Sinusoidal models

Recall that, in our introduction to signal processing, we saw that we could use the Fourier series to generate any periodic signal from a sum of sinusoids (Section 10.1.2)  

The techniques known as sinusoidal models use this as their basic building block and perform speech modification by finding the sinusoidal components for a waveform and carrying out modification by altering the parameters of the above equation, namely the amplitudes, phases and frequencies. This has some advantages over models such as TD-PSOLA in that it allows adjustments in the frequency domain. Frequency-domain 

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Another model which is a strong candidate for musical signals is the sinusoidal model, which has been used effectively for speech applications ([McAulay and Quatieri, 1986b] and chapter 9 of this book). A constrained form of the sinusoidal model is implicitly at the heart of short-time spectral attenuation (STSA) methods of noise reduction (see section 4.5.1). The model is also a fundamental assumption of the pitch variation algorithms presented in section 4.6. In its general form the signal can be expressed as ... 

Time scaling. Because the phase-vocoder (the short-time Fourier transform) gives access to the implicit sinusoidal model parameters, the ideal time-scale operation described by Eq. (7.3) can be implemented in the same framework. Synthesis time-instants / are usually set at a regular interval / +1 - / = R. From the series of synthesis time-instants / analysis time-instants / are calculated according to the desired time warping function tua = T l(t ). The short-time Fourier transform of the time-scaled signal is then ... 

An early approach to music processing, referred to as additive synthesis [Moorer, 1977], used the sinusoidal model of a quasi-periodic music note... 

George and Smith, 1992] George, E. B. and Smith, M. J. T. (1992). Analysis-by-synthesis/Overlap-add sinusoidal modeling applied to the analysis and synthesis of musical tones. J. Audio Eng. Soc., 40(6) 497-516. 

Kates, 1994] Kates, J. (1994). Speech enhancement based on a sinusoidal model. Journal of Speech and Hearing Research, 37 449-464. 

Macon and Clements, 1996] Macon, M. and Clements, M. (1996). Speech concatenation and synthesis using an overlap-add sinusoidal model. In Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing., volume 1, pages 361-364, Atlanta, GA. 

Marques and Almeida, 1988] Marques, J. and Almeida, L. (1988). Sinusoidal modeling of speech Representation of unvoiced sounds with narrowband basis functions. 

Quatieri and Danisewicz, 1990] Quatieri, T. and Danisewicz, R. (1990). An approach to co-channel talker interference suppression using a sinusoidal model for speech. IEEE Trans, on Acoustics, Speech, and Signal Processing, ASSP-38(l) 56-69. 

T.F. Quatieri and R.J. McAulay Audio Signal Processing Based on Sinusoidal Analysis/Synthesis. One of the basic paradigms of digital audio analysis, coding (i.e. analysis/synthesis) and synthesis systems is the sinusoidal model. It has been used for many systems from speech coding to music synthesis. The chapter contains the unified view of both the basics of sinusoidal analysis/synthesis and some of the applications. 

In the sinusoidal model above, an ideal time-scaled signal corresponding to the signal described by Eq.(7.1) would be... 

The reader can refer to chapter 9.2.2 for an alternative presentation of the short-time Fourier transform, in the context of sinusoidal modeling. 

The preceding analysis views the problem of solving for the sine-wave amplitudes and phases in the frequency domain. Alternatively, the problem can be viewed in the time domain. It has been shown that [Quatieri and Danisewicz, 1990], for suitable window lengths, the vectors a andJ3 that satisfy Equation (9.75) also approximate the vectors that minimize the weighted mean square distance between the speech frame and the steady state sinusoidal model for summed vocalic speech with the sinusoidal frequency vector . Specifically, the following minimization is performed with respect to a andJ3... 

Other models of hepatic clearance include the sinusoidal model and the dispersion model. All these... 

The compressive strengths of the composites obtained increased and their temperature dependencies decreased with increasing fiber length, fiber-volume fraction, and density of the matrix foam. More specifically, the compressive strengtii of the composite was found to be proportional to that of the matrix and increased linearly with increased fiber-volume fraction in the experimental range employed (below 2% by volume). This result could be explained by Swift s sinusoidal model, assuming that the adhesion between fiber and matrix foam is perfect. 

In principle we could perform Fourier analysis to find the model parameters, but for reasons explained below, it is in fact advantageous to follow a different procedure that is geared towards our synthesis goals. For purposes of modifying pitch, it is useful to perform the analysis in a pitch synchronous manner and in fact one of the main advantages of sinusoidal modelling is that the accuracy of this does not have to be as high as that for PSOLA [420], [293], [520]. 

Given the parameters of the model, we can reconstruct a time domain waveform for each frame by use of the synthesis Equation 14.3. Figure 14.6 shows a real and resynthesised frame of speech. An entire waveform can be resynthesised by overlapping and adding the frames just as with the PSOLA method (in fact the use of overlap add techniques was first developed for conjunction with sinusoidal models). 

Sinusoidal models use a harmonie model and decompose each frame into a set of harmonics of an estimated fundamental frequency. The model parameters are the amplitudes and phases of the harmonics. With these, the value of the fundamental can be changed while keeping the same basic spectral envelope. 

Harmonic Noise models are similar to sinusoidal models, except that they have an additional noise component which allows accurate modelling of noisy high frequency portions of voiced speech and all parts of unvoiced speech. 

MBROLA is a PSOLA like technique which uses sinusoidal modelling to decompose each frame and from this resynthesise the database at a constant pitch and phase, thus alleviating many problems in inaccurate epoch detection. 

Abrantes, a. j.. Marques, J. S., and Trancoso, I. M. Hybrid sinusoidal modeling of speech without voicing decision. In Proceedings of Eurospeech 1991 (1991). 

Macon, M. W. Speech synthesis based on sinusoidal modeling. PhD thesis, Georgia Tech, 1996. 

ZOLFAGHARI, P., Nakatani, T., AND Irino, T. Glottal closure instant synchronous sinusoidal model for high quality speech analysis/s3mthesis. In Proceedings of Eurospeech... 

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