Fourier Analysis: Unlocking the Secrets of Signals (Part-1: Fourier Series)

Fourier analysis, named after the renowned French mathematician and physicist Jean-Baptiste Joseph Fourier (1768-1830), is a powerful mathematical tool that has revolutionized the understanding and manipulation of signals and spectra. With its ability to decompose complex signals into simpler sinusoidal components, Fourier analysis provides a fundamental framework for exploring the frequency content and spectral characteristics of various phenomena. In this comprehensive article and few following articles, we will delve into the fascinating world of Fourier analysis, examining its principles, applications, and the profound impact it has had across diverse fields of science and engineering.

Fourier Series:

At the core of Fourier analysis lies the concept of the Fourier series, which states that any periodic function can be expressed as a sum of sine and cosine functions with different frequencies and amplitudes. By decomposing a function into its constituent harmonics, the Fourier series allows for precise analysis and manipulation of periodic signals.

Considering x(t) to be a periodic signal with period T, i.e.

Then x(t) can be decomposed as a sum of sine and cosine as below:

Where,

and the coefficients are given by

Hence, any periodic signal is a sum of DC or average term and integral harmonics of sine and cosine waves.

Example-1:

Consider a bipolar square wave as given below:

Using the formulas given in the previous section the coefficients A? and B? are given as follows for n = 0 to 50:

All these add up ideally from n = 0 to infinity to give the bipolar square wave. The sum waveform from n = 0 to 50 is given below:

Here some ripples are there since I didn't add all the harmonics till infinity but truncated till n = 50. The ripple will be higher if the harmonics are truncated far below as shown:

Example-2:

Consider a sawtooth wave as given below:

Using the formulas given in the previous section the coefficients A? and B? are given as follows for n = 0 to 50:

All these add up ideally from n = 0 to infinity to give the bipolar square wave. The sum waveform from n = 0 to 50 is given below:

Here some ripples are there since I didn't add all the harmonics till infinity but truncated till n = 50. The ripple will be higher if the harmonics are truncated far below as shown:

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Marvelous Balance of Harmonics:

You can clearly see that the ripples are reducing if n is increased. This is the beauty of the Fourier Series. All the harmonics till infinity add up exactly as they needed to be to generate the exact square waveform. This is beyond human imagination but it is there in nature and we are experiencing this always.

Applications of Fourier Analysis:

The applications of Fourier analysis are vast and pervasive. In telecommunications, Fourier analysis enables the efficient encoding and decoding of signals, leading to high-speed data transmission. In image processing, Fourier analysis aids in compression, noise reduction, and image enhancement techniques. It plays a vital role in medical imaging, facilitating the identification and diagnosis of abnormalities in the human body. Moreover, Fourier analysis finds applications in physics, providing insights into the behavior of electromagnetic waves, quantum phenomena, and the study of wave propagation. We will explore these applications, showcasing the diverse domains in which Fourier analysis has made significant contributions.

Beyond Engineering: Fourier Analysis in Science and Mathematics:

Beyond engineering disciplines, Fourier analysis extends its influence into physics, mathematics, and other scientific fields. It serves as a foundation for solving differential equations, allowing for the understanding of various dynamical systems. In physics, Fourier analysis elucidates the behavior of waves, harmonics, and resonance phenomena in natural systems.

To be Continued in the next edition … ??