of Microprocessors

550 MUSICAL ApPLICATIONS OF MICROPROCESSORS
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Fig. 16-3. Illustration of time-frequency resolution limitation. (A) Actual synthesized
tones. (8) Narrowband spectral analysis. (C) Wideband
spectral analysis.
pitch. Note that onset and termination of the major features is somewhat
smeared. The second spectrogram is a wideband analysis with an analysis
bandwidth of approximately 300 Hz. Here the time resolution is so good
(approximately 3 msec) that individual cycles of the fundamental frequency
are resolved, which causes the vertical bands. However, only the formants
show up; the harmonics are too closely spaced relative to 300 Hz to be
resolved.
Returning to the array of rectangles that makes up a digital sound
spectrogram, one can consider a narrowband analysis to correspond to rectangles
that are wide and short, while a wideband analysis uses narrow and tall
rectangles. The area covered by a rectangle remains constant and is equal to
approximately unity (hertz times time). The actual surface area covered on
the plot depends on the scale factor chosen for frequency and time axes. An
analog spectrogram is subject to the same limitation except that the rectangles
become somewhat diffuse, overlapping ellipses. By using interpolation
in two dimensions, a digital spectrogram can be made to appear identical to
an analog one.
Although it would seem that the narrowband and wideband plots could
be combined into a single figure with both good frequency and time resolution,
it is not possible to do so unambiguously. This is illustrated in Fig.
16-3 in which two nearly simultaneous tone bursts are analyzed. The narrowband
analysis easily separates the two tones but smears the leading and
trailing edges such that it is impossible to tell which occurred first. The
wideband analysis has very sharp leading and trailing edges, but now the two
tones are merged together so it is still not possible to say which came first.