Fundamental Frequency Estimation Research Articles

Speech analysis and synthesis using an AM–FM modulation model

In this paper, the AM–FM modulation model is applied to speech analysis, synthesis and coding. The AM–FM model represents the speech signal as the sum of formant resonance signals each of which contains amplitude and frequency modulation. Multiband filtering and demodulation using the energy separation algorithm are the basic tools used for speech analysis. First, multiband demodulation analysis (MDA) is applied to the problem of fundamental frequency estimation using the average instantaneous frequency as estimates of pitch harmonics. The MDA pitch tracking algorithm is shown to produce smooth and accurate fundamental frequency contours. Next, the AM–FM modulation vocoder is introduced, which represents speech as the sum of resonance signals. A time-varying filterbank is used to extract the formant bands and then the energy separation algorithm is used to demodulate the resonance signals into the amplitude envelope and instantaneous frequency signals. Efficient modeling and coding (at 4.8–9.6 kbits/sec) algorithms are proposed for the amplitude envelope and instantaneous frequency of speech resonances. Finally, the perceptual importance of modulations in speech resonances is investigated and it is shown that amplitude modulation patterns are both speaker and phone dependent.

Ab initio geometry, quartic force field, and vibrational frequencies for P4

We have computed the equilibrium bond length and quartic force field for tetrahedral P4 using coupled-cluster methods and a variety of extended basis sets. Our calculations include the effect of correlating core, as well as valence, electrons. As with earlier P4 studies, our computed results show significant differences with current experimental estimates of both bond length and fundamental vibrational frequencies: possible reasons for this disagreement are discussed, with particular reference to the effects of the elevated temperatures used in gas-phase spectroscopic studies of P4. Our calculated endothermicity relative to two P2 molecules appears to be well converged with respect to correlation treatment and basis set, but again differs substantially from present experimental values, which we speculate are underestimates.

Real-time fundamental frequency estimation by least-square fitting

For real-time applications, a fundamental frequency estimation algorithm must be able to obtain accurate estimates from short signal segments. Characterization of the error function of fitting a sinusoid to the signal segment allows its spectrum to be deduced and the algorithm to be implemented efficiently. Musical signals are discussed in particular.

Fundamental frequency estimation of speech signals using a nonlinear observer

This paper describes a new method of estimating the fundamental frequency of speech signals using an adaptive Fourier analysis (AFA) algorithm based on the presumption of signal parameters using a nonlinear observer. The recursive presumption of signal parameters such as fundamental frequency can be effectively implemented by using the proposed AFA algorithm. Nonlinear observers can be used to measure nonlinear parameters which are included in observed signals. Recursive estimation procedures make it possible to follow the slow changes of signal parameters to be measured. In real-time signal analysis, the recursive algorithm is preferred to the conventional Fourier transformation because of this property. The AFA algorithm, which is based on the recursive discrete Fourier transform (RDFT), is a recursive algorithm for the simultaneous estimation of the Fourier coefficients and instantaneous fundamental frequency of speech signals. This AFA algorithm is applied to speech signals with an arbitrary length of the transform window. Further efforts are required for better convergence speed of the proposed algorithm, but this method is mostly capable of adapting and tracking nonlinear signals such as speech sound.

Estimation of fundamental frequencies of perhalogenated ethylenes in terms of molecular fragment contributions

A model, in which a molecule is subdivided into a number of subunits, is used to calculate approximately fundamental frequencies of perhalogenated ethylenes in terms of contributions of subunits and clusters of subunits. Different levels of approximation are defined by a subunit additivity scheme, a linear subunit pair and subunit triple scheme as well as by a nonlinear coupled subunit pair scheme. Algebraic dependencies of the approximate formulas for distinct ethylene derivatives are used as criteria for testing the different schemes. The method is applied to the fundamental frequencies of normal vibrations which, with respect to symmetry, correlate uniquely in the series of substituted ethylenes considered, otherwise the method is applied to corresponding sums of frequencies. For the fundamental frequencies of the 27 ethylene derivatives C2FnClmBr4−n−m, n+m≤4, completely known from a normal coordinate analysis, the main results are: (a) The out-of-plane frequencies correlate uniquely in these molecules, three of the in-plane frequencies and two sums formed from the remaining ones correlate as well. (b) The simple additivity scheme with three parameters for each set of correlating frequencies yields a sufficiently exact fit of the out-of-plane frequencies; their mean absolute deviation is about 5 cm−1. (c) The linear subunit pair approximation with twelve parameters for each set is sufficiently exact for all the correlating frequencies and sums of frequencies, respectively; their mean absolute deviation is about 1 cm−1.

Fundamental frequency estimation of musical signals using a two-way mismatch procedure

Fundamental frequency (F0) estimation for quasiharmonic signals is an important task in music signal processing. Many previously developed techniques have suffered from unsatisfactory performance due to ambiguous spectra, noise perturbations, wide frequency range, vibrato, and other common artifacts encountered in musical signals. In this paper a new two-way mismatch (TWM) procedure for estimating F0 is described which may lead to improved results in this area. This computer-based method uses the quasiharmonic assumption to guide a search for F0 based on the short-time spectra of an input signal. The estimated F0 is chosen to minimize discrepancies between measured partial frequencies and harmonic frequencies generated by trial values of F0. For each trial F0, mismatches between the harmonics generated and the measured partial frequencies are averaged over a fixed subset of the available partials. A weighting scheme is used to reduce the susceptibility of the procedure to the presence of noise or absence of certain partials in the spectral data. Graphs of F0 estimate versus time for several representative recorded solo musical instrument and voice passages are presented. Some special strategies for extending the TWM procedure for F0 estimations of two simultaneous voices in duet recordings are also discussed.

Conditioned constant-Q spectra for improved frequency tracking in the presence of interfering signals

A technique is presented for improving the performance of a method for estimating the time-varying fundamental frequency of a signal corrupted by interference components. The original method involves the calculation and comparison of frequency-pair ratios in each short-time spectrum of the corrupted signal [M. Piszczalski and B. A. Galler, ‘‘Predicting musical pitch from component frequency ratios,’’ J. Acoust. Soc. Am. 66, 710–720 (1979)]. As the strength of the interference components is increased, the frequency estimates from this method become more unreliable. To obtain improved performance, a procedure based upon principal components analysis is used to enhance the harmonic signal’s spectral components prior to the calculation of frequency-pair ratios. The resulting improvement in fundamental frequency estimation is demonstrated through examples involving music signals.

Maximum likelihood estimation of signals in autoregressive noise

Time series modeling as the sum of a deterministic signal and an autoregressive (AR) process is studied. Maximum likelihood estimation of the signal amplitudes and AR parameters is seen to result in a nonlinear estimation problem. However, it is shown that for a given class of signals, the use of a parameter transformation can reduce the problem to a linear least squares one. For unknown signal parameters, in addition to the signal amplitudes, the maximization can be reduced to one over the additional signal parameters. The general class of signals for which such parameter transformations are applicable, thereby reducing estimator complexity drastically, is derived. This class includes sinusoids as well as polynomials and polynomial-times-exponential signals. The ideas are based on the theory of invariant subspaces for linear operators. The results form a powerful modeling tool in signal plus noise problems and therefore find application in a large variety of statistical signal processing problems. The authors briefly discuss some applications such as spectral analysis, broadband/transient detection using line array data, and fundamental frequency estimation for periodic signals. >

Separation of concurrent harmonic sounds: Fundamental frequency estimation and a time-domain cancellation model of auditory processing

Signal-processing methods and auditory models for separation of concurrent harmonic sounds are reviewed, and a processing principle is proposed that cancels harmonic interference in the time domain. The principle is first formulated in signal processing terms as a time-domain comb filter. The critical issue of fundamental frequency estimation is investigated and an algorithm is proposed. Tested on a restricted database of natural voiced speech, the algorithm successfully found estimates correct within 3% of an octave for 90% of all frames. Next, the principle is formulated in physiological terms. A hypothetical ‘‘neural comb filter’’ is described, based on neural delay lines and inhibitory synapses, and tested using auditory-nerve fiber discharge data obtained in response to concurrent vowels [A. R. Palmer, J. Acoust. Soc. Am. 88, 1412–1426 (1990)]. Processing successfully suppresses the correlates of either vowel in the response of fibers that respond to both, allowing the other vowel to be better represented. The filter belongs to the class of ‘‘cancellation models’’ for which predictions can be made concerning the outcome of certain psychoacoustic experiments. These predictions are discussed in relation to recent experimental results obtained elsewhere.

Maximum-likelihood harmonic matching for fundamental frequency estimation

This presentation deals with the estimation of fundamental frequency (f0) of pseudoperiodic sound signals with important results for polyphonic frequency tracking, and voice separation. Given a set of candidate partials in the signal, the estimation of f0 is taken in the sense of finding the optimal period duration(s) according to a criterion of maximum-likelihood harmonic matching. Excellent results have been obtained on large databases of speech (40 mn) and music [B. Doval and X. Rodet, Proc. IEEE-ICASSP, Toronto, May (1991)]. The algorithm has been implemented at IRCAM to run in real time for live performance frequency tracking. Developments are in several directions. A combined estimation of f0 and of a spectral envelope improves both estimations. Most important is the estimation of the ‘‘apriori’’ distributions of the different random variables on a learning set. Finally, a hidden Markov model tracks f0 trajectories between adjacent frames. The first experiments of polyphonic frequency tracking and voice separation are very promising. The model can be transposed directly to the maximum-likelihood estimation of several harmonic sounds since it already considers more than one f0 value. a)Presently on sabbatical at Ctr. for New Music and Audio Technol. (CNMAT), Univ. of California at Berkeley, 1750 Arch St., Berkeley, CA 94709.

Frequency tracking of solo and duet passages using a harmonic two-way mismatch procedure

Estimation of the fundamental frequency (F0) of quasiharmonic signals is an important task in musical acoustics. Many existing techniques have unsatisfactory performance due to octave errors, noise perturbations, and other artifacts encountered in practice. Our work in this area, begun in 1986, uses the quasiharmonic assumption to guide a search for an estimated F0, based on the McAulay–Quatieri short-time spectrum measurement procedure. Estimation of F0 involves a two-way mismatch calculation where the discrepancy between the measured partial frequencies and a harmonic search grid is used to refine the F0 estimate. Inaccuracies present in the measured partial frequencies are averaged over all the available partials, thereby improving the F0 prediction. A heuristic weighting scheme is used to help reduce the susceptibility of the procedure to noise or partials missing in the spectral data. The frequency search strategy has also been extended to fundamental frequency estimation of duet (two voice) polyphony. Results for F0 tracking of solo and duet recorded musical instrument passages will be demonstrated.

Fundamental frequency determination of speech and singing voice−A review

Fundamental frequency determination methods and algorithms (PDAs) can be grouped into two major classes: time-domain PDAs and short-term analysis PDAs. The short-term analysis PDAs leave the signal domain by a short-term transformation. They supply a sequence of average fundamental frequency estimates from consecutive frames. The individual algorithm is characterized by the short-term transform it applies: autocorrelation, distance functions (average magnitude difference function), least-squares method, harmonic analysis, and multiple spectral transform (cepstrum) are the most common algorithms. The time-domain methods, on the other hand, track the signal period by period. Extraction and isolation of the fundamental harmonic, and investigation of the temporal signal structure are the two extremes between which most of these PDAs are found. Special attention is given to the time-variant aspect of frequency determination and to postprocessing methods that allow fundamental frequency tracking over time. Among these, simple smoothing and list correction procedures are found that remove isolated errors as well as sophisticated schemes that apply dynamic optimization or stochastic modeling.

The influence of spectral composition of complex tones and of musical experience on the perceptibility of virtual pitch.

Fifty-eight trained musicians and 58 naive listeners were instructed to adjust pure tones, generated by a computer-soundsampler system (ATARI MEGA/STE, AKAI S1100), to the perceived fundamental frequencies of alternatively presented harmonic complexes. Stimuli varied according to the number of their partials, the relative distance to the missing fundamental, and spectral density. Every person had to fulfill 112 different items. The degree of task difficulty was determined by the average of deviations of subjects’ performances from the greatest common submultiples of the complexes. It is shown that each of the considered variables has significant influence on fundamental frequency estimation. Moreover musicians tend to tune fundamentals to low (average: −17 Cent) whereas for nonmusicians the opposite trend is observed (average: +15 Cent). Results were related to performances on the subtests ‘‘Virtual/Spectral Test’’ [M. E. Albrecht, doctoral thesis (1972)] and ‘‘Interval-Classification’’ (Preisler, 1991) by a statistical ANOVA design. Interactions between performance scores are discussed with respect to their theoretical background. [Work supported by the Austrian Fonds for Scientific Research.]

Peak-picking fundamental period estimation for hearing prostheses.

A real-time peak-picking fundamental period estimation device is described which is used in advanced hearing prostheses for the totally and profoundly deafened. The operation of the peak picker is compared with three well-established fundamental frequency estimation techniques: the electrolaryngograph, which is used as a "standard" hardware implementations of the cepstral technique, and the Gold/Rabiner parallel processing algorithm. These comparisons illustrate and highlight some of the important advantages and disadvantages that characterize the operation of these techniques. The special requirements of the hearing prostheses are discussed with respect to the operation of each device, and the choice of the peak picker is found to be felicitous in this application.

A logarithmic approach to fundamental frequency error measurement in speech

The octave is a logarithmic unit of measure that offers significant advantages when applied to frequency error measurement. The octave provides an inherent normalization of frequency errors, displays harmonic errors, weights reciprocal errors equally, and can easily be interpreted in either the time or frequency domain. In this letter, the octave is compared with three commonly used frequency error measures. To illustrate the advantages of the octave as an error measure, its application to fundamental frequency estimation error in speech is discussed.

Excitation synchronous formant analysis

Speech signals can be efficiently parametrised by the resonant frequencies of the vocal tract known as formants. The automatic analysis of the signal into a suitable set of formant parameters has however proved to be a difficult problem, particularly for female speech. The technique of excitation synchronous formant analysis has been proposed as an improved method of formant analysis [5]. The paper considers the performance of this technique, particularly where the analysis interval is over the closed phase of the larynx. The improved performance of closed-phase formant analysis is demonstrated by comparison with pitch-synchronous and fixed-frame formant analysis. The closed-phase region is determined first using a laryngograph signal and secondly using a modified form of the Gold-Rabiner fundamental-frequency estimator, using only the acoustic waveform. The improved performance of closed-phase formant analysis is also demonstrated by a better ability to follow the transient features of the signal, with fewer missed or extra formants, and better formant continuity. The ability to follow formant transitions during glides (e.g. w, r, l) and in voiced segments following plosives is particularly apparent. These improvements are illustrated in various phonetic contexts. The technique has been tested for sensitivity to analysis position, which is important when the glottal closures are determined from the acoustic waveform. This method of formant analysis is currently being applied to the development of speech synthesis by rule, and to provide a set of features for phonetic recognition.

A method for the extraction of spectral peaks and its application to fundamental frequency estimation of speech signals

This paper describes a method to estimate the temporal patterns of the fundamental frequency of speech signals. This method makes use of the first and the second derivatives of a short-time power spectrum with respect to frequency. The time window of the spectrum is designated as i exp(− BiT) for i⩾0, and an effective algorithm to calculate the derivatives is proposed. An algorithm to determine voiced intervals is also proposed. The results of the application to the speech signals of Japanese vowels and the ten digit utterances show the effectiveness of the proposed method for obtaining a smoothed pitch contour.

A new algorithm for speech fundamental frequency estimation

In this correspondence, we present a method of detecting the frequency based on the recognition of fundamental peaks of a low-pass filtered (f c = 0.7 kHz) speech signal. Average overall recognition accuracy of approximately 99 percent for the training speech sample and 97.5 percent for a test speech sample were achieved. The experimental verification of the developed FPR (fundamental peak recognition) method also included an objective comparison to Miller's data reduction method, and a subjective performance evaluation using the standard cepstrum method. The FPR classification scheme is computationally efficient and easily implementable with relatively slow 8-bit microprocessors.

Programs for the estimation of fundamental frequency, amplitude, and voicing of speech

Programs for the estimation of fundamental frequency, amplitude, and voicing of speech

Clinical estimation of fundamental frequency: The 3m plastiform magnetic tape viewer

A procedure and method is described to determine the frequency or pitch of tape-recorded voice samples, while keeping the cost of measurement within the reach of practicing clinicians. The method described reduces the uncertainty of pitch determination by comparison techniques and improves the accuracy of clinical estimation of fundamental vocal frequency and pitch. The accuracy, reliability, cost, and ease with which the method can be used makes it a viable tool to be used by the clinician.