As for the phase of the
spectrum, what do we expect? We have chosen
the
sinusoid phase offset to be zero. The window is
causal and
symmetric about its middle. Therefore, we expect a
linear phase term
with slope

samples (as discussed in connection with the
shift theorem in §
7.4.4).
Also, the window transform has
sidelobes which cause a phase of

radians to switch in and out. Thus, we expect to see samples of a
straight line (with slope

samples) across the
main lobe of the
window transform, together with a switching offset by

in every
other sidelobe away from the main lobe, starting with the immediately
adjacent sidelobes.

In Fig.
8.9(a), we can see the negatively sloped line
across the main lobe of the window transform, but the sidelobes are
hard to follow. Even the unwrapped phase in Fig.
8.9(b)
is not as clear as it could be. This is because a phase jump of

radians and

radians are equally valid, as is any odd multiple
of

radians. In the case of the unwrapped phase, all phase jumps
are by

starting near frequency

.
Figure
8.9(c) shows what could be
considered the ``canonical'' unwrapped phase for this example: We see
a
linear phase segment across the main lobe as before, and outside the
main lobe, we have a continuation of that linear phase across all of
the positive sidelobes, and only a

-radian deviation from that
linear phase across the negative sidelobes. In other words, we see a
straight linear phase at the desired slope interrupted by temporary
jumps of

radians. To obtain unwrapped phase of this type, the
unwrap function needs to alternate the sign of successive
phase-jumps by

radians; this could be implemented, for example,
by detecting jumps-by-

to within some numerical tolerance and
using a bit of state to enforce alternation of

with

.
To convert the expected phase slope from

``radians per
(rad/sec)'' to ``radians per cycle-per-sample,'' we need to multiply
by ``radians per cycle,'' or

. Thus, in
Fig.
8.9(c), we expect a slope of

radians
per unit normalized frequency, or

radians per

cycles-per-sample, and this looks about right, judging from the plot.
Figure 8.9:
Spectral phase and two different phase unwrappings.
Raw spectral phase and its interpolation
Unwrapped spectral phase and its interpolation
Canonically unwrapped spectral phase and its interpolation
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