McAulayQuatieri
Method
Typical
sort time Fourier analysis involves windowing pieces of a continuous
time signal and computing its Continuous Time Fourier Transforms
to get a local in time frequency composition.
Windows must be chosen that minimizes the amount of included
frequency content from outside the window in time (sidelobe interference).
Manipulating the information involves standard methods (lowpass
filters, etc). Synthesis
is accomplished by computing Inverse Fourier Transforms of each
section and special additive techniques are utilized to recombine
the altered windows.
In
1986 Robert McAulay and Thomas Quatieri proposed a new method of
analysis/synthesis for continuous time speech signals which attempted
to develop a reconstruction process that would result in an “as
close as possible” approximation of the original
signal.
They modeled speech signals as two components.
The first was an excitation signal which consisted of a sum
of sinusoids with timevarying amplitudes and frequencies, as well
as an initial phase offset. The second component is the voice track
which is modeled as a timevariant filter with timevarying magnitudes
and phase. These two
components are combined and expressed as:
Equation
1
To
find expressions for these sinusoids they derived a new technique
to analyze the signal. Using
overlapping windowing methods similar to standard short time analysis,
the MQ method computes Fourier transforms of the individual windows.
The peak frequencies of each window (the partials) are found
and their amplitudes and phases are extracted.
The partials for each window are linked to those in the following
window in order to develop a trend in the progression of frequencies
(their amplitude and phases).
We call each progression a track.
The birth of a track occurs when there does not exist a partial
in the previous window with which to connect one in the current
window. Conversely,
a death track occurs when a partial does not exist in the following
window with which to connect one in the current window (Figure 1).
Figure
1
To
generate the sign waves each point is connected smoothly by interpolating
between them using a cubic function.
This gives us continuous functions that describe the progression
of the amplitude, phase, and frequency of our signal.
We use these to construct a sign wave and then weighted by
a triangle window with a width of twice that of the window.
The
MQ Model has outstanding results and reproduces inaudibly different
signals when applied to a wide variety of quasiharmonic sounds.
Perhaps its greatest advantage is the small amount of data
required to perform this process.
To reproduce a signal using standard Fourier techniques information
about a great many coefficients must be retained.
To reconstruct perfectly an infinite number must be used.
With the MQ method information about several time varying
sinusoids must be stored, and little else.
One
of the flaws in the MQ method is how it represents noise.
Noise shows up as tracks that span only a small number of
windows. It is difficult
to represent these short tracks using sinusoids so other methods
must be developed (see section entitled BandwidthEnhancement).
