In this blog post, we’ll explain how the Fourier transform decomposes signals and how it’s applied in real-life scenarios, such as speech recognition.

Basic Concepts of the Fourier Transform

In 2012, MIT Technology Review named the FFT (Fast Fourier Transform) one of the most influential technologies. The Fourier transform is, in fact, a core tool used in nearly all electronic devices, including communication devices like smartphones, cameras that produce sharp photos, and MP3 equalizers. In electrical engineering, the Fourier transform is so important that it has long been a central topic in the curriculum.
The basic idea behind the Fourier transform is to decompose a physical signal into simpler fundamental units. Here, a “signal” refers to anything whose amplitude varies over time. Things like sound or light, whose amplitude and intensity can change, are examples of signals. These fundamental units are sine waves, and each sine wave has its own unique frequency.
To understand sine waves and frequency, imagine shaking a long string. If you move your hand slowly, the rope moves up and down slowly, creating a waveform with wide intervals; if you move your hand quickly, the intervals between the waves become closer together. If it moves up and down once per second, it’s called 1 Hz; if it moves twice per second, it’s called 2 Hz. In other words, frequency is the number of repetitions per second and is denoted in Hz.
The analogy that a signal is composed of sine waves of various frequencies is similar to mixing ingredients in cooking. Just as chicken, water, gochujang, and corn syrup go into dakdoritang, a single signal also contains different sine waves—such as 1 Hz, 2 Hz, and 3 Hz—mixed in varying proportions. Since the original signal is a mixture of these components, it is difficult to analyze directly.
Salt, sand, and iron filings serve as examples of methods for separating mixtures. By dissolving them in water and filtering the solution—then separating what remains, what sticks to a magnet, and what is left behind—we can determine the characteristics of each individual substance. Similarly, the Fourier transform decomposes a complex signal into sine wave components, revealing the amplitude and frequency of each component. By examining the separated components, it becomes easier to understand what the signal is composed of.

Principles and Practical Applications of the Fourier Transform

The reason the Fourier transform uses sine waves as its basic units is that sine waves have a unique relationship with one another—namely, orthogonality. When sine waves of different frequencies are multiplied and integrated, the result is 0; when sine waves of the same frequency are multiplied and normalized, the result is a specific value (usually 1). By utilizing this property, we can isolate the amplitude of specific frequency components in a signal.
For example, suppose a signal contains a 1 Hz component with an amplitude of 1, a 2 Hz component with an amplitude of 2, and a 3 Hz component with an amplitude of 3. If we multiply this signal by a 1 Hz sine wave and integrate, the result is 1; if we multiply it by a 2 Hz sine wave, the result is 2; and if we multiply it by a 3 Hz sine wave, the result is 3. By performing this inner product with the signal at various frequencies, we can determine the magnitude of each frequency component. The Fourier transform is a mathematical tool that systematizes this process.
A prime example of a field where the Fourier transform is widely used is speech recognition. The human voice consists of waves generated in the air by the vibration of the vocal cords, and a microphone converts these air vibrations into electrical signals. Since the electrical signal retains the exact form of the original speech signal, a computer applies the Fourier transform to this signal to analyze its frequency components.
Each person’s voice has a unique frequency distribution, and each phoneme (e.g., /p/, /t/, /s/, /a/, etc.) also has its own distinct frequency components. Speech recognition systems analyze these separated frequency components to determine which phonemes are present and who is speaking. This capability has led to real-world applications such as smartphone voice assistants and has been utilized in services like Siri.
Electrical engineering is the field of study that converts low-level data into electrical signals, analyzes and processes those signals to transform them into high-level information, and researches methods to convey this information to humans or act on their behalf. In this sense, the Fourier transform is an essential tool that allows us to view signals from a frequency perspective, enabling us to solve problems from a new angle.
Research into the Fourier transform and algorithms like the FFT—which perform these operations more quickly—remains active today. As technology advances, the scope of applications for the Fourier transform will continue to expand, and the ability to decompose and recombine signals—much like “alchemy”—will continue to form the foundation of much electronic technology in the future.