The Bohman window is a smooth window function used in digital signal processing (DSP). It is designed as the convolution of two half-duration cosine lobes, resulting in a very smooth taper toward zero at both ends with continuous first derivatives.
Unlike cosine-based windows such as Hamming or Hann, the Bohman window uses a unique mathematical formulation that combines a triangular window with a cosine cycle, producing excellent sidelobe roll-off characteristics.
where n ranges from -(N-1)/2 to (N-1)/2. The first and last elements of the Bohman window are forced to zero, ensuring smooth transitions at the boundaries.
The Bohman window provides excellent spectral leakage reduction with very fast sidelobe falloff, making it particularly suitable for applications requiring high dynamic range.
Intuition Behind the Bohman Window
The Bohman window is constructed by convolving two cosine lobes of half duration. In the time domain, this translates to the product of a triangular window and a single cycle of a cosine, with an additional term that zeroes the first derivative at the boundaries.
This construction results in an exceptionally smooth window shape with continuous derivatives, which minimizes abrupt changes at the edges and provides superior spectral performance compared to many classical windows.
The Bohman window features a very smooth polynomial-cosine hybrid shape with continuous first derivatives at boundaries, excellent spectral leakage reduction, fast sidelobe roll-off (∼18 dB/octave), and moderate main lobe width. It is useful when smooth transitions and reduced edge discontinuities are required, particularly in applications demanding high dynamic range and excellent suppression of distant frequency components.
- Main lobe width: Approximately 0.12–0.13 normalized frequency. This is slightly wider than Parzen, meaning somewhat lower frequency resolution.
- First sidelobe level: Around -46 dB – very good suppression of the nearest sidelobe.
- Sidelobe roll-off: Extremely fast – approximately 18 dB per octave or more, significantly faster than most cosine-based windows.
- Higher sidelobes: Drop below -80 dB very quickly, making the Bohman window excellent for applications requiring high dynamic range.
Analysis: The Bohman window offers one of the fastest sidelobe roll-off rates among common window functions. While windows like Blackman–Harris have slightly lower first sidelobes (down to -70 dB), the Bohman window's superior falloff means that distant frequency components are suppressed much more effectively. The rectangular window has a very narrow main lobe (~0.04) but terrible sidelobes (only -13 dB), causing severe spectral leakage.
Typical applications: Use the Bohman window when you need excellent suppression of distant frequency components and can tolerate a moderate reduction in frequency resolution. Its fast sidelobe roll-off makes it particularly valuable in spectral analysis, antenna pattern synthesis, and high-dynamic-range measurements where distant interference must be minimized.
The Bohman window excels in applications where fast roll-off and excellent distant sidelobe suppression are critical. The Bohman window's construction via convolution of cosine lobes gives it a unique spectral signature that many engineers prefer for high-dynamic-range measurements.
The Bohman window should be used when one needs clean spectral estimation with minimal distant interference, and when the slightly wider main lobe compared to Parzen is acceptable. It is particularly popular in antenna design, high-dynamic-range spectral analysis, and precision measurements where distant frequency components must be suppressed below -80 dB.