In a previous paper, we discussed estimation of the parameters of a single tone from a finite number of noisy discrete-time observations. In this paper, we extend the discussion to include several tones. The Cramer-Rao bounds are derived and their properties examined. Estimation algorithms are discussed and characterized.

Multiple tone parameter estimation from discrete-time observations

Muscle tremors reflect rhythmical motor unit (MU) activities. Therefore, the MU firing patterns and synchrony determine the properties of the parkinsonian force tremor (FT) and the neurogenic components of associated limb tremors. They may also be indicative of the neural mechanisms of tremor genesis which to date remain uncertain. We examined these MU behaviours during isometric contractions of a finger muscle in 19 parkinsonian subjects. Our results reveal that the parkinsonian FT is abnormally large. Like the physiological FT, it is accompanied by in-phase rhythms in all MU activities. However, there exist two important differences. Firstly, the synchrony during the parkinsonian FT is stronger than the normal one and therefore contributes to the FT enhancement. Secondly, the synchronous MU components partly represent rhythmical sequences of spike doublets and triplets whose incidences directly reflect the differences of the MU firing rates to the FT frequency. According to our analyses, the latter frequency coincides with the MU recruitment rate. Consequently, the numerous medium- and small-sized active MUs contribute rhythmical twitch doublets and triplets, i.e. large force pulses, to the parkinsonian FT. The impact of this effect on the FT amplitude is found to predominate over the impact of the augmented synchrony. Importantly, apart from the rule governing the occurrence of doublets/triplets, the mean interspike intervals within such spike events are fairly fixed around 50 ms. Such regularities in MU activities may reflect properties of the neural input underlying the FT, and thus represent a basis for more focused studies of the generator(s) of parkinsonian tremors.

https://physoc.onlinelibrary.wiley.com/doi/pdf/10.1113/jphysiol.2009.173989

Tremor‐related motor unit firing in Parkinson's disease: implications for tremor genesis

The firing behavior of motor units (MUs) of the first dorsal interrosseus muscle of the hand was examined during both constant-force and varying-force (sinusoidal or broadband random variations) isometric contractions in healthy adults. The emphasis was on the analysis of MU synchrony with an efficient and sensitive method. In static contractions, widespread and strong MU firing correlations, with the MUs in phase, were present at the frequency of muscle tremor, when the tremor was regular (narrowband) and large. MU correlations could also exist in contractions where the tremor of a subject was irregular (broadband) overall, but they were generally weak. These correlations were at the frequency of the subject's regular tremor, and the corresponding distinct tremor component was sometimes discernible within the broad tremor-band. In contrast, the MUs did not show any such correlations in the case of purely irregular and small tremor. On the basis of these observations, it is concluded that the rhythms in the force contributions of the last- recruited, large MUs, which fire near their threshold rate, compose the broadband frequency content of physiological muscle tremor in every contraction. Within this band, there is an additional distinct tremor component when MU correlations are present. For widespread and strong MU correlations, this component dominates and constitutes the observed regular tremor. In dynamic contractions, the firing of all MUs was modulated in the frequency band of both the sinusoidal and the complex variations of the force. The MU modulations showed a time-lead over the force variations and were strongly correlated both to these variations and among themselves. Thus widespread and strong correlations of MU firing modulations seem to provide a mechanism for generation of time-varying voluntary force, under general dynamic conditions. Finally, when regular tremor was present in dynamic contractions, widespread and fairly strong MU correlations also existed at the tremor frequency. It is concluded that at least two mechanisms can cause widespread MU synchrony, and they can act in parallel. They involve two types of correlated inputs to the α-motoneurons (presumably from the muscle spindles and the cortex), whose effects combine at the level of the membrane potential of the cells.

https://www.physiology.org/doi/pdf/10.1152/jn.1999.82.5.2839

Occurrence of Widespread Motor-Unit Firing Correlations in Muscle Contractions: Their Role in the Generation of Tremor and Time-Varying Voluntary Force

The paper analyses power-spectrum estimates obtained by fast-Fourier-transform techniques Distributions are obtained for data, augmented, where necessary, by sequences of zeros, and the effect of data smoothing on the reliability of the estimates is considered The effect of segment averaging is analysed and a joint probability distribution is derived for the resulting spectrum estimates The number of degrees of freedom per estimate can then be directly determined 1st- and 2nd-order moments of logarithmic spectra are derived which lead to confidence bands on the spectral estimates Frequency-domain smoothing is then considered, and it is shown, that, for specified lengths of Gaussian random data, this, unlike data smoothing, does not lead to a reduction in the number of degrees of freedom Finally, the general case of frequency smoothing followed by adjacent estimate averaging is analysed A factor is proposed for assessing loss of stability of such estimates Computer results are given which demonstrate the effects of several data windows and sets of frequency-smoothing coefficients Results in the appendixes show that loss in degrees of freedom is related to the eigenvalues of a specific covariance matrix

Probability distributions for discrete Fourier spectra

An analysis is presented for the joint probability density functions of power spectra of random processes computed under various conditions of data/frequency smoothing Spectral correlation functions are determined which illustrate the effect of smoothing on adjacent spectral ordinates The technique of segment averaging of spectra is explored and bivariate probability density functions established for correlated spectral ordinates These distributions are shown to be generalizations of the well known Rician probability density functions Finally, the case of spectra obtained as moving average values is considered and expressions are derived for the spectral correlation functions in terms of the parameter of the smoothing sequences and the averaging length It is shown that spectra obtained in this manner introduce a distortion of the spectral correlation function Several computed results are included to illustrate the analysis

Joint density functions for digital spectra

Essentially related to the problem of spectrum estimation of random data by fast-Fourier-transform techniques is the problem of establishing efficient data windows or data-smoothing procedures for minimising leakage effects. Statistics of spectral estimates derived from finite data lengths indicate that the estimates are related to the true spectrum by a frequency convolution with a Fejer window. It is well known that leakage of power in the spectrum occurs from the nonnegligible sidelobes of this window. The paper presents an investigation into the application of data windows in spectral analysis. Consequent effects on the bias (amplitude) and the variance of the spectral estimates are analysed and compensating factors are specified. Criteria are further developed for comparing the increase in bandwidth of spectral analysis caused by different data-smoothing sequences. As the bandwidth of analysis is dependent on the bandwidth of the spectral window associated with a data window to the frequency tion of the window bandwidth leads to a measure of the loss of frequency resolution entailed by tapering the data. Although no universal definition exists for the bandwidth of a system or a time series, four different expressions are included for the bandwidth of a spectral window which relate the parameters of a data window to the frequency spread of its associated spectral window. Three families of data windows are considered, and computed results are presented which allow the choice of a data window as an optimum compromise between critical factors involved in the design of an efficient spectrum-estimation procedure.

J.M. Nightingale

Papers

Data windows for digital spectral analysis