of Microprocessors

564 MUSICAL ApPLICATIONS OF MICROPROCESSORS
Although the preceding windows are continuous with zero at both
ends, there is a slope discontinuity at the ends. One window that provides for
continuity of slope (and continuity for all derivatives as well) is the cosine be!!
or hanning window. Actually, it is just the point-by-point square of the
half-sine window but, due to a trig identity, is the same shape as a full cosine
cycle shifted up to the axis and turned upside down. The response curve in
Fig. 16-11D exhibits a double width central lobe and fitst sidelobe of - 32
dB, which is still unacceptable, although the second and succeeding ones are
fine.
The last window shown is called a Hamming window and consists of a
very judicious combination of a hanning and a rectangular window. Essentially
the relative contributions are scaled so that the first sidelobes cancel
and the rest partially cancel. The result is a curve with all sidelobes at -42
dB or less, which is accomplished with a central lobe only twice as wide as
the rectangular window. For most audio spectral/analysis work, this is the
optimum window. For specialized applications such as signal detection in the
presence of overpowering noise, other windows with sidelobes of -80 dB
and better are available at the expense of an even wider central lobe.
Performing the Analysis
At this point, we are ready to outline the procedure necessary to go
from an indefinite-length sample string to a sequence ofspectral frames using
the FFT. For illustration purposes, the example ofa 20-ks/s sample rate, 512­
point FFT, 39-Hzfrequency resolution, and 78-s/s spectral sample rate will
be continued. The crux of the analysis procedure is how the continuous
stream ofsamples will be broken up into 512 sampLe records for the FFT. One
possibility is to simply take the 512 samples from the input, transform them,
take the next 512, transform, etc. If this is done, windowing of the records
may greatly attenuate or miss significant waveform detaiLs that occur when
the window amplitude is near zero. Furthermore, only 39 spectrums wilL be
computed per 20,000 samples (l sec).
Overlap can be used to insure that all parts of the waveform are seen and
increase the spectral sample rate as well. The general idea is the exact inverse
of the method outlined in Chapter 13 in which direct FFT synthesis was
discussed. For two-to-one overlap, one would only take 256 samples from the
string for each spectral frame. The other 256 in the record would be left over
from the previous frame. The process can be likened to a 512-sample shift
register. Each frame time 256 new samples would be shifted in and the oLdest
256 wouLd be shifted out and thrown away. Other overlap factors (which
need not pe integers) for both higher- and lower-spectrum sample rates are
possible simply by aLtering the number of samples shifted in.
There are a couple of complications in the computation that can lead to
a lot of partially redundant data arrays. When a window is applied to