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What is the relationship between the Fourier transform and the Fourier series?

Published in Signal Analysis 4 mins read

The Fourier transform is a generalization of the Fourier series, extending the powerful concept of frequency domain analysis from periodic functions to virtually any non-periodic function.

Understanding the Fourier Series

The Fourier series is a mathematical representation that allows a periodic function to be expressed as a sum of simple oscillating sine and cosine waves. This makes it an invaluable tool for analyzing and approximating periodic signals found in various fields, such as electrical engineering, acoustics, and mechanical vibrations. When a periodic signal is decomposed into its Fourier series, the result is a discrete spectrum of frequencies, where each frequency component corresponds to a specific harmonic of the fundamental frequency of the periodic signal.

Understanding the Fourier Transform

In contrast, the Fourier transform is a mathematical tool designed to decompose a non-periodic function into its individual frequency components. Unlike the Fourier series, which yields a discrete set of frequencies, the Fourier transform produces a continuous spectrum of frequencies. This continuous spectrum is essential for analyzing transient signals, impulses, or any signal that does not repeat itself over time. It finds extensive use in fields like image processing, quantum mechanics, and communications.

The Core Relationship: Generalization and Duality

The fundamental relationship between the Fourier transform and the Fourier series lies in the fact that the Fourier transform can be viewed as a more encompassing concept. Essentially, the Fourier transform is the continuous analogue and generalization of the discrete Fourier series.

Here's how they are intimately related:

  • Generalization to Aperiodicity: The Fourier series applies exclusively to functions that repeat themselves periodically. The Fourier transform extends this analysis to functions that are not periodic (aperiodic). One way to conceptualize this is by imagining a periodic function whose period approaches infinity; as the period grows infinitely large, the function effectively becomes non-periodic, and its Fourier series coefficients merge into a continuous spectrum, which is precisely what the Fourier transform describes.
  • Dirac Delta Functions: For a periodic function, its Fourier transform will not be a continuous curve but rather a series of impulses or Dirac delta functions. These impulses are located at the discrete frequencies that correspond to the harmonics of the Fourier series, and their "strengths" (amplitudes) are directly proportional to the Fourier series coefficients. This demonstrates how the continuous Fourier transform inherently contains the discrete information provided by the Fourier series for periodic signals.
  • Duality: There is also a form of duality: the Fourier transform of a discrete series (like a sequence of Fourier series coefficients) results in a periodic function in the frequency domain, mirroring how a periodic function in the time domain yields discrete frequencies.

Key Differences Summarized

While closely related, their applications are determined by the nature of the signal being analyzed.

Feature Fourier Series Fourier Transform
Function Type Periodic functions (repeating signals) Non-periodic (aperiodic) functions
Frequency Output Discrete spectrum (sum of distinct frequencies) Continuous spectrum (spectrum over all frequencies)
Representation Sum of sines and cosines Integral over continuous frequencies
Application Analysis of repetitive signals (e.g., AC circuits) Analysis of transient signals (e.g., impulses)

Practical Applications and Significance

Both mathematical tools are indispensable in engineering and science for understanding and manipulating signals.

  • Fourier Series Applications:
    • Audio Synthesis: Reconstructing complex sound waves from individual pure tones.
    • Electrical Engineering: Analyzing the behavior of AC circuits and power signals.
    • Vibration Analysis: Identifying resonant frequencies in mechanical systems.
  • Fourier Transform Applications:
    • Image Processing: Filtering noise, edge detection, and compression (e.g., JPEG).
    • Signal Processing: Designing filters, analyzing communication signals, and data compression.
    • Quantum Mechanics: Describing the momentum representation of a particle's wave function.
    • Medical Imaging: Used in MRI (Magnetic Resonance Imaging) to convert raw data into images.

In essence, the Fourier series provides the harmonic content of repeating patterns, while the Fourier transform offers a comprehensive frequency breakdown for any signal, acting as a powerful continuum that encompasses the discrete nature of its series counterpart.