02_Classic FFT

1. Rectangle (Dirichlet) Window 

The spectral window for the DFT window sequence is given in

Understanding the Rectangle Window and the Dirichlet Kernel in the DFT

When we apply a rectangle window (sometimes called a “boxcar” or “uniform” window) in digital signal processing, its frequency response is described by the Dirichlet kernel. This is a fundamental concept in the Discrete Fourier Transform (DFT).

What does this mean for your spectrum?

• Mainlobe Width:
  The Dirichlet kernel creates a central peak (mainlobe) in the frequency domain. The distance between the two zero crossings around this mainlobe is exactly 2 DFT bins. In other words, the mainlobe is “2 bins wide.”

• Sidelobe Level:
  The first sidelobe—the first “bump” you see next to the mainlobe—reaches a height that is about 13 dB lower than the mainlobe’s peak. This is a key number to remember: with the rectangle window, energy from strong tones can “leak” into nearby frequencies, but the first sidelobe is always significantly lower than the mainlobe.

This graph compares the ideal square wave with its Fourier series approximations using a finite number of terms. As more sine wave harmonics are added, the approximation becomes closer to the true square wave, but overshoots (Gibbs phenomenon) still appear near the discontinuities. The formula shown in the graph expresses how each approximation is constructed as a sum of odd harmonics.

These oscillations—typically observed near discontinuities—are known as the "Gibbs phenomenon." To reduce this oscillatory behavior, various window functions are often applied during the Fourier approximation process.

2. Triangle (Fejer, Bartlet) Window 

• Zero-Padding

1. Array Example

• Original signal (length 5):
  x = [2, 1, 0.5, 0, −1]

• After zero-padding (extended to length 12):
  x_padded = [2, 1, 0.5, 0, −1, 0, 0, 0, 0, 0, 0, 0]
      (Zeros are added at the end to increase the length)

2. Formula (Mathematical Definition)

• N : original signal length

• L : length after zero-padding

3. DFT/FFT Computation Formula

• x_padded[n] : zero-padded signal

• L: total length (FFT size)

4. Effect (Interpretation)

• Zero-padding does not add new information to the signal.

• By increasing the FFT size, you get more frequency bins.

• This results in a smoother and higher-resolution spectrum, as if you interpolated between the original spectral points.

• The frequency axis becomes more detailed, but the fundamental spectral features do not change

• Fejér Sum/ Cesàro Average

: When approximating a signal with a Fourier series, partial sums often produce oscillations near discontinuities—this is known as the Gibbs phenomenon. Fejér summation reduces these oscillations by averaging multiple partial sums, resulting in a much smoother and more accurate approximation.

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