### Loudness Spectrogram Examples

We now illustrate a particular Matlab implementation of a loudness
spectrogram developed by teaching assistant Patty Huang, following
[87,182,88]
with slight modifications.^{8.9}

#### Multiresolution STFT

Figure 7.4 shows a *multiresolution STFT* for the same
speech signal that was analyzed to produce Fig.7.2. The
bandlimits in Hz for the five combined FFTs were
, where the last two (in
parentheses) were not used due to the signal sampling rate being only
kHz. The corresponding window lengths in milliseconds were
, where, again, the last two are not needed for
this example. Our hop size is chosen to be 1 ms, giving 75% overlap
in the highest-frequency channel, and more overlap in lower-frequency
channels. Thus, all frequency channels are *oversampled* along
the time dimension. Since many frequency channels from each FFT will
be combined via smoothing to form the ``excitation pattern'' (see next
section), temporal oversampling is necessary in all channels to avoid
uneven weighting of data in the time domain due to the hop size being
too large for the shortened effective time-domain windows.

#### Excitation Pattern

Figure 7.5 shows the result of converting the MRSTFT
to an *excitation pattern*
[87,182,108].
As mentioned above, this essentially converts the MRSTFT into a better
approximation of an *auditory filter bank* by non-uniformly
resampling the frequency axis using auditory interpolation kernels.

Note that the harmonics are now clearly visible only up to approximately 20 ERBs, and only the first four or five harmonics are visible during voiced segments. During voiced segments, the formant structure is especially clearly visible at about 25 ERBs. Also note that ``pitch pulses'' are visible as very thin, alternating, dark and light vertical stripes above 25 ERBs or so; the dark lines occur just after glottal closure, when the voiced-speech period has a strong peak in the time domain.

#### Nonuniform Spectral Resampling

Recall sinc interpolation of a discrete-time signal [270]:

And recall asinc interpolation of a sampled spectrum (§2.5.2):

We see that resampling consists of an inner-product between the given
samples with a continuous *interpolation kernel* that is sampled
where needed to satisfy the needs of the inner product operation. In
the time domain, our interpolation kernel is a scaled sinc function,
while in the frequency domain, it is an asinc function. The
interpolation kernel can of course be horizontally *scaled* to
alter bandwidth [270], or more generally
*reshaped* to introduce a more general *windowing* in the
opposite domain:

- Width of interpolation kernel (main-lobe width)
1/width-in-other-domain
- Shape of interpolation kernel gain profile (window) in other domain

Getting back to non-uniform resampling of audio spectra, we have that
an auditory-filter frequency-response can be regarded as a
frequency-dependent *interpolation kernel* for nonuniformly
resampling the STFT frequency axis. In other words, an auditory
filter bank may be implemented as a non-uniform resampling of the
uniformly sampled frequency axis provided by an ordinary FFT, using
the auditory filter shape as the interpolation kernel.

When the auditory filters vary systematically with frequency, there
may be an equivalent *non-uniform frequency-warping* followed by
a *uniform sampling* of the (warped) frequency axis. Thus, an
alternative implementation of an auditory filter bank is to apply an
FFT (implementing a uniform filter bank) to a signal having a properly
*prewarped spectrum*, where the warping is designed to approximate
whatever auditory frequency axis is desired. This approach is
discussed further in Appendix E. (See also §2.5.2.)

#### Auditory Filter Shapes

The topic of auditory filter banks was introduced in §7.3.1 above.
In this implementation, the auditory filters were synthesized from the
*Equivalent Rectangular Bandwidth (ERB) frequency scale*, discussed
in §E.5. The auditory filter-bank shapes are a function of
level, so, ideally, the true physical amplitude level of the input
signal at the ear(s) should be known. The auditory filter shape at 1
kHz in response to a sinusoidal excitation for a variety of amplitudes
is shown in Fig.7.6.

#### Specific Loudness

Figure 7.7 shows the *specific loudness* computed from the
excitation pattern of Fig.7.5. As mentioned above, it is a
compressive nonlinearity that depends on level and also frequency.

#### Spectrograms Compared

Figure 7.8 shows all four previous spectrogram figures in a two-by-two matrix for ease of cross-reference.

#### Instantaneous, Short-Term, and Long-Term Loudness

Finally, Fig.7.9 shows the *instantaneous loudness*,
*short-term loudness*, and *long-term loudness* functions
overlaid, for the same speech sample used in the previous plots.
These are all single-valued functions of time which indicate the
relative loudness of the signal on different time scales. See
[88] for further discussion. While the lower
plot looks reasonable, the upper plot (in sones) predicts only three
audible time regions. Evidently, it corresponds to a very low
listening level.^{8.10}

The instantaneous loudness is simply the sum of the specific loudness
over all frequencies. The short- and long-term loudnesses are derived
by smoothing the instantaneous loudness with respect to time using
various psychoacoustically motivated time constants
[88]. The smoothing is nonlinear because the
loudness tracks a rising amplitude very quickly, while decaying with a
slower time constant.^{8.11} The loudness of a brief sound is
taken to be the local maximum of the short-term loudness curve. The
long-term loudness is related to loudness memory over time.

The upper plot gives loudness in *sones*, which is based on
loudness perception experiments [276]; at 1 kHz and
above, loudness perception is approximately logarithmic above 50 dB
SPL or so, while below that, it tends toward being more linear. The
lower plot is given in *phons*, which is simply sound pressure
level (SPL) in dB at 1 kHz [276, p. 111]; at other
frequencies, the amplitude in phons is defined by following an
``equal-loudness curve'' over to 1 kHz and reading off the level there
in dB SPL. This means, *e.g.*, that all pure tones have the same
perceived loudness when they are at the same phon level, and the dB
SPL at 1 kHz defines the loudness of such tones in phons.

**Next Section:**

Cyclic FFT Convolution

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Loudness Spectrogram