Engineering Math Series 19: Fourier Transform and Frequency Domain

While Fourier series was a tool for periodic functions, the Fourier transform is a tool for analyzing more general signals from a frequency perspective. It can be considered virtually the basic language in signal processing, communications, and image processing.

Time Domain and Frequency Domain

In the time domain, you look at when and how a signal changes. In the frequency domain, you look at what oscillation components the signal is composed of. It is the same signal viewed from different perspectives.

The Fourier transform is usually defined as

$F(\omega) = \int_{-\infty}^{\infty} f(t)e^{-i\omega t}\,dt$

This can be understood as measuring "how much the signal $f(t)$ resembles the complex oscillation of frequency $\omega$."

Intuition

If a signal strongly contains a certain frequency component, the transform value at that frequency will be large. Conversely, if it is barely present, the value will be small.

The Fourier transform is a lens that looks at the function not all at once but decomposed by frequency.

Representative Example

The fact that the Fourier transform of a Gaussian function is again a Gaussian is very famous. While the detailed calculation is long, the key message is that "a function that spreads widely in the time domain is relatively narrow in the frequency domain, and conversely a very short pulse in the time domain spreads widely in the frequency domain."

At the introductory level, you frequently see the example where a rectangular pulse has a sinc-shaped spectrum. A sharply cut signal in the time domain has a complex tail in the frequency domain.

Differentiation and Convolution

One of the most important properties of the Fourier transform is that differentiation becomes multiplication.

$\mathcal{F}\{f'(t)\} = i\omega F(\omega)$

This property makes it easier to handle differential equations in the frequency domain.

Another important property is that convolution becomes multiplication.

$\mathcal{F}\{f*g\} = F(\omega)G(\omega)$

This greatly simplifies the analysis of signals passed through filters.

Engineering Applications

Audio and Speech

Analyzing sound by frequency components makes it easier to understand pitch, noise, and harmonic structure.

Communications

Concepts like channel bandwidth, modulation, and spectral efficiency are inseparable from frequency-domain analysis.

Image Processing

Images viewed from a 2D Fourier perspective split into low-frequency and high-frequency components, and blur or sharpening can be interpreted as frequency filters.

Common Mistakes

Viewing time domain and frequency domain as different problems

They are two perspectives of the same signal. A problem that is difficult in one may become easy in the other.

Understanding the frequency axis only abstractly

Frequency actually represents how fast something oscillates. The physical meaning must be continuously connected.

Only memorizing formulas and missing properties

In practice, properties like differentiation, shifting, scaling, and convolution are used more often than the definition.

One-Line Summary

The Fourier transform is a tool for decomposing non-periodic signals into frequency components for analysis.

Next Post Preview

In the next post, based on the Fourier perspective, we will organize what partial differential equations are and why we classify them at the introductory level.

References

• Erwin Kreyszig, Advanced Engineering Mathematics, 10th Edition
• Ronald N. Bracewell, The Fourier Transform and Its Applications
• Alan V. Oppenheim, Ronald W. Schafer, Discrete-Time Signal Processing