Fourier Series: Trigonometric Form

Representing complex, repeating patterns as simpler, oscillatory components is a cornerstone of engineering analysis. The trigonometric Fourier series provides the mathematical machinery to do exactly this, transforming any periodic signal into a sum of basic sine and cosine waves. Mastering this decomposition is essential for fields ranging from signal processing and communications to vibrations analysis and acoustics, as it reveals the fundamental frequencies that constitute a complex waveform.

The Core Idea: Decomposing Periodic Signals

A periodic function $f(t)$ is one that repeats its values at regular intervals, such that $f(t+T)=f(t)$ for all $t$. The constant $T$ is the fundamental period. The trigonometric Fourier series states that a wide class of such periodic functions can be expressed as an infinite sum of sine and cosine functions whose frequencies are integer multiples of the fundamental frequency $f_0=1/T$ (or fundamental angular frequency ${\omega }_0=2\pi f_0=2\pi /T$).

The general form of the series is:

$$f(t)=a_0+\sum _{n=1}^{\infty }\left[a_n\cos (n{\omega }_{0}t)+b_n\sin (n{\omega }_{0}t)\right]$$

This equation is a recipe: take a constant offset, add a cosine wave at the fundamental frequency, add a sine wave at the fundamental frequency, then add cosine and sine waves at every integer multiple (harmonics), each with its own specific weight. The magic lies in calculating those weights—the Fourier coefficients $a_0$, $a_n$, and $b_n$.

Calculating the Fourier Coefficients

The coefficients are not arbitrary; they are computed by projecting the function $f(t)$ onto each of the basis functions (the constant, cosine, and sine terms). This is achieved through integration over one complete period, exploiting the orthogonality properties of sines and cosines. The formulas are:

The DC (or average) component: $$a_0=\frac{1}{T}{\int }_{T}f(t)dt$$ This coefficient represents the average value of the function over one period.

The cosine coefficients: $$a_n=\frac{2}{T}{\int }_{T}f(t)\cos (n{\omega }_{0}t)dt,n\ge 1$$

The sine coefficients: $$b_n=\frac{2}{T}{\int }_{T}f(t)\sin (n{\omega }_{0}t)dt,n\ge 1$$

The integration limits ${\int }_T$ denote integration over any convenient interval of length $T$, such as from $0$ to $T$ or from $-T/2$ to $T/2$. The factor of 2 for $a_n$ and $b_n$ (but not for $a_0$) arises from the average value of the square of a sine or cosine over a period.

A Worked Example: The Square Wave

Consider an odd-symmetry square wave with period $T$, amplitude $A$, and defined over one period as: $f(t)=\{\begin{array}{ll}A,& 0<t<T/2\\ -A,& T/2<t<T\end{array}$

Let's compute its trigonometric Fourier series.

1. DC Offset ($a_0$): The average value over one period is clearly zero. The integral confirms: $a_0=\frac{1}{T}\left({\int }_0^{T/2}Adt+{\int }_{T/2}^T(-A)dt\right)=0$.
2. Cosine Coefficients ($a_n$): Due to the odd symmetry of $f(t)$ multiplied by the even symmetry of $\cos (n{\omega }_{0}t)$, the integrand $f(t)\cos (n{\omega }_{0}t)$ is odd over a symmetric interval. Therefore, all $a_n=0$.
3. Sine Coefficients ($b_n$): We compute:

$$b_n=\frac{2}{T}\left({\int }_0^{T/2}A\sin (n{\omega }_{0}t)dt+{\int }_{T/2}^T(-A)\sin (n{\omega }_{0}t)dt\right)$$ Solving the integrals yields: $$b_n=\frac{2A}{n\pi }\left(1-\cos (n\pi )\right)=\{\begin{array}{ll}\frac{4A}{n\pi },& \text{n odd}\\ 0,& \text{n even}\end{array}$$

Thus, the square wave's series is: $$f(t)=\frac{4A}{\pi }\left(\sin ({\omega }_{0}t)+\frac{1}{3}\sin (3{\omega }_{0}t)+\frac{1}{5}\sin (5{\omega }_{0}t)+\cdots \right)$$ This reveals the square wave contains only odd harmonics of the fundamental frequency, with amplitudes diminishing as $1/n$.

From Coefficients to Amplitude and Phase Spectrum

The trigonometric form $a_0+\sum (a_n\cos (n{\omega }_{0}t)+b_n\sin (n{\omega }_{0}t))$ directly shows the presence of sine and cosine components. However, we can often gain clearer physical insight by combining each sine-cosine pair at a given harmonic into a single sinusoid with a magnitude and phase.

Using trigonometric identities, for each $n\ge 1$: $$a_n\cos (n{\omega }_{0}t)+b_n\sin (n{\omega }_{0}t)=C_n\cos (n{\omega }_{0}t-{\varphi }_n)$$ where:

• Harmonic Amplitude: $C_n=\sqrt{a_n^2+b_n^2}$
• Harmonic Phase Angle: ${\varphi }_n={\tan }^{-1}\left(\frac{b_n}{a_n}\right)$ (taking care with quadrant)

The constant $a_0$ remains the DC offset. In this amplitude-phase form, the series becomes: $$f(t)=a_0+\sum _{n=1}^{\infty }{C}_n\cos (n{\omega }_{0}t-{\varphi }_n)$$ This representation tells you precisely "how much" of each frequency $n{\omega }_0$ is present ($C_n$) and "when" it starts relative to a pure cosine (${\varphi }_n$). Engineers commonly plot $C_n$ vs. frequency ($n{\omega }_0$) as the amplitude spectrum and ${\varphi }_n$ vs. frequency as the phase spectrum, which together form the signal's frequency-domain portrait.

Convergence and the Gibbs Phenomenon

The Fourier series converges to $f(t)$ at all points where the function is continuous. At points of finite discontinuity (like the jumps in the square wave), the series converges to the average of the function's limit from the left and right. A fascinating and important artifact occurs near these discontinuities: the Gibbs phenomenon.

When approximating a discontinuous function with a finite number of terms $N$ (a partial sum), the series exhibits an overshoot and ripple near the jump. Crucially, as $N$ increases, the overshoot does not disappear; it approaches a constant value of about 9% of the jump height. The width of the oscillatory region narrows, however, so in the infinite limit, the overshoot is confined to a single point of discontinuity. This is a vital consideration in filter design, where truncating a series in the frequency domain can cause ringing artifacts in the time-domain signal.

Common Pitfalls

1. Incorrect Integration Limits or Period: The most frequent error is misidentifying the fundamental period $T$ or integrating over an interval that is not exactly one period long. Always verify the function satisfies $f(t+T)=f(t)$ and set your integration bounds to span exactly that $T$. For a function defined piecewise, be meticulous in breaking the integral at the discontinuity points.
2. Misapplying Symmetry: Recognizing function symmetry (even, odd, half-wave) can drastically simplify calculations. An even function ($f(-t)=f(t)$) will have all $b_n=0$. An odd function ($f(-t)=-f(t)$) will have all $a_n=0$ (including $a_0$). Failing to spot symmetry leads to unnecessary computation. Remember, symmetry must be evaluated relative to the origin of your chosen period.
3. Confusing Forms: Be precise about which form of the series you are using (trigonometric vs. amplitude-phase vs. complex exponential). The coefficient formulas differ. For the standard trigonometric form, remember $a_0$ has a factor of $1/T$, while $a_n$ and $b_n$ for $n\ge 1$ have a factor of $2/T$.
4. Ignoring Convergence Conditions (Dirichlet Conditions): While many engineering functions satisfy them, it's important to know the requirements for a Fourier series to exist: the function must be absolutely integrable over one period, have a finite number of maxima/minima and discontinuities in a period. A function like $f(t)=\tan (t)$ over a period containing $\pi /2$ would violate these.

Summary

• The trigonometric Fourier series decomposes a periodic function into a DC component plus an infinite sum of sine and cosine harmonics at integer multiples of the fundamental frequency.
• The coefficients $a_0$, $a_n$, and $b_n$ are determined by integrating the product of the function with the corresponding basis function (1, $\cos (n{\omega }_{0}t)$, or $\sin (n{\omega }_{0}t)$) over a single period.
• The series can be equivalently expressed in amplitude-phase form $C_n\cos (n{\omega }_{0}t-{\varphi }_n)$, where $C_n=\sqrt{a_n^2+b_n^2}$ is the harmonic amplitude and ${\varphi }_n={\tan }^{-1}(b_n/a_n)$ is its phase, providing a direct link to frequency spectrum plots.
• Symmetry properties (even, odd) immediately determine which sets of coefficients are zero, significantly simplifying analysis.
• Understanding convergence behavior, including the Gibbs phenomenon at discontinuities, is critical for practical applications, especially when dealing with finite, truncated series approximations.