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.

Raw spectral phase and its interpolation
Unwrapped spectral phase and its interpolation
Canonically unwrapped spectral phase and its interpolation |

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University