Summary of Generalized Hamming Windows

Definition:

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

where

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

Transform:

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

where

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

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 Hann^4.7