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**TFTS'97 (IEEE Time-Frequency and Time-Scale Workshop 97), Coventry, Grande Bretagne, août 1997**

Copyright © IEEE 1997

In consequence, it is not surprising that additive analysis and synthesis of musical signals have recently received a great deal of attention. Even though the, now classical, additive sinusoidal analysis is based on rather simple principles, it has been very successful. The first goal of this paper is to examine this method, the reasons of its success and its weaknesses. The second goal is to try to extrapolate from our conclusions in order to propose new reasearch directions for musical signals analysis.

In section 2, we present additive sinusoidal analysis. The main drawbacks of
the classical method are then explained in section 3 and some important
improvements are briefly exposed. In section 4 we try to understand the
advantages of additive analysis as well as its limitations. In section 5, under
the term *Elementary Waveforms*, we present some new directions better
suited for the analysis of musical sound signals.

With a

cA first important assumption underlying (often implicitly) sinusoidal models is that c_{i }(t)=a_{i }(t)cos(_{i}(t))

That

Speech and musical sounds always have random components, often heard as a noise superposed for instance on the harmonic part. A second assumption often underlying a sinusoidal model is that the number I of sinusoidal partials is limited. Therefore, a purely sinusoidal model with slowly varying parameters can hardly represent all of a real signal x(t) and needs to be completed with a non-sinusoidal residual part r(t):

r(t) = x(t) - s(t)Another argument in favor of a non-sinusoidal residual part r(t) is that the residual should be considered as a random signal in case of transformations such as time compression or expansion. In consequence, classical representations of random signals are better suited for the residual. It is common to represent only the short-time magnitude frequency content of the residual by a spectral envelope G(t,) [Serra 89]. If n(t) is a white gaussian noise and G(t,) is the Fourier Transform of a time-varying impulse response g([theta],t), then the model of the residual is:

This filtering can be implemented in the time domain or in the frequency domain. If R(,t) and N(,t) are the Short-Time Fourier Transforms of r(t) and n(t) respectively, then:

Fig. 1. Peaks found in successive analysis frames are grouped into tracks,i.e. sinusoidal partials.

Let x[n] be the analysed discrete signal, h[n] the analysis window and
X(n,_{k}) its STFT at time n and frequency
_{k}:

, (1)Since we look at the STFT of a given frame around time n, let us drop index n and write X(

|X(A peak at index supposedly indicates the presence of a sinusoidal partial at a near-by frequency (_{-1})|<|X(_{})|>|X(_{+1})|

The relatively simple detection and estimation procedure described here above has weaknesses which are detailed in section 3. However, it should be underlined that it has known a great success because it is fast and very robust. It can deal with the hundreds of sinusoidal partials encountered in musical signals and does not suffer from model-order determination and numerical or computational limitations which are common, for example, in parametric methods [Laroche & Rodet 89b], [Laroche 93a].

The second step of sinusoidal parameter estimation is the grouping of
successive analysis frame peaks into tracks. This tracking is usually based on
a heuristic approach [McAulay & Quatieri 86], [Serra 89] which matches
peaks in successive frames while allowing *deaths* and *births* of
tracks. It simply grows trajectories, iteratively frame after frame, in the
direction of increasing time: for each frame successively, the tracks which
still appear in the current frame are possibly continued provided there is a
convenient peak in the next frame according to an optimal frequency match. When
needed, some tracks are terminated and new tracks arise. We will not detail
this algorithm here since we prefer a well grounded statistical approach
presented in section 3.4.

Estimation of the spectral envelope of the residual signal around time t,
G(,t), can be done with any usual AR estimation technique [Kay 88]. Such
a technique provides the P coefficients a_{p}(t) of an all-pole filter
with magnitude transfer function G(,t). The coefficients
a_{p}(t) are well suited for time-domain filtering of a noise n(t) at
the synthesis stage. In practice, the a_{p}(t) are only estimated
around successive times t_{l} ,l=1, 2, 3,..., with a step
t_{l+1}-t_{l} on the order of 5 to 20 milliseconds. Cepstral
estimation can also be used on sliding window STFT R(,t) of r(t).
Cepstral estimation provides P cepstral coefficients c_{p}(t) which are
well suited for frequency domain filtering of a noise n(t) at the synthesis
stage. When estimating the spectral envelope G(,t), a nonlinear
frequency scale, such as the *Mel* or the *Bark* scale, is appealing
since is reflects some properties of human perception. Some authors [Serra 89],
[Goodwin 96] have proposed to simply represent the magnitude short-time
spectrum |R(,t)| by its mean value in channels distributed on such a
nonlinear scale. This representation also is well suited for frequency domain
filtering at the synthesis stage, since it requires only a mutiplication of the
STFT of the noise n(t) by |R(,t)|.

Each maximum |()| indicates a sinusoidal partial candidate at frequency . An estimation of the amplitude A and of the phase of the partial can then be derived. Defining at a norm for H and X by:

and

we obtain:

Note that this computation also provides a measure v

A sinusoidal likeness measure (SLM) v

In [Laroche 89a] a time-domain method is developped to estimate complex
amplitudes (i. e. real amplitudes and phase deviations), when mean frequency
values are known. Note that phase deviation is then equivallent to frequency
variation. The model of the complex amplitude of the i^{th} sinusoidal
partial is a low order (e.g. 3) polynomial of time n:

The model [n] of the signal is a sum of I sinusoidal partials. With z

, (2)Let be the column vector =([1],[2],...,[N])

Then we can write equation (2):

=Minimisation ofB

leads to:

The method has been applied successfully by [Laroche 89a] to find the amplitude and phase variations of sinusoidal partials of percussive sound signals, after the mean frequencies were found by Prony's method. This simultaneous determination of amplitude and frequency variations would be a useful complement of the classical sinusoidal analysis where mean frequency values are estimated on spectral peaks.B=[^{h}]^{-1}^{h}

Naturally, sinusoidal partial parameter evolution also appears in the STFT,
leading to a distorsion of peaks from the pure-sinusoid peak shape as
mentionned in section 3.1. The distorsion caused by linear frequency modulation
(LFM) and exponential amplitude modulation (EAM), in a given STFT
X(_{k})=X(n,_{k}), is examined in [Masri 96]. In
the case of a pure sinusoid, the phase of the FFT bins,
Arg{X(_{k})}, in the vicinity of the corresponding peak, is
constant. In the case of modulation, this phase shows a variation
(_{r}) which is a function of the frequency
_{r} relative to the peak center. Experimental measures show
that LFM, up to 16 bins of modulation per frame, causes a variation
_{f}(_{r}) which is increasing with
_{r} for small |_{r}|. Similarly, EAM, up to 6 dB
of modulation per frame, causes a variation
_{a}(_{r}) which is proportional to
_{r} for small |_{r}|. Therefore, when only one
type of modulation appears and is not too large, it can be estimated from the
phase spectrum. Furthermore, for small |_{r}|, the phase
variations due to LFM and EAM are additive. But, since
_{f}(_{r}) has even symmetry and
_{a}(_{r}) has odd symmetry, their cumulative
effect produces a global variation
_{f}(_{r})+_{a}(_{r})
with distinctive shapes according to the slope signs of
_{f}(_{r}) and
_{a}(_{r}). Therefore, simultaneous LFM and EAM
can be estimated from the phase spectrum. The method has been successfully
applied on simulated audio signals. However, for real musical signals, where
additive interference from the more prominent peaks affects the phase profile
accross less prominent neighbouring peaks, the described implementation is not
resilient enough [Masri 96]. But it seems that the extracted slope information
could be profitable at the partial tracking stage.

L > q/fwhere L is in seconds, q depends on the window main lobe width and is on the order of 3.5. Let us take for example a harmonic sound with a fundamental frequency of 110 Hz (which is heard as the note A2). Its sinusoidal partials are 110 Hz apart and L should be greater than 3.5/110, i.e. 32 ms. In polyphonic sound signals, partials can be even much closer and the length L should be accordingly larger. Note that the minimum frequency distance between sinusoidal partials is often unknown. A large window is a great inconvenience when sinusoidal parameters vary substantially over a time segment L. In particular, fast transitions such as consonnants or percussive attacks are smoothed. The problem is even worse for sinusoidal partials the frequency of which varies substantially. As an example, if the fundamental frequency varies by d Hertz on the window length L, then the i

The method presented in [Depalle & Tromp 96] and [Depalle & Hélie 97] remedies these difficulties by using a parametric model of the STFT of the signal and by allowing the window length to be as short as 2/f.Let us still write X(

Therefore, the model of the Fourier Transform is:

where H() is the Fourier Transform of the analysis window h(n). The method consists in identifying the parameters for which the model best fits the observation X(

On the contrary, the procedure described in [Depalle & al. 93a], [Depalle & al. 93b], copes with these problems by globally optimizing the set of tracks. The peak tracking problem is formulated in terms of a Hidden Markov Model (HMM) [Rabiner 86]. The optimization is performed in a given time interval T according to a statistical criterion of slope continuity for all the sinusoidal parameters. Therefore, the optimal set of trajectories is found as the highest probability state sequence, by means of the Viterbi algorithm [Rabiner 86]. Note that the use of parameter slopes rather than parameter values, while being consistant with the first assumption of a sinusoidal model (see section 2.1), enables one to track time-varying partials as easily as constant ones, and solves the problem of detecting crossing trajectories.

We shall only indicate here a few features of the algorithm. Since the number
of tracks can be in the hundreds, the biggest problem is to reduce
computational complexity. Therefore, the Viterbi algorithm is applied on a
window length of T frames, which slides frame by frame, and some constraints on
index combinations, maximum number of tracks, etc., are added. Furthermore, the
algorithm considers only the possible combinations of peaks between successive
frames. Sinusoidal parameters are used to compute state transition
probabilities [Depalle & al. 93] which favour slope continuity and disfavour spurious peaks. At time m, there are h_{m }peaks P_{m}[i], 1 < i h_{m}. Each track is labelled by an
index greater than zero. The problem is to associate an index D_{m}[i],
1 < i h_{m,} to each peak P_{m}[i]_{.} When a
peak P_{m}[i] is considered as a spurious one, it is associated with a
null index D_{m}[i] = 0. A state S_{m} is defined by an ordered
pair of vectors (D_{m-1}, D_{m}) and the observation is defined
by an ordered pair of integers (h_{m-1}, h_{m). }The optimal
sequence of states S_{m}=(D_{m-1}, D_{m}) is found by
means of the Viterbi algorithm, which maximises the joint probability of state
and observation sequences leading to a globally optimal solution. Then the
tracks are defined by the sequence of vectors D_{m} from the state
sequence.

This algorithm has been implemented at IRCAM by G. Garcia. Other computational cost reductions have been applied. In particular, the Viterbi algorithm has been replaced by a more efficient one taking advantage of the factorised structure of transition probabilities and eliminating computational redundancy. IRCAM's HMM tracking algorithm has been successfully used for sound analysis, processing and synthesis for research and for musical creation. As an example, it is possible to analyse polyphonic music comprising simultaneously several instruments, chords and percussion sounds. Examples will be played at the conference.

However, there are sound signals for which sinusoidal analysis does not seem
so well adapted, typically signals where excitation departs from periodicity.
Curiously enough, sinusoidal analysis is also used for speech signals even
though they often fall in the last category. The classical model of glottal
speech production [Fant 70] consists of short pulses filtered through the vocal
tract. Firstly, variations of vocal tract transfer function can be appreciable
at the time scale of three glottal periods. Secondly, time locations of pulses
can be far from periodical. We already noted in section 3.3 that high rank
partials cause difficulties even for small fundamental variations. Figure 2
shows another case, i.e. a speech waveform resulting from irregular pulses, as
often occurs at the end of a sentence (it is sometimes called *vocal
fry*). Sinusoids make sens when, in a given frequency band, a waveform
repeats periodically at least three times. On the contrary, signals like the
one in figure 2 suggest the use of other methods based on waveforms better
localised in time when needed, sometimes called Elementary Waveforms (see
section 5).

Finally, the standard noise source and filter model represents non-sinusoidal and random components in a very unsatisfactory way [Goodwin 96]. Moreover, the fact that two totally different analysis techniques are needed is a weakness which leads to difficulties since the separation of sinusoidal and random components is not based on any solid grounding.

Figure 2. A speech waveform resulting from irregular pulses, as often occurs at the end of a sentence.

In [Liénard 87], a narrow band-pass filter bank is used to ensure locality in frequency. The signal at the output of each filter is segmented at successive minima of its amplitude envelope. Each segment is considered as an EW. The method has been used for speech analysis and synthesis.

Note that sinusoidal analysis starts with a STFT at arbitrary regularly spaced times, then look for specific peak patterns in the STFT. On the other hand, some EW analyses start with arbitrary regularly spaced band-pass filtering, then look for specific patterns in filter outputs. Matching Pursuit, presented here below, does not favour time or frequency but, at each step, looks for a time and a frequency elementary waveform position, as well as a scale, which are optimal according to the properties of the signal under analysis.

Pursuit algorithms, such as Matching Pursuit (MP) [Mallat & Zhang 93] or
Basis Pursuit [Chen & Donoho 95] have been designed to overcome these
difficulties. The decomposition vectors are selected among a *redundant*
family, called a *dictionary*, of EWs well localised in frequency and
time. In MP, the EWs which constitute the dictionary have three parameters, a
scale factor s, a time position u and a modulation frequency (note
that, unlike in Wavelets, scale and frequency *are* independant). With
=(s,u,):

is a Gaussian fonction with unit norm. A MP is an iterative algorithm which decomposes a signal x over dictionary vectors as follows. Let us write R

,where is the set of all possible values for and C(x,g

Finally, the signal is represented as:

In [Mallat & Zhang 93], the correlation function C is the inner product C(x,g

Fig. 3. Time-frequency distribution of a G5 sharp piano note, obtained with HRMP

The time-frequency distribution of a G5 sharp piano note, obtained with HRMP, is displayed in figure 3. One can easily distinguish long horizontal lines due to large-scale vectors well defined in frequency around 830 Hz, 1660 Hz, etc.. They correspond to the damped sinusoidal quasi-harmonic modes of the string. On the contrary, vertical lines corresponding to small-scale transient structures are visible at the attack and at the release of the damper of the piano. This example shows how HMRP provides a time-frequency representation adapted to the specificities of sound signals. The elements of this representation are easily related to perceptually important structures such as fast transients, or sustained sinusoidal partials.

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