Exponential Fourier Series

Electrical Circuit Analysis > The Fourier Series

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in Mar 2026

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Introduction

A compact way of expressing the Fourier series is to put it in the exponential form. The traditional sine‑cosine form of a periodic function

$$f(t)=a_{0}+\sum_{n=1}^{\infty}\left(a_{n}\cos n\omega_{0}t + b_{n}\sin n\omega_{0}t\right)$$

can be rewritten using complex exponentials.

This conversion is achieved by expressing the sine and cosine functions in terms of exponentials using Euler’s identity.

Euler’s Identities and Exponential Form

Using Euler’s identities:

$$\cos n\omega_{0}t = \frac{1}{2}\left[e^{jn\omega_{0}t} + e^{-jn\omega_{0}t}\right], \quad \sin n\omega_{0}t = \frac{1}{2j}\left[e^{jn\omega_{0}t} - e^{-jn\omega_{0}t}\right]$$

Substituting these into the original Fourier series and combining terms gives:

$$f(t)=a_{0} + \frac{1}{2}\sum_{n=1}^{\infty}\left[(a_{n}-jb_{n})e^{jn\omega_{0}t} + (a_{n}+jb_{n})e^{-jn\omega_{0}t}\right]$$

Complex Coefficients

Define new coefficients so that:

$$c_{0}=a_{0}, \quad c_{n}=\frac{a_{n}-jb_{n}}{2}, \quad c_{-n}=\frac{a_{n}+jb_{n}}{2}$$

Then the series becomes:

$$f(t)=c_{0}+\sum_{n=1}^{\infty}\left(c_{n}e^{jn\omega_{0}t}+c_{-n}e^{-jn\omega_{0}t}\right)$$

which can be expressed compactly as:

$$f(t)=\sum_{n=-\infty}^{\infty}c_{n}e^{jn\omega_{0}t}$$

This is the exponential Fourier series representation of $f(t)$.

Direct Coefficient Formula

The complex Fourier coefficient $c_{n}$ can be obtained directly from $f(t)$ using:

$$c_{n}=\frac{1}{T}\int_{0}^{T}f(t)e^{-jn\omega_{0}t}dt$$

where $T$ is the period of the function and $\omega_{0}=2\pi/T$.

The amplitude and phase of these complex coefficients form the complex frequency spectrum of $f(t)$.

Exponential Fourier Series Interpretation

The exponential Fourier series represents a periodic function in terms of complex exponentials $e^{jn\omega_{0}t}$. This form is often more compact and mathematically convenient than the sine–cosine form, especially in engineering analysis involving phasors and signals.

The magnitude and phase of each coefficient $c_{n}$ describe the contribution of each frequency component in $f(t)$.

Power and RMS Value

Using the coefficients $c_{n}$, the rms value of a periodic signal $f(t)$ can be expressed as:

$$F_{\text{rms}}^{2}=\sum_{n=-\infty}^{\infty}|c_{n}|^{2}$$

This result is a restatement of the Parseval’s theorem, which relates the total power in the time domain to the sum of the squared magnitudes of its Fourier coefficients.

Example 1: Periodic Exponential

Consider a periodic function given by $f(t)=e^{t}$ where $f(t+2\pi)=f(t)$.

Since $T=2\pi$, $\omega_{0}=1$.

Then

$$c_{n}=\frac{1}{2\pi}\int_{0}^{2\pi}e^{t}e^{-jnt}dt=\frac{1}{2\pi(1‑jn)}\left[e^{2\pi}‑1\right]$$

The exponential Fourier series becomes:

$$f(t)=\sum_{n=-\infty}^{\infty}\frac{85}{1‑jn}e^{jnt}$$

Example 2: Sawtooth Wave

For a sawtooth wave defined as $f(t)=t$ for $0 \lt t \lt 1$ with period $T=1$:

$\omega_{0}=2\pi$.

Applying the coefficient formula yields:

$$c_{n}=\frac{j}{2n\pi}, \quad n\neq0$$ $$c_{0}=0.5$$

Thus:

$$f(t)=0.5+\sum_{n\neq0}\frac{j}{2n\pi}e^{j2n\pi t}$$

Example 1: Find the exponential Fourier series expansion of the periodic function $ f(t)=e^{t}, 0 \lt t \lt 2 \pi $ with $ f(t+2 \pi)=f(t) $.

Solution: Since $ T=2 \pi, \omega_{0}=2 \pi / T=1 $. Hence,

$$\begin{array}{l}c_{n}=\frac{1}{T} \int_{0}^{T} f(t) e^{-j n \omega \omega_{0} t} d t=\frac{1}{2 \pi} \int_{0}^{2 \pi} e^{t} e^{-j n t} d t \\=\left.\frac{1}{2 \pi} \frac{1}{1-j n} e^{(1-j n) t}\right|_{0} ^{2 \pi}=\frac{1}{2 \pi(1-j n)}\left[e^{2 \pi} e^{-j 2 \pi n}-1\right] \\\end{array}$$

But by Euler's identity,

$$e^{-j 2 \pi n}=\cos 2 \pi n-j \sin 2 \pi n=1-j 0=1$$

Thus,

$$c_{n}=\frac{1}{2 \pi(1-j n)}\left[e^{2 \pi}-1\right]=\frac{85}{1-j n}$$

The complex Fourier series is

$$f(t)=\sum_{n=-\infty}^{\infty} \frac{85}{1-j n} e^{j n t}$$

We may want to plot the complex frequency spectrum of $ f(t) $. If we let $ c_{n}=\left|c_{n}\right| \angle \theta_{n} $, then

$$\left|c_{n}\right|=\frac{85}{\sqrt{1+n^{2}}}, \quad \theta_{n}=\tan ^{-1} n$$

By inserting in negative and positive values of $ n $, we obtain the amplitude and the phase plots of $ c_{n} $ versus $ n \omega_{0}=n $, as in Fig. 4.

Fig. 4: The complex frequency spectrum of the function in Example 1: (a) amplitude spectrum, (b) phase spectrum.

Example 2: Find the complex Fourier series of the sawtooth wave in Fig. 5. Plot the amplitude and the phase spectra.

Fig. 5: For Example 2.

Solution: From Fig. 5, $ f(t)=t, 0 \lt t \lt 1, T=1 $ so that $ \omega_{0}=2 \pi / T=2 \pi $. Hence,

$$c_{n}=\frac{1}{T} \int_{0}^{T} f(t) e^{-j n \omega o s t} d t=\frac{1}{1} \int_{0}^{1} t e^{-j 2 n \pi t} d t \tag{2.1}$$

But

$$\int t e^{a t} d t=\frac{e^{a t}}{a^{2}}(a x-1)+C$$

Applying this to Eq. (2.1) gives

$$\begin{aligned}c_{n} &=\left.\frac{e^{-j 2 n \pi t}}{(-j 2 n \pi)^{2}}(-j 2 n \pi t-1)\right|_{0} ^{1} \\&=\frac{e^{-j 2 n \pi}(-j 2 n \pi-1)+1}{-4 n^{2} \pi^{2}}\end{aligned} \tag{2.2}$$

Again,

$$e^{-j 2 \pi n}=\cos 2 \pi n-j \sin 2 \pi n=1-j 0=1$$

so that Eq. (2.2) becomes

$$c_{n}=\frac{-j 2 n \pi}{-4 n^{2} \pi^{2}}=\frac{j}{2 n \pi}$$

This does not include the case when $ n=0 $. When $ n=0 $,

$$c_{0}=\frac{1}{T} \int_{0}^{T} f(t) d t=\frac{1}{1} \int_{0}^{1} t d t=\left.\frac{t^{2}}{2}\right|_{1} ^{0}=0.5$$

Hence,

$$f(t)=0.5+\sum_{\substack{n=-\infty \\ n \neq 0}}^{\infty} \frac{j}{2 n \pi} e^{j 2 n \pi t}$$

and

$$\left|c_{n}\right|=\left\{\begin{array}{ll}\frac{1}{2|n| \pi}, & n \neq 0 \\0.5, & n=0\end{array}, \quad \theta_{n}=90^{\circ}, \quad n \neq 0\right.$$

By plotting $ \left|c_{n}\right| $ and $ \theta_{n} $ for different $ n $, we obtain the amplitude spectrum and the phase spectrum shown in Fig. 6.

Fig. 6: For Example 2: (a) amplitude spectrum, (b) phase spectrum.

Conclusion

The exponential Fourier series is a powerful method to represent a periodic signal as a sum of complex exponentials. By converting traditional sine–cosine Fourier coefficients into complex coefficients $c_{n}$, this form provides a compact and mathematically convenient way to analyze periodic functions, especially in signal processing and electrical engineering.

It also forms the basis for understanding frequency spectra, power distribution, and spectral decomposition of signals using complex analysis.

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