Spectrum Analysis Windows

In *spectrum analysis* of naturally occurring audio signals, we
nearly always analyze a short *segment* of a signal, rather than
the whole signal. This is the case for a variety of reasons.
Perhaps most fundamentally, the *ear*
similarly Fourier analyzes only a short
segment of audio signals at a time (on the order of 10-20 ms worth).
Therefore, to perform a spectrum analysis having time- and
frequency-resolution comparable to human hearing, we must limit
the time-window accordingly.
We will see that the proper way to extract a ``short time segment'' of
length
from a longer signal is to multiply it by a *window
function* such as the *Hann window*:

(4.1) |

We will see that the main benefit of choosing a good Fourier analysis window function is minimization of side lobes, which cause ``cross-talk'' in the estimated spectrum from one frequency to another.

The study of spectrum-analysis windows serves other purposes as well. Most immediately, it provides an array of useful window types which are best for different situations. Second, by studying windows and their Fourier transforms, we build up our knowledge of Fourier dualities in general. Finally, the defining criteria for different window types often involve interesting and useful analytical techniques.

In this chapter, we begin with a summary of the rectangular window, followed by a variety of additional window types, including the generalized Hamming and Blackman-Harris families (sums of cosines), Bartlett (triangular), Poisson (exponential), Kaiser (Bessel), Dolph-Chebyshev, Gaussian, and other window types.

- The Rectangular Window

- Generalized Hamming
Window Family
- Hann or Hanning or Raised Cosine
- Matlab for the Hann Window
- Hamming Window
- Matlab for the Hamming Window
- Summary of Generalized Hamming Windows
- The MLT Sine Window

- Blackman-Harris Window Family
- Blackman Window Family
- Classic Blackman
- Matlab for the Classic Blackman Window
- Three-Term Blackman-Harris Window
- Frequency-Domain Blackman-Harris
- Power-of-Cosine Window Family

- Spectrum Analysis of an Oboe Tone
- Rectangular-Windowed Oboe Recording
- Hamming-Windowed Oboe Recording
- Blackman-Windowed Oboe Recording
- Conclusions

- Bartlett (``Triangular'') Window

- Poisson Window
- Hann-Poisson Window

- Slepian or DPSS Window

- Kaiser Window
- Kaiser Window Beta Parameter
- Kaiser Windows and Transforms
- Minimum Frequency Separation vs. Window Length
- Kaiser and DPSS Windows Compared

- Dolph-Chebyshev Window
- Matlab for the Dolph-Chebyshev Window
- Example Chebyshev Windows and Transforms
- Chebyshev and Hamming Windows Compared
- Dolph-Chebyshev Window Theory

- Gaussian Window and Transform

- Optimized Windows

- Optimal Window Design by
`linprog`- Linear Programming (LP)
- LP Formulation of Chebyshev Window Design
- Symmetric Window Constraint
- Positive Window-Sample Constraint
- DC Constraint
- Sidelobe Specification
- LP Standard Form
- Remez Exchange Algorithm
- Monotonicity Constraint
- L-Infinity Norm of Derivative Objective
- L-One Norm of Derivative Objective
- Summary

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University