of the cosine terms and are calculated using the formula:
  • an = (2/T) * ∫(0 to T) f(t) * cos(nω0t) dt, for n ≥ 1
• The sine coefficients bn represent the amplitudes of the sine terms and are calculated using the formula:
  • bn = (2/T) * ∫(0 to T) f(t) * sin(nω0t) dt, for n ≥ 1

Methods for calculating Fourier coefficients

• Fourier coefficients can be calculated analytically for simple periodic functions
  • Examples of simple periodic functions include square waves, triangular waves, and sawtooth waves
  • Analytical calculations involve evaluating the integrals using techniques such as trigonometric identities and integration by parts
• For more complex signals, numerical methods can be used to calculate the Fourier coefficients
  • Numerical methods involve discretizing the signal and approximating the integrals using techniques such as the trapezoidal rule or Simpson's rule
  • Software tools such as MATLAB and Python libraries (NumPy, SciPy) can be used to calculate Fourier coefficients numerically

Reconstructing periodic signals

Fourier series synthesis

• Given the Fourier coefficients (a0, an, and bn) and the fundamental frequency (ω0) of a periodic signal, it is possible to reconstruct the original signal using the Fourier series formula
  • The reconstructed signal is an approximation of the original signal
  • The accuracy of the reconstruction improves as more terms (harmonics) are included in the series
• The number of terms required for an accurate reconstruction depends on the complexity of the original signal and the desired level of accuracy
  • Signals with sharp transitions or discontinuities may require more terms for accurate reconstruction
  • Smooth signals can often be reconstructed with fewer terms

Partial sums and Gibbs phenomenon

• Partial sums of the Fourier series can be used to analyze the contribution of individual harmonics to the overall signal
  • A partial sum includes only a finite number of terms from the Fourier series
  • Plotting partial sums can provide insights into how the signal is constructed from its frequency components
• Gibbs phenomenon may occur when reconstructing a signal with discontinuities using a Fourier series
  • Gibbs phenomenon results in oscillations near the discontinuities that do not diminish as more terms are added to the series
  • The oscillations are a consequence of the Fourier series trying to approximate the discontinuity with continuous functions
  • Techniques such as signal windowing and using alternative basis functions (e.g., wavelets) can be used to mitigate Gibbs phenomenon

Even vs odd functions in Fourier series

Properties of even and odd functions

• Even functions are symmetric about the y-axis, satisfying the condition f(-t) = f(t) for all values of t
  • Examples of even functions include cosine functions and even powers of t (e.g., t^2, t^4)
  • For even functions, the Fourier series contains only cosine terms (an coefficients), and the sine coefficients (bn) are zero
• Odd functions are symmetric about the origin, satisfying the condition f(-t) = -f(t) for all values of t
  • Examples of odd functions include sine functions and odd powers of t (e.g., t, t^3)
  • For odd functions, the Fourier series contains only sine terms (bn coefficients), and the cosine coefficients (an) are zero
• If a periodic function is neither even nor odd, its Fourier series will contain both cosine and sine terms

Exploiting symmetry in Fourier analysis

• The even and odd properties of functions can be used to simplify the calculation of Fourier coefficients by exploiting the symmetry of the integrals
  • For even functions, the integrals involving sine terms evaluate to zero, simplifying the calculation of an coefficients
  • For odd functions, the integrals involving cosine terms evaluate to zero, simplifying the calculation of bn coefficients
• Decomposing a periodic function into its even and odd components can provide insights into the signal's properties and simplify the analysis of the Fourier series
  • The even component of a function is given by [f(t) + f(-t)]/2
  • The odd component of a function is given by [f(t) - f(-t)]/2
  • The original function can be reconstructed by adding its even and odd components