Fourier Analysis

Fourier Analysis refers to the tool used to compress complex data into a series of trigonometric or exponential functions. The primary purpose of this tool is to decompose a complex signal into a function of sin and cosine waves. In finance, Fourier analysis signify using mathematical techniques given by Fourier transform theory to analyze financial data in terms of frequency components. 

By decomposing a complex signal into its constituent sinusoidal components, it gives important information about the signal's frequency content and phase relationships, among other properties. In addition, it solves the time issue within the data set, particularly in audio processing. It highlights the relationship between a signal's behavior in time and its corresponding frequency spectrum and amplitude.

• Fourier Analysis is a mathematical tool used to chop complex data (or samples) into different components using time and frequency. 
• In 1922, the French mathematician Jean-Baptiste Joseph Fourier introduced this concept in the book titled "The Analytical Theory of Heat."
• There are two major variants of this analysis, namely Fourier Series and Fourier Transform. Fourier Transform is further divided into three main types, Discrete Fourier Transform (DFT), Fast Fourier Transform (FFT), and Inverse Fourier Transform (IFT). 
• Its application is visible in various fields like signal transformation, image processing, quantum mechanics, spectral analysis, and many others.

Fourier Analysis is a vital mathematical concept that breaks complex wave signals into sin and cosine functions. It helps understand how a frequency behaves in different time and frequency models. Also, it breaks down complex data into a collection of functions, where each represents its share in the original function. Hence, if one wants to decode a complex audio signal from a radio, they may conduct a Fourier analysis experiment to determine the different components.

The classical Fourier analysis dates back to the early 19th century. A French mathematician, Jean-Baptiste Joseph Fourier, introduced it in his book The Analytical Theory of Heat in 1922. While working on heat conduction, he realized that any function, regardless of its complexity, could be represented as a sum of sinusoidal functions. Hence, it has various practical applications.

This concept finds a place in technical analysis, options pricing, derivative structuring, and financial forecasting. It helps with overall investment planning and decision-making, particularly when expert-level analysis is needed. For instance, periods of market volatility can be analyzed effectively using this mathematical technique. It can indicate the levels of market volatility (high or low) by decomposing data and offering analysts chunks of useful information.

The classical Fourier analysis experiment works on the principle of time and frequency. Users can employ it to understand the components of a signal. Though the process of signal detection differs, the decomposition of a signal is the most important step in this activity.

Signals react and exhibit specific behaviors in two domains: time and frequency. Time signifies how a signal evolves, while the frequency domain focuses on how the signal behaves (or reacts) at different frequencies. For instance, the characteristics of a sound sample can be decoded through analysis. The time domain or zone of a sound sample depicts how it changes with time when pressure is created. When analyzed in the frequency domain, the same sample represents different frequency waveforms.

The Fourier series and Fourier transform help switch between these domains, offering valuable knowledge and insights about the characteristics of signals. Certain assumptions influence the results of this analysis, though. It assumes that major events are, by default, periodic. As a result, transient behavior or aperiodicity cannot be accurately interpreted with the Fourier transform. To resolve this problem, several variations of the Fourier transform have been developed.

It has different variants, with most types falling under the purview of Fourier series and Fourier transform. Let us study them.

#1 - Fourier Series

The first variant of discrete Fourier analysis includes the Fourier series. As the name suggests, this series is periodic. Also, it is well-behaved, which can be represented as a sum of the trigonometric functions in the form of sin and cosine. However, this behavior can vary in time as well as frequency domains. For instance, the discrete Fourier analysis series is continuous and periodic in the time domain. However, it is discrete in the frequency domain. In short, it is aperiodic and discrete in the frequency domain. The following is the equation for this series:

Here, a₀, a_n, and b_n refer to the coefficients that are further used to determine the amplitudes of the sine waves. T represents the trigonometric function of the series.

#2 - Fourier Transform

In the case of non-periodic functions, the Fourier transform is useful. It helps eliminate unnecessary periodic terms and variations within the event. In short, it converts the time function to frequency components. As a result, it behaves differently compared to the Fourier series. This transform is aperiodic and discrete in both time and frequency domains.

Let us look at the examples of Fourier analysis to understand the concept in detail.

Example #1

Suppose Laura, an investor, wants to determine when she should move her investments in the stock market. For this, she decides to use Fourier analysis. She decomposes the data using software, which highlights the seasonal patterns (including how prices move) of the specific stock market in which she is interested. 

Through this analysis, she outlines the periods suitable for buying and selling stocks based on historical data. By using this technique, she is able to specifically pinpoint which periods are useful for selling certain stocks, and the time that is predicted as suitable for buying certain stocks. This helps her maximize her returns in this market.

Example #2

Suppose Jenny is a sound engineer working in the music industry. She creates music for various artists and musicians. Jenny has decoded some old audio samples and created a piece of art out of it. She also received an audio piece from her friend, Sam. This audio had amazing notes, but the recording was unclear and problematic. Therefore, she installed the right software and started a fast Fourier analysis transform as part of FFT.

Jenny was able to decompose or break down the audio sample into chunks. Now, she had separate frequencies of different sections (or constituent frequencies), namely Part 1, 2, 3, 4..., to 10. From these sections, she analyzed each soundwave in different spectrums and picked a few of them. With these few, Jenny created a new beat, and finally, music. In short, FFT helped Jenny decode a noisy sample and create new output from it. In this case, she successfully employed FFT for music creation.

Example #3

According to a November 2023 report, Kyocera Corporation, a well-known fine ceramics, electronics, and industrial tools manufacturer, has introduced Silicon Nitride (SN) light in its products. It will enhance the performance of the Fourier Transform Infrared (FTIR) spectrometers in the heaters.

The inclusion of SN will improve the higher spectral emissivity, which produces clear spectral peaks to ensure more stable performance, an extended duty cycle, and reduced noise in spectral data. Quality control processes will also improve due to this development. This is a good example of the application of Fourier Analysis in the field of spectroscopy.

The following are the use cases and applications of Fourier analysis equations in the real-world. Let us study them.

• Technical analysis: It is used to identify patterns and trends in financial time series data, which can be used to make trading decisions.
• Signal transformation: It is used in audio and sound engineering since it helps users transform noisy signals into chunk pieces from the time domain to frequency waveform. Radio stations or satellite stations use this tool for signal identification. 
• Spectral analysis: It is useful in conducting spectral analysis. It allows users to see the different wavelengths or oscillations within a data set. Applications are seen in oceanography, astronomy, geophysics, atmospheric science, and others. 
• Image processing: With the help of Fourier analysis, users can easily process images using software. It helps denoise, sharpen, and modify the wavelengths of the images through time and frequency domains. 
• Frequency detection: It is useful to understand the channel characteristics and frequencies in telecommunication technology. This improves signal quality, transmission, and reception. 
• Quantum mechanics: In addition to the above uses, individuals can also apply it to quantum mechanics. It enables the identification of minute particles and their wavelengths with the Fourier series. It helps understand the behavior of quantum systems and particles.

This article has been a guide to what is Fourier Analysis. Here, we explain the concept along with its examples, applications, and variants. You may also find some useful articles here -

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