Fourier Transform

Normal Modes of Vibration:
- Euler examined the motion of a vibrating string and identified what are known as “normal modes.” These are specific patterns of vibration that the string can exhibit.
- Each normal mode corresponds to a particular frequency of oscillation, and these modes are harmonically related sinusoidal functions of the position along the string.
Linear Combination of Normal Modes:
- Euler noted that if the configuration of the vibrating string at some point in time is a linear combination of these normal modes, then the configuration at any subsequent time will also be a linear combination of the same normal modes.
- This means that the shape of the string’s vibration can be described as a sum of these sinusoidal functions, each with its own amplitude and phase.
Evolution of Coefficients:
- Euler showed that the coefficients (amplitudes) of the linear combination at a later time can be calculated in a straightforward manner from the coefficients at an earlier time.
- This implies that the dynamics of the vibrating string can be understood by analysing how the coefficients of the normal modes evolve over time.

This observation from Euler laid the ground work for Jean Baptiste Joseph Fourier.

Fourier Transform:

Fourier extended the representation of periodic signals to aperiodic signals using the Fourier transform, which expresses signals as integrals of sinusoids.
This extension has become a cornerstone of signal analysis in both continuous and discrete-time domains.

Fourier analysis is crucial in numerous scientific and engineering disciplines, including the study of vibrations, heat diffusion, alternating currents, wave motion, and signal transmission.
The tools of Fourier analysis enable the examination of LTI systems’ responses to sinusoidal inputs, making it invaluable in fields like electrical engineering and telecommunications.