Signal Processing Basics

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Nov 24, 2013 (3 years and 6 months ago)

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Dr. Paul A. Wetzel


Department of Biomedical Engineering

Virginia Commonwealth University

March 10
th
, 2004

Signal Processing Basics

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
(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


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

The resulting
complex but periodic
waveform

Summation of
sinusoidal components
results in a complex
waveform

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 10
u
V, 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

V
in

V
out

Circuit Diagram of a
First Order LPF

LPF Frequency Response

Filters
(cont)


A High Pass Filter (HPF) removes or attenuates low frequency
components from the signal.


Steady state portions decay to zero


Rapid transitions are accentuated


Acts as a differentiator

V
in

V
out

Circuit Diagram of a
First Order HPF

HPF Frequency Response

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

Transducer

Anti
-
Alaising
Filter

Amplifier


Analog
Multiplexor


Sample &
Hold

Program
Sequencer

Quantizer

Coder

A/D Converter

Digital
Output
Signal

Analog
Input
Signal

uP

Control

Additional
Analog
Signals

Analog
Output
Signal

D/A Converter

Data Recovery Filter

Amplifier

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
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


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


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

Frequency Foldover Effects Due
to
Inadequate

Sample Rate

Elimination of Foldover
Due to
Adequate

Sampling Rate

Analysis Techniques


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.

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

Ensemble Average

Signal w/Noise

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