Chapter 14 — Signal Analysis and Fourier Series
Chapter Overview (click to expand)
Fourier series analysis reveals that any periodic signal can be decomposed into a sum of sinusoids at harmonically related frequencies, transforming complex waveform analysis into a manageable collection of single-frequency problems. This chapter develops the Fourier series formula, introduces the frequency-domain spectrum as a visualization tool, and applies harmonic analysis to understand how filters and amplifiers shape real signals. **Key Takeaways** 1. Any periodic signal can be expressed as a sum of sinusoids (Fourier series) with frequencies that are integer multiples of the fundamental frequency. 2. The frequency spectrum — a plot of harmonic amplitudes and phases versus frequency — provides an alternative and often more informative view of a signal than its time-domain waveform. 3. Because linear circuits apply their transfer function independently to each harmonic, Fourier analysis lets you predict the exact output waveform of a filter or amplifier for any periodic input.14.1 Introduction
Every circuit you have analysed so far has a well-defined response to a sinusoidal input: the transfer function \(H(j\omega)\) scales the amplitude and shifts the phase. But real-world signals — speech, music, clock pulses, sensor readings — are rarely pure sinusoids. They are complex waveforms whose shape changes from moment to moment. How do you predict what a filter or amplifier will do to such a signal?
The key is a remarkable result discovered by Jean-Baptiste Joseph Fourier around 1807: any periodic signal can be expressed as a sum of sinusoids. This is the Fourier series. Once a signal is expressed as a sum of sinusoids, linearity means you can apply the transfer function to each sinusoid independently and add the results. The Fourier series thereby transforms a difficult problem — system response to an arbitrary waveform — into a family of easy problems, each solved with the phasor techniques already in your toolkit.
The Fourier series also provides a second, equally important viewpoint: the frequency domain. Rather than plotting a signal versus time, you plot its amplitude and phase versus frequency. This frequency-domain picture — called the spectrum — reveals which frequencies carry the signal's energy. The spectrum is the language of filter design, audio engineering, communications, and signal processing. Learning to read and compute spectra is a foundational skill for any circuits engineer.
This chapter builds the Fourier series from first principles, develops the coefficient integrals, introduces spectral representations, exploits waveform symmetry to simplify calculations, analyses the canonical waveforms — square, triangle, and sawtooth — and connects everything to practical applications in audio and filter design.
14.2 The Fourier Series Formula
A signal \(f(t)\) is periodic with period \(T\) if \(f(t + T) = f(t)\) for all \(t\). The Fourier series represents any such signal as a DC offset plus an infinite sum of cosine and sine terms:
where \(\omega_0 = 2\pi / T\) is the fundamental angular frequency in radians per second. Each pair \(\{a_n \cos(n\omega_0 t),\ b_n \sin(n\omega_0 t)\}\) constitutes the nth harmonic.
Fourier Coefficients
The three coefficient formulas follow from the orthogonality of sines and cosines over one period. Integrating \(f(t)\) against each basis function isolates the corresponding coefficient.
DC component (average value):
Cosine coefficients (for \(n = 1, 2, 3, \ldots\)):
Sine coefficients (for \(n = 1, 2, 3, \ldots\)):
The limits of integration can be any convenient interval of length \(T\), such as \([-T/2,\ T/2]\). Choose whichever limits make the integrand simplest.
Amplitude–Phase Form
It is often more intuitive to combine the cosine and sine terms into a single sinusoid with amplitude \(c_n\) and phase \(\phi_n\):
The equivalent amplitude–phase Fourier series is then:
The amplitudes \(c_n\) are the heights of the lines in the amplitude spectrum; the angles \(\phi_n\) form the phase spectrum.
Key Insight — Linearity and the Fourier Series
Because the Fourier series decomposes a signal into individual sinusoids, and because circuit analysis with phasors handles one sinusoid at a time, the response of any linear circuit to a periodic input is found by applying the transfer function \(H(j\omega)\) to each harmonic individually and summing the outputs. This is the practical power of the Fourier series.
14.3 Fundamental Frequency and Harmonics
The fundamental frequency \(f_0\) is the reciprocal of the period:
It is the lowest frequency component of the signal and determines its repetition rate. In music, the fundamental sets the perceived pitch.
Harmonics are sinusoidal components at integer multiples of \(f_0\). The \(n\)th harmonic has frequency \(nf_0\) and angular frequency \(n\omega_0\):
| Harmonic number \(n\) | Frequency | Name |
|---|---|---|
| 1 | \(f_0\) | Fundamental (1st harmonic) |
| 2 | \(2f_0\) | 2nd harmonic |
| 3 | \(3f_0\) | 3rd harmonic |
| 4 | \(4f_0\) | 4th harmonic |
| \(n\) | \(nf_0\) | \(n\)th harmonic |
Harmonic content refers to the set of amplitudes \(\{c_n\}\) across all harmonics. It is harmonic content that distinguishes the timbre of two instruments playing the same note at the same loudness: a violin and a clarinet both at concert A (440 Hz) have the same fundamental but radically different harmonic amplitudes.
Even vs. Odd Harmonics
Harmonics with \(n = 1, 3, 5, \ldots\) are called odd harmonics; those with \(n = 2, 4, 6, \ldots\) are even harmonics. Many common waveforms — square and triangle waves — contain only odd harmonics, a direct consequence of their symmetry (see Section 14.5).
Diagram: Harmonic Explorer — Fourier Waveform Composition
14.4 Spectrum Representation
The frequency spectrum is a plot of the amplitude (and optionally phase) of each Fourier component against frequency. For a periodic signal with period \(T\), energy exists only at the discrete frequencies \(0,\ f_0,\ 2f_0,\ 3f_0,\ldots\) — this is called a discrete line spectrum. Each line's height equals the amplitude \(c_n\) of the corresponding harmonic.
Reading a Spectrum Plot
| Feature in spectrum | Meaning in time domain |
|---|---|
| Single line at \(f_0\) | Pure sinusoid at the fundamental frequency |
| Lines at \(f_0, 3f_0, 5f_0, \ldots\) only | Odd harmonics only — indicates symmetry |
| Lines at \(f_0, 2f_0, 3f_0, \ldots\) | All harmonics — asymmetric waveform |
| Rapidly falling amplitudes | Smooth, slowly changing waveform |
| Slowly falling amplitudes | Sharp transitions, many harmonics needed |
| DC line at \(f = 0\) | Non-zero average value |
Two spectra are needed for a complete frequency-domain description of a periodic signal:
- Amplitude spectrum: Plot of \(c_n\) vs. \(nf_0\). This shows how signal power is distributed among harmonics.
- Phase spectrum: Plot of \(\phi_n\) vs. \(nf_0\). Important when waveform shape must be exactly reconstructed.
For many engineering purposes — especially filter design and power analysis — only the amplitude spectrum is needed. The phase spectrum matters when signal delay is significant, as in audio and communications systems.
Parseval's Theorem
The total average power of a periodic signal equals the sum of the powers in its DC component and each harmonic: \[\frac{1}{T}\int_0^T [f(t)]^2\, dt = a_0^2 + \frac{1}{2}\sum_{n=1}^{\infty}(a_n^2 + b_n^2) = a_0^2 + \frac{1}{2}\sum_{n=1}^{\infty} c_n^2\] This means you can calculate total power either from the waveform directly or from the spectrum — both methods give the same answer.
Diagram: Signal Parameters — Amplitude, Frequency, and Phase Spectrum
14.5 Waveform Symmetry
Waveform symmetry is the single most useful shortcut in Fourier analysis. Before computing any integral, inspect the waveform for symmetry. Each type of symmetry eliminates a class of coefficients, turning an infinite integral computation into a finite one.
Even Symmetry
A waveform has even symmetry if it is symmetric about the vertical axis:
Because the product of an even function with the odd function \(\sin(n\omega_0 t)\) is odd and integrates to zero over a symmetric interval:
Examples: cosine wave, triangle wave (centred at peak), even rectangular pulse train.
Odd Symmetry
A waveform has odd symmetry if it is antisymmetric about the vertical axis:
By the same orthogonality argument, the product of an odd function with the even function \(\cos(n\omega_0 t)\) integrates to zero, and the average value is also zero:
Examples: sine wave, odd square wave (centred at zero crossing), sawtooth through origin.
Half-Wave Symmetry
A waveform has half-wave symmetry if the second half of each period is the negative of the first half:
When this condition holds, the contributions from even-numbered harmonics cancel over a full period:
Examples: square wave, triangle wave, any waveform that looks like its own negative shifted by half a period.
Symmetry Summary Table
| Symmetry | Condition | Effect on Fourier series |
|---|---|---|
| Even | \(f(-t) = f(t)\) | \(b_n = 0\) (cosine terms only) |
| Odd | \(f(-t) = -f(t)\) | \(a_0 = 0\), \(a_n = 0\) (sine terms only, no DC) |
| Half-wave | \(f(t+T/2) = -f(t)\) | \(a_n = b_n = 0\) for even \(n\) (odd harmonics only) |
Tip — Check Symmetry First
Always inspect the waveform for all three symmetries before integrating. A waveform can have more than one type simultaneously — for example, a square wave that is both odd and half-wave symmetric. Each identified symmetry halves (or more) the number of coefficients you must actually compute.
14.6 Common Waveforms and Their Spectra
Three waveforms appear so often in electronics that their Fourier series are worth memorising. Each has a characteristic spectral signature that immediately suggests which harmonics dominate and how quickly they decay.
Square Wave
A square wave of amplitude \(A\) alternates between \(+A\) and \(-A\) with 50 % duty cycle. It has odd symmetry and half-wave symmetry, so only odd sine terms survive.
Spectral characteristics:
- Only odd harmonics present (\(n = 1, 3, 5, \ldots\))
- Amplitudes \(c_n = 4A/(n\pi)\) decay as \(1/n\) — slow decay
- Many harmonics needed to represent the sharp corners accurately
Triangle Wave
A triangle wave of amplitude \(A\) ramps linearly between \(+A\) and \(-A\). Depending on placement, it can have even or odd symmetry; it always has half-wave symmetry.
Spectral characteristics:
- Only odd harmonics present
- Amplitudes \(c_n \propto 1/n^2\) — rapid decay
- The \(1/n^2\) decay reflects the smoother waveform (no discontinuities in \(f(t)\) itself, only in its derivative)
Sawtooth Wave
A sawtooth wave of amplitude \(A\) ramps linearly from \(-A\) to \(+A\) then resets abruptly. It lacks half-wave symmetry and (centred at the origin) has odd symmetry only, so all harmonics appear.
Spectral characteristics:
- All harmonics present (both odd and even)
- Amplitudes \(c_n = 2A/(n\pi)\) decay as \(1/n\)
- The sudden reset discontinuity requires all harmonics; eliminating even harmonics would produce a triangle wave instead
Waveform Comparison
| Waveform | Harmonics present | Amplitude decay | Smoothness |
|---|---|---|---|
| Sine | Fundamental only (\(n = 1\)) | — | Perfectly smooth |
| Square | Odd only (\(n = 1, 3, 5, \ldots\)) | \(1/n\) | Sharp corners |
| Triangle | Odd only (\(n = 1, 3, 5, \ldots\)) | \(1/n^2\) | Smooth (no discontinuity in \(f\)) |
| Sawtooth | All (\(n = 1, 2, 3, \ldots\)) | \(1/n\) | Sharp reset |
Drag harmonic amplitude sliders or pick a preset waveform (Square, Sawtooth, Triangle). Watch the time-domain composite and frequency spectrum update in real time.
14.7 Applications
Audio Timbre and Musical Instruments
Every musical instrument produces a distinctive harmonic profile. A flute playing concert A (440 Hz) is nearly a pure sine — its spectrum shows one dominant line. A violin on the same note has strong 2nd, 3rd, and 4th harmonics that give it richness. A clarinet, owing to its cylindrical bore, emphasises odd harmonics almost exclusively. Fourier analysis quantifies these differences and enables digital synthesis, equalisation, and timbre modelling.
Amplifier Distortion Analysis
When a sinusoidal signal passes through a nonlinear amplifier, the output contains harmonics not present in the input. Total harmonic distortion (THD) quantifies this:
A THD below 0.1 % is inaudible; high-fidelity audio amplifiers typically specify THD below 0.01 %. Fourier analysis of the output spectrum directly reveals which harmonics the amplifier generates and at what levels.
Filter Design for Harmonic Control
Because a low-pass filter attenuates high-frequency harmonics, it smooths sharp waveforms. A square wave passed through a low-pass filter with cutoff frequency between the 3rd and 5th harmonics loses its sharp corners and takes on a sinusoidal shape. Anti-aliasing filters in analogue-to-digital converters (ADCs) exploit this: by removing harmonics above the Nyquist frequency before sampling, they prevent aliasing artefacts. The Fourier series tells the designer exactly which harmonics fall above the cutoff and what their amplitudes are, allowing an accurate prediction of the filtered output.
Power Systems — Harmonic Pollution
Nonlinear loads on the power grid (variable-speed motor drives, switched-mode power supplies, fluorescent lighting) draw non-sinusoidal currents that inject harmonics back into the 50/60 Hz supply. The 3rd, 5th, and 7th harmonics (150, 250, 350 Hz in a 50 Hz system) cause overheating of neutral conductors, transformer core losses, and interference with sensitive equipment. Fourier analysis of the line current spectrum is the standard diagnostic tool for power quality assessment.
14.8 Worked Example — Fourier Coefficients of a Rectangular Pulse Train
Compute the Fourier series of the following rectangular pulse train:
Step 1 — Identify Symmetry
The waveform is neither even nor odd (it is not symmetric about \(t = 0\)), and it does not satisfy \(f(t + T/2) = -f(t)\) because the second half is zero, not \(-A\). Therefore no symmetry shortcuts apply; all three coefficient integrals must be evaluated.
Step 2 — DC Coefficient \(a_0\)
The DC component equals the average value: the waveform is \(A\) for half the period and 0 for the other half, so the average is \(A/2\). ✓
Step 3 — Cosine Coefficients \(a_n\)
All cosine coefficients vanish because \(\sin(n\pi) = 0\) for every integer \(n\).
Step 4 — Sine Coefficients \(b_n\)
Evaluating for the first several harmonics:
| \(n\) | \((-1)^n\) | \(1 - (-1)^n\) | \(b_n\) |
|---|---|---|---|
| 1 | \(-1\) | 2 | \(2A/\pi\) |
| 2 | \(+1\) | 0 | 0 |
| 3 | \(-1\) | 2 | \(2A/(3\pi)\) |
| 4 | \(+1\) | 0 | 0 |
| 5 | \(-1\) | 2 | \(2A/(5\pi)\) |
Even harmonics vanish; odd harmonics have \(b_n = 2A/(n\pi)\).
Step 5 — Complete Fourier Series
Interpretation
The result is essentially a DC offset \(A/2\) plus a scaled square wave. This makes intuitive sense: a square wave toggling between \(+A/2\) and \(-A/2\) has a zero average; adding a DC level of \(A/2\) shifts it to toggle between \(A\) and \(0\) — exactly our pulse train. The factor \(2A/\pi\) vs. the square-wave factor \(4A/\pi\) reflects the different amplitude definition used here (\(0\) to \(A\) vs. \(-A\) to \(+A\)).
14.9 Summary
This chapter developed the Fourier series as the bridge between time-domain waveforms and frequency-domain spectra. The key results are:
- Any periodic signal can be decomposed into a DC term plus sinusoids at integer multiples of the fundamental frequency \(f_0 = 1/T\).
- The three coefficient integrals — for \(a_0\), \(a_n\), and \(b_n\) — follow from the orthogonality of sines and cosines. The amplitude–phase form \(c_n = \sqrt{a_n^2 + b_n^2}\) is the natural representation for spectrum plots.
- Even symmetry eliminates all \(b_n\); odd symmetry eliminates \(a_0\) and all \(a_n\); half-wave symmetry eliminates all even harmonics. Checking these before integrating saves enormous effort.
- Square and sawtooth waves have amplitudes decaying as \(1/n\) (slow); triangle waves decay as \(1/n^2\) (fast). Smoother waveforms require fewer harmonics.
- The frequency spectrum of a periodic signal is a discrete line spectrum at \(0, f_0, 2f_0, \ldots\). Amplitude and phase spectra together completely characterise the signal in the frequency domain.
- Applications include audio timbre analysis, amplifier THD measurement, filter design for harmonic control, and power-system harmonic assessment.
📝 Test your understanding: Go to Chapter 14 Quiz →