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Frequency-Domain Analysis with DFTs
Author(s): Gary B. Hughes
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Book Description

This monograph addresses the fundamental concepts of frequency spectrum analysis using discrete Fourier transforms. Topics range from the discretization of measured physical phenomenon to the interpretation of periodogram spectral content in terms of the Fourier basis sinusoids. The text also discusses peculiarities that arise due to discretization, such as leakage and aliasing, and why de-trending and windows mitigate such effects. Included examples explain the details of implementation and interpretation, with MATLAB source code provided for a number of frequency-spectrum-analysis tools.

Book Details

Date Published: 19 December 2017
Pages: 230
ISBN: 9781510616097
Volume: PM282

Table of Contents
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Table of Contents

1 Introduction
1.1 Time-Varying Signals
1.2 The Frequency Domain

2 Preliminary Background for the Fourier Analysis of Mathematical Functions
2.1 Fourier's Theorem
2.2 Conditions of Fourier's Theorem
2.3 A Review of General Sinusoid Features
2.4 Building Periodic Functions from Restricted-Domain-Function Snippets
2.5 Frequencies of Fourier Series Terms
2.6 Sum of Sinusoids with the Same Frequency
2.7 Determining Rectangular (Real) Fourier Series Terms
2.8 Euler's Formula and the Circular Form of Fourier's Theorem
2.9 Determining Circular (Complex) Fourier Series Terms
2.10 Interpreting Circular Fourier Components for Real-Valued Functions
2.11 The Fourier Transform and Its Relationship to Fourier Series Expansions

3 Fourier Transforms for Discretely Sampled Data
3.1 Discrete Sampling of Continuous Signals
3.2 Discrete Fourier Transform
3.3 Fast Fourier Transform
3.4 Periodogram with Peak-Amplitude Normalization
3.5 Signal Reconstruction from DFT Sinusoidal Components
3.6 MATLAB Source Code: DFT Amplitude Spectral Density
3.7 Notes on Validation and Verification of the MATLAB Source Code

4 Pre-Processing and Other Tricks of the DFT Trade
4.1 Mean Subtraction and Signal De-trending
4.2 Data Padding and Interpolation for FFTs
4.3 Nyquist Limits, Aliasing, and Oversampling
4.4 Power Leakage and Window Functions
4.5 MATLAB Source Code: Data Windows
4.6 Narrow Windows and Exploring Changes in Frequency Content through Time
4.7 MATLAB Source Code: Frequency Content through Time
4.8 DFT Spectrum Variability and Overlapping Data Segments
4.9 MATLAB Source Code: Overlapping Data Segments
4.10 Sensor Integration Time Effects
4.11 Phase, Cross-Spectra, and Coherence
4.12 MATLAB Source Code: Phase and Coherence

5 Signal Analysis Applications
5.1 Filters in the DFT Frequency Domain
5.2 Characterizing, Modeling, and Synthesizing Noise
5.3 Accelerometers and Random Vibration Testing
5.4 Spectral Analysis and Causal Relationships in Correlated Signals
5.5 Some Remarks about DFT Frequency-Domain Analysis

Appendix: MATLAB Source Code: Validation and Verification


Fourier transforms provide a mechanism for translating suitable mathematical functions between the time domain and the frequency domain. Many excellent references describe the theoretical basis for using Fourier transforms to analyze the frequency content of mathematical functions. Fourier transforms have also been adapted for applied scenarios, such as estimating the frequency content of discretely sampled signals. Procedures for discrete frequency-domain analysis using Fourier methods are, however, laden with subtle foibles. Many of the quirks that are unique to the analysis of discretely sampled signals elude the insight of the typical, rigorous Fourier development that is so elegantly valuable for analyzing mathematical functions. In my experience, many mathematically oriented references leave the details of practical implementation for readers to discover on their own. The main objective of this book is to provide a practical guide for the implementation of Fourier transform methods to perform frequency-domain analysis of discretely sampled time-series signals.

The topic of frequency-domain analysis is initially motivated by presenting a basic example: examining the frequency content of a synthetic line voltage that is created by summing the sinusoidal signals and a noise component. The composite sum is displayed as a time-series signal, where the sinusoidal components are not necessarily easy to identify in the time domain. A discrete Fourier transform (DFT) peak-amplitude spectral density profile (periodogram) of the composite signal is presented that shows how the sinusoidal contributions are explicitly revealed in the frequency domain.

Creating and interpreting the DFT periodogram requires specific knowledge of Fourier transform theory, combined with techniques that are tailored for discretely sampled signals. Salient mathematical concepts that are pertinent to Fourier analysis are explored, with the aim of providing enough theory to fully understand and interpret the Fourier frequency spectra of mathematical functions. The theory is followed by precise explanations of essential practical concepts that are required for analyzing discretely sampled time-series signals using Fourier transform methods. A discussion of common DFT preprocessing techniques, such as de-trending, data padding, data windowing, and remedies for certain discretization issues, is presented. MATLAB source code is provided that implements all of the ideas discussed in the text. (MATLAB is a registered trademark of The Mathworks, Inc.) Some validation and verification activities for the source code are explored. The source code is then put to good use in several applied examples that illustrate some of the potent capabilities of DFT frequency-domain analysis.

Gary B. Hughes
November 2017

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