Fourier Transforms and Signal-Spectrum Methods

Introduction

Fourier analysis provides a systematic way to represent a signal in terms of its frequency components. Instead of describing a vibration, sensor voltage, image, or control-system response only as a function of time or position, Fourier methods reveal the amplitudes and phases of the sinusoidal components that compose it. This change of representation is especially valuable in mechatronics, where mechanical motion, electrical signals, digital controllers, sensors, actuators, and embedded processors interact.

A time-domain signal shows when an event occurs. Its spectrum shows which frequencies are present and with what strength. Filtering, noise removal, vibration diagnosis, modulation, image processing, system identification, and fault detection can often be expressed more simply in the frequency domain.

This chapter develops the continuous Fourier transform (CFT), the discrete Fourier transform (DFT), the fast Fourier transform (FFT), convolution, correlation, sampling, aliasing, and practical spectral interpretation.

Learning Objectives

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

1. Explain the relationship between time-domain signals and frequency spectra.
2. Apply the continuous and discrete Fourier transform definitions.
3. Interpret magnitude and phase spectra with correct units.
4. Use the convolution and correlation theorems.
5. Determine sampling requirements and identify aliasing.
6. Compute and interpret an FFT-based spectrum.
7. Select suitable windows and distinguish spectral leakage from genuine frequency content.
8. Apply Fourier methods to sensor, vibration, filter, and image-processing problems.
9. Evaluate the practical limitations and assumptions of Fourier analysis.

1. Signals and Their Spectral Representation

A signal may be continuous or discrete, deterministic or random, periodic or nonperiodic. Examples include motor current, accelerometer output, encoder position, temperature, acoustic pressure, and an image intensity field.

A periodic signal repeats after a period T. Its fundamental frequency is f₀ = 1/T in hertz, and angular frequency is ω₀ = 2π/T in radians per second. A Fourier series represents a periodic signal as a sum of harmonically related sinusoids. A nonperiodic signal generally has a continuous range of frequency components, so it is analyzed using a Fourier transform rather than a Fourier series.

For a sinusoid

A is amplitude, f is frequency in hertz, t is time in seconds, and φ is phase in radians. The frequency-domain description identifies f, A, and φ. A time shift changes phase but not magnitude: if y(t) = x(t - t₀), then the spectrum is multiplied by e^{-j2πf t₀}.

The complex exponential e^jθ = cos θ + j sin θ is central to Fourier analysis. Here j = √−1. Complex notation permits amplitude and phase to be handled together.

2. Continuous Fourier Transform

Using frequency f in hertz, the Fourier transform pair is

x(t) is the time-domain signal and X(f) is its complex spectrum. The forward transform has units of signal-units × seconds, or signal-units per hertz. The inverse transform reconstructs x(t), provided the signal satisfies suitable integrability or generalized-function conditions.

The magnitude spectrum is |X(f)|, and the phase spectrum is ∠X(f). For a real-valued signal, positive- and negative-frequency components are conjugate symmetric:

where the asterisk denotes complex conjugation. Consequently, the two-sided spectrum contains redundant information for real signals, although the negative-frequency representation is mathematically useful.

Important transform pairs

The impulse δ(t) is not an ordinary finite-valued function. It is defined by its sampling property ∫x(t)δ(t−t₀)dt = x(t₀).

Parseval’s relation

Fourier transformation preserves energy in the following sense:

This relation assumes the signal has finite energy. The left side has units of signal-squared × seconds; the right side has the same units because X(f) has units of signal × seconds and df has units of 1/second.

3. Convolution and Linear Time-Invariant Systems

The convolution of two continuous signals is

where τ is a dummy time variable. If a linear time-invariant (LTI) system has impulse response h(t), its output for input x(t) is

The convolution theorem states

where H(f) is the system frequency response. Thus, convolution in time becomes multiplication in frequency. The magnitude response |H(f)| describes gain or attenuation, while ∠H(f) describes phase shift.

For a sinusoidal input at frequency f, an LTI system changes amplitude by |H(f)| and phase by ∠H(f), without changing the frequency. This explains why frequency response is central to filter and control-system analysis.

The reverse relationship is multiplication in time:

The precise scaling depends on the chosen Fourier-transform convention; with the convention used here, no extra factor appears in the first convolution theorem, while multiplication in time produces convolution in frequency.

Worked Example 1: First-order low-pass response

Consider the low-pass transfer function

where f_c is the cutoff frequency. Its magnitude and phase are

For f = 100 Hz and f_c = 50 Hz,

|H| = 1/√(1+2²) = 1/√5 ≈ 0.447. The output sinusoid therefore has approximately 44.7% of the input amplitude and a phase lag of −tan⁻¹(2) ≈ −63.4°.

4. Correlation and Spectral Measurement

Correlation measures similarity as one signal is shifted relative to another. The cross-correlation of x(t) and y(t) may be defined as

For real signals, the complex conjugate may be omitted. Autocorrelation is

The correlation theorem connects correlation to spectra:

Correlation is useful for detecting a known waveform in noise, estimating time delay, synchronizing sensors, and identifying periodic components. Unlike convolution, correlation compares one signal with a shifted and usually conjugated version of another.

The power spectral density (PSD) describes how signal power is distributed with frequency. For a stationary random process, an idealized relationship is

PSD units are signal-units-squared per hertz. A PSD is not the same as the squared magnitude of one finite record unless normalization and estimation procedures are specified.

5. Discrete Fourier Transform

A digital system stores N samples x[0], x[1], …, x[N−1]. The DFT is

The inverse DFT is

The DFT treats the N-sample record as one period of a periodic discrete-time sequence. This assumption is important: if the first and last sample values do not join smoothly, discontinuities occur at the artificial boundary and produce spectral leakage.

If the sampling frequency is f_s samples per second, the frequency represented by bin k is

For a real-valued signal, bins above the Nyquist frequency are interpreted as negative frequencies. The Nyquist frequency is

The frequency spacing, or bin width, is

where T_R = N/f_s is the record duration. Increasing N by sampling for a longer time improves frequency spacing; merely increasing the sampling rate does not necessarily improve frequency resolution.

Amplitude scaling

For an unwindowed real sinusoid whose frequency lies exactly on a DFT bin, the positive-frequency single-sided amplitude estimate is approximately

for a non-DC, non-Nyquist bin. The factor of two combines equal positive- and negative-frequency contributions. It must not be applied to the DC bin or, when applicable, the Nyquist bin. Windowing changes this scaling through coherent gain.

Worked Example 2: DFT frequency grid

A vibration signal is sampled at f_s = 2000 Hz for N = 1000 samples. The record duration is

T_R = N/f_s = 1000/2000 = 0.5 s.

The frequency spacing is

∆f = f_s/N = 2000/1000 = 2 Hz.

A component at 148 Hz corresponds to bin

k = f/∆f = 148/2 = 74.

If the signal also contains a component at 151 Hz, the two frequencies are separated by only 3 Hz, so their separation is comparable to the 2-Hz grid spacing. Whether they can be reliably distinguished also depends on record length, window choice, noise, and relative amplitudes.

6. The Fast Fourier Transform

The FFT is not a different transform; it is a family of algorithms for evaluating the DFT efficiently. Direct DFT evaluation requires approximately N² complex operations. A radix-2 FFT exploits symmetry and periodicity to reduce the computational order to approximately N log₂N operations when N is a power of two.

The radix-2 decomposition separates the input into even and odd samples:

where W_N = e^−j2π/N, E[k] is the DFT of the even-indexed samples, and O[k] is the DFT of the odd-indexed samples. These operations form the familiar butterfly structure.

In engineering software, an FFT workflow should specify:

1. Sampling rate and sensor units.
2. Record length and duration.
3. Detrending or removal of the mean.
4. Window function.
5. One-sided or two-sided spectrum.
6. Amplitude, power, or PSD normalization.
7. Frequency range and interpretation of peaks.

A peak is not automatically a fault signature. It may result from electrical interference, structural resonance, leakage, sensor saturation, or an incorrectly scaled spectrum.

7. Sampling, Reconstruction, and Aliasing

Sampling converts a continuous signal x(t) into values x[n] = x(nT_s), where T_s is the sampling interval and f_s = 1/T_s. Ideal uniform sampling can be represented by multiplying x(t) by an impulse train. In the frequency domain, this creates repeated copies of X(f) centered at integer multiples of f_s.

If the original signal is strictly band-limited to |f| ≤ B, exact reconstruction is possible in ideal conditions when

This is the Nyquist sampling condition. The inequality is strict for practical signals because a filter cannot have an infinitely sharp transition and because components exactly at the boundary are not robustly represented.

An ideal reconstruction formula is the sinc interpolation equation

where sinc(u) = sin(πu)/(πu). This formula assumes ideal band limitation, infinite-duration data, exact samples, and an ideal reconstruction filter.

Aliasing

Aliasing occurs when frequencies above the Nyquist frequency are represented as lower frequencies after sampling. For a sinusoid at frequency f_0, the observed alias frequency can be found by folding f_0 into the interval [0, f_s/2]. Equivalently, choose an integer m so that

lies between 0 and f_s/2.

For f_s = 1000 Hz and f_0 = 700 Hz, choosing m = 1 gives f_alias = |700−1000| = 300 Hz. Once aliased, the original 700-Hz component cannot be recovered from the sampled data alone.

An anti-aliasing low-pass filter must be placed before the ADC. It attenuates frequency components that could otherwise fold into the measurement band. The filter cutoff must account for the desired signal bandwidth, transition band, ADC rate, filter order, and required attenuation.

8. Windows, Leakage, and Spectral Resolution

The finite record can be modeled as x_R(t) = x(t)w(t), where w(t) is a window. Multiplication by w(t) in time produces convolution of the true spectrum with the window spectrum. This spreads energy from a frequency component into neighboring frequencies, a phenomenon called spectral leakage.

A rectangular window has narrow main-lobe width but relatively high side lobes. Hann, Hamming, Blackman, and flat-top windows reduce side-lobe leakage to different degrees, generally at the cost of a wider main lobe. The correct choice depends on the measurement goal.

A window does not create additional information. Zero-padding interpolates the plotted spectrum between DFT frequencies, making peaks appear smoother, but it does not improve the fundamental resolution determined mainly by record duration and window main-lobe width.

9. Fourier Methods for Images

An image is a two-dimensional discrete signal x[m,n]. Its 2-D DFT is

The inverse transform contains the factor 1/(MN). Low spatial frequencies represent slowly varying illumination and broad structures. High spatial frequencies represent edges, fine texture, and abrupt intensity changes.

A 2-D low-pass filter can reduce noise but may blur edges. A high-pass filter emphasizes edges but can amplify noise. Practical image processing often uses masks, Gaussian filters, or frequency-domain convolution. As in one-dimensional signals, boundary assumptions and discontinuities can generate artifacts; padding and appropriate boundary handling are therefore important.

10. Applied Case: Motor-Bearing Condition Monitoring

Background

Consider an illustrative industrial case in which a three-axis accelerometer is mounted on a motor housing. The objective is to detect a developing bearing defect before it causes unplanned downtime.

Problem

The motor rotates at 25 Hz. A bearing defect is expected to generate vibration near 150 Hz, together with harmonics and broadband noise. The measurement system samples at 2000 Hz. Engineers must determine whether a spectral peak is a genuine defect signature.

Analysis and approach

A 2-second record gives N = 4000 samples and frequency spacing

∆f = 1/T_R = 1/2 = 0.5 Hz.

The Nyquist frequency is 1000 Hz, comfortably above the 150-Hz band of interest. An analog anti-aliasing filter is installed before the ADC. The mean is removed, a Hann window is applied, and an FFT is calculated. Because acceleration is real-valued, a single-sided amplitude spectrum is displayed from 0 to 1000 Hz.

The spectrum shows peaks at 150 Hz, 300 Hz, and 450 Hz. Their relationship to the known rotational speed supports the bearing-defect hypothesis, but does not prove it. The analyst compares the result with a baseline spectrum from a healthy bearing, checks sensor mounting, examines the time waveform, and repeats the measurement under similar load conditions. Cross-correlation with a tachometer pulse can also estimate phase or order-related timing.

Outcome

Suppose the 150-Hz amplitude increases consistently over several operating cycles while the motor speed remains stable. Maintenance personnel schedule a controlled bearing inspection rather than waiting for catastrophic failure. If the speed varies substantially, a fixed-frequency spectrum may smear the defect energy; order tracking or time-frequency analysis would then be more appropriate.

Learning points

The case illustrates that reliable diagnosis requires more than pressing an FFT button. Sampling, anti-aliasing, record length, window correction, calibration, repeatability, operating condition, and knowledge of machine physics all influence the conclusion.

11. Limitations, Professional Practice, and Emerging Tools

Fourier analysis assumes that frequency content is meaningful over the selected observation interval. Transient or rapidly changing signals may require short-time Fourier transforms (STFT), wavelets, or other time-frequency methods. The STFT applies a window that moves through time, producing a spectrum indexed by both time and frequency. Short windows improve time localization but reduce frequency resolution; long windows do the opposite.

Practical limitations include sensor noise, quantization, clipping, missing samples, timing jitter, nonuniform sampling, insufficient record length, leakage, and inappropriate normalization. Units must be preserved: acceleration amplitude may be expressed in m/s² or g, while an acceleration PSD has units such as (m/s²)²/Hz.

In safety-critical mechatronic systems, spectral algorithms should be validated against calibrated test signals and documented acceptance criteria. AI and machine-learning methods can classify spectra or detect anomalies, but they do not remove the need for correct sampling, trustworthy labels, explainable features, cybersecurity, and human oversight. A model trained on one machine and operating condition may fail under another. Data governance should address sensor ownership, maintenance records, privacy where human activity is measured, and traceability of automated decisions.

Best practice is to preserve raw data when feasible, record the complete acquisition configuration, use calibration checks, separate training and test data, and report uncertainty rather than presenting every spectral peak as certain evidence.

Chapter Summary

Fourier transforms express signals as frequency components. The continuous transform is suited to theoretical analysis of continuous nonperiodic signals, while the DFT represents a finite sampled record and the FFT computes the DFT efficiently. Magnitude and phase spectra describe amplitude and timing effects. Convolution converts LTI-system analysis into multiplication of spectra, and correlation supports delay estimation and pattern detection.

Sampling must satisfy the Nyquist condition for the intended bandwidth, and anti-aliasing filters are required in practical data-acquisition systems. DFT frequency spacing depends on record duration, while leakage depends on the finite observation window and boundary discontinuity. Window selection involves a trade-off among amplitude accuracy, main-lobe width, and side-lobe suppression. Fourier methods also support image filtering, vibration diagnosis, sensor monitoring, and digital filter design. Sound engineering interpretation requires attention to calibration, scaling, units, operating conditions, uncertainty, and the limitations of stationary spectral analysis.

Key Terms

Fourier transform: Integral representation of a signal in terms of complex sinusoids.

Spectrum: Frequency-domain representation of a signal.

Magnitude spectrum: Absolute value of the complex Fourier spectrum.

Phase spectrum: Argument or angle of the complex Fourier spectrum.

Fourier series: Harmonic representation of a periodic signal.

Convolution: Integral or sum describing the output of an LTI system.

Correlation: Measure of similarity between shifted signals.

DFT: Finite-sample discrete Fourier transform.

FFT: Efficient algorithm for computing a DFT.

Sampling frequency: Number of samples acquired per second.

Nyquist frequency: One-half of the sampling frequency.

Aliasing: Irreversible frequency misrepresentation caused by insufficient sampling.

Anti-aliasing filter: Analog filter that attenuates unwanted high-frequency components before sampling.

Spectral leakage: Spreading of spectral energy caused by finite-record effects.

Window: Weighting sequence applied to a finite signal record.

Frequency resolution: Separation between adjacent DFT frequency bins.

Power spectral density: Distribution of signal power per unit frequency.

LTI system: Linear time-invariant system characterized by an impulse response or frequency response.

Zero-padding: Appending zeros before an FFT to obtain a denser plotted frequency grid without adding information.

Review Questions

Short Answer Questions

1. Distinguish between a time-domain signal and its frequency spectrum.
2. State the continuous Fourier transform pair used in this chapter.
3. What is the physical meaning of the magnitude and phase of a frequency response?
4. Define aliasing and explain why it cannot generally be removed after sampling.
5. Why does zero-padding not fundamentally improve frequency resolution?

Long Answer Questions

1. Derive the relationship between convolution in time and multiplication in the frequency domain, and explain its significance for LTI systems.
2. Explain the DFT frequency grid, including the roles of sampling frequency, record length, record duration, and Nyquist frequency.
3. Compare rectangular, Hann, Hamming, Blackman, and flat-top windows in terms of leakage, main-lobe width, and amplitude measurement.
4. Describe a complete FFT-based workflow for measuring vibration from an industrial machine.
5. Explain how the 2-D Fourier transform can be used for image smoothing and edge enhancement, including boundary-related limitations.

Analytical / Application Questions

1. A sensor is sampled at 1200 Hz and contains a 850-Hz sinusoidal component. Determine its aliased frequency and explain the result.
2. An FFT uses 4096 samples at 4096 Hz. Determine the record duration, frequency spacing, and Nyquist frequency. Explain whether a 0.25-Hz separation between two tones is likely to be resolved.
3. A motor spectrum contains a strong peak near 50 Hz and smaller peaks near 150 Hz and 250 Hz. Develop a measurement plan to decide whether the smaller peaks are mechanical harmonics, leakage, electrical interference, or sensor artifacts.

Multiple Choice Questions with Answers

Question 1

For a continuous Fourier transform using frequency in hertz, which expression is the forward transform?

A. X(f) = ∫x(t)e^−j2πftdt

B. X(f) = ∫x(t)e^j2πftdt

C. X(f) = ∑x[n]e^−jπfn

D. X(f) = x(t) * h(t)

Correct Answer: A. X(f) = ∫x(t)e^−j2πftdt

Question 2

An LTI system has frequency response H(f) and input spectrum X(f). What is the output spectrum?

A. Y(f) = X(f)+H(f)

B. Y(f) = X(f)H(f)

C. Y(f) = X(f)/H(f)

D. Y(f) = X(f)−H(f)

Correct Answer: B. Y(f) = X(f)H(f)

Question 3

A signal is sampled at 1000 Hz. What is its Nyquist frequency?

A. 250 Hz

B. 500 Hz

C. 1000 Hz

D. 2000 Hz

Correct Answer: B. 500 Hz

Question 4

A 700-Hz sinusoid is sampled at 1000 Hz. What alias frequency appears below the Nyquist frequency?

A. 100 Hz

B. 200 Hz

C. 300 Hz

D. 700 Hz

Correct Answer: C. 300 Hz

Question 5

A record contains 2000 samples acquired at 1000 Hz. What is the DFT frequency spacing?

A. 0.5 Hz

B. 1 Hz

C. 2 Hz

D. 2000 Hz

Correct Answer: A. 0.5 Hz

Question 6

Which operation is computed efficiently by the FFT?

A. The continuous Fourier integral for every possible signal

B. The discrete Fourier transform of a finite sequence

C. The analog anti-aliasing filter

D. The physical sampling process

Correct Answer: B. The discrete Fourier transform of a finite sequence

Question 7

What is the principal effect of spectral leakage?

A. It removes all negative frequencies

B. It spreads energy from a frequency component into neighboring bins

C. It increases the sampling frequency

D. It guarantees exact amplitude measurement

Correct Answer: B. It spreads energy from a frequency component into neighboring bins

Question 8

For a real-valued time signal, which spectral property normally holds?

A. X(−f)=X(f)

B. X(−f)=−X(f)

C. X(−f)=X*(f)

D. X(−f)=0

Correct Answer: C. X(−f)=X*(f)

Question 9

Which component should normally be used before an ADC to reduce aliasing?

A. Analog anti-aliasing low-pass filter

B. Digital high-pass filter after sampling only

C. FFT display interpolator

D. Correlation detector

Correct Answer: A. Analog anti-aliasing low-pass filter

Question 10

A two-dimensional image’s high spatial frequencies are most strongly associated with which features?

A. Uniform illumination only

B. Very broad intensity gradients only

C. Edges and fine texture

D. The image’s average value only

Correct Answer: C. Edges and fine texture