Hamming Window
The Hamming window is determined by choosing
in
(3.17) (with
) to cancel the largest side lobe
[101].4.4 Doing this results in the values
![\begin{eqnarray*}
\alpha &=& \frac{25}{46} \approx 0.54\\ [5pt]
\beta &=& \frac{1-\alpha}{2} \approx 0.23.
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img375.png)
The peak side-lobe level is approximately
dB for the Hamming
window [101].4.5 It happens that this
choice is very close to that which minimizes peak side-lobe level
(down to
dB--the lowest possible within the generalized
Hamming family) [196]:
![]() |
(4.19) |
Since rounding the optimal



The Hamming window and its DTFT magnitude are shown in
Fig.3.10. Like the Hann window, the Hamming window is
also one period of a raised cosine. However, the cosine is raised so
high that its negative peaks are above zero, and the window has
a discontinuity in amplitude at its endpoints (stepping
discontinuously from 0.08 to 0). This makes the side-lobe roll-off
rate very slow (asymptotically
dB/octave). On the other hand,
the worst-case side lobe plummets to
dB,4.6which is the purpose of the Hamming window. This is 10 dB better than
the Hann case of Fig.3.9 and 28 dB better than the
rectangular window. The main lobe is approximately
wide,
as is the case for all members of the generalized Hamming family
(
).
Due to the step discontinuity at the window boundaries, we expect a
spectral envelope which is an aliased version of a
dB per octave
(i.e., a
roll-off is converted to a ``cosecant roll-off'' by
aliasing, as derived in §3.1 and illustrated in
Fig.3.6). However, for the Hamming window, the
side-lobes nearest the main lobe have been strongly shaped by the
optimization. As a result, the nearly
dB per octave roll-off
occurs only over an interior interval of the spectrum, well between
the main lobe and half the sampling rate. This is easier to see for a
larger
, as shown in
Fig.3.11, since then the optimized side-lobes nearest
the main lobe occupy a smaller frequency interval about the main
lobe.
Since the Hamming window side-lobe level is more than 40 dB down, it
is often a good choice for ``1% accurate systems,'' such as 8-bit
audio signal processing systems. This is because there is rarely any
reason to require the window side lobes to lie far below the signal
quantization noise floor. The Hamming window has been extensively
used in telephone communications signal processing wherein 8-bit
CODECs were standard for many decades (albeit
-law encoded).
For higher quality audio signal processing, higher quality windows may
be required, particularly when those windows act as lowpass filters
(as developed in Chapter 9).
Next Section:
Matlab for the Hamming Window
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Matlab for the Hann Window