of Microprocessors

SOURCE-SIGNAL ANALYSIS 571
component is actually being seen (as opposed to random noise), then all of
the frequency measures will yield the same value. A high spectral sample rate
is necessary because otherwise the phase shift for a distant band such as 1+2
in the diagram may exceed 180 0
per frame and give a false result. For the
Hamming window, an analysis overlap factor of four or more would give a
sufficiently high spectral sample rate.
Another constraint is that frequency resolution must be high enough so
that every component of the signal is completely resolved. If the components
are not completely separated, serious errors in frequency and amplitude
measurement are likely. For a Hamming window, this means that frequency
components can be no closer than four times the reciprocal of the analysis
record duration. In our analysis example (20-ks/s Fs, 512-sample analysis
record), this evaluates to about 150 Hz. The only reasonable way to improve
frequency resolution is to use longer analysis records, which, of course,
degrade time resolution. Alternatively, if all of the frequency components are
nearly equal in amplitude, a rectangular window having one-half the
bandwidth of the Hamming window can be considered.
Once the frequency of a component has been determined, its amplitude
must be calculated to complete the analysis. The simplest method is to note
the band with the greatest response to the component and use its response
amplitude. The error incurred when doing this is small enough (1.5 dB
maximum for a Hamming window) that it can be ignored in many applications.
For greater accuracy, the curved top of the equivalent bandpass filter
can be considered and a correction factor derived based on the difference
between the dominant band's center frequency and the actual signal frequency.
One could also rms sum (square root of the sum of squares) the
responding bands to get a single amplitude measurement for the component.
Of course, all of the analysis bands will show some response due to noise or
leakage. However, only those with sufficiently high outputs and reasonable
frequency correlation among adjacent bands should actually be considered as
detecting a valid signal component.
Spectral Shape Anal)'sis
Many natural sounds can be modeled as an oscillator driving a filter as
in Fig. 16-14. The oscillator's waveform is called the excitation function and is
normally rich in harmonics. The filter is called the system function and is
typically rather complex having several resonant peaks and possibly some
notches as well. The spectrum of the output sound is the point-by-point
product of the excitation function ~pectrum and the amplitude response of
the filter as in Fig. 16-14B. In musical applications, the frequency of the
excitation function is the primary variable, since it determines the pitch of
the resulting tone. The waveform may also change some, typically acquiring
additional upper harmonic amplitude as its overall amplitude increases. Al-