1.1 Continuous-time Fourier series

Definition of Fourier series

The continuous-time Fourier series (CTFS) decomposes any periodic signal into a weighted sum of harmonically related complex exponentials (or equivalently, sinusoids). This decomposition is the starting point for nearly all frequency-domain analysis in signal processing: once you can express a signal in terms of its frequency components, you can filter it, pass it through LTI systems, or compress it by keeping only the dominant terms.

Two equations define the entire framework: the analysis equation (signal → coefficients) and the synthesis equation (coefficients → signal).

Representation of periodic signals

A signal $x(t)$ is periodic with period $T$ if $x(t) = x(t + T)$ for all $t$. The Fourier series expresses such a signal as an infinite sum of complex exponentials whose frequencies are integer multiples of the fundamental frequency:

$\omega_0 = \frac{2\pi}{T}$

The $n$-th component oscillates at frequency $n\omega_0$. Because every component frequency is an integer multiple of $\omega_0$, they are all harmonically related, which guarantees the sum is itself periodic with period $T$.

Synthesis equation

The synthesis equation reconstructs $x(t)$ from its Fourier series coefficients:

$x(t) = \sum_{n=-\infty}^{\infty} c_n \, e^{jn\omega_0 t}$

Each term $c_n e^{jn\omega_0 t}$ is a complex exponential at the $n$-th harmonic. The coefficient $c_n$ carries both the amplitude and phase of that component. Summing over all $n$ (positive, negative, and zero) recovers the original signal.

Analysis equation

To find the coefficients, you project $x(t)$ onto each basis function $e^{jn\omega_0 t}$:

$c_n = \frac{1}{T} \int_{0}^{T} x(t) \, e^{-jn\omega_0 t} \, dt$

This is an inner-product operation. You're measuring how much of the $n$-th harmonic is "present" in $x(t)$. The integration interval can be any contiguous interval of length $T$; using $[0, T)$ or $[-T/2, \, T/2)$ is a matter of convenience.

The resulting $c_n$ is in general a complex number. Its magnitude $|c_n|$ gives the amplitude of the $n$-th harmonic, and its angle $\angle c_n$ gives the phase.

Properties of Fourier series

These properties let you manipulate signals in the frequency domain without re-evaluating the analysis integral every time. In the notation below, $x(t) \leftrightarrow c_n$ means "$x(t)$ has Fourier series coefficients $c_n$."

Linearity

If $x_1(t) \leftrightarrow c_{1n}$ and $x_2(t) \leftrightarrow c_{2n}$ (both with the same period $T$), then:

$a\,x_1(t) + b\,x_2(t) \;\leftrightarrow\; a\,c_{1n} + b\,c_{2n}$

This follows directly from the linearity of integration. It means you can analyze complicated signals by breaking them into simpler pieces, finding each piece's coefficients, and combining the results.

Time shifting

Delaying a signal by $t_0$ multiplies each coefficient by a phase factor:

$x(t - t_0) \;\leftrightarrow\; c_n \, e^{-jn\omega_0 t_0}$

The magnitudes $|c_n|$ stay the same, so a time shift changes only the phases of the harmonics, not their amplitudes. This is why the magnitude spectrum is shift-invariant.

Time scaling

Compressing or stretching $x(t)$ in time changes the period. If $x(t)$ has period $T$ and coefficients $c_n$, then $x(at)$ has period $T/|a|$ and a new fundamental frequency $|a|\,\omega_0$. The coefficient values $c_n$ themselves do not change; what changes is the set of harmonic frequencies they correspond to. (The $1/|a|$ scaling that appears in the Fourier transform pair does not apply to the Fourier series coefficients directly, because the normalization by $1/T$ in the analysis equation absorbs it when the period changes.)

Time reversal

$x(-t) \;\leftrightarrow\; c_{-n}$

Reversing the signal in time reverses the index of the coefficients. For a real-valued signal, conjugate symmetry ($c_{-n} = c_n^*$) means time reversal is equivalent to conjugating the coefficients.

Conjugate symmetry

For any real-valued signal $x(t)$:

$c_{-n} = c_n^*$

This has practical consequences:

• $|c_{-n}| = |c_n|$ (the magnitude spectrum is even)
• $\angle c_{-n} = -\angle c_n$ (the phase spectrum is odd)
• $c_0$ is always real (it's the DC/average value)

You only need to compute coefficients for $n \geq 0$; the negative-index coefficients follow immediately.

Parseval's theorem

Parseval's theorem equates the average power of a periodic signal in the time domain to the sum of squared magnitudes of its coefficients:

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

The left side is the average power computed from the waveform. The right side distributes that same power across individual harmonics. This is useful for determining how much power resides in specific frequency bands and for verifying coefficient calculations.

Fourier series coefficients

Calculation of coefficients

The procedure for finding $c_n$:

1. Identify the period $T$ and compute $\omega_0 = 2\pi / T$.
2. Choose a convenient integration interval of length $T$ (e.g., $[0, T)$ or $[-T/2, T/2)$).
3. Evaluate the analysis integral: $c_n = \frac{1}{T} \int_{0}^{T} x(t) \, e^{-jn\omega_0 t} \, dt$
4. Simplify using symmetry whenever possible. For example, if $x(t)$ is even, the sine components vanish and you only need cosine integrals.

Trigonometric vs exponential form

The exponential (complex) form uses complex exponentials summed over all integers:

$x(t) = \sum_{n=-\infty}^{\infty} c_n \, e^{jn\omega_0 t}$

The trigonometric (real) form uses cosines and sines summed over non-negative integers:

$x(t) = a_0 + \sum_{n=1}^{\infty} \bigl(a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t)\bigr)$

The exponential form is more compact and easier to manipulate algebraically. The trigonometric form is sometimes more intuitive because $a_n$ and $b_n$ are real numbers that directly give cosine and sine amplitudes.

Relationship between coefficients

For a real-valued signal, the two sets of coefficients are related by:

• $a_0 = c_0$
• $a_n = c_n + c_{-n} = 2\,\text{Re}\{c_n\}$
• $b_n = j(c_n - c_{-n}) = -2\,\text{Im}\{c_n\}$

Going the other direction:

• $c_0 = a_0$
• $c_n = \frac{1}{2}(a_n - jb_n)$ for $n \geq 1$
• $c_{-n} = c_n^*$ for $n \geq 1$

Symmetry of coefficients

For real-valued signals:

• Conjugate symmetry: $c_{-n} = c_n^*$
• Even signals ($x(t) = x(-t)$): all $c_n$ are real, and $b_n = 0$
• Odd signals ($x(t) = -x(-t)$): all $c_n$ are purely imaginary, and $a_n = 0$

Recognizing these symmetries before computing can cut your work roughly in half.

Convergence of Fourier series

Not every periodic signal has a well-behaved Fourier series. Convergence conditions tell you when the series actually represents the signal faithfully.

Dirichlet conditions

The Dirichlet conditions are sufficient (not necessary) for convergence. A periodic signal $x(t)$ has a convergent Fourier series if, over one period:

1. $x(t)$ is absolutely integrable: $\int_0^T |x(t)|\,dt < \infty$
2. $x(t)$ has a finite number of discontinuities.
3. $x(t)$ has a finite number of maxima and minima.

When these hold, the Fourier series converges to $x(t)$ at every point of continuity. At a discontinuity, it converges to the midpoint of the left- and right-hand limits: $\frac{1}{2}[x(t^-) + x(t^+)]$.

Gibbs phenomenon

When you truncate the Fourier series to $N$ terms near a discontinuity, you'll see oscillatory overshoots and undershoots. This is the Gibbs phenomenon.

The peak overshoot is approximately 9% of the jump size, and this percentage does not decrease as you add more terms. What does happen is that the oscillations get squeezed into a narrower region around the discontinuity. In the limit $N \to \infty$, the overshoot is confined to an infinitesimally thin region, but it never disappears for any finite truncation.

Uniform vs pointwise convergence

• Pointwise convergence: the partial sum converges to $x(t)$ at each individual point $t$, but the rate of convergence can vary across the period.
• Uniform convergence: the partial sum converges to $x(t)$ at the same rate everywhere. The maximum error over the entire period goes to zero.

For continuous periodic signals satisfying the Dirichlet conditions, the Fourier series converges uniformly. For signals with discontinuities, convergence is only pointwise because the Gibbs phenomenon prevents the maximum error from vanishing at any finite truncation order.

Fourier series for common signals

Working through standard waveforms builds intuition about how time-domain shape maps to frequency content.

Square wave

A square wave of amplitude $A$, period $T$, and 50% duty cycle (symmetric about the origin) has only odd harmonics:

$c_n = \begin{cases} \dfrac{2A}{jn\pi} & n \text{ odd} \$6pt] 0 & n \text{ even} \end{cases}$

The amplitudes fall off as $1/n$, which is relatively slow. This slow decay reflects the sharp discontinuities in the waveform and is why you need many terms to get a decent approximation (and why Gibbs phenomenon is prominent here).

Sawtooth wave

A sawtooth wave rising linearly from $-A$ to $A$ over one period has both even and odd harmonics:

$c_n = \frac{A}{jn\pi}, \quad n \neq 0$

Amplitudes again decay as $1/n$. Like the square wave, the sawtooth has jump discontinuities, so convergence is slow and Gibbs phenomenon appears.

Triangular wave

A triangular wave of amplitude $A$ and period $T$ has only odd harmonics, but the amplitudes decay as $1/n^2$:

$c_n = \begin{cases} \dfrac{-4A}{n^2\pi^2} & n \text{ odd} \$6pt] 0 & n \text{ even} \end{cases}$

The faster $1/n^2$ decay comes from the fact that the triangular wave is continuous (no jumps). Smoother signals have coefficients that decay faster, so fewer terms are needed for a good approximation.

Full-wave rectified sine

Taking $|A\sin(\omega_0 t)|$ produces a signal with period $T/2$ (twice the original fundamental frequency). Its Fourier series contains a DC term and even harmonics of the original frequency:

$c_0 = \frac{2A}{\pi}, \qquad c_n = \frac{2A}{\pi(1 - 4n^2)} \quad \text{for } n \neq 0$

where $n$ indexes harmonics of the new fundamental $2\omega_0$. The coefficients decay as $1/n^2$ because the waveform is continuous.

Applications of Fourier series

Filtering in frequency domain

Once a signal is decomposed into harmonics, you can selectively modify individual components. Multiplying each $c_n$ by a frequency-dependent gain $H(n\omega_0)$ implements a filter:

• Low-pass: keep low-$n$ coefficients, attenuate high-$n$
• High-pass: attenuate low-$n$, keep high-$n$
• Band-pass: keep coefficients in a specific range of $n$

This is far simpler than trying to design the equivalent operation directly in the time domain.

Approximation of signals

Truncating the Fourier series to $N$ terms gives the best $N$-term approximation in the least-squares sense. This is the foundation of signal compression: if the coefficients decay quickly, a small number of terms captures most of the signal's energy (by Parseval's theorem). You can quantify the approximation quality by checking what fraction of total power is captured by the retained terms.

Analysis of LTI systems

For an LTI system with frequency response $H(\omega)$, the response to a periodic input is straightforward:

1. Decompose the input: $x(t) = \sum c_n \, e^{jn\omega_0 t}$
2. Each harmonic passes through the system independently: the $n$-th harmonic gets scaled by $H(jn\omega_0)$
3. The output coefficients are $d_n = c_n \cdot H(jn\omega_0)$
4. Synthesize the output: $y(t) = \sum d_n \, e^{jn\omega_0 t}$

This works because complex exponentials are eigenfunctions of LTI systems.

Solving differential equations

For ODEs or PDEs with periodic boundary conditions or periodic forcing:

1. Express the unknown solution as a Fourier series with undetermined coefficients.
2. Substitute into the differential equation.
3. Exploit the orthogonality of the complex exponentials to obtain an algebraic equation for each $c_n$.
4. Solve for the coefficients individually.

This converts a differential equation into an (often simpler) set of algebraic equations. The technique is widely used in heat conduction, vibration analysis, and electromagnetic field problems.

Relationship with other transforms

Fourier series vs Fourier transform

The Fourier series handles periodic signals and produces a discrete set of coefficients $c_n$ at frequencies $n\omega_0$. The Fourier transform handles aperiodic signals and produces a continuous spectral density $X(\omega)$.

You can connect them by letting $T \to \infty$: the harmonic spacing $\omega_0 = 2\pi/T$ shrinks to zero, the discrete coefficients merge into a continuous function, and the Fourier series sum becomes the inverse Fourier transform integral. Formally, $T \cdot c_n \to X(\omega)$ as $T \to \infty$.

Discrete-time Fourier series

The discrete-time Fourier series (DTFS) is the counterpart for discrete-time periodic sequences with period $N$. The key difference: because discrete-time complex exponentials are periodic in frequency, the DTFS has only $N$ distinct coefficients ($n = 0, 1, \ldots, N-1$), not infinitely many. The analysis and synthesis equations mirror the continuous-time case but use finite sums instead of integrals.

Fourier series vs Laplace transform

The Laplace transform uses a complex frequency variable $s = \sigma + j\omega$, which allows it to handle signals with exponential growth or decay and to incorporate initial conditions. The Fourier series is essentially the Laplace transform evaluated along the imaginary axis ($s = j\omega$) and restricted to periodic signals. If a signal is periodic and stable, both tools give equivalent frequency-domain information, but the Laplace transform is more general for transient and stability analysis.