Summary of Generalized Hamming Windows

Definition:

$\displaystyle w_H(n) = w_R(n) \left[ \alpha + 2 \beta \cos \left( \Omega_M n \right) \right], \quad n \in {\bf Z}, \; \Omega_M \isdef \frac{2 \pi}{M}$ (4.20)

where

$\displaystyle w_R(n) \isdefs \left\{\begin{array}{ll} 1, & \left\vert n\right\vert\leq\frac{M-1}{2} \\ [5pt] 0, & \hbox{otherwise} \\ \end{array} \right.$ (4.21)

Transform:

$\displaystyle W_H( \omega ) \isdefs \alpha W_R( \omega ) + \beta W_R( \omega - \Omega_M ) + \beta W_R( \omega + \Omega_M ), \quad \omega\in[-\pi,\pi)$ (4.22)

where

$\displaystyle W_R(\omega) = M\cdot \hbox{asinc}_M(\omega) \isdefs \frac{\sin\left(M\frac{\omega}{2}\right)}{\sin\left(\frac{\omega}{2}\right)}$ (4.23)

Common Properties

  • Rectangular + scaled-cosine window
  • Cosine has one period across the window
  • Symmetric ( $ \Rightarrow$ zero or linear phase)
  • Positive (by convention on $ \alpha $ and $ \beta $ )
  • Main lobe is $ 4\Omega_M$ radians per sample wide, where $ \Omega_M\isdef 2\pi/M$
  • Zero-crossings (``notches'') in window transform at intervals of $ \Omega_M$ outside of main lobe

Figure 3.12 compares the window transforms for the rectangular, Hann, and Hamming windows. Note how the Hann window has the fastest roll-off while the Hamming window is closest to being equal-ripple. The rectangular window has the narrowest main lobe.

Figure 3.12: Comparison of window transforms for the rectangular, Hann, and Hamming windows.
\includegraphics[width=\twidth]{eps/RectHannHamm}

Rectangular window properties:

  • Abrupt transition from 1 to 0 at the window endpoints
  • Roll-off is asymptotically $ -6$ dB per octave (as $ T\rightarrow 0$ )
  • First side lobe is $ -13$ dB relative to main-lobe peak

Hann window properties:

  • Smooth transition to zero at window endpoints
  • Roll-off is asymptotically -18 dB per octave
  • First side lobe is $ -31$ dB relative to main-lobe peak

Hamming window properties:

  • Discontinuous ``slam to zero'' at endpoints
  • Roll-off is asymptotically -6 dB per octave
  • Side lobes are closer to ``equal ripple''
  • First side lobe is $ 41$ dB down = $ 10$ dB better than Hann4.7


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The MLT Sine Window
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Matlab for the Hamming Window