Fourier Series and Periodic Signal Analysis

Introduction

Many physical signals are periodic: mains voltage, rotating-machine vibration, pulse-width-modulated waveforms, sampled clock signals, and musical tones. Although such signals may not be sinusoidal, they can often be represented as weighted sums of sinusoids. Fourier series provides this representation and converts periodic-signal analysis from a time-domain problem into a frequency-domain problem.

This chapter develops the trigonometric and complex forms of the Fourier series, explains the conditions under which the representation is valid, derives the principal coefficient formulas, and applies them to linear time-invariant (LTI) systems. Particular attention is given to symmetry, convergence, power, spectra, Gibbs oscillations, numerical computation, and engineering interpretation.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Explain why periodic signals can be decomposed into harmonically related sinusoids.
2. Derive and use trigonometric and exponential Fourier-series coefficients.
3. Exploit even, odd, and half-wave symmetry to simplify calculations.
4. Interpret line spectra, phase spectra, average value, and signal power.
5. Predict the response of an LTI system to a periodic input using its frequency response.
6. Analyze convergence and understand the Gibbs phenomenon.
7. Apply Fourier-series methods to a practical communication or power-electronics problem.

1. Periodic Signals and Harmonic Structure

A continuous-time signal x(t) is periodic if there is a positive number T₀ such that

The smallest positive T₀ satisfying this relation is the fundamental period. The fundamental angular frequency is

where ω₀ is measured in radians per second. Frequencies nω₀, where n is an integer, are called harmonics. The zeroth harmonic is the constant or dc component.

A sinusoid with frequency ω₀ is periodic with period T₀. Every sinusoid with frequency nω₀ is also periodic with period T₀ or a divisor of it. In contrast, two sinusoids whose frequency ratio is irrational do not form a periodic sum.

A periodic waveform is therefore naturally associated with a discrete frequency spectrum. Unlike the Fourier transform of an isolated aperiodic signal, which is generally continuous, the Fourier series consists of spectral lines at integer multiples of ω₀.

2. Trigonometric Fourier Series

Under appropriate convergence conditions, a periodic signal can be represented as

Here a₀ is the dc value, while aₙ and bₙ are the cosine and sine coefficients. Over any interval of length T₀, such as [t₁, t₁ + T₀], the coefficients are

The interval can begin at any time because the integrands are periodic over a complete period. The formulas assume that x(t) is integrable over one period. For the usual pointwise convergence result, x(t) should be piecewise smooth, or more generally satisfy Dirichlet-type conditions: it has a finite number of discontinuities and extrema in one period and is absolutely integrable there.

The coefficient a₀ is the average value, not twice the dc coefficient. Some texts instead write the series as a₀/2 plus the harmonic terms. Both conventions are valid, but they must not be mixed.

Amplitude–phase form

The nth harmonic can be written as

where

and, with correct quadrant selection,

Thus

Cₙ has the same units as x(t), and φₙ is in radians. The atan2 function is preferable to tan⁻¹(bₙ/aₙ), since it preserves the quadrant.

3. Orthogonality and Derivation of the Coefficients

The coefficient formulas follow from orthogonality over one period. For positive integers m and n,

For n ≥ 1, the squared norms are

where all integrals are over any complete period. Multiplying the series by cos(mω₀t), integrating, and using orthogonality isolates aₘ. Multiplication by sin(mω₀t) similarly isolates bₘ. Multiplication by one isolates a₀ because all nonconstant sinusoidal terms have zero average.

This is an important general principle: Fourier coefficients are projections of a signal onto mutually orthogonal basis functions.

4. Symmetry as an Analytical Tool

Symmetry can eliminate half or more of the integrations.

Even signals

If x(−t) = x(t), x(t) is even. Its product with sin(nω₀t) is odd, so bₙ = 0. Therefore,

For an even signal, integration over a symmetric interval [−T₀/2, T₀/2] gives

Odd signals

If x(−t) = −x(t), x(t) is odd. Its average and cosine coefficients vanish: a₀ = 0 and aₙ = 0. Hence only sine terms remain, and

Half-wave symmetry

If x(t + T₀/2) = −x(t), the signal has half-wave symmetry. Its dc term and all even harmonics vanish. Only odd n may occur. This property is particularly useful for square and certain pulse waveforms.

A time shift does not change harmonic magnitudes but changes phase. If y(t) = x(t − t_d), then the nth complex coefficient is multiplied by exp(−jnω₀t_d), so the phase changes by −nω₀t_d.

5. Complex Exponential Fourier Series

Euler's identities allow the trigonometric series to be expressed compactly as

where

The index n ranges over positive, negative, and zero integers. For a real-valued x(t), the coefficients have conjugate symmetry:

where * denotes complex conjugation. Consequently, the negative-frequency spectrum contains no independent information, although retaining it makes algebra and system analysis more natural.

The relation between the two coefficient conventions is

and, for n ≥ 1,

The complex form is especially useful with LTI systems because exponentials are eigenfunctions of such systems.

6. Worked Example: A Symmetric Square Wave

Consider the periodic square wave of amplitude A and period T₀ defined over one period by

with periodic extension. It is odd and has half-wave symmetry. Therefore a₀ = aₙ = 0, and only odd sine harmonics remain. For n ≥ 1,

Evaluation gives

Since ω₀T₀ = 2π,

Thus bₙ = 4A/(nπ) for odd n and bₙ = 0 for even n. The series is

The 1/n decay means that discontinuous waveforms contain substantial high-frequency energy. At a discontinuity, the Fourier series converges to the midpoint of the two one-sided values, which is zero for this waveform.

7. Parseval's Relation and Signal Power

For a real periodic signal, the mean-square value is

In complex form,

If x(t) is a voltage across a resistance R, average power is

provided x(t) is the voltage and R is a constant positive resistance. For the square wave above, x²(t) = A² except at isolated transition points, so the mean-square value is A² and the rms value is A.

Parseval's relation is both a theoretical identity and a practical validation tool. A numerical Fourier calculation should yield nearly the same power in time and frequency domains, subject to truncation and numerical error.

8. Fourier Series through an LTI System

Let an LTI system have frequency response H(jω). If

then, under conditions that permit term-by-term system operation, the steady-state output is

Thus each harmonic is independently scaled by |H(jnω₀)| and shifted by ∠H(jnω₀). The output coefficient is

This result is central to filter analysis. A low-pass filter preserves the dc component and low harmonics while attenuating high harmonics, thereby smoothing a waveform. A band-pass filter can isolate a selected harmonic, and a notch filter can suppress a particular spectral line.

For a stable first-order low-pass system with

where ω_c is the cutoff angular frequency, the nth harmonic has magnitude factor

and phase −tan⁻¹(nω₀/ω_c). This formula applies to the stated first-order system and assumes sinusoidal steady state.

9. Convergence, Gibbs Phenomenon, and Truncation

For a piecewise smooth periodic signal, the Fourier series converges at a point t to:

• x(t), when x is continuous at t;
• [x(t⁺) + x(t⁻)]/2, when x has a finite jump.

A finite partial sum cannot reproduce a discontinuity exactly. Near a jump, it exhibits overshoot and ringing known as the Gibbs phenomenon. Increasing the number of terms narrows the ringing region but does not eliminate the limiting overshoot percentage, approximately 9% of the jump magnitude for ordinary partial sums.

In numerical work, abrupt truncation can be improved by spectral-window or summability techniques such as Fejér averaging. These reduce oscillations but may broaden transitions. The appropriate method depends on whether edge sharpness or low ripple is more important.

The decay rate of coefficients provides regularity information. A waveform with a jump commonly has coefficients proportional to 1/n; a continuously differentiable waveform generally has faster decay. Therefore spectral decay can reveal discontinuities and smoothness.

10. Numerical and Experimental Computation

For measured data, one period may be sampled at N equally spaced points. The discrete Fourier transform (DFT) estimates spectral coefficients. Its frequency bins are separated by 1/T₀ when the record contains exactly one period, or more generally by the reciprocal of the observation duration.

Important practical issues include:

• Sampling: The sampling frequency must exceed twice the highest frequency of interest to avoid aliasing.
• Record length: An incomplete or noninteger number of periods causes spectral leakage.
• Windowing: Windows reduce leakage but alter amplitude and resolution.
• Quantization: Finite ADC resolution introduces noise and distortion.
• Synchronization: Triggering at a consistent phase improves repeatability.
• Calibration: Sensor gain, probe loading, and instrument bandwidth affect coefficients.

A DFT is not automatically a Fourier series. A Fourier series describes a periodic extension of a finite-period signal; a DFT describes finite sampled data under an implicit periodic extension. Confusing these models leads to incorrect interpretation of leakage and harmonics.

11. Applied Case: Harmonic Reduction in a PWM Inverter

Background

This illustrative engineering case concerns a single-phase PWM inverter supplying a motor-drive filter. The switching waveform is periodic, but it contains a fundamental component and many switching-related harmonics.

Problem

The load requires a nearly sinusoidal voltage at fundamental angular frequency ω₀. Excessive harmonic content causes motor heating, acoustic noise, electromagnetic interference, and additional filter stress.

Analysis

One switching-period waveform is measured and represented by complex Fourier coefficients Dₙ. The load-side filter has response H(jnω₀). The predicted output coefficient is Eₙ = DₙH(jnω₀). The harmonic distortion can be quantified by total harmonic distortion (THD):

where V₁,rms is the rms fundamental voltage and Vₙ,rms is the rms nth-harmonic voltage. The summation must use the harmonics included in the measurement bandwidth.

Approach and solution

The design team first identifies dominant spectral lines, then selects an LCL or LC filter whose attenuation is strong at switching harmonics but acceptable at the fundamental. The predicted filtered spectrum is compared with measured output. Component tolerances and resonance damping are included because an undamped filter may amplify frequencies near resonance.

Outcome

The Fourier model permits rapid comparison of switching strategies and filter values without repeatedly solving the complete time-domain circuit. It also reveals whether a reduction in one harmonic merely transfers energy to another. Final compliance must still be verified by measurement using applicable power-quality and electromagnetic-compatibility standards.

Learning points

Fourier series links waveform design, filter response, thermal stress, and electromagnetic compatibility. It should be used with a validated system model, calibrated instruments, and clearly defined bandwidth and operating conditions.

12. Limitations, Design Judgment, and Contemporary Directions

Fourier series is most natural for steady periodic behavior. It is less suitable for transients, drifting frequency, nonstationary signals, or strongly time-varying systems. Fourier coefficients also describe what frequencies are present but do not by themselves identify when a transient occurred. Short-time Fourier analysis, wavelets, or other time-frequency methods may be more appropriate in those situations.

In modern signal-processing systems, Fourier coefficients may be estimated adaptively from streaming data. Machine-learning systems can use harmonic features for fault diagnosis, power-quality classification, and biomedical monitoring. However, automated classification does not remove the need for sampling theory, uncertainty analysis, traceable calibration, and protection of sensitive measurement data. In safety-critical applications, model validation and explainability are essential.

Best practice is to state the period, time origin, coefficient convention, measurement bandwidth, convergence assumptions, and truncation rule. Always check symmetry before integrating, verify power with Parseval's relation, and distinguish physical harmonics from artifacts caused by leakage, aliasing, sensor saturation, or noise.

Chapter Summary

Fourier series represents a periodic signal as a dc component plus harmonically related sine and cosine components. Orthogonality produces the coefficient formulas, while symmetry often eliminates many coefficients. The complex exponential form provides an efficient representation and makes LTI-system analysis direct: each input coefficient is multiplied by the system frequency response at the corresponding harmonic.

Amplitude–phase spectra show harmonic magnitude and phase; Parseval's relation connects spectral coefficients with mean-square value and power. Discontinuities lead to slow coefficient decay and Gibbs oscillations. Numerical implementation requires careful attention to sampling, record length, leakage, windowing, and calibration. In engineering practice, Fourier-series analysis supports filter design, waveform reconstruction, power-quality evaluation, communications, vibration monitoring, and harmonic mitigation, but it must be applied with explicit assumptions and awareness of its limitations.

Key Terms

Periodic signal: A signal satisfying x(t + T₀) = x(t) for a positive period T₀.

Fundamental period: The smallest positive period of a periodic signal.

Fundamental frequency: The frequency 1/T₀, or angular frequency ω₀ = 2π/T₀.

Harmonic: A sinusoidal component at an integer multiple of the fundamental frequency.

Fourier series: A representation of a periodic signal using harmonically related sinusoids.

DC component: The average or zero-frequency component of a signal.

Trigonometric coefficients: The cosine coefficients aₙ and sine coefficients bₙ.

Complex coefficient: The coefficient Dₙ in the exponential Fourier series.

Orthogonality: The zero inner product between distinct basis functions over a complete period.

Even signal: A signal satisfying x(−t) = x(t).

Odd signal: A signal satisfying x(−t) = −x(t).

Half-wave symmetry: The property x(t + T₀/2) = −x(t).

Line spectrum: A discrete plot of harmonic magnitude or phase versus frequency.

Parseval's relation: An equality connecting time-domain mean square and spectral coefficient power.

Gibbs phenomenon: Persistent ringing near a jump in a truncated Fourier representation.

Spectral leakage: Energy spreading between frequency bins due to finite or noncoherent observation.

THD: The rms value of harmonic components relative to the rms fundamental.

Frequency response: The complex gain H(jω) of an LTI system for sinusoidal input.

Review Questions

Short Answer Questions

1. Define the fundamental period and fundamental angular frequency of a periodic signal.
2. Why are sine and cosine functions useful as a basis for Fourier representation?
3. State the symmetry consequences for an even signal.
4. What does the complex conjugate symmetry D₋ₙ = Dₙ* imply for a real signal?
5. State the value to which a Fourier series converges at a finite jump discontinuity.

Long Answer Questions

1. Derive the trigonometric Fourier-series coefficient formulas using orthogonality.
2. Explain even, odd, and half-wave symmetry and show how each reduces computation.
3. Derive the Fourier series of a symmetric square wave and discuss its convergence.
4. Explain how an LTI system modifies the Fourier coefficients of a periodic input.
5. Discuss sampling, leakage, windowing, and aliasing in experimental Fourier analysis.

Analytical / Application Questions

1. A periodic waveform has T₀ = 2 ms and coefficients a₃ = 0.8 V and b₃ = −0.6 V. Determine the third-harmonic amplitude and phase in cosine amplitude–phase form.
2. A first-order low-pass system has ω_c = 10ω₀. Determine the magnitude attenuation of the fifth harmonic and explain its engineering significance.
3. Design an analysis procedure for diagnosing excessive harmonic distortion in a PWM inverter, including measurement, Fourier calculation, filtering interpretation, and validation steps.

Multiple Choice Questions with Answers

Question 1

For a periodic signal with period T₀, the fundamental angular frequency is:

A. πT₀

B. 2π/T₀

C. T₀/(2π)

D. 1/(2πT₀)

Correct Answer: B. 2π/T₀

Question 2

If x(t) is even, which Fourier-series coefficients must be zero?

A. All aₙ

B. Only a₀

C. All bₙ

D. Only the odd bₙ

Correct Answer: C. All bₙ

Question 3

A signal with half-wave symmetry has:

A. Only even harmonics

B. No dc component and no even harmonics

C. Only a dc component

D. Equal cosine and sine coefficients

Correct Answer: B. No dc component and no even harmonics

Question 4

For the symmetric square wave of amplitude A, the coefficient of sin(3ω₀t) is:

A. 2A/(3π)

B. 4A/(3π)

C. 3A/(4π)

D. A/(3π)

Correct Answer: B. 4A/(3π)

Question 5

At a finite jump discontinuity, a Fourier series converges to:

A. The right-hand value only

B. The left-hand value only

C. The larger one-sided value

D. The average of the two one-sided values

Correct Answer: D. The average of the two one-sided values

Question 6

For an LTI system, the output complex Fourier coefficient at harmonic n is:

A. Dₙ + H(jnω₀)

B. Dₙ/H(jnω₀)

C. DₙH(jnω₀)

D. H(jω₀)/Dₙ

Correct Answer: C. DₙH(jnω₀)

Question 7

The mean-square value of a periodic signal in complex Fourier coefficients is:

A. ∑|Dₙ|

B. ∑|Dₙ|²

C. |∑Dₙ|²

D. D₀²

Correct Answer: B. ∑|Dₙ|²

Question 8

For a first-order low-pass system with ω_c = 10ω₀, the magnitude response at the fifth harmonic is:

A. 1/√1.25

B. 1/√26

C. 1/√101

D. 1/5

Correct Answer: A. 1/√1.25

Question 9

Gibbs phenomenon is most directly associated with:

A. A discontinuity represented by a truncated Fourier series

B. A purely constant signal

C. A signal with no harmonics

D. Perfect removal of spectral leakage

Correct Answer: A. A discontinuity represented by a truncated Fourier series

Question 10

Spectral leakage is most likely when:

A. The observation contains an integer number of periods

B. The signal is sampled at zero frequency

C. The record contains a noninteger number of periods

D. The signal has only a dc component

Correct Answer: C. The record contains a noninteger number of periods