### Additive Synthesis Analysis

In order to reproduce a given signal, we must first *analyze* it
to determine the amplitude and frequency *trajectories* for each
sinusoidal component. We do not need phase information (
in (10.18)) during steady-state segments, since phase is normally
not perceived in steady state tones [293,211].
However, we do need phase information for analysis frames containing an
*attack transient*, or any other abrupt change in the signal.
The phase of the sinusoidal peaks controls the position of time-domain
features of the waveform within the analysis frame.

#### Following Spectral Peaks

In the analysis phase, sinusoidal peaks are measured over time in a sequence of FFTs, and these peaks are grouped into ``tracks'' across time. A detailed discussion of various options for this can be found in [246,174,271,84,248,223,10,146], and a particular case is detailed in Appendix H.

The end result of the analysis pass is a collection of amplitude and
frequency envelopes for each spectral peak versus time. If the time
advance from one FFT to the next is fixed (5ms is a typical choice for
speech analysis), then we obtain *uniformly sampled amplitude and
frequency trajectories* as the result of the analysis. The sampling
rate of these amplitude and frequency envelopes is equal to
the *frame rate* of the analysis. (If the time advance between
FFTs is
ms, then the frame rate is defined as
Hz.) For resynthesis using inverse FFTs, these data may be
used unmodified. For resynthesis using a bank of sinusoidal
oscillators, on the other hand, we must somehow
*interpolate* the envelopes to create envelopes at the signal
sampling rate (typically
kHz or higher).

It is typical in computer music to *linearly interpolate* the
amplitude and frequency trajectories from one frame to the next
[271].^{11.10} Let's call the piecewise-linear upsampled envelopes
and
, defined now for all
at the normal signal
sampling rate. For steady-state tonal sounds, the phase may be
discarded at this stage and redefined as the integral of the
instantaneous frequency when needed:

When phase must be matched in a given frame, such as when it is known to contain a transient event, the frequency can instead move quadratically across the frame to provide

*cubic phase interpolation*[174], or a second linear breakpoint can be introduced somewhere in the frame for the frequency trajectory (in which case the area under the triangle formed by the second breakpoint equals the added phase at the end of the segment).

#### Sinusoidal Peak Finding

For each sinusoidal component of a signal, we need to determine its
frequency, amplitude, and phase (when needed). As a starting point,
consider the windowed complex sinusoid with *complex* amplitude
and frequency
:

(11.20) |

As discussed in Chapter 5, the transform (DTFT) of this windowed signal is the convolution of a frequency domain delta function at [ ], and the transform of the window function, , resulting in a shifted version of the window transform . Assuming is odd, we can show this as follows:

Hence,

At , we have

If we scale the window to have a dc gain of 1, then the peak magnitude
equals the amplitude of the sinusoid, *i.e.*,
, as shown in Fig.10.8.

If we use a zero-phase (even) window, the phase at the peak equals the
phase of the sinusoid, *i.e.*,
.

#### Tracking Sinusoidal Peaks in a Sequence of FFTs

The preceding discussion focused on estimating sinusoidal peaks in a
*single frame* of data. For estimating sinusoidal parameter
*trajectories* through time, it is necessary to *associate
peaks* from one frame to the next. For example, Fig.10.9
illustrates a set of *frequency trajectories*, including one with
a missing segment due to its peak not being detected in the third
frame.

Figure 10.10 depicts a basic analysis system for tracking spectral
peaks in the STFT [271]. The system tracks peak
amplitude, center-frequency, and sometimes phase. Quadratic
interpolation is used to accurately find spectral magnitude peaks
(§5.7). For further analysis details, see Appendix H.
Synthesis is performed using a bank of amplitude- and phase-modulated
oscillators, as shown in Fig.10.7. Alternatively, the
sinusoids are synthesized using an *inverse FFT*
[239,94,139].

**Next Section:**

Sines + Noise Modeling

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Additive Synthesis (Early Sinusoidal Modeling)