Spectral analysis o biosignals Biosignal processing I, 573S Autun 07
Introduction Oscillations o biosignals oten have intuitive physiological interpretation breathing rate, heart rate, alpha rhyth, Frequencies o interest are low, typically: EEG <30 Hz, EMG <00Hz, ECG <300Hz, MRI <3Hz Careul signal preprocessing needed due to artiacts and noise oveent, blins, uscle activity, Wide-sense stationarity assuption WSS First- and second order statics reain the sae over the signal: the ean and the autocorrelation unction and spectru Spectral analysis is challenged by non-stationarities Changing conditions due to body oveents, reactions o body to various eternal/internal ipulses, edication, injury etc.
on-paraetric ethods or power-spectral density PSD estiation no assuption is ade o the signal generating process such as being a stochastic process: AR, MA, ARMA, which are paraetric ethods Discrete Fourier Transor, DFT, is the ost coon ethod Can be coputed eectively with Fast Fourier Transor, FFT oten, signal is divided into segents, spectru is estiated or each, and inally averaging o the segent-wise spectrus
Periodogra Periodogra estiate o PSD can be achieved through Fourier transoring the autocorrelation unction o the signal PSD can also be coputed ro DFT o the signal: j e S 0 n n j e n X X S 0 n n n Delay Sω ω
Signal windowing Multiplication o signal saples by window unction saples Oten recoended or power spectru estiation Several alternatives; sall dierences in practice Haing and Hanning oten used
Windowed power spectru Signal is ultiplied saple-wise by the window unction w Then, DFT is taen and squared Windowed periodogra Window unction Signal Windowed signal S M ME w n0 n w n e i jn i 4 w4 E w M M n0 w n
Eect o windowing to PSD Convolution theore: Multiplication in tie doain corresponds to convolution in requency doain and vice versa Spectral leaing eect: due to windowing, power leas ro actual requencies to neighboring requencies Also soothes the spectru Coplicates interpretation o the spectru shape Leaing with the rectangular window Leaing with the Blacan window
Eplanation o spectral leaage in PSD P PSD W o a window unction DFT Window unction Original signal DFT i 4 w4 Windowed signal P PSD X o: t = Asinwt Multiplication in tie-doain i nwn DFT P Convolution in requency-doain X*W PSD estiate o a digital signal n
Spectru o window unction rectangular W R sin / sin / Bartlet W B sin / sin / Hanning Hanning Haing
Eect o zero padding in spectral resolution Zero padding: appending zero saples at the end o signal segent in order to ae the length or FFT Only the nuber o true signal saples deterines the spectral resolution Zero-padding only introduces interpolation o requency saples True signal saples Padded zeros 04 = 0 saples Only two spectral peas are present when dierent nubers o zeros have been padded at the end o window
Disadvantages o non-paraetric ethods require long signals or good requency resolution spectral leaage due to windowing side-lobes cause energy igration and spectral soothing ay as wea signal coponents paraetric odel-based ethods have none o these disadvantages but assue a paraetric odel o a signal such as stochastic processes: AR, MA, ARMA
Additional issues Spectru estiation with unevenly sapled signals: e.g., Lob algorith on-stationary dynaics: baseline luctuation Linear / nonlinear detrending baseline estiation and subtraction decide irst what is signal and what is noise Filter bans, wavelets, epirical ode decoposition, and soe other tie-doain decoposition techniques can be used instead i ast variation o the signal needs to be analyzed
Spectrogra estiation Copute windowed periodogra or each possibly partly overlapping signal segent Display the side-by-side in vertical orientation Shows the tie dynaics o requency content o the signal Short-Tie Fourier Transor STFT Frequency Hz 30 5 0 5 0 5 3 4 5 Tie s Frequency Hz 30 5 0 5 0 5 3 4 5 Tie s
Welch procedure or averaging periodogras The signal is segented into K consecutive partially overlapping parts o lenght M saples, spectru is coputed or each, and the average is taen Results in a soothed PSD estiate SW i M ME w n0 n w n e i jn E w M M n0 w n S W K SW i K i Segent windows: partial overlap
AR Spectru Estiate Autoregressive odel AR ro stochastic process theory A paraetric odel with order p, all-poles odel Suitable or signals with spectru with sharp peas and wide valleys Model order p should be at least the total nuber o spectral peas in order to odel the all including peas in both positive and negative requencies Oten recoended: /3 p /, is signal length in saples However, this depends on the application: prior nowledge o the spectral content is iportant For eaple: p=0 is enough or tachogra analysis even or -hour signal segents It is the ost oten used stochastic odel o signals t p a t t e t Sˆ p aˆ E p p e j
Eaple: Event related changes in spectru by tie-varying AR
Characteristics o PSD / Mean requency s : sapling requency Median requency with the largest such that Variance where is the requency saple inde corresponding to / 0 s X E 0 X E ed s 0 X E 0 / 0 s X E
Characteristics o PSD / Sewness Kurtosis Fraction o power in requency band : and are indees corresponding to requencies and, correspondingly Spectral power ratio 3 / 0 3 s X E 4 / 0 4 s X E, X E E,, 4 3 E E