19.1 Fourier series for periodic signals

Fourier series break down periodic signals into simple sine and cosine waves. This powerful tool helps us understand complex waveforms by splitting them into basic building blocks.

In this section, we'll learn how to represent signals using Fourier series. We'll explore the math behind it and see how it applies to real-world signals like square waves.

Periodic Signals and Fourier Series

Defining Periodic Signals

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• Periodic signals repeat at regular intervals called the period $T$
• Mathematically, a signal $x(t)$ is periodic if $x(t) = x(t + T)$ for all $t$, where $T$ is the period
• Examples of periodic signals include sine waves, square waves, and sawtooth waves
• Periodic signals can be represented as a sum of sinusoidal components using Fourier series

Fourier Series Representation

• Fourier series represents a periodic signal as an infinite sum of sinusoidal components
• Each component has a specific frequency, amplitude, and phase
• The fundamental frequency $f_0$ is the lowest frequency component and equals the reciprocal of the period ($f_0 = 1/T$)
• Harmonics are integer multiples of the fundamental frequency ($f_n = n \cdot f_0$, where $n = 1, 2, 3, \ldots$)
  • The first harmonic is the fundamental frequency itself
  • Higher harmonics contribute to the shape and complexity of the periodic signal

Fourier Series Representation

Fourier Series Coefficients

• Fourier series coefficients determine the amplitude and phase of each sinusoidal component
• The DC component $a_0$ represents the average value of the signal over one period
  • $a_0 = \frac{1}{T} \int_{0}^{T} x(t) dt$
• The coefficients $a_n$ and $b_n$ represent the amplitudes of the cosine and sine components, respectively
  • $a_n = \frac{2}{T} \int_{0}^{T} x(t) \cos(2\pi n f_0 t) dt$
  • $b_n = \frac{2}{T} \int_{0}^{T} x(t) \sin(2\pi n f_0 t) dt$

Trigonometric and Complex Exponential Forms

• The trigonometric form of the Fourier series is:
  • $x(t) = a_0 + \sum_{n=1}^{\infty} \left(a_n \cos(2\pi n f_0 t) + b_n \sin(2\pi n f_0 t)\right)$
• The complex exponential form of the Fourier series is:
  • $x(t) = \sum_{n=-\infty}^{\infty} c_n e^{j2\pi n f_0 t}$
  • The coefficients $c_n$ are complex numbers that combine the information from $a_n$ and $b_n$
  • $c_n = \frac{1}{T} \int_{0}^{T} x(t) e^{-j2\pi n f_0 t} dt$

Fourier Series Properties

Parseval's Theorem

• Parseval's theorem relates the energy of a periodic signal to its Fourier series coefficients
• The total energy of a periodic signal over one period is equal to the sum of the squared magnitudes of its Fourier coefficients
  • $\frac{1}{T} \int_{0}^{T} |x(t)|^2 dt = |a_0|^2 + \frac{1}{2} \sum_{n=1}^{\infty} (|a_n|^2 + |b_n|^2)$ (trigonometric form)
  • $\frac{1}{T} \int_{0}^{T} |x(t)|^2 dt = \sum_{n=-\infty}^{\infty} |c_n|^2$ (complex exponential form)
• This theorem is useful for analyzing the energy distribution among the frequency components of a periodic signal

Gibbs Phenomenon

• Gibbs phenomenon occurs when a Fourier series approximates a discontinuous periodic signal
• Near the discontinuities, the Fourier series approximation exhibits oscillations (overshoots and undershoots)
• As more terms are added to the Fourier series, the oscillations become narrower but do not decrease in amplitude
• The maximum overshoot is approximately 9% of the jump discontinuity, regardless of the number of terms used
• Gibbs phenomenon is important to consider when using Fourier series to approximate signals with sharp transitions (square waves or sawtooth waves)