Biomedical Instrumentation Signals and Noise Chapter 5 in Introduction to Biomedical Equipment Technology By Joseph Carr and John Brown.

Biomedical Instrumentation

Signals and NoiseChapter 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 =

• Inverse Fourier =

deFtf tj

)(2

1)(

dtetfF tj

)(

2

1)(

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

Sine Wave in time domain f(t) = sin(23t)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Magnitude

Time (Sec)

Time (sec) 1 sec0 1 2 3 4 5 6 7 8

1

Frequency (Hz)

Eg. v = Vm sin(2ωt)

Every Signal can be described as a series of sinusoids

Signal with DC Component

)sin()sin()( ttty 32322341

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

• range of frequencies contained in signal Absolute bandwidth

• width of spectrum Effective bandwidth

• 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 boards are at least

12 bits or 212 = 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

Red = magnitudeBlack = timing interval

Quantization Noise

20 levels

4 levels

Red = magnitudeBlack = timing interval

Nyquist Sampling Theorem Error in Signals

1 Sec

30 samples / 1 sec = 30 Hertz 10 samples / 1 sec = 10 Hertz

1 Sec

Signal that is digitized into computer 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)

-Fm 0 Fm Fs-Fm Fs Fs+ 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.

where• Vn = noise Voltage (V)• k = Boltzman’s constant

• Boltzman’s constant = 1.38 x 10 -23Joules/Kelvin

• T = temperature in Kelvin• R = resistance in ohms (Ώ)• B = Bandwidth in Hertz (Hz)

kTRBVn 42

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)

(1KHz)

• = 1.6 x 10-14 V2

• Vn = 1.26 x10 –7 V = 0.126 uV

Eg of Thermal Noise

• Vn = 4 (R/1Kohm) ½ units nV/(Hz)1/2

• Given R = 1 M find noise

• Vn = 4 (1 x 106 / 1x 103) ½ units nV/ (Hz) ½

• = 126 nV/ (Hz) ½

• Given BW = 1000 Hz find Vn with units of V

• Vn = 126 nV/ (Hz) ½ * (1000 Hz)1/2 = 400 nV = 0.4 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

where

• In = noise current (amps)

• q = elementary electric charge

• = 1.6 x 10-19 Coulombs

• I = Current (amp)

• B = Bandwidth in Hertz (Hz)

qIBIn 22 qIBIn 2

Eg: Shot Noise

Given I = 10 mA Given B = 100 Hz to 1200 Hz = 1100 Hz In

2= 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

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

kTBG

NkTBGFn

Noise Factor

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

• Where FN = total noise • Fn = noise factor at stage n ; • G(n-1) = Gain at stage n-1

12121

3

1

21 ...

1...

11

n

nN GGG

F

GG

F

G

FFF

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.

• Note that the book has a typo in equation 5-27 where Gn should be G(n-1)

5.12

10

1612

NF

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

G1

β

Σ+

+Vin E Vo

11

1

111

11

11

1

1

G

G

V

V

VGGV

VGVGV

VGVGV

VVGV

VVE

EGV

in

o

ino

inoo

oino

oino

oin

o

Use of Feedback to reduce Noise

G1 G2ΣΣ

Β

Vin VoV1 V1G1

Vn = Noise

V2 V2G2

B Vo+

+ +

22

12

112

1

GVV

VGVVV

VGVV

VVV

o

noin

n

oin

Use of Feedback to reduce Noise

G1 G2ΣΣ

Β

Vin VoV1 V1G1

Vn = Noise

V2 V2G2

B Vo+

+ +

nino

ninoo

noino

noino

VGVGGGGV

VGVGGVGGV

VGVGGVGGV

GVGVVV

221211

22121

22121

21

Use of Feedback to reduce Noise

Thus Vn is reduced by Gain G1

Note Book forgot V in equation 5-35

G1 G2ΣΣ

Β

Vin VoV1 V1G1

Vn = Noise

V2 V2G2

B Vo+

+ +

1211

21

211

2

1

1

211

21

211

221

G

VV

GG

GGV

GG

VG

G

G

GG

VGGV

GG

VGVGGV

nino

nino

nino

Derivation:

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• Sn = 20 log (Vin/Vn) = 20 log (5uV/100uV) = -26dB

Processing SNR• Ave Sn = 20 log (Vin/Vn/N1/2)

= 20 log (5u/100u / (1000)1/2) = 4 dB Processing gain = 4 – (- 26) = 30 dB

Noise Reduction by Signal Averaging

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