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Chapters

See Also

Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Spectrum Analysis Windows
       Blackman-Harris Window Family
          Power-of-Cosine Window Family

Chapter Contents:

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Power-of-Cosine Window Family


Definition:

$\displaystyle w(n)=w_R(n) \cos^P\left( \frac{ \pi n}{M} \right) $

where $ P$ is a nonnegative integer.


Properties:

  • The first $ P$ terms of the window's Taylor expansion, evaluated at the endpoints are identically 0.
  • Roll-off rate $ \approx 6(P+1)$ dB/octave.


Special Cases:

  • $ P=0 \Rightarrow$ Rectangular window
  • $ P=1 \Rightarrow$ MLT sine window
  • $ P=2 \Rightarrow$ Hann window (``raised cosine'' = ``$ \cos^2$'')
  • $ P=4 \Rightarrow$ Alternative Blackman (maximized roll-off rate)

Thus, $ \cos^P$ windows parametrize $ L$-term Blackman-Harris windows (for $ L=P/2+1$) which are configured to use all degrees-of-freedom to maximize roll-off rate.

Previous: Frequency-Domain Implementation of the Blackman-Harris Family
Next: Example: Spectrum Analysis of an Oboe Tone

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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