976 Journal ofNeurology, Neurosurgery, and Psychiatry 1990;53:976-981 J Neurol Neurosurg Psychiatry: first published as 10.1136/jnnp.53.11.976 on 1 November 1990. Downloaded from Spectral analysis of tremor: understanding the results Michael Gresty, David Buckwell Abstract Nature of the tremor record: amplitude and Spectral analysis of a tremor record can frequency modulation sometimes produce a spectrum with Tremor is an approximately sinusoidal oscilla- multiple components of significant tion of a part of the body. The amplitude of amplitude. The problem is to determine the oscillation may fluctuate and this is whether the presence of several peaks termed amplitude modulation "AM". The represents the coexistence of separate frequency of the oscillation also varies. This is tremor mechanisms or be a consequence termed frequency jitter in physiology but here of fluctuations in the frequency or we will use the universal term frequency amplitude of a single tremor. The spec- modulation "FM". The tremor waveform trum of a tremor whose frequency or may be distorted, "harmonic distortion", amplitude vary and are independent has which arises, in part, because of mechanical the recognisable pattern of a central factors, and in part reflects the specific inten- carrier frequency with sidebands of sity and sequence in Wvhich various muscles equal amplitudes distributed sym- are activated during the tremor cycle. These metrically around the carrier. However, characteristics are illustrated in fig 1 showing if tremor amplitude and frequency the tremor in flexion/extension of the out- fluctuate and are not independent, stretched hand in a subject with essential (frequency proportional to amplitude or tremor. The AM, FM and harmonic distor- frequency inversely proportional to tion of tremor may vary in an unlimited num- amplitude), the spectrum has a pattern ber of ways; thus for example, the amplitude of sidebands which are asymmetrical may vary in a predictable way, waxing and in amplitudes and may resemble the waning at regular intervals ("spindling"); it spectrum of the combined signal from may maintain an almost constant level or different independent oscillators. The fluctuate randomly, or according to some investigation of sidebands in spectra has combination of these. been neglected in tremor studies and For some tremors, frequency and amplitude multiple irregular peaks on a tremor are related as in the example of essential spectrum have sometimes been used tremor given in fig 1. The superimposed wrongly as evidence for the coexistence cycles of tremor show that as the time length of multiple tremor mechanisms or of a cycle of the tremor decreases the frequency components assumed to be amplitude also decreases; thus for this tremor, http://jnnp.bmj.com/ concurrent. amplitude is inversely proportional to frequency within certain constraints. This Medical Research characteristic of tremor has not been studied Council Human As early as 1904 Gordon Holmes recognised Movement and that individual types of tremor, such as the Balance Unit, Institute rest tremor of Parkinson's disease ofNeurology, National or Hospital, Queen cerebellar action tremor, appear to be on October 3, 2021 by guest. Protected copyright. Square, London relatively fixed in frequency and speculated M Gresty that the frequency of a tremor is specific to its D Buckwell pathophysiological origin.' In contrast to This article is dedicated to Mr Edward Trinder, Chief frequency, the amplitude of tremor and the Research Officer of the shape of the tremor are subject to wide Medical Research Council Neuro-otology Unit who fluctuations. In recent times, precise 0-1 retired in the Autumn of measurement of the characteristics of tremor, 1989. His career was marked by great kindness to staff including frequency, has become widely under his care and available through spectral analysis which has Isb cf (2nd Hs) outstanding technical contributions to been used to quantify tremor in clinical trials understanding deafness and or as the bases of arguments concerning the ............... ........................ vertigo. nature of tremor mechanisms. Unfortunately, Correspondence to: 0 25 Halz Dr Gresty, Medical Research the results of spectral analysis can be as com- Council Human Movement plicated in themselves as the raw tremor Figure 1 Accelerometric recordings and averaged and Balance Unit, Institute spectrum ofpostural tremor of the hand in a 43 year old ofNeurology, National records. We draw attention to the problems of male with essential tremor. Individual cycles of tremor Hospital, Queen Square, interpreting spectral analyses of tremor are shown superimposed by triggeringfrom an London WCIN 3BG, United records and in particular to the circumstances oscilloscope. As amplitude increases, the period of the Kingdom tremor cycle elongates. The acceleration is scaled in units Received 12 January 1990. under which multiple peaks may appear in of "g" (earth gravitationalfield). lsb: lower sideband, Accepted 10 April 1990 tremor spectra. cf: carrierfrequency, Hs: harmonics.., Spectral analysis of tremor: understanding the results 977 J Neurol Neurosurg Psychiatry: first published as 10.1136/jnnp.53.11.976 on 1 November 1990. Downloaded from "baseband signal". The spectrum of an (i) F-A (i) F-1/A amplitude modulated signal consists of a peak usb at the carrier frequency equal amplitude FM o "sidebands" which are frequency components cf cf equally spaced above and below the carrier lsb frequency. If the amplitude of the tremor varies slowly over time (implying that the baseband signal is (AM * FM) low frequency) the sidebands merge with the a -I peak of the carrier frequency as shown in fig 3. If the tremor spindles, for example fluctuates in amplitude every 2s, there are two distinct + b I sidebands in the spectrum spaced 0-5 Hz above and below the carrier frequency. This is shown in graphs a, b, c of fig 2 which are AM spectra - 180° IIfc I I derived by amplitude modulation ofthe carrier frequency and of the upper sideband and the lower sideband of the FM signal. abc Frequency modulation and FM spectra: Tremor (AM * FM) varies in frequency about a typical centre frequency termed the "carrier frequency". The way in which the frequency varies can be Figure 2 A, B Graphical representation of the described by a waveform, which can be regular frequency components offrequency and amplitude or irregular, and is termed the "baseband modulated tremor signalsforfrequency proportional to amplitude F - A andfrequency inversely proportional to signal". The spectrum of this signal consists of amplitude F - 1/A. Each of the bar pictures is a a peak at the carrier frequency of the tremor spectrum. The horizontal axis in each case represents and equal amplitude "sidebands" which are frequency. The vertical bars represent the individual terms of the Sine series which makes up a signal (see frequency components equally spaced above appendix). Thefrequency of each Sine component is and below the carrier frequency. The greater shown by its location on thefrequency axis. The the variability in frequency ofa signal, the more coefficient of each term, that is, the size of each component, is shown by the length of the bars. The sidebands are produced. For example, the positive or negative sign of the coefficient is shown by the frequency of Parkinsonian postural or rest directions of the bars; 0°for + ve and - 180° for -ve. tremor varies within narrow limits of + /- The simplest case is shown ofsinusoidalfrequency and 0-5 Hz which amplitude modulation. Frequency modulation of a produces only two important sinusoidal signal produces thefrequency components sidebands. shown in the spectra labelled "FM" where "cf' is carrier If the frequency of a tremor signal varies frequency, "usb" is upper sideband and "lsb" is lower slowly over time (implying that the baseband sideband. Amplitude modulation takes place on each of the components of thefrequency modulated signal signal is low frequency) the sidebands merge producing thefrequency components shown in spectra with the peak ofthe carrier frequency as shown labelled "AM*FM a, b, c". Thus: "a" is the spectrum of in 3. amplitude modulation of the FM carrierfrequency, "b" fig If the tremor jitters, for example, is the spectrum ofamplitude modulation of the FM lower fluctuates in frequency in two second epochs sideband and "c" is the spectrum of amplitude modulation (05 Hz), then distinct sidebands appear on the of the FM upper sideband. Note that each of these spectrum spaced 0-5 Hz above and below the http://jnnp.bmj.com/ resulting spectra have a central carrier frequency with an upper and a lower sideband. Since all these components carrier frequency giving multiple peaks in coexist the resulting signal is their algebraic sum shown in spectrum. Important to the present argument the lowest traces labelled "Z;a,b,c (AM*FM)". Bars of is that the spectra ofboth the FM signal and the equal amplitude but in opposite directions cancel when AM signal are the same except that the FM added together. Thusfor FM - AM (fig 2A) the lower sidebands of spectra a and b cancel each other. For signal has a phase shift of 1800 in the lower FM 1/AM (fig 2B) the upper sidebands of spectra a sideband. This is shown by the downwards and c cancel each other. direction of the lower sideband of the FM on October 3, 2021 by guest. Protected copyright. spectra in fig 2 (see appendix). Combined FM and AM spectra: If the and yet is of crucial importance, for interac- frequency modulation and amplitude modula- tion between amplitude and frequency can tion of a signal are unrelated then the spectrum give rise to an ambiguous spectrum. typically has numerous upper and lower side- bands symmetrical in amplitude and evenly spaced in frequency about the carrier fre- Spectral analysis of combined FM and AM quency. For the special cases where amplitude waveforms and frequency are related (an example is shown The spectra of AM and FM waveforms are in the upper raw data traces of fig 1 where explained in the appendix and shown graph- cycles of tremor with higher amplitudes have ically in fig 2.