Fourier analysis in square wave transform

Last edited time
Jan 20, 2024 03:23 AM
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To derive the amplitude of the fundamental component of a square wave's Fourier series, we start with the basic concepts of Fourier analysis. Fourier analysis allows us to represent a periodic function as a sum of sines and cosines. A square wave, due to its symmetry and periodicity, can be represented as a series of sine waves with specific amplitudes.
A square wave \( \) with period \( \) and amplitude \( \) can be defined as follows:
  • \( \) for \( \)
  • \( \) for \( \)
The Fourier series representation of a periodic function \( \) is given by:
\[ \]
where \( \) is the fundamental frequency, and \( \) are Fourier coefficients given by:
\[ \] \[ \] \[ \]
For a square wave, all \( \) are zero because of the symmetry of the function. The formula for \( \) simplifies due to the square wave's characteristics. We can compute the coefficient for the fundamental frequency (n=1) and note that only odd harmonics (n=1, 3, 5, ...) exist due to the shape of the square wave. Let's calculate the coefficient \( \) for the fundamental frequency (n=1):
\[ \]
For \( \), this becomes:
\[ \]
Given that \( \), let's compute this integral.
The calculation yields that the coefficient \( \) for the fundamental frequency of the square wave's Fourier series is \( \), provided that \( \) (the period of the square wave) is non-zero and finite. This result is a piecewise function, but in practical terms, for a square wave with a defined period, it simplifies to \( \).
This means the amplitude of the fundamental component (the first harmonic) of a square wave with peak amplitude \( \) is \( \). This fundamental component is a sine wave with the same frequency as the square wave and an amplitude that is \( \) times the square wave's peak amplitude. This relationship is a key aspect of Fourier analysis and is particularly important in signal processing and communications.
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