5.1 Fourier series expansion for periodic signals

Periodic signals repeat at regular intervals and can be broken down into simpler sinusoidal components. This breakdown, called a Fourier series, represents the signal as a sum of sine and cosine waves with different frequencies and amplitudes.

The Fourier series helps us understand and analyze complex periodic signals. By calculating the coefficients for each component, we can reconstruct the original signal with varying levels of accuracy, depending on how many terms we include in the series.

Periodic Signals and Fourier Series Expansion

Properties of periodic signals

• Repeat at regular intervals (period $T$) satisfying $x(t) = x(t + nT)$, where $n$ is an integer
• Fundamental period $T$ is the smallest positive value that satisfies the periodicity condition
• Fundamental frequency $f_0$ is the reciprocal of the fundamental period $f_0 = \frac{1}{T}$
• Can be decomposed into a sum of sinusoids with frequencies that are integer multiples of the fundamental frequency (harmonics)
  • Example: Square wave consists of odd harmonics of the fundamental frequency

Fourier series for periodic signals

• Represents a periodic signal as an infinite sum of sinusoids with frequencies that are integer multiples of the fundamental frequency
• Consists of a DC component (average value), cosine terms, and sine terms representing oscillatory components at different frequencies
• General form of the Fourier series for a periodic signal $x(t)$:
  • $x(t) = a_0 + \sum_{n=1}^{\infty} (a_n \cos(n \omega_0 t) + b_n \sin(n \omega_0 t))$
    • $a_0$ is the DC component
    • $a_n$ and $b_n$ are the Fourier series coefficients
    • $\omega_0 = 2\pi f_0$ is the fundamental angular frequency
• Example: Sawtooth wave has a Fourier series with both cosine and sine terms

Derivation of Fourier coefficients

• Fourier series coefficients ($a_0$, $a_n$, and $b_n$) calculated using the following formulas:
  1. DC component: $a_0 = \frac{1}{T} \int_{-T/2}^{T/2} x(t) dt$
  2. Cosine coefficients: $a_n = \frac{2}{T} \int_{-T/2}^{T/2} x(t) \cos(n \omega_0 t) dt$
  3. Sine coefficients: $b_n = \frac{2}{T} \int_{-T/2}^{T/2} x(t) \sin(n \omega_0 t) dt$
• Integration limits can be adjusted to any interval of length $T$ (e.g., $[0, T]$)
• Coefficients determine the amplitude and phase of each sinusoidal component in the Fourier series
  • Example: For a square wave, $a_0$ is the average value, and $a_n$ decreases as $\frac{1}{n}$ for odd $n$, while $b_n = 0$

Signal reconstruction via Fourier series

• Periodic signal reconstructed by summing the DC component and sinusoidal components
  • Sinusoidal components obtained by multiplying Fourier series coefficients with corresponding cosine and sine terms
• Reconstructed signal is an approximation of the original signal
  • Accuracy improves as more terms are included in the Fourier series
• In practice, a finite number of terms ($N$) are used for reconstruction:
  • $x(t) \approx a_0 + \sum_{n=1}^{N} (a_n \cos(n \omega_0 t) + b_n \sin(n \omega_0 t))$
• Choice of $N$ depends on desired accuracy and complexity of the signal
  • Higher $N$ results in better approximations but increases computational complexity
  • Example: Reconstructing a square wave with 10 terms provides a good approximation, while 100 terms yields a nearly perfect reconstruction