Trigonometric Fourier Series

Introduction

The trigonometric Fourier series represents a periodic signal as a sum of sines and cosines at harmonically related frequencies. This is often the most intuitive form of the Fourier series for students encountering the topic for the first time, because sines and cosines are real-valued functions that you can directly visualize as oscillating waveforms. While the exponential form is mathematically more compact, the trigonometric form connects directly to physical quantities — the amplitude and phase of each harmonic oscillation.

For B.Tech students, the trigonometric form is particularly useful when dealing with real signals in circuit analysis, vibration analysis, and power systems, where knowing the magnitude and phase of each harmonic is directly meaningful.

The Trigonometric Form

A periodic signal $x(t)$ with period $T_0$ and fundamental frequency $\omega_0 = 2\pi/T_0$ can be expressed as:

$$x(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty}[a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t)]$$

The Fourier Coefficients

DC component (average value): $$a_0 = \frac{2}{T_0}\int_0^{T_0} x(t)\, dt$$

(Note: Some books define $a_0$ without the 2, using $a_0/2$ in the series. The convention here uses $a_0/2$ in the formula.)

Cosine coefficients: $$a_n = \frac{2}{T_0}\int_0^{T_0} x(t)\cos(n\omega_0 t)\, dt, \quad n = 1, 2, 3, \ldots$$

Sine coefficients: $$b_n = \frac{2}{T_0}\int_0^{T_0} x(t)\sin(n\omega_0 t)\, dt, \quad n = 1, 2, 3, \ldots$$

The integration can be performed over any complete period — for example, $[-T_0/2, T_0/2]$ is often more convenient for symmetric signals.

Exploiting Symmetry

Signal symmetry can dramatically simplify Fourier coefficient calculations. Recognizing symmetry before integrating saves enormous computational effort.

Even Symmetry: $x(t) = x(-t)$

For even signals (symmetric about $t = 0$):

• All $b_n = 0$ (no sine terms, because sine is odd)
• $a_n = \frac{4}{T_0}\int_0^{T_0/2} x(t)\cos(n\omega_0 t)\, dt$ (integrate over half period, multiply by 2)

Examples: triangular wave, $|\sin(t)|$, even pulse trains.

Odd Symmetry: $x(t) = -x(-t)$

For odd signals (antisymmetric about $t = 0$):

• All $a_n = 0$ (no cosine terms, no DC)
• $b_n = \frac{4}{T_0}\int_0^{T_0/2} x(t)\sin(n\omega_0 t)\, dt$

Examples: sawtooth wave (antisymmetric version), square wave centered at origin.

Half-Wave Symmetry: $x(t) = -x(t + T_0/2)$

If the second half of each period is the negative of the first half:

• Only odd harmonics exist ($a_n = b_n = 0$ for even $n$)
• Integration needed only over half a period

Examples: square wave, triangular wave (if centered properly), any waveform with matching positive and negative half-cycles.

Quarter-Wave Symmetry

If the signal has BOTH even (or odd) symmetry AND half-wave symmetry, integration reduces to just one quarter period:

$$a_n = \frac{8}{T_0}\int_0^{T_0/4} x(t)\cos(n\omega_0 t)dt \quad \text{(even + half-wave, odd } n \text{ only)}$$

The Amplitude-Phase (Compact) Form

The trigonometric form can be rewritten by combining the sine and cosine at each frequency:

$$x(t) = C_0 + \sum_{n=1}^{\infty} C_n \cos(n\omega_0 t + \phi_n)$$

where: $$C_0 = a_0/2 \quad \text{(DC component)}$$ $$C_n = \sqrt{a_n^2 + b_n^2} \quad \text{(amplitude of } n\text{th harmonic)}$$ $$\phi_n = -\arctan\left(\frac{b_n}{a_n}\right) \quad \text{(phase of } n\text{th harmonic)}$$

This form directly reveals the amplitude spectrum $C_n$ vs $n$ and the phase spectrum $\phi_n$ vs $n$.

Relationship to Exponential Form

The connection between trigonometric coefficients ($a_n$, $b_n$) and exponential coefficients ($c_n$):

$$c_n = \frac{a_n - jb_n}{2}, \quad c_{-n} = \frac{a_n + jb_n}{2}$$

$$a_n = c_n + c_{-n} = 2\text{Re}(c_n)$$ $$b_n = j(c_n - c_{-n}) = -2\text{Im}(c_n)$$

And: $C_n = 2|c_n|$, $\phi_n = \angle c_n$

Worked Examples

Example 1: Square Wave (Odd Symmetry)

A square wave with amplitude $A$, period $T_0$, odd symmetry:

$$x(t) = \begin{cases} A & 0 < t < T_0/2 \\ -A & T_0/2 < t < T_0 \end{cases}$$

Since it has odd symmetry (when centered): $a_n = 0$ for all $n$.

$$b_n = \frac{4}{T_0}\int_0^{T_0/2} A\sin(n\omega_0 t)dt = \frac{4A}{n\pi}[1-\cos(n\pi)]$$

$$= \begin{cases} 4A/(n\pi) & n \text{ odd} \\ 0 & n \text{ even} \end{cases}$$

Result: $x(t) = \frac{4A}{\pi}\left[\sin(\omega_0 t) + \frac{\sin(3\omega_0 t)}{3} + \frac{\sin(5\omega_0 t)}{5} + \cdots\right]$

Example 2: Full-Wave Rectified Cosine (Even Symmetry)

$x(t) = |\cos(\omega_0 t)|$, period $T_0/2$ (fundamental frequency $2\omega_0$).

Since it's even: $b_n = 0$. After integration:

$$x(t) = \frac{2}{\pi} + \frac{4}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{4n^2-1}\cos(2n\omega_0 t)$$

Example 3: Trapezoidal Wave

A trapezoidal wave is between a square wave and triangular wave (finite rise time). Its coefficients combine sinc-like behavior with $1/n$ or $1/n^2$ decay depending on the slope.

Power Spectrum and Total Harmonic Distortion

The power in the $n$th harmonic is:

$$P_n = \frac{C_n^2}{2} = \frac{a_n^2 + b_n^2}{2} = 2|c_n|^2 \quad (n \geq 1)$$

Total Harmonic Distortion (THD) measures how much a signal deviates from a pure sinusoid:

$$\text{THD} = \frac{\sqrt{\sum_{n=2}^{\infty}C_n^2}}{C_1} = \frac{\sqrt{P_2 + P_3 + \cdots}}{\sqrt{P_1}}$$

In power systems, THD indicates the quality of the AC supply — lower THD means cleaner power.

Common Fourier Series Results (Reference)

Key Takeaways

• The trigonometric form uses real-valued sines and cosines — intuitive for physical signals
• Even signals have only cosine terms; odd signals have only sine terms
• Half-wave symmetry eliminates even harmonics — a powerful simplification
• The amplitude-phase form $C_n\cos(n\omega_0 t + \phi_n)$ directly shows the spectrum
• Always exploit symmetry before computing integrals — it can reduce work by 75%
• THD quantifies harmonic contamination relative to the fundamental