A Quadrature Signals Tutorial: Complex, But Not Complicated

Understanding the 'Phasing Method' of Single Sideband Demodulation

Complex Digital Signal Processing in Telecommunications

Introduction to Sound Processing

Introduction of C Programming for DSP Applications

Spectrum Analysis Windows

Generalized Hamming Window Family

Hamming Window

**Search Spectral Audio Signal Processing**

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The *Hamming window* is determined by choosing in
(3.1) (with
) to cancel the largest side lobe
[97].^{4.1} Doing this results in the values

The peak side-lobe level is approximately dB for the Hamming
window [97].^{4.2} 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) [185]:

The Hamming window and its DTFT magnitude are shown in
Fig.3.4. 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* leaving the window (stepping
discontinuously from 0.08 to 0). This makes the side-lobe roll-off
rate very slow. On the other hand, the worst-case side lobe plummets
to dB,^{4.3} which is the purpose of the Hamming
window. This is 10 dB better than the Hann case of
Fig.3.3 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 §4.5 and illustrated in
Fig.4.12). 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.3.5, 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 have been standard for many years (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 8).

Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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