Signal Processing Basics

Dr. Paul A. Wetzel Department of Biomedical Engineering Virginia Commonwealth University March 10 th , 2004

Concepts

• Origin of Signals • Representation of Signals • Data Acquisition • Sampling Theorem (Nyquist) • Analog to Digital Conversion • Digital to Analog Conversion • Signal Processing Techniques • Examples

Signals and Systems

(cont)

• Signals contain information that can be used to explain the underlying physiological mechanisms of a specific event or system.

• Signals must generally be acquired then analyzed to extract the desired information • Interpretation of a physical process based on observation of a signal or how a process affects the characteristic of the signal.

Characteristics of Signals

• Signals can be defined as: • Continuous – A continuum of space or time – Continuous variable functions • Discrete – Discrete points in time or space – Represented as sequences of numbers • Biological signals are almost always continuous

Characteristics of Signals

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Biological signals can be: • Deterministic – Defined by mathematical functions or rules • Periodic signals are deterministic (sums of sinusoids) • Transient signals can be deterministic • Random – Are described by statistical or distribution properties – Stationary signals remain the same over time • Statistical • Frequency spectra

Characteristics of Signals

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Periodic Sinusoid Damped Sinusoidal/Transient

Characteristics of Signals

(cont)

Real biological signals are not necessarily deterministic • Unpredictable noise • Non-stationary – Change in cardiac waveform over time • Identification of stationary segments of random signals is an important part of signal processing and pattern analysis

Time and Frequency Domain Relationships

• All signals have frequency domain components • Physiological and time domain signals can often be decomposed into a summation of sinusoidal frequency component waveforms.

• Time domain waveforms can be synthesized by proper addition of frequency domain components.

• The frequency content of a signal contributes to the shape and time domain behavior of the signal.

• Modification of a signal in the time domain will affect the frequency content of the signal in the frequency domain.

• Modification of the frequency domain components or frequency spectra will affect the shape/characteristics of a signal in the time domain.

Fourier

Analysis

Summation of sinusoidal components results in a complex waveform The resulting complex but periodic waveform

• Decomposition of a periodic signal into a sum of sinusoidal functions.

Fourier

Synthesis

• A square wave function with sharp edges can be synthesized by summing an infinite number of odd harmonic sinusoidal waveforms of differing amplitudes and phases.

Frequency Content of Biomedical Signals

• Electroencephalogram (EEG) • Frequency range: DC – 100 Hz • Electromyogram (EMG) • Frequency range: 10 – 200 Hz Signal range: Dependant on Electrode Placement • Electrocardiogram (ECG) • Frequency range: 0.05 – 200 Hz Signal range: Fetal 10uV, 5 mV Adult • Heart Rate • Frequency range: 45 – 200+ beats/min • Blood Pressure • Frequency range: DC – 200 Hz 40 – 300 mm Hg (arterial) 0 – 15 mm Hg (venous) • Breathing Rate • Frequency range: 12 – 40 breaths/min

Filters

Filters allow signals of certain frequencies to pass but attenuate the passing of other frequencies. Filters are an important aspect of signal processing.

• Low Pass Filters – pass low frequencies and attenuate high frequency components above a cutoff frequency.

• High Pass Filters – pass high frequencies and attenuate low frequency components less than the cutoff frequency.

• Band Pass Filters – pass frequencies between a lower and upper frequency cutoff point. Signals with frequencies above and below the cutoff are attenuated.

• Notch Filters – Attenuate or reject frequencies between a lower cutoff and an upper cutoff frequency.

Filters

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• A Low Pass Filter (LPF) removes or attenuates high frequency components from the signal. • Sharp transitions become more rounded and smoothed • Rapid transitions become slower LPF Frequency Response V in V out Circuit Diagram of a First Order LPF

Filters

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• A High Pass Filter (HPF) removes or attenuates low frequency components from the signal. • Steady state portions decay to zero • Rapid transitions are accentuated

HPF Frequency Response

• Acts as a differentiator V in V out Circuit Diagram of a First Order HPF

Signal Processing Basics

• Effects of measuring devices and noise on characterization and interpretation of the signal.

• Developing skills for identifying and separating the desired and unwanted components of a signal. • Uncovering the nature of the phenomena responsible for generating the signal based on identification and interpretation of an appropriate model for the signal.

• An understanding of the properties of a physical system based on a signal used to stimulate the system and the response of the system.

Data Acquisition and Recovery System

Analog Input Signal

Transducer

Analog Output Signal

Amplifier Anti-Alaising Filter Additional Analog Signals Amplifier Data Recovery Filter Analog Multiplexor Program Sequencer uP Control D/A Converter A/D Converter Sample & Hold Quantizer Coder Digital Output Signal

Data Acquisition Stages

• Transducer • Convert from some physical modality to some electrical signal • Linearity/Calibration effects • Amplifier • Linear, Logarithmic, Computational • Input impedance • Common mode rejection • Amplify signal to maximize range of A/D • Anti-Aliasing Filter (LPF) • Limit high frequency components • Noise reduction • Analog Multiplexor • Sequentially switch between multiple signal inputs • A/D Converter (Successive approximation type most common for biomedical signal acquisition) • Sample and Hold • Quantizer – Transform a CT signal into a set of discrete output states – Resolution (FSV/2 • Converter errors • Converter speed

n

) where n = number of bits – Quantization errors – minimize by increasing nit number • Coder – Process of assigning a digitally coded word to each of the output states

Sampling Theorem

• The signal must be bandlimited • When sampling a continuous time signal it must be sampled at a frequency at least twice the signal’s highest frequency component.

• The minimum sampling frequency is called the Nyquest sampling rate or the Nyquist sampling frequency.

– If the highest frequency in a signal is 50 Hz the signal must be sampled at a minimum rate of 100 Hz.

• Sampling at less than the Nyquist rate results in alaising and distortion of the signal.

– If a signal with a 100 Hz component is sampled at 150 Hz an alias frequency of 50 Hz will result.

Effects of Sampling Rate on Sinusoidal Signals

• Accuracy of waveform reproduction increases with higher sampling rates.

• A minimum of two samples per cycle are required to completely reproduce the sinusoidal waveform.

• The Nyquest theorem states that to accurately reproduce a signal: • The signal must be bandlimited • The sampling rate must be at least twice the highest frequency component of the signal.

Alaising Due to Undersampling

• • Inadequate sampling rates results in aliasing and frequency folding in the frequency domain. To minimize alaising, the input signal must be bandlimited and a sufficient sampling frequency must be used.

Antialaising Prefilters

Ideal Low Pass Prefiltering Stage

An Ideal Sampler

• The analog signal is periodically sampled every T seconds.

• Time is discretized in units of the sampling interval T. • Sampling results in a severe chopping of the original signal.

• The accuracy of reproduction of the signal is highly dependent on the sampling rate. Where f s = 1/T

Sampling Process

Ideal Sampling Practical Sampling Changes in aperture width  can lead to measurement uncertainty

Quantization of Data

Only discrete values are coded • The quantized sample can take only one of 2 b possible values • The spacing between levels is called the quantization resolution. • Quantization error is the error that results from using the quantized signal instead of the true signal.

• Quantization can be reduced by increasing resolution.

Frequency Domain Effects Due to Sampling

Elimination of Foldover Due to Adequate Sampling Rate Frequency Foldover Effects Due to Inadequate Sample Rate

Analysis Techniques

• Fourier Analysis • Spectral Analysis • Correlation • Signal Averaging/Smoothing • Differentiation/Integration • Digital Filtering • System Identification • Wavelet Analysis • Neural Networks • Fuzzy Logic

Pure 100 Hz sinusoidal signal w/o noise.

Fourier Analysis

Conditions

• Function should be periodic • Finite number of discontinuities • Finite number of min/max • Integral < infinity Fourier analysis of the signal reveals the 100 Hz component.

Noisy signal with 100 Hz component

Fourier Analysis

Fourier analysis of the signal reveals the 100 Hz component.

Ensemble Averaging

Signal w/Noise Ensemble Average Fiducial Marks • Signal to Noise Ratio (SNR) improves by • Assumptions: known  m where m is equal to the size of the ensemble average. • Signal must be repetitive but not necessarily periodic • Noise must be random and not correlated to the signal • The temporal position of each waveform must be

Ensemble Averaging