Fourier Transform Wavelet Transform and Frequency analysis in

Fourier Transform, Wavelet Transform and Frequency analysis in Biomedical Signal Processing Liang-Yu Shyu, Ph. D. BME Department of Biomedical Engineering, Chung Yuan Christian University

Introduction • Signal is a pattern of variations of some form. • The purpose of signal processing is to obtain valuable information from the signal. • In biomedical, these information include: – Time period, i. e. pre-ejection period – Characteristic feature, i. e. dicrotic notch – Specific event, i. e. p 50 in the auditory EP – Rate, etc. BME Department of Biomedical Engineering, Chung Yuan Christian University

• Some signal characteristics are easy to distinguish. • Some information are easy to obtain too. R-R interval Diastolic pressure Systolic pressure BME Department of Biomedical Engineering, Chung Yuan Christian University

• However, some characteristic and info are not easy to pint point and/or extract. For example: QRS interval R-R interval Diastolic pressure Systolic pressure sin(3 pt)+0. 3 sin(7 pt)+0. 5 sin(18 pt) Dicrotic notch sin(3 pt)+0. 3 sin(7 pt)+0. 3 sin(9 pt)+0. 5 sin(18 pt) BME Department of Biomedical Engineering, Chung Yuan Christian University

Biological Signal Measurement Control Feedback Stimulus Sensor Transducer noise BME Signal conditioning equipment Display Recording, data processing and transmission of data Department of Biomedical Engineering, Chung Yuan Christian University

Characteristic of Bio-Signal • Bio-signals are almost always continuous signals. • Bio-signals are neither pure deterministic nor pure random. • There is no standard signal, there is only reference signal. • The variation of ‘normal’ bio-signal is very large. BME Department of Biomedical Engineering, Chung Yuan Christian University

Fourier Series ak: the kth Fourier coefficient of x(t) We can synthesize any periodic signal by using a sum of harmonically related sinusoids. BME Department of Biomedical Engineering, Chung Yuan Christian University

Synthesis of a Square Wave a 0 -a 3 a 0 -a 13 a 0 -a 25 BME Gibbs phenomenon Department of Biomedical Engineering, Chung Yuan Christian University

Fourier Transform x(t) 1 -T/2 -2 p/T BME T/2 X(j. W) • x(t) is a stable, nonperiodic signal. • X(j. W) is a continuous function of W. Department of Biomedical Engineering, Chung Yuan Christian University

Continuous vs. Discrete • Continuous transformation is not suitable for digital computation. • To take the advantage of digital computing, signal must be digitized and transformation must be in discrete form. • Signal digitization must follow the sampling theorem. BME Department of Biomedical Engineering, Chung Yuan Christian University

Analog to Digital Conversion • The two main processes in A/D are sampling and quantization. • Sampling converts continuous signal into discrete sequence. Aliasing BME Department of Biomedical Engineering, Chung Yuan Christian University

Nyquist Sampling Theorem • If the sampling rate is too low, the higher frequencies are reflected into the lower frequency range. This is known as aliasing. • For a bandlimited signal, the original signal can be completely recovered without distortion if it is sampled at a rate of at least twice the highest frequency. BME Department of Biomedical Engineering, Chung Yuan Christian University

Frequency Ranges of Bio-signal • Electromyography (EMG)---- dc~10000 Hz • Electrocardiography (ECG) ----- -0. 01~250 Hz • Electroencephalography (EEG) -- dc~150 Hz • Indirect blood pressure ------ dc~60 Hz • Respiratory rate ---------- 0. 1~10 Hz • Body temperature --------- dc~0. 1 Hz BME Department of Biomedical Engineering, Chung Yuan Christian University

Quantization • Quantization digitizes the continuous amplitudes into discrete value and generates quantization error. • Most A/D approximate the discrete samples with 8, 10, 12, or 16 bits. • A quantizer with N bits is capable of representing a total of 2 N possible amplitude values. • The resolution of an A/D is determined by the full voltage range of input signal divided by the possible number of amplitude values. BME Department of Biomedical Engineering, Chung Yuan Christian University

Discrete Fourier Transform Computation of DFT is O(N 2) BME Department of Biomedical Engineering, Chung Yuan Christian University

Fast Fourier Transform • The fast Fourier transform (FFT) is a faster way to compute DFT. • The output of FFT and DFT algorithms are the same, when the input is the same. • The ratio of computing time for the FFT and DFT is FFT computing time N log N DFT computing time = 2 N 2 • The number of data samples, N, must be a power of 2, for FFT to be efficient. BME Department of Biomedical Engineering, Chung Yuan Christian University

Spectrum Estimation BME Department of Biomedical Engineering, Chung Yuan Christian University

Spectrum of ECG BME Department of Biomedical Engineering, Chung Yuan Christian University * Pictures was adapted from “Biomedical Digital Signal Processing”, by Willis J. Tompkins

Spectrum of Blood Pressure Signal noise BME Department of Biomedical Engineering, Chung Yuan Christian University

Heart Rate Variability 1/6 • In 1733, Hales reported the variation in heart rate is correlated with respiration and blood pressure. • Hyndman et al. , used spectrum analysis to study the heart rate variability, in 1975. They identified three components in the HRV spectrum: – Low frequency (0. 04 Hz): correlated with heat regulation – Mid frequency (0. 1 -0. 12 Hz): related to baro-reflector – High frequency (0. 3 Hz): respiration frequency BME Department of Biomedical Engineering, Chung Yuan Christian University Hyndman BW, Gregory JR. “Spectral analysis of sinus arrhythmia during mentall loading”. Ergonomics Vol. 18, pp. 255 -

Heart Rate Variability 2/6 • At present, it is commonly accepted that the HRV spectrum contains three major components: – The low frequency component (LF, 0. 04 -0. 15 Hz): synchronous with respiration; – The high frequency component (HF, 0. 15 -0. 4 Hz) : mediated by vagus and cardiac sympathetic nerves; – The third component: the very low frequency (VLF, 0. 01 -0. 04 Hz) component: is less defined. BME Department of Biomedical Engineering, Chung Yuan Christian University “Heart Rate Variability Standards of Measurement, Physiological Interpretation, and Clinical Use “ Circulation. 1996; 93: 1043 -1065

Heart Rate Variability 3/6 • A reduced HRV has been observed consistently in patients with cardiac failure. • Depressed HRV, after MI, may reflect a decrease in vagal activity directed to the heart, which leads to prevalence of sympathetic mechanisms and to cardiac electrical instability. • In diabetic patients without evidence of autonomic neuropathy, reduction of the absolute power of LF and HF during controlled conditions was also reported. BME Department of Biomedical Engineering, Chung Yuan Christian University “Heart Rate Variability Standards of Measurement, Physiological Interpretation, and Clinical Use “ Circulation. 1996; 93: 1043 -1065

Heart Rate Variability 4/6 A B C ECG BP filter d/dt D Threshold LP filter QRS detection BME Department of Biomedical Engineering, Chung Yuan Christian University | |

Heart Rate Variability 5/6 I 1 I 2 a a/I 2 BME t 1 I 3 I 4 b I 5 c t 2 b/I 3+c/I 4 Department of Biomedical Engineering, Chung Yuan Christian University IEEE trans Bio-med Eng, vol 33, no 9, pp 900 -904, 1986

Heart Rate Variability 6/6 LF BME HF Department of Biomedical Engineering, Chung Yuan Christian University

Shortcoming of FT • Fourier transform is not adapted to analyze information that is localized in time. • To overcome this difficulty, short-time Fourier transform (STFT) was proposed. • STFT divides time into equal-time intervals and individually analyzes them using the Fourier transform. BME Department of Biomedical Engineering, Chung Yuan Christian University

Short-Time Fourier Transform BME Department of Biomedical Engineering, Chung Yuan Christian University

HRV Using Short-Time Fourier Transform Start R-R 5 Hz re-sampling 256 point windowing Zero pad to 1024 points Shift 64 points FFT No End BME Time-frequency spectrum Yes End of signal? Department of Biomedical Engineering, Chung Yuan Christian University

HRV Using Short-Time Fourier Transform BME Department of Biomedical Engineering, Chung Yuan Christian University

BME Department of Biomedical Engineering, Chung Yuan Christian University

BME Department of Biomedical Engineering, Chung Yuan Christian University http: //www. ling. lu. se/research/speechtutorial/tutorial. html

Wavelet Transform • STFT has trade off between. W time and frequency W 4 W resolution: 0 3 W 0 – poor frequencies and good time resolution for the low 2 W frequency, 2 W 0 – good frequency and poor time resolution for the high W W 0 frequencies. 0 0 W 0 t of different size for high • Wavelet uses windows t and low frequency. • Therefore, it can improve the frequency resolution. T 1 BME T 2 T 3 2 T 1 T 2 Department of Biomedical Engineering, Chung Yuan Christian University

Wavelet Transform Where a is scale factor, b is time shift BME Department of Biomedical Engineering, Chung Yuan Christian University

Wavelets Daubechies 4 wavelet BME Harr wavelet Daubechies 20 wavelet Department of Biomedical Engineering, Chung Yuan Christian University

Discrete Wavelet Transform • Dsicrete wavelet transform (DWT) uses only discrete set of the wavelet scales and decomposes the signal into mutually orthogonal set of wavelets. • When the scaling factor of the dilation is discretized to a dyadic sequence, i. e, a {2 j} then it is known as the Dyadic Wavelet Transform. BME Department of Biomedical Engineering, Chung Yuan Christian University

Dyadic Wavelet Transform BME Department of Biomedical Engineering, Chung Yuan Christian University

Spline Wavelet Quadratic Spline Compactly Supported Wavelet BME Department of Biomedical Engineering, Chung Yuan Christian University

BME Department of Biomedical Engineering, Chung Yuan Christian University IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 51, NO. 4, APRIL 2004

Denoising • Smoothing removes high frequencies and retains low ones. • Denoising attempts to remove whatever noise is present and retain whatever signal is present regardless of the signal’s frequency content. BME Department of Biomedical Engineering, Chung Yuan Christian University

Denoising BME Department of Biomedical Engineering, Chung Yuan Christian University COMPUTING IN SCIENCE & ENGINEERING, May/June 2000

Detection of ECG Characteristic Points Using Wavelet • In 1995, Li et al. used quadratic spline wavelet with compact support and one vanishing moment to construct equivalent filters. • The QRS detection rate is 99. 85%. BME Department of Biomedical Engineering, Chung Yuan Christian University IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 42, NO. 1, JANUARY 1995

ECG Characteristic Features Extraction 1/3 N N N N P N N N S 1 f W 2 1 W 2 2 f W 2 3 f 2 4 f W 2 5 f W 2 6 f W f ECG With Low-Frequency Drift BME Department of Biomedical Engineering, Chung Yuan Christian University

ECG Characteristic Features Extraction 2/3 P N N N P N S 1 f W 22 f W 23 f W 24 f W 25 f W 26 f ECG With High-Frequency Interference BME Department of Biomedical Engineering, Chung Yuan Christian University

ECG Characteristic Features Extraction 3/3 S 1 f start abs(W 3 f ) 2 end Area Width abs(W 4 f ) 2 BME Department of Biomedical Engineering, Chung Yuan Christian University

ECG Classification Using Fuzzy Neural Network Output Layer 1 NOR 0 o Hidden Layer m Rule Layer k PVC j Membership Layer m 1 0 m 2 Input Layer BME Department of Biomedical Engineering, Chung Yuan Christian University i

ECG Classification Using Fuzzy Neural Network - Results BME Department of Biomedical Engineering, Chung Yuan Christian University

Hand Motion Identification start BME end Department of Biomedical Engineering, Chung Yuan Christian University

EMG Characteristic Features 1/2 BME Department of Biomedical Engineering, Chung Yuan Christian University

EMG Characteristic Features 2/2 BME Department of Biomedical Engineering, Chung Yuan Christian University

Hand Motion Identification Results 1 2 3 4 5 6 7 8 9 10 11 90% 92% 89% 85% 94% 97% 92% 99% 97% 90% 99% BME Department of Biomedical Engineering, Chung Yuan Christian University 93. 1 %

Hand Motion Identification EMG Envelop Length EMG 1 Threshold for EMG Envelop Energy New Information EMG 2 BME Department of Biomedical Engineering, Chung Yuan Christian University

Stationary Wavelet Transform BME Department of Biomedical Engineering, Chung Yuan Christian University

ICA Channel Reduction (a) BME (b) Department of Biomedical Engineering, Chung Yuan Christian University

Results Average Training Epochs Convergence Probability Procedures Average Discrimination Rate Without channel reduction 93. 7± 12. 5 % * 482. 3± 168. 6 * 83. 7± 11% 0 % Channel reduction using ICA 87. 9± 16. 5 %*, # 1297. 5± 804. 8 87. 5± 12. 6 *, # % 42. 8 % Channel reduction by selecting channels 1, 3, 5, 7 84. 5± 28. 6 % * 1888. 2± 2587 * 75. 1± 29. 1 %* 42. 8 % Channel reduction by selecting channels 2, 4, 6, 7 83. 7± 28. 8 % # 1792. 8± 2580 %# 71. 7± 31% 42. 8 % * BME Data Reduction Department of Biomedical Engineering, Chung Yuan Christian University

Future Health Care BME Department of Biomedical Engineering, Chung Yuan Christian University http: //lifeguard. stanford. edu/lifeguard_writeup_medium. pdf

Future Health Care In the future, when the health care hardware becomes wearable, the signal processing techniques need to be fast, efficiency and robust in order to achieve real-time monitoring. BME Department of Biomedical Engineering, Chung Yuan Christian University http: //darbelofflab. mit. edu/ring_sensor. htm

Thank you for your attention! BME Department of Biomedical Engineering, Chung Yuan Christian University

Bioelectric Phenomena BME Department of Biomedical Engineering, Chung Yuan Christian University “Color Atlas of Physiology, ” Wolf-Rudiger Gay and Barbara Gay Action potential

BME Department of Biomedical Engineering, Chung Yuan Christian University Propagation of action potential

BME Department of Biomedical Engineering, Chung Yuan Christian University “醫用電子學，”黃豪銘 Electro and skin interface

Electrocardiography (ECG) Limb leads BME Chest leads Department of Biomedical Engineering, Chung Yuan Christian University “Medical Instrumentation, ” 3 ed, John G. Webster Standard leads

BME Department of Biomedical Engineering, Chung Yuan Christian University “How to Quickly and Accu 5 rately Master ECG Interpretation, ” Dale Davis 12 Leads ECG

BME Department of Biomedical Engineering, Chung Yuan Christian University “Color Atlas of Physiology, ” Wolf-Rudiger Gay and Barbara Gay ECG