### 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

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 to two significant digits gives , the Hamming window can be considered the ``Chebyshev Generalized Hamming Window'' (approximately). Chebyshev-type designs normally exhibit

*equiripple*error behavior, because the worst-case error (side-lobe level in this case) is minimized. However, within the generalized Hamming family, the asymptotic spectral roll-off is constrained to be at least dB per octave due to the form (3.17) of all windows in the family. We'll discuss the true Chebyshev window in §3.10 below; we'll see that it is not monotonic from its midpoint to an endpoint, and that it is in fact impulsive at its endpoints. (To peek ahead at a Chebyshev window and transform, see Fig.3.31.) Generalized Hamming windows can have a step discontinuity at their endpoints, but no impulsive points.

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,^{4.6}which 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