Theory

The Parzen window

ValkanPavlov Theory 21 April 2026

The Parzen window is a smooth window function used in digital signal processing (DSP). It is designed to gradually taper the signal toward zero at both ends, reducing discontinuities.

Unlike cosine-based windows, the Parzen window uses a piecewise polynomial shape, resulting in a very smooth transition and good spectral behavior.

\[ w(n) = \begin{cases} 1 - 6\left(\frac{|n - \frac{N-1}{2}|}{\frac{N-1}{2}}\right)^2 + 6\left(\frac{|n - \frac{N-1}{2}|}{\frac{N-1}{2}}\right)^3, & 0 \le |n - \frac{N-1}{2}| \le \frac{N-1}{4} \\ 2\left(1 - \frac{|n - \frac{N-1}{2}|}{\frac{N-1}{2}}\right)^3, & \frac{N-1}{4} < |n - \frac{N-1}{2}| \le \frac{N-1}{2} \end{cases} \]

The Parzen window provides smooth edge behavior, which helps reduce spectral leakage while maintaining a continuous shape.

The Blackman–Harris window

ValkanPavlov Theory 20 April 2026

The Blackman–Harris window is a highly optimized cosine-sum window function used in digital signal processing (DSP). It is designed to achieve extremely strong sidelobe suppression and very low spectral leakage, making it one of the most aggressive windows in the cosine-sum family. It is commonly used in precision FFT analysis where accurate amplitude detection and dynamic range are more important than sharp frequency resolution.

\[ w(n) = 0.35875 - 0.48829 \cos\left(\frac{2\pi n}{N-1}\right) + 0.14128 \cos\left(\frac{4\pi n}{N-1}\right) - 0.01168 \cos\left(\frac{6\pi n}{N-1}\right), \quad 0 \le n \le N-1 \]

The Nuttall window

ValkanPavlov Theory 20 April 2026

The Nuttall window is a cosine-sum window function used in digital signal processing (DSP). It is designed to provide very strong sidelobe suppression and smooth spectral behavior.

It is commonly used in FFT analysis where high dynamic range and minimal spectral leakage are required.

\[ w(n) = a_0 a_1 \cos\left(\frac{2\pi n}{N-1}\right) a_2 \cos\left(\frac{4\pi n}{N-1}\right) a_3 \cos\left(\frac{6\pi n}{N-1}\right) \]

Typical coefficients: a₀ = 0.355768, a₁ = 0.487396, a₂ = 0.144232, a₃ = 0.012604

The Flat Top window

ValkanPavlov Theory 20 April 2026

The Flat Top window is a specialized window function designed for highly accurate amplitude measurements in the frequency domain. Unlike other windows that optimize for spectral resolution or leakage, the Flat Top window is optimized for amplitude accuracy, making it ideal for FFT-based measurements where knowing the exact amplitude of a frequency component is more important than distinguishing closely spaced frequencies. Its frequency response has an extremely flat passband, which minimizes the amplitude error that occurs when a signal frequency falls between FFT bins (scalloping loss).

\[ w(n) = a_0 - a_1 \cos\left(\frac{2\pi n}{N-1}\right) + a_2 \cos\left(\frac{4\pi n}{N-1}\right) - a_3 \cos\left(\frac{6\pi n}{N-1}\right) + a_4 \cos\left(\frac{8\pi n}{N-1}\right), \quad 0 \le n \le N-1 \]

Typical coefficients for the 5-term Flat Top window are:

\[ a_0 = 0.21557895, \quad a_1 = 0.41663158, \quad a_2 = 0.277263158, \quad a_3 = 0.083578947, \quad a_4 = 0.006947368 \]

These coefficients are carefully chosen to create a frequency response that is nearly flat across the width of each FFT bin, ensuring that the measured amplitude is accurate to within ±0.01 dB regardless of where the signal frequency falls within the bin.

The Blackman window

ValkanPavlov Theory 20 April 2026

The Blackman window is a smooth window function used in digital signal processing (DSP). It is designed to reduce spectral leakage by strongly tapering the signal toward zero at both ends. It belongs to the family of cosine-sum windows and uses an additional cosine term compared to simpler windows like the Hann window. This results in significantly better sidelobe suppression, making it useful for analyzing signals with large dynamic range where weak components must be detected near strong ones.

\[ w(n) = 0.42 - 0.5\cos\left(\frac{2\pi n}{N-1}\right) + 0.08\cos\left(\frac{4\pi n}{N-1}\right), \quad 0 \le n \le N-1 \]