Fourier Series and Fourier Transforms

Introduction

Mechatronic systems generate signals from sensors, actuators, controllers, machines, and communication interfaces. These signals are often easier to understand in the frequency domain than in the time domain. A vibration signal, for example, may appear irregular when plotted against time, yet its frequency spectrum can clearly reveal shaft speed, gear-mesh frequencies, imbalance, or bearing damage.

Fourier analysis provides the mathematical framework for this change of viewpoint. Fourier series represent periodic signals as sums of sinusoids. The Fourier transform extends the idea to nonperiodic or finite-energy signals by representing them as a continuous distribution of frequencies. Together, these tools support vibration analysis, digital filtering, communications, instrumentation, image processing, and system identification.

The central principle is that many signals can be decomposed into elementary oscillations. Each oscillation is described by amplitude, frequency, and phase. The frequency-domain representation does not replace the time-domain signal; it provides a complementary description that often makes hidden structure visible.

Learning Objectives

By the end of this chapter, a student should be able to:

1. Explain the relationship between time-domain and frequency-domain descriptions.
2. Derive and use trigonometric Fourier series coefficients.
3. Recognize symmetry and use it to simplify Fourier calculations.
4. Interpret amplitude, phase, harmonics, bandwidth, and spectral density.
5. Use the complex Fourier series and Fourier transform with correct conventions.
6. Apply transform properties to differentiation, convolution, modulation, and filtering.
7. Explain sampling, aliasing, leakage, windowing, and spectral resolution.
8. Analyze a practical sensor or vibration signal and recommend an appropriate method.

1. Signals, Periodicity, and Frequency

A continuous-time signal is a function x(t), where t is time measured in seconds. A signal is periodic if there is a positive number T such that

for every t. The smallest positive value of T is the fundamental period. The fundamental frequency is

where f₀ is measured in hertz (Hz), or cycles per second. Angular frequency is

and is measured in radians per second. The distinction is important: f describes cycles per second, whereas ω describes radians per second.

A sinusoid can be written as

where A is amplitude, ω is angular frequency, and φ is phase in radians. Its frequency is f = ω/(2π). A phase shift changes the time alignment of a waveform but not its frequency.

A periodic waveform may contain a constant component and sinusoids at integer multiples of f₀. These multiples are called harmonics. The first harmonic is the fundamental frequency f₀; the second harmonic is 2f₀; the nth harmonic is nf₀.

2. The Trigonometric Fourier Series

Under the usual Dirichlet conditions—roughly, that a periodic signal is absolutely integrable over one period and has only a finite number of discontinuities and extrema in that period—a periodic signal can be represented by

The coefficients are calculated over any complete period:

Here n is a positive integer, t₀ is any starting point, and T is the period. The coefficient a₀/2 is the average or dc value of the signal. The factor 2/T follows from the orthogonality of sine and cosine functions over a complete period.

The nth harmonic can alternatively be expressed in amplitude-phase form:

where

and, with the correct quadrant selected,

Thus,

The function atan2 is preferred to ordinary arctangent because it correctly identifies the quadrant of the phase.

Orthogonality

For positive integers m and n, sine and cosine functions of different harmonic orders are orthogonal over a complete period. For example,

and

These relations allow each coefficient to be isolated by multiplying the series by a selected sine or cosine and integrating.

3. Symmetry and Efficient Coefficient Calculation

Symmetry can substantially reduce the work required to obtain a Fourier series.

• An even signal satisfies x(−t) = x(t). All bₙ coefficients are zero, so only cosine terms and the average remain.
• An odd signal satisfies x(−t) = −x(t). The average and all aₙ coefficients are zero, so only sine terms remain.
• A half-wave antisymmetric signal satisfies x(t + T/2) = −x(t). Its dc component and all even harmonics vanish.
• A quarter-wave symmetric waveform may permit integration over only one quarter of a period, with appropriate multiplication.

These conclusions assume that the chosen time origin preserves the stated symmetry. Shifting the origin can change sine and cosine coefficients, although the physical harmonic amplitudes remain consistent when phase is interpreted correctly.

4. Worked Example: Square-Wave Harmonics

Consider the bipolar square wave of amplitude V and period T:

with periodic repetition. This signal is odd, so a₀ = 0 and aₙ = 0. Its sine coefficients are

Using odd symmetry, the product x(t)sin(nω₀t) is even:

Integration gives

Therefore,

The square wave contains only odd harmonics. The amplitude of the nth odd harmonic decreases as 1/n. This slow decrease explains why a square wave has substantial high-frequency content and why sharp edges are difficult to reproduce with a bandwidth-limited system.

At a jump discontinuity, the Fourier series converges to the midpoint of the jump. Near a discontinuity, finite partial sums exhibit the Gibbs phenomenon: oscillations remain near the edge even as more harmonics are included. Increasing the number of terms narrows the oscillatory region but does not eliminate its approximate overshoot percentage.

5. Complex Fourier Series

The trigonometric series can be written more compactly using complex exponentials. Euler’s identity is

where j = √(−1). The complex Fourier series is

with coefficients

For a real-valued signal, C₋ₙ = Cₙ* (the complex conjugate). The correspondence with the trigonometric coefficients is

and, for n > 0,

The complex form is especially useful in communications, control, and transform theory because multiplication by e^{jωt} represents frequency translation.

6. Power and Parseval’s Relation

Fourier coefficients permit signal power to be calculated in the frequency domain. For a real periodic signal, the mean-square value is

This is Parseval’s relation for the trigonometric series. If x(t) is a voltage across a resistance R, average power is

provided R is constant and the voltage is measured in volts. Parseval’s relation then shows how the total power is distributed among dc and harmonic components.

7. The Fourier Transform

Fourier series are appropriate for periodic signals. A nonperiodic signal can be viewed, conceptually, as a periodic signal whose period tends to infinity. The discrete harmonic lines then become a continuous spectrum.

Using the angular-frequency convention, the Fourier transform pair is

Here x(t) is the time-domain signal and X(ω) is its spectrum. If x(t) has units U, then X(ω) has units U·s because the integration variable is time. The inverse factor 1/(2π) belongs to this angular-frequency convention. Other conventions use frequency f and place factors of 2π differently; conventions must not be mixed.

The magnitude spectrum is |X(ω)| and the phase spectrum is ∠X(ω). For real x(t),

so the magnitude is even and the phase has conjugate symmetry, apart from phase discontinuities caused by phase wrapping.

Common transform pairs

Here δ is the Dirac delta distribution and u(t) is the unit-step function. These are generalized functions, not ordinary finite-valued functions.

8. Important Fourier Transform Properties

The transform properties explain why Fourier analysis is valuable for engineering systems.

Linearity

If x₁(t) ⇌ X₁(ω) and x₂(t) ⇌ X₂(ω), then

for constants α and β.

Time shifting

A time delay changes phase but not magnitude.

Frequency shifting

Multiplication by a complex sinusoid translates the spectrum. Multiplication by cos(ω₀t) creates two shifted copies, each scaled by one-half.

Differentiation

If x(t) and its derivative vanish sufficiently at infinity,

Thus differentiation emphasizes high frequencies. For the second derivative,

This is important in vibration and mechanical dynamics, but it also explains why numerical differentiation amplifies high-frequency noise.

Convolution

The convolution of x(t) and h(t) is

and satisfies

For a linear time-invariant system, x(t) is the input, h(t) is the impulse response, and y(t) = x(t)*h(t). Therefore,

where H(ω) is the frequency response. Filtering is multiplication in the frequency domain.

Parseval’s relation for transforms

Under the transform convention used here,

This applies when x(t) is square-integrable. It equates time-domain energy with frequency-domain energy.

9. Filtering and Frequency Response

An ideal low-pass filter passes frequencies below a cutoff and rejects frequencies above it. Real filters transition gradually and introduce phase shift. A first-order low-pass filter with time constant τ has transfer function

Its magnitude and phase are

The −3 dB angular cutoff is ωc = 1/τ, or fc = 1/(2πτ) in hertz. The −3 dB point means the amplitude magnitude is 1/√2 of its low-frequency value, while power is one-half.

A high-pass filter can remove sensor drift and dc offset, whereas a low-pass filter can suppress measurement noise. Band-pass filters are useful when a machine fault occupies a known frequency range. Filter choice must account for signal bandwidth, phase distortion, transient response, computational delay, and stability.

10. Sampling and the Discrete Fourier Transform

A continuous signal sampled at frequency fs produces samples x[k] = x(kTₛ), where Tₛ = 1/fs. The Nyquist frequency is fs/2. If the signal contains no frequency components above B Hz and is adequately band-limited, exact reconstruction requires

This is the sampling theorem. The strict inequality allows a practical transition band; components exactly at or above fs/2 are unsafe in real systems.

When a frequency exceeds fs/2, it appears at a false lower frequency. This is aliasing. An anti-aliasing low-pass filter is therefore applied before an analog-to-digital converter.

For N samples, the discrete Fourier transform (DFT) is

The inverse DFT is

The frequency spacing is

where Tobs = NTₛ is the observation duration. Frequency resolution is therefore improved by observing a longer record, not merely by increasing the sampling rate.

The fast Fourier transform (FFT) is an efficient algorithm for computing the DFT. It produces the same mathematical result as the DFT but reduces computational effort, especially when N is a power of two. Zero-padding creates a denser plotted frequency grid but does not increase the true resolution or recover information absent from the measured record.

Leakage and windows

If a sampled record does not contain an integer number of cycles, truncation creates discontinuities at its boundaries. Energy from one frequency spreads into neighboring DFT bins; this is spectral leakage. Window functions reduce boundary discontinuities at the cost of wider main lobes and altered amplitude calibration.

A rectangular window gives narrow main-lobe width but high sidelobes. Hann and Hamming windows provide better sidelobe suppression for many measurements. The window must be selected according to whether the priority is frequency separation, amplitude accuracy, or detection of weak nearby components.

11. Spectral Interpretation in Mechatronics

Fourier results should be interpreted in relation to the physical system.

• A strong line at shaft rotational frequency often indicates imbalance or misalignment.
• Gear-mesh frequencies may occur near the number of teeth multiplied by rotational frequency, often with sidebands caused by modulation.
• A broad high-frequency increase may indicate friction, impacts, or sensor noise.
• Resonance appears as a large response near a natural frequency, but the measured peak depends on excitation, damping, sensor location, and filter response.
• Periodic encoder errors generate harmonics tied to the mechanical cycle.
• Motor-current spectra can reveal electrical supply frequencies, slotting effects, and mechanical faults.

Amplitude units must be preserved. An acceleration spectrum in m/s² is not automatically a velocity spectrum. In the frequency domain, velocity and displacement are related to acceleration by

and

for nonzero ω, assuming consistent Fourier conventions and zero-initial-condition interpretations. Integration suppresses high-frequency components but can magnify low-frequency drift, so practical processing often includes detrending and carefully designed high-pass filtering.

12. Illustrative Case: Condition Monitoring of a Motor-Driven Pump

Background

Consider an illustrative industrial case involving a three-phase motor driving a centrifugal pump. An accelerometer is mounted on the bearing housing. The shaft rotates at 1,800 rpm.

Problem

The pump exhibits increasing vibration, but a time trace alone does not identify the cause. Engineers must distinguish imbalance, misalignment, bearing damage, and electrical interference.

Analysis

The rotational frequency is

A measurement is sampled at fs = 2,000 Hz for N = 4,000 samples. The observation duration is

and the DFT bin spacing is

The Nyquist frequency is 1,000 Hz. An analog anti-aliasing filter is selected below this limit, with allowance for a transition band. The data are detrended, a Hann window is applied, and the single-sided amplitude spectrum is computed with proper scaling.

Suppose the spectrum contains a dominant line at 30 Hz, a smaller line at 60 Hz, and no significant bearing-fault band. The 30 Hz component is consistent with rotational imbalance; the 60 Hz component may arise from waveform asymmetry or misalignment. If sidebands appear around a gear-mesh frequency, modulation or gear damage becomes more likely.

Approach and outcome

The maintenance team first verifies sensor mounting, calibration, and tachometer speed. It then compares horizontal, vertical, and axial directions. If the 30 Hz component is dominant in the radial direction and decreases after balancing, imbalance is supported. If axial vibration and harmonics remain high, coupling misalignment should be investigated. The spectrum is therefore used as diagnostic evidence, not as an automatic diagnosis.

Learning points

1. Mechanical interpretation requires a known operating speed.
2. Sampling and anti-aliasing determine whether spectral evidence is trustworthy.
3. Frequency resolution depends on record length.
4. Windowing and amplitude correction affect quantitative conclusions.
5. A spectrum should be combined with phase, time waveform, process conditions, and inspection data.

13. Limitations, Safety, and Professional Practice

Fourier methods assume or exploit linearity and stationarity more readily than many real machines permit. A changing-speed motor produces smeared spectral lines because frequency varies during the record. Order tracking, tachometer resampling, or time-frequency methods may then be more suitable. Transient impacts are poorly represented by a single stationary spectrum; short-time Fourier transforms or wavelets can reveal when frequency content occurs.

Noise, sensor saturation, mounting resonance, calibration errors, and electrical interference can produce misleading peaks. Engineers should document sampling rate, record length, window, overlap, filter settings, scaling convention, sensor units, and operating state. Automated or AI-based fault classifiers should retain traceability, validation data, uncertainty estimates, and human review. A false maintenance alarm may cause unnecessary shutdown, while a missed fault may create safety and financial consequences.

Data governance also matters. Industrial sensor data may reveal production rates, equipment health, or operator behavior. Access control, secure storage, calibration records, and responsible retention are part of competent engineering practice. Fourier analysis is mathematically neutral; the safety of its use depends on measurement quality and engineering judgment.

14. Best-Practice Workflow

A reliable frequency-domain investigation generally follows this sequence:

1. Define the physical question and expected frequency range.
2. Select and calibrate a sensor with suitable bandwidth and dynamic range.
3. Choose a sampling rate above the required bandwidth and install anti-aliasing protection.
4. Acquire a sufficiently long record under documented operating conditions.
5. Remove offsets and trends only when justified.
6. Select a window appropriate to the measurement objective.
7. Compute and scale the DFT or FFT consistently.
8. Check units, frequency axis, one-sided versus two-sided conventions, and noise floor.
9. Compare spectral features with machine speed, design frequencies, and independent evidence.
10. Report uncertainty and limitations before making a maintenance or control decision.

Chapter Summary

Fourier series represent periodic signals as a dc component plus harmonically related sine and cosine terms. Their coefficients follow from orthogonality, and symmetry can eliminate many calculations. The complex Fourier series provides a compact exponential representation.

The Fourier transform extends frequency analysis to nonperiodic signals. Its major properties show that time delay changes phase, differentiation emphasizes high frequencies, modulation shifts spectra, and convolution in time becomes multiplication in frequency. These results explain the operation of linear filters and frequency-response methods.

Sampling converts continuous signals into discrete data. The sampling rate must exceed twice the highest relevant frequency for band-limited reconstruction, and anti-aliasing filtering is essential. The DFT and FFT provide practical spectra, but frequency resolution depends on observation time. Leakage, windowing, scaling, and sensor quality must be considered.

In mechatronics, spectra assist vibration diagnosis, control design, communications, and sensor analysis. A spectrum is evidence that must be interpreted with physical knowledge, operating conditions, calibration, and safety-conscious engineering practice.

Key Terms

Fourier series: Representation of a periodic signal as harmonically related sinusoids.

Fourier transform: Continuous frequency-domain representation of a generally nonperiodic signal.

Fundamental frequency: The reciprocal of the fundamental period.

Angular frequency: Frequency expressed in radians per second, ω = 2πf.

Harmonic: An integer multiple of the fundamental frequency.

Amplitude spectrum: Magnitude of a frequency-domain representation.

Phase spectrum: Phase angle associated with spectral components.

Orthogonality: Integral property that allows Fourier coefficients to be isolated.

Complex Fourier coefficient: Coefficient multiplying a complex exponential in a Fourier series.

Parseval’s relation: Equality between time-domain mean-square value or energy and its frequency-domain equivalent.

Convolution: Integral operation describing the output of a linear time-invariant system.

Frequency response: Transfer function evaluated along the frequency axis.

Bandwidth: Frequency range containing significant signal or system response.

Sampling frequency: Number of samples acquired per second.

Nyquist frequency: One-half of the sampling frequency.

Aliasing: Misidentification of a frequency caused by insufficient sampling.

DFT: Discrete Fourier transform of a finite sequence of samples.

FFT: Efficient algorithm for computing the DFT.

Spectral leakage: Spreading of spectral energy caused by finite or noncoherent observation.

Window function: Weighting applied to a finite record to control leakage.

Review Questions

Short Answer Questions

1. Define the fundamental period, fundamental frequency, and angular frequency of a periodic signal.
2. Why are the sine coefficients zero for an even periodic signal?
3. State the Fourier transform property relating convolution in time to multiplication in frequency.
4. What is aliasing, and how does an anti-aliasing filter reduce it?
5. Why does zero-padding not improve the true frequency resolution of a measurement?

Long Answer Questions

1. Derive the trigonometric Fourier coefficients using orthogonality over one complete period.
2. Explain the Fourier transform properties of time shifting, differentiation, and frequency shifting, including their engineering interpretations.
3. Derive the Fourier series of a bipolar square wave and explain why only odd harmonics occur.
4. Describe the complete procedure for obtaining a reliable vibration spectrum from an accelerometer measurement.
5. Compare rectangular, Hann, and Hamming windows in terms of leakage, main-lobe width, and practical use.

Analytical / Application Questions

1. A periodic machine signal has a period of 0.04 s. Determine its fundamental frequency and angular frequency, and identify the frequencies of its second and fifth harmonics.
2. A data-acquisition system samples a signal at 1,000 Hz. Explain what happens to a 700 Hz sinusoid and specify the aliased frequency.
3. An acceleration spectrum contains a strong component at 25 Hz and a second component at 50 Hz in a motor operating at 1,500 rpm. Interpret these components cautiously and propose additional measurements needed before diagnosing the fault.

Multiple Choice Questions with Answers

Question 1

For a periodic signal with period T, which expression gives its fundamental angular frequency?

A. ω₀ = T/(2π)

B. ω₀ = 2π/T

C. ω₀ = 1/(2πT)

D. ω₀ = 2πT

Correct Answer: B. ω₀ = 2π/T

Question 2

Which Fourier-series coefficients vanish for an even signal, assuming the time origin preserves even symmetry?

A. All aₙ coefficients

B. Only a₀

C. All bₙ coefficients

D. Only the odd bₙ coefficients

Correct Answer: C. All bₙ coefficients

Question 3

A square wave of amplitude V with odd symmetry contains which harmonics in its ideal Fourier series?

A. Only even harmonics

B. Only odd harmonics

C. Only the dc component

D. All harmonics with equal amplitude

Correct Answer: B. Only odd harmonics

Question 4

What is the Fourier-transform result for a time delay x(t−t₀)?

A. X(ω−t₀)

B. X(ω)e^{jωt₀}

C. X(ω)e^{-jωt₀}

D. X(ω)/t₀

Correct Answer: C. X(ω)e^{-jωt₀}

Question 5

For a first-order low-pass filter H(jω)=1/(1+jωτ), what is the magnitude at ω=1/τ?

A. 0

B. 1/2

C. 1/√2

D. √2

Correct Answer: C. 1/√2

Question 6

A signal is sampled at 2,000 Hz. What is its Nyquist frequency?

A. 500 Hz

B. 1,000 Hz

C. 2,000 Hz

D. 4,000 Hz

Correct Answer: B. 1,000 Hz

Question 7

A 2-second record is sampled at 2,000 Hz. What is the DFT frequency-bin spacing?

A. 0.25 Hz

B. 0.5 Hz

C. 1 Hz

D. 2 Hz

Correct Answer: B. 0.5 Hz

Question 8

Which operation most directly reduces aliasing before analog-to-digital conversion?

A. Applying an anti-aliasing low-pass filter

B. Zero-padding the sampled record

C. Increasing the display resolution

D. Changing the phase reference

Correct Answer: A. Applying an anti-aliasing low-pass filter

Question 9

For a linear time-invariant system, if Y(ω) is the output spectrum, which relationship is correct?

A. Y(ω)=X(ω)+H(ω)

B. Y(ω)=X(ω)H(ω)

C. Y(ω)=X(ω)/H(ω) in every case

D. Y(ω)=X(ω)−H(ω)

Correct Answer: B. Y(ω)=X(ω)H(ω)

Question 10

A motor rotates at 1,800 rpm. What is its rotational frequency?

A. 18 Hz

B. 30 Hz

C. 60 Hz

D. 1,800 Hz

Correct Answer: B. 30 Hz