Download Biomedical Instrumentation

Biomedical
Instrumentation
Signals and Noise
Chapter 5 in
Introduction to Biomedical Equipment
Technology
By Joseph Carr and John Brown
Types of Signals

Signals can be represented in time or
frequency domain
Types of Time Domain Signals





Static = unchanging over long period of time essentially a
DC signal
Quasistatic = nearly unchanging where the signal
changes so slowly that it appears static
Periodic Signal = Signal that repeats itself on a regular
basis ie sine or triangle wave
Repetitive Signal = quasi periodic but not precisely
periodic because f(t) /= f(t + T) where t = time and T =
period ie is ECG or arterial pressure wave
Transient Signal = one time event which is very short
compared to period of waveform
Types of Signals:






A. Static = non-changing
signal
B. Quasi Static = practically
non-changing signal
C. Periodic = cyclic pattern
where one cycle is exactly the
same as the next cycle
D. Repetitive = shape of the
cycle is similar but not
identical (many BME signals
ECG, blood pressure)
E. Single-Event Transient =
one burst of activity
F. Repetitive Transient or
Quasi Transient = a few
bursts of activity
Fourier Series

All continuous periodic signals can be
represented as a collection of harmonics of
fundamental sine waves summed linearly.
• These frequencies make up the Fourier Series

Definition
•
•
Fourier = F ( ) 
1
2


f (t )e  jt dt

1
Inverse Fourier = f (t ) 
2

 jt
F
(

)
e
d


Eg. v = Vm sin(2ωt)




v = instantaneous amplitude of sin wave
Vm = Peak amplitude of sine wave
ω = angular frequency = 2π f
T = time (sec)
Fourier Series found using many frequency selective
filters or using digital signal processing algorithm known
as FFT = Fast Fourier Transform
1
0.8
0.6
1
0.4
0.2
Magnitude

0
-0.2
-0.4
-0.6
-0.8
-1
0
0.1
0.2
0.3
0.4
0.5
Time (Sec)
0.6
0.7
0.8
0.9
1
Time (sec) 1 sec
Sine Wave in time domain f(t) = sin(23t)
0 1 2 3 4 5 6 7 8
Frequency (Hz)
Every Signal can be described as a series
of sinusoids
Signal with DC Component
y(t )  1  4 3 sin( 2t )  2 3 sin( 2 3t )
Time vs Frequency Relationship



Signals that are infinitely continuous in the
frequency domain (nyquist pulse) are finite in
the time domain
Signals that are infinitely continuous in the time
domain are finite in the frequency domain
Mathematically, you cannot have a finite time
and frequency limited signal
Time vs Frequency
Spectrum & Bandwidth



Spectrum
•
Absolute bandwidth
•
width of spectrum
Effective bandwidth
•
•
•

range of frequencies contained in signal
Often just bandwidth
Narrow band of frequencies containing most of the energy
Used by Engineers to gain the practical bandwidth of a signal
DC Component
•
Component of zero frequency
Biomedical Examples of
Signals

ECG vs Blood Pressure
•
•
Pressure Waveform has a slow
rise time then ECG thus need
less harmonics to represent the
signal
Pressure waveform can be
represented in with 25
harmonics whereas ECG needs
70-80 harmonics
ECG
Biomedical Examples of Signals

Square wave theoretically has infinite number of
harmonics however approximately 100 harmonics
approximates signal well
Time (sec)
Odd or Even Function
Even function when f(t) = f(-t)
Odd function –f(t) = f(-t)
Analog to Digital Conversion


Digital Computers cannot accept Analog
Signal so you need to perform and Analog to
digital Conversion (A/D conversion)
Sampled signals are not precisely the same
as original.
•
The better the sampling frequency the better the
representation of the signal

Two types of error with digitalization.
• Sampling Error
• Quantization Error
Sampling Rate

Sample Rate must follow Nyquist’s
theorem.
• Sample rate must be at least 2 times the
maximum frequency.
Quantization Error

When you digitize the
signal you do so with
levels based on the
number of bits in your
DAC (data acquisition
board)
• Example is of a 4 bit 24 or
16 level board
• Most boards12 are
at least
12 bits or 2 = 4096 levels
• The “staircase” effect is
call the quantization noise
or digitization noise
Quantization Noise

Quantization noise = difference from
where analog signal actually is to where
the digitization records the signal
Quantization Noise
20 levels
Red = magnitude
Black = timing interval
4 levels
Red = magnitude
Black = timing interval
Nyquist Sampling Theorem Error
in Signals
1 Sec
30 samples / 1 sec = 30 Hertz
Signal that is digitized into computer
1 Sec
10 samples / 1 sec = 10 Hertz
Signal that is digitized into computer
Spectral Information: Sampling
when Fs > 2Fm

Sampling is a form of amplitude
modulation
• Spectral Information appears not only
around fundamental frequency of carrier but
also at harmonic spaced at intervals Fs
(Sampling Frequency)
-Fs-Fm -Fs
-Fs+ Fm
-Fm
0
Fm
Fs-Fm
Fs
Fs+ Fm
Spectral Information: Sampling
when Fs < 2Fm

Aliasing occurs when Fs< 2Fm where
you begin to see overlapping in
frequency domain.
-Fm
0
Fm

Problem: if you try to filter the signal you
will not get the original signal
• Solution use a LPF with a cutoff frequency to
•

pass only maximum frequencies in waveform
Fm not Fs
Set sampling Frequency Fs >=2Fm
Shows how very fast sampled frequency
if sampled incorrectly can be a slower
frequency signal
Noise

Every electronic component has noise
• thermal noise
• shot noise
• distribution noise (or partition noise)
Thermal Noise





Thermal noise due to agitation of
electrons
Present in all electronic devices and
transmission media
Cannot be eliminated
Function of temperature
Particularly significant for satellite
communication
thermal noise

thermal noise is caused by the thermal
motion of the charge carriers; as a result
the random electromotive force appears
between the ends of resistor;
Johnson Noise, or Thermal Noise, or
Thermal Agitation Noise

Also referred to as white noise because of
gaussian spectral density.

2
Vn
 4kTRB where
• Vn = noise Voltage (V)
• k = Boltzman’s constant
• Boltzman’s constant = 1.38 x 10
• T = temperature in Kelvin
• R = resistance in ohms (Ώ)
• B = Bandwidth in Hertz (Hz)
-23Joules/Kelvin
Eg. of Thermal Noise
• Given R = 1Kohm
• Given B = 2 KHz to 3 KHz = 1 KHz
• Assume: T = 290K (room Temperature)
• Vn2 = 4KTRB units V2
• Vn2= (4) (1.38 x 10 –23J/K) (290K) (1 Kohm)
•
•V
(1KHz)
= 1.6 x 10-14 V2
–7 V = 0.126 uV
n = 1.26 x10
Eg of Thermal Noise
• V = 4 (R/1Kohm) ½ units nV/(Hz)1/2
• Given R = 1 MW find noise
• V = 4 (1 x 106 / 1x 103) ½ units nV/ (Hz) ½
• = 126 nV/ (Hz) ½
• Given BW = 1000 Hz find V with units of V
• V = 126 nV/ (Hz) ½ * (1000 Hz)1/2 = 400 nV = 0.4
n
n
n
n
uV
Shot noise

Shot noise appears because the current
through the electron tube (diode, triode
etc.) consists of the separate pulses
caused by the discontinuous electrons;
• This effect is similar to the specific sound
when the buckshot is poured out on the floor
and the separate blows unite into the
continuous noise;
Shot Noise

Shot Noise: noise from DC current flowing
in any conductor

2
In
•
•
•
•
•
 2qIB
where
In = noise current (amps)
q = elementary electric charge
= 1.6 x 10-19 Coulombs
I = Current (amp)
B = Bandwidth in Hertz (Hz)
I n  2qIB
Eg: Shot Noise



Given I = 10 mA
Given B = 100 Hz to 1200 Hz = 1100 Hz
In2= 2q I B =
= 2 (1.6 x 10 –19Coulomb) ( 10 X10 –3A)(1100 Hz)
= 3.52 x10 –18 A2
In = (3.52 x10–18 A2) ½ = 1.88 nA
Noise cont

Flicker Noise also known as Pink Noise or 1/f noise
is the lower frequency < 1000Hz phenomenon and
is due to manufacturing defects
• A wide class of electronic devices demonstrate so
called flicker effect or wobble (=trembling), its
intensity depends on frequency as 1/f, ~1, in the
wide band of frequencies;
• For example, flicker effect in the electron tubes is
caused by the electron emission from some
separate spots of the cathode surface, these spots
slowly vary in time; at the frequencies of about 1
kHz the level of this noise can be some orders
higher then thermal noise.
distribution noise

Distribution noise (or partition noise)
appears in the multi-electrode devices
because the distribution of the charge
carriers between the electrodes bear
the statistical features;
Signal to Noise Ratio = SNR
 SNR
= Signal/ Noise
• Minimum signal level detectable
at the output of an amplifier is the
level that appears above noise.
Signal to Noise Ratio = SNR
 Noise
Power Pn
• Pn = kTB, where
•Pn =noise power in watts
•k = Boltzman’s constant
• Boltzman’s constant = 1.38 x 10 -23Joules/Kelvin
• T = temperature in Kelvin
• B = Bandwidth in Hertz (Hz)
Internal and External Noise



Internal Noise
External Noise
Total Noise Calculation
Internal Noise

Internal Noise: Caused by thermal
currents in semiconductor material
resistances and is the difference between
output noise level and input noise level
External Noise

External Noise: Noise produced by
signal sources also called source noise;
cause by thermal agitation currents in
signal source
External Noise

Total Noise Calculation = square root of
sum of squares Vne = (Vn2+(InRs)2) ½
necessary because otherwise positive
and negative noise would cancel and
mathematically show less noise that what
is actually present
Noise Factor

Noise Factor = ratio of noise from real
resistance to thermal noise of an ideal
resistor
Noise Factor

Fn = Pno/Pni evaluated at T = 290oK (room
temperature) where
• Pno = noise power output and
• Pni = noise power input
Noise Factor

Pni =kTBG where
• G = Gain;
• T = Standard Room temperature = 290oK
• K = Boltzmann’s Constant = 1.38 x10-23J/oK
• B = Bandwidth (Hz)
Noise Factor

Pno = kTBG + ΔN where
• ΔN = noise added to system by network or
amplifier

kTBG  N 
Fn 

kTBG
N
kTBG
Noise Figure


Noise Figure : Measure of how close
is an amplifier to an ideal amplifier
NF = 10 log (Fn) where
• NF = Noise Figure (dB)
• Fn = noise factor (previous slide)
Noise Figure

Friis Noise Equation: Use when you have a
cascade of amplifiers where the signal and
noise are amplified at each stage and each
component introduces its own noise.
• Use Friis Noise Equation to calculated total Noise
Fn  1
F2  1 F3  1
FN  F1 

 ... 
G1
G1G2
G1G2 ...Gn1
• Where FN = total noise
• Fn = noise factor at stage n ;
• G(n-1) = Gain at stage n-1

Example: Given a 2 stage amplifier where A1
has a gain of 10 and a noise factor of 12 and
A2 has a gain of 5 and a noise factor of 6.
FN
•

6  1
 12 
 12.5
10
Note that the book has a typo in equation 5-27
where Gn should be G(n-1)
Noise Reduction Strategies
1. Keep source resistance and amplifier input
resistance low (High resistance with increase
thermal noise)
2. Keep Bandwidth at a minimum but make
sure you satisfy Nyquist’s Sampling Theory
3. Prevent external noise with proper ground,
shielding, filtering
4. Use low noise at input stage (Friis Equation)
5. For some semiconductor circuits use the
lowest DC power supply
Feedback Control Derivation
Vo  G1  E
Vin
+
E
Σ
G1
+
β
Vo
E  Vin  Vo
Vo  G1Vin  Vo 
Vo  G1Vin  G1Vo
Vo  G1Vo  G1Vin
Vo 1  G1   G1Vin
Vo
G1

Vin 1  G1 
Use of Feedback to reduce Noise
Vn = Noise
Vin +
Σ
+
V1
B Vo
V1G1 +
G1
Σ
V2
G2
V2G2 Vo
Β
V 1  Vin  Vo
V 2  V 1G1  Vn
V 2  Vin  Vo G1  Vn
Vo  V 2G 2
Use of Feedback to reduce Noise
Vn = Noise
Vin +
Σ
+
V1
B Vo
V1G1 +
G1
Σ
V2
G2
V2G2 Vo
Β
Vo  Vin  Vo G1  Vn G 2
Vo  G1G 2Vin   G1G 2 Vo   G 2Vn 
Vo  G1G 2 Vo  G1G 2Vin  G 2Vn
Vo 1  G1G 2    G1G 2Vin  G 2Vn
Use of Feedback to reduce Noise
Derivation:
Vn = Noise
Vin +
Σ
+
V1
B Vo
V1G1 +
G1
Σ
V2
G2
Β
Thus Vn is reduced by Gain G1
Note Book forgot V in equation 5-35
Vo 
V2G2
Vo
G1G 2Vin  G 2Vn 
1  G1G 2  

G1G 2Vin
G 2Vn
G1 
Vo 

1  G1G 2   G1  1  G1G 2  
Vo 
Vn 
G1G 2

V

1  G1G 2    in G1 
Noise Reduction by Signal Averaging


Un processed SNR Sn =20 log (Vin/Vn)
Processed SNR Ave Sn = 20 log (Vin/Vn/ N1/2)
• Where
• SNR Sn = unprocessed SNR
• SNR Ave Sn = time averaged SNR
• N = # repetitions of signals
• Vin = Voltage of Signal
• Vn = Voltage of Noise

Processing Gain = Ave Sn – Sn in dB
Noise Reduction by Signal Averaging

Ex: EEG signal of 5 uV with 100 uV of
random noise
• Find the unprocessed SNR, processed SNR
with 1000 repetitions and the processing Gain
Noise Reduction by Signal Averaging

Unprocessed SNR

Processing SNR
• Sn = 20 log (Vin/Vn) = 20 log (5uV/100uV) = -26dB
• Ave Sn = 20 log (Vin/Vn/N1/2)
= 20 log (5u/100u / (1000)1/2) = 4 dB

Processing gain = 4 – (- 26) = 30 dB
Review
 Types of Signals (Static, Quasi Static,
Periodic, Repetitive, Single-Event
Transient, Quasi Transient)
 Time vs Frequency
•
•
•

Fourier
Bandwidth
Alaising
Sampled signals: Quantization, Sampling
and Aliasing
Review
 Noise:Johnson, Shot, Friis Noise
 Noise Factor vs Noise Figure
 Reduction of Noise via
•
•
•
5 different Strategies {keep resistor values
low, low BW, proper grounding, keep 1st
stage amplifier low (Friis Equation),
semiconductor circuits use the lowest DC
power supply}
Feedback
Signal Averaging
Homework




Read Chapter 6
Chapter 3 Problems: #16, 17, 21
Chapter 4 Questions and Problems: # 5, 18,
19, 21, 22
Chapter 5 Homework Problems: 4, 6, 7, 8,
10, 11, 12, 13