Up/down-sampling & interpolation Centre for Doctoral Training in Healthcare Innovation

Transcription

1 Up/down-sampling & interpolation Centre for Doctoral Training in Healthcare Innovation Dr. Gari D. Clifford, University Lecturer & Director, Centre for Doctoral Training in Healthcare Innovation, Institute of Biomedical Engineering, University of Oxford

2 Downsampling Upsampling Time Frequency Interpolation HRV examples Dealing with noise and missing data ECG Derived Respiration

3 Why downsample? Either You need computational efficiency (too much data to process not enough RAM / time) Or You want to remove higher frequencies (and then don t need to process the higher frequency data)

4 Sometimes, we need to reduce the sampling rate of a signal. This is downsampling. (The clue is in the name.) A popular misnomer downsampling is (wrongly) used synonymously with decimation

5 Of course, when we downsample to a new sampling rate f s, we run the risk of lowering our sampling rate below the Nyquist-Shannon limit. This can lead to aliasing: Therefore, we should apply a low-pass filter prior to downsampling, to ensure that our signal contains no significant power above f s / 2.

6 Either Your data are missing some important feature (say the QRS complex is cut off due to low sampling) and you want to restore the feature Or You must use a model to restore data Model can be simple or linear, cubic, realistic Your data are unevenly sampled and your technique requires an evenly sampled time series (e.g. PSD estimation)

7 Downsampling is related to decimation you average out data to reduce noise or compress signal. Upsampling is a form of interpolation estimating data in missing points Add zero samples to scale time axis (leads to scaling of frequency axis by factor 1/N) Must also apply a filter to prevent aliasing! Both resampling and anti-aliasing are low-pass filters So the more restrictive filter (with the smallest bandwidth) can be used in place of both the resampling and antialias filters

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10 16Hz downsample and 256Hz upsample

11 Y = RESAMPLE(X,P,Q) resamples the sequence in vector X at P/Q times the original sample rate using a polyphase implementation: Upsample P times, downsample Q times Y is P/Q times the length of X (or the ceiling of this if P/Q is not an integer). P and Q must be positive integers. RESAMPLE applies an anti-aliasing (lowpass) FIR filter to X during the resampling process, and compensates for the filter's delay.

12 Interpolation at nodes Sample and hold (Nearest neighbour / piecewise constant interpolation) Extra high and low frequencies conservative approach Linear interpolation Extra high and low frequencies more accurate & dangerous? Polynomial interpolation Fit curves - more accurate & dangerous? Spline interpolation Better model with more constraints but unstable -> ringing Trigonometric interpolation (e.g. by the FFT) See interpft.m and appendix in slides Krigging / Interpolation via Gaussian processes Minmax interpolation Model-based (e.g. the IPFM for heart rate) Spectral analysis without interpolation (Lomb-Scargle Periodogram) See: interp1.m

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15 Invent a regularly spaced grid / define a new set of abscissae Look at actual points either side of each new abscissa Imagine a straight line between the original points New ordinate value is the value of the intersection with abscissa Simple, but not a particularly good (smooth) representation of the underlying process though

16 Interpolation using high-order polynomials often ill-behaved results. Cubic splines: cubic polynomials approximate curve between each of the N abscissae. # cubic polynomials can approximate a curve between two points, additional constraints placed on cubic polynomials result unique. The 1 st & 2 nd derivatives of each cubic polynomial constrained to abscissae. all internal cubic polynomials are well defined, with slope and curvature of approximating polynomials continuous across the abscissae.

17 Equating the right hand side of the previous two equations gives a quadratic in (x-xn) This defines the conditions for an optimal cubic polynomial fit to the data. However, since the first and last polynomials do not have adjoining cubic polynomials additional constraints must be introduced for these two. The most common approach is to adopt a not-a-knot condition. This condition forces the third derivative of the first and second cubic polynomials to be identical, Likewise for the last and second to last cubic polynomials Finally you compute derivative of these functions

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20 Hz Sine wave (6Hz sampling) * Downsampled (~1Hz) - - Linear interp -+- Cubic spline

21 Peak at 0.25Hz Sample and hold introduces lots of high frequency noise Linear interp less, but more high freq noise than cubic spline All methods introduce low frequency noise too why?

22 Hz Sine wave (6Hz sampling) * Downsampled (~1Hz) - - Linear interp -+- Cubic spline Sample and hold introduces flat regions (low freq) and corners (high freq) Linear interpolation does the same, although not as pronounced Cubic spline interpolation smoothes the edges, with only small amounts of low frequency and high frequency noise

23 If there is too much missing data, the cubic spline becomes unstable constraints become too loose Interpolation/resampling frequency has to be chosen wisely Not too high to avoid instability Too low and you will cut corners off the data Also - look out for noise in the time series remove anomalous intervals! Extrapolation outside of boundaries?

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25 Amplitude Modulation QRS height (or area) Due to observational axis changes Frequency Modulation RSA Due to parasympathetic neural modulation

26 ECG lead (top), HR (middle) and respiration (lower) How can we measure respiratory rate from this?

27 Concentrate on deriving it from peaks of ECG Form time series Need to resample why?

28 Derive features (t,x) what are they? Select resample rate what is a good rate? Remove noisy beats why? how? Select resampling interpolation method which one? Estimate respiration rate? How? Time domain - how? Frequency domain how?

29 Use your peak detector to derive respiration: From ECG peaks And from RR intervals if you have the time Compare

30 Friesen et al. (1990) Types of detector Amplitude threshold 1 st derivative 2 nd derivative Digital filters (Energy) Responses to Electromyographic noise Mains Respiration Baseline wander Composite noise

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32 Y = INTERPFT(X,N) returns a vector Y of length N obtained by interpolation in the Fourier transform of X. Assume x(t) is a periodic function of t with period p, sampledat equally spaced points, X(i) = x(t(i)) where T(i) = (i-1)*p/m, i = 1:M, M = length(x). Then y(t) is another periodic function with the same period and qy(j) = y(t(j)) where T(j) = (j-1)*p/n, j = 1:N, N = length(y). If N is an integer multiple of M, then Y(1:N/M:N) = X. Example: % Set up a triangle-like signal signal to be interpolated y = [0:.5:2 1.5:-.5:-2-1.5:.5:0]; % equally spaced factor = 5; % Interpolate by a factor of 5 m = length(y)*factor; x = 1:factor:m; xi = 1:m; yi = interpft(y,m); plot(x,y,'o',xi,yi,'*') legend('original data','interpolated data') %Compare to resample: ym = resample(y,factor,1) ;