of Microprocessors

SOURCE-SIGNAL ANALYSIS 569
from an amplitude-frequency "source" form into an amplitude-phase and
ultimately sine-cosine "objeer" form that was compatible with the FFT.
However, most of the spectrum modifications that have just been discussed
can be performed directly on the sine-cosine or amplitude-frequency form of
the spectrum. Resynthesis should therefore be simplified over the method
given in Chapter 13.
One desirable property of an analysis-synthesis system is transparency,
that is, no signal distortion in the absence spectral modification. Distortionless
reconstruction can, in fact, be done very simply using unmodified spectral
data from the analysis. First, each spectral frame is inverse transformed to
recover the windowed sample record that created the frame. Next, the window
is divided out giving the original sequence of samples. Finally, the
overlap is removed to obtain the original continuous stream of samples. Any
difference between the original and reconstructed data is due to roundoff
error in the arithmetic, most of it probably from the window removal (division
by small numbers at the edges of the window) step.
However, if modifications are made, two things are likely to happen.
First, the inverse-transformed modified records cannot be expected to butt
together nicely and continuously like unmodified records do. Second, the
inverse transform of a modified frame may show little if any evidence of a
window because the precise amplitude and phase relations necessary for the
window-shaped envelope will have been altered. Inverse windowing, therefore,
is likely to distort the record and actually emphasize discontinuities
between the records.
Thus, the problem of eliminating interrecord discontinuities is similar
to that encountered in FFT synthesis from scratch. The same solution, that
is, windowing and synthesis overlap, used for direct synthesis is reasonably
adequate for resynthesis. The disadvantage of applying this method is that an
unmodified spectrum will no longer provide precisely the original data, since
the resynthesis manipulation itself amounts to a modification. Methods are
available to reduce resynthesis error to zero but they ar.e complex and amount
to essentially infinite overlapping (overlap factor=record length) of the synthesis.
As mentioned in Chapter 13, a four-to-one synthesis overlap provides
results clearly superior to two-to-one overlap and therefore is preferred in
conjunction with four-to-one analysis or careful time interpolation of twoto-one
analysis data.
Parameter Extraction
Instead of immediate resynthesis, one may wish to further analyze the
spectral data in an effort to extract one or more of the fundamental parameters
of sound. These parameters may then be used to direct a conventional
synthesis process or may be stored for later recall with the advantage that a
handful of fundamental parameters can often replace hundreds of spectral
components without loss of information significant to the application.