Short-time FFT, Multi-taper analysis & Filtering in SPM12

Transcription

1 Short-time FFT, Multi-taper analysis & Filtering in SPM12 Computational Psychiatry Seminar, FS 2015 Daniel Renz, Translational Neuromodeling Unit, ETHZ & UZH

2 Overview Refresher Short-time Fourier transform Filter design Multitapering Filtering of EEG data in SPM12 Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 2

3 Recap: Fourier Transform (FT) Our goal: To analyze event related signal changes in the time-frequency domain. FT expands a function in terms of sinusoids (or complex exponentials) to fully reveal all frequency components. FT does not have any time resolution. Cohen TalkOSCON/blob/master/images/prism.png Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 3

4 Recap: Fourier Transform (FT) _transform_time_and_frequency_domains_%28small%29.gif Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 4

5 Recap: Inner product (dot product) Measures covariance / similarity / non-orthogonality between two vectors (the inner product of orthogonal vectors is zero). Can be extended to functions. Any two sinusoids of different frequency are orthogonal. This means that sinusoids can be used as basis functions. We can think of Fourier transformation at frequency ω as the inner product of the signal with a complex-valued oscillation. As the basis functions are orthogonal, the Fourier-transform gives a perfect representation of the frequency spectrum. How to extend this to have temporal resolution? ge/image/dot_product_projection _unit_vector.png Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 5

6 How to get a time-frequency representation Model-free non-parametric methods: Short-time Fourier, Wavelet, Hilbert. We can view all three methods as convolutions with complex oscillatory kernels [Bruns 2003]. Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 6

7 How to get a time-frequency representation Model-free non-parametric methods: Short-time Fourier, Wavelet, Hilbert. We can view all three methods as convolutions with complex oscillatory kernels [Bruns 2003]. Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 7

8 Recap: Convolution We can think of convolution as the extension of the inner product (dot product) to incorporate timelags. Compute the inner product of signal with kernel (window) at beginning of signal, then move across the signal, computing products on the way. Outcome: time series of one signal weighted by another signal that slides along the first signal. repeat Cohen 2014 Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 8

9 Recap: Convolution Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 9

10 Recap: Convolution Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 10

11 Short-time Fourier Transform (STFT) Segment signal in narrow time segments. Taper (apply window on segment) each segment. Apply Fourier transform to each segment. repeat Cohen 2014 Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 11

12 Short-time Fourier Transform (STFT) What shape should the window function have? Boxcar, Gaussian,... How wide should it be? Should be narrow enough to consider windowed signal stationary. Narrow windows do not offer good localization in the frequency domain. Uncertainty principle: There is always a trade-off. Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 12

13 Windows in STFT Should not use boxcar function as a window, as this will lead to unwanted artifacts. The discontinuities at the ends of the signal are the problem. Idea: smooth signal to zero at the end points. Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 13

14 Windows in STFT Should not use boxcar function as a window, as this will lead to unwanted artifacts. Many possible windows, e.g. Hann, Hamming, Gaussian. A Gaussian has the same localization in time and frequency domain. Often Hann is preferred, because it tapers the data fully to zero. Use overlapping segments to compensate for signal loss (often around 90% overlap). Cohen 2014 Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 14

15 Recap: Wavelets Complex Morlet wavelet: Complex-valued oscillation multiplied by Gaussian-shaped envelope. Variance of Gaussian envelope scales with frequency of oscillation. The wavelet is convolved with the signal of interest. This effectively implements a band-pass filter. No fundamental difference to STFT with Gaussian window, but more convenient to use if frequency-dependent temporal resolution is required. Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 15

16 Recap: Hilbert analysis Window on frequencies only (via band-pass filter). Then extend the signal to the complex plane (analytic signal) via Hilbert transform. One can show that this, too, is equivalent to convolution of the signal with a complex oscillatory kernel [Bruns 2003]. In practice, Hilbert analysis allows for direct control over filter properties as compared to wavelets. But it s also more complicated. Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 16

17 Comparing STFT, Wavelet and Hilbert Formally, they are very close (convolutions with complex-valued kernels). Practical differences: Computation speed, ease-of-use Conventional shape of kernel: Gaussian for Wavelets, Hamming for STFT, the equivalent of an ideal bandpass impulse response for Hilbert. Bruns 2004: Fourier with Hamming window, Hilbert with ideal bandpass filter (smooth flanks) and complex Morlet wavelet. Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 17

18 Comparing STFT, Wavelet and Hilbert The interpretation of power is the same for STFT, filter-hilbert and wavelet convolution. The interpretation of phase changes Hilbert / Wavelet methods give instantaneous phase (phase at each time point). STFT gives one phase value per time window, referring to the phase of the sinusoid in that window. STFT phase in window Cohen 2014 Hilbert / Wavelet phase time series Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 18

19 Recap: Filtering We can think of filtering as a clever kind of averaging performed as a convolution with a filter s impulse reponse function (Thus, STFT can also be viewed as a filtering procedure). And this is equivalent to multiplication in the frequency domain. Slides from J. Heinzle Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 19

20 Recap: Filtering Can use filters to cut out a band of frequencies or cut out everything but a band of frequencies. But not perfectly: Precise cutoffs in the time domain (box-car function!) result in problems in the frequency domain, and vice versa. Slides from J. Heinzle Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 20

21 Spectral Leakage Steep slopes in time domain cause low main-lob width (good!) and high side-lobes (bad!) in the frequency domain. Fundamental trade-off between main-lobe width and side-lobe amplitude. Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 21

22 Multitapering Idea: Apply several tapers (window functions) that have slightly different temporal characteristics and then average the power spectra. This increases the SNR, so it s useful for situations of low SNR ratio, such as higherfrequency activity or single-trial analysis. Cohen 2014 Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 22

23 Multitapering: Slepian tapers If we use orthogonal tapers, windowed signals provide independent spectral estimates. This reduces estimation bias of power. Discrete prolate spheroidal sequence (DPSS, also known as Slepian sequence) is the orthogonal sequence that has the optimal main lobe energy concentration (main lobe energy/ total energy). Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 23

24 Multitapering More tapers means more smoothing. How much smoothing is appropriate? Trade-Off between SNR and accuracy. In MATLAB, the function dpss will compute a number of tapers, based on the desired spectral smoothing and the length of the time segment. [desired segment length] * [frequency bandwidth of main lobe] = 2* [number of tapers]. Should only use tapers with energy concentration > 90%. Good official tutorial: Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 24

25 Filter design Can optimize two out of three requirements: filter order (lower=better), main lobe width (narrower=better) and side-lobe amplitude (lower=better). Filter order refers to the complexity of the filter. In FIR filters, it specifies the number of time points in the signal that are used to compute the filter output (so higher order = more temporal smoothing). In IIR filters, it s more complicated, but higher order filters can quickly get unstable. Butterworth filter IIR filter that is maximally flat in the pass-band (no ripples). This means that the transition band is really wide. Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 25

26 Summary By Multitapering we mean taking the average of several STFT on the same signal with orthogonal windows. This improves SNR at the cost of more smoothing. STFT, Wavelet and Hilbert can all be thought of as convolving the signal with complex oscillatory kernels. Which one you want to use depends on your requirements STFT by default has a fixed resolution across frequencies and enables multitapering Wavelets provide higher temporal resolutions for higher frequencies. Hilbert is flexible, as it gives complete control over filter design. Designing a good filter is difficult and always depends on your specific goals. The butterworth filter is a good general-purpose choice, as it has a flat pass-band. Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 26

27 Filtering of EEG data in SPM12 Daniel Renz, Short-time FFT, Multi-tapering & Filtering in SPM12 27