19.1 Fourier series for periodic signals

Periodic Signals and Fourier Series

Fourier series let you decompose any periodic signal into a sum of simple sine and cosine waves. This matters because analyzing a complex waveform directly is hard, but analyzing individual sinusoids is straightforward. Once you can break a signal into its frequency components, you can predict how circuits (filters, amplifiers, etc.) will respond to it.

Defining Periodic Signals

A periodic signal repeats at regular intervals. The time it takes to complete one full cycle is the period $T$. Formally, a signal $x(t)$ is periodic if:

$x(t) = x(t + T) \text{ for all } t$

Common examples: sine waves, square waves, sawtooth waves, and triangle waves. Any of these can be represented as a sum of sinusoidal components using a Fourier series.

Fourier Series Representation

A Fourier series expresses a periodic signal as an infinite sum of sinusoids. Each sinusoidal component has its own frequency, amplitude, and phase.

• The fundamental frequency $f_0$ is the lowest frequency in the series and equals the reciprocal of the period: $f_0 = 1/T$. You can also express this as an angular frequency: $\omega_0 = 2\pi f_0$.
• Harmonics are integer multiples of the fundamental: $f_n = n \cdot f_0$, where $n = 1, 2, 3, \ldots$
  • The first harmonic ($n = 1$) is the fundamental frequency.
  • Higher harmonics shape the waveform. A square wave, for instance, needs many odd harmonics to approximate its sharp edges, while a pure sine wave has only the fundamental.

Fourier Series Coefficients

The coefficients tell you how much of each sinusoidal component is present in the signal.

DC component $a_0$: This is just the average value of the signal over one period.

$a_0 = \frac{1}{T} \int_{0}^{T} x(t) \, dt$

If your signal is symmetric about zero (like a standard square wave centered at the origin), $a_0 = 0$.

Cosine coefficients $a_n$ and sine coefficients $b_n$: These capture how much of each harmonic is present.

$a_n = \frac{2}{T} \int_{0}^{T} x(t) \cos(2\pi n f_0 t) \, dt$

$b_n = \frac{2}{T} \int_{0}^{T} x(t) \sin(2\pi n f_0 t) \, dt$

A useful shortcut: if the signal has even symmetry ($x(t) = x(-t)$), all the $b_n$ coefficients are zero, so you only need cosines. If it has odd symmetry ($x(t) = -x(-t)$), all the $a_n$ coefficients are zero, so you only need sines. Recognizing symmetry can cut your work in half.

How to Compute Fourier Coefficients (Step-by-Step)

1. Identify the period $T$ and fundamental frequency $f_0 = 1/T$.
2. Write out the mathematical expression for $x(t)$ over one period.
3. Check for symmetry (even or odd) to determine if any coefficients are automatically zero.
4. Compute $a_0$ by integrating $x(t)$ over one period and dividing by $T$.
5. Compute $a_n$ and $b_n$ using the integral formulas above for each harmonic $n$.
6. Plug the coefficients into the trigonometric Fourier series formula.

Trigonometric and Complex Exponential Forms

Trigonometric form uses real-valued sines and cosines:

$x(t) = a_0 + \sum_{n=1}^{\infty} \left(a_n \cos(2\pi n f_0 t) + b_n \sin(2\pi n f_0 t)\right)$

This form is intuitive because you can directly read off the amplitude of each sine and cosine component.

Complex exponential form is more compact and often easier to work with mathematically:

$x(t) = \sum_{n=-\infty}^{\infty} c_n \, e^{j2\pi n f_0 t}$

Here the coefficients $c_n$ are complex numbers that encode both amplitude and phase:

$c_n = \frac{1}{T} \int_{0}^{T} x(t) \, e^{-j2\pi n f_0 t} \, dt$

The relationship between the two forms: $c_0 = a_0$, and for $n \geq 1$, $c_n = \frac{1}{2}(a_n - jb_n)$ and $c_{-n} = c_n^*$ (the complex conjugate). The complex form is preferred in most EE courses because it handles phase naturally and extends cleanly into the Fourier transform.

Fourier Series Properties

Parseval's Theorem

Parseval's theorem connects the time domain and frequency domain: the average power of a periodic signal equals the sum of the powers in each of its frequency components.

In trigonometric form:

$\frac{1}{T} \int_{0}^{T} |x(t)|^2 \, dt = |a_0|^2 + \frac{1}{2} \sum_{n=1}^{\infty} (a_n^2 + b_n^2)$

In complex exponential form:

$\frac{1}{T} \int_{0}^{T} |x(t)|^2 \, dt = \sum_{n=-\infty}^{\infty} |c_n|^2$

Why this matters: it lets you figure out how much of a signal's power sits at each frequency. For example, if you're designing a filter to clean up a noisy square wave, Parseval's theorem tells you exactly how much power you'd lose by cutting off higher harmonics.

Gibbs Phenomenon

When you approximate a signal with sharp discontinuities (like a square wave) using a finite number of Fourier terms, you'll notice ringing near the edges. This is the Gibbs phenomenon.

• The oscillations appear as overshoots and undershoots right at the discontinuity.
• Adding more terms makes the ripples narrower, but the peak overshoot stays at about 9% of the jump size, no matter how many terms you include.
• This 9% overshoot never goes away with more terms. It's a fundamental limitation of representing discontinuities with smooth sinusoids.

For a square wave that jumps between $-1$ and $+1$ (a jump of 2), the overshoot peaks at roughly $\pm 1.09$. You'll see this clearly if you plot partial sums with 10, 50, or 100 terms: the spike stays the same height but gets thinner. Keep this in mind when working with signals that have sharp transitions, since the Fourier series will never perfectly reconstruct those edges with a finite number of terms.