Signal Processing Basics
Dr. Paul A. Wetzel
Department of Biomedical Engineering
Virginia Commonwealth University
March 10th, 2004
Concepts
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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 (cont)
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 (cont)
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 (cont)
• 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
Vin
Vout
Circuit Diagram of a
First Order LPF
Filters (cont)
• A High Pass Filter (HPF) removes or attenuates low frequency
components from the signal.
• Steady state portions decay to zero
HPF Frequency Response
• Rapid transitions are accentuated
• Acts as a differentiator
Vin
Vout
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
A/D Converter
Transducer
Amplifier
Analog
Input
Signal
Anti-Alaising
Filter
Additional
Analog
Signals
Analog
Multiplexor
Sample &
Hold
Quantizer
Program
Sequencer
Coder
uP Control
Analog
Output
Signal
Amplifier
Data Recovery Filter
D/A Converter
Digital
Output
Signal
Data Acquisition Stages
• Transducer
• Convert from some physical modality to some electrical signal
• Linearity/Calibration effects
• Amplifier
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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/2n) 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
• Converter errors
• Converter speed
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
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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 fs = 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 2b 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
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Fourier Analysis
Spectral Analysis
Correlation
Signal Averaging/Smoothing
Differentiation/Integration
Digital Filtering
System Identification
Wavelet Analysis
Neural Networks
Fuzzy Logic
Fourier Analysis
Pure 100 Hz
sinusoidal signal
w/o noise.
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Fourier analysis of
the signal reveals
the 100 Hz
component.
Conditions
Function should be
periodic
Finite number of
discontinuities
Finite number of
min/max
Integral < infinity
Fourier Analysis
Noisy signal with
100 Hz component
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 m where m is
equal to the size of the ensemble average.
• Assumptions:
• 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
known
Ensemble Averaging