Please provide a detailed explanation of the Gibbs phenomenon
The Gibbs phenomenon is a phenomenon that occurs in signal processing when approximating a discontinuous function using Fourier series. Specifically, when attempting to reconstruct a discontinuous function, there will be overshoots or ringing artifacts near the point of discontinuity.

To illustrate this phenomenon, consider the step function f(x) defined as:

f(x) = -1, for x < 0 1, for x >= 0

When we attempt to reconstruct this function using Fourier series, we will obtain a series that converges pointwise at every point except for x = 0. At x = 0, the series converges to the average of the left and right limits, which is 0.5.

However, as we increase the number of terms in the series, we will observe that the overshoots (i.e., the magnitude of the amplitude) close to the point of discontinuity increases, and the ringing becomes more pronounced.

The Gibbs phenomenon can have significant impact on signal reconstruction as it causes a distortion of the reconstructed signal. In some cases, the ringing can be large enough to mask the original signal, making it difficult to analyze the data.

In conclusion, the Gibbs phenomenon is a well-known phenomenon that occurs in signal processing when approximating a discontinuous function using Fourier series. It is characterized by overshoots or ringing artifacts near the point of discontinuity. While the phenomenon is well understood, it is important to keep in mind its potential impact on signal reconstruction.
Please simplify into 2 paragraphs and make it easier to understand.
Imagine you are trying to draw a square wave using only sine waves. The first few sine waves will do a good job of approximating the square wave, but as you add more sine waves, the approximation will start to look more and more like a triangle wave. This is because the sine waves are not able to perfectly reproduce the sharp corners of the square wave.

The Gibbs phenomenon is the name given to this overshoot of the Fourier series at discontinuities. It can be reduced by using a technique called windowing.