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 = Inverse Fourier =

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(2  3t) Time (sec) 1 sec Frequency (Hz) Eg. v = V m sin(2ωt)

Every Signal can be described as a series of sinusoids

Signal with DC Component

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 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 2 4 or 16 level board Most boards are at least 12 bits or 2 12 = 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 = magnitude Black = timing interval Quantization Noise 20 levels

4 levels Red = magnitude Black = 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

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 FmFs-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 V n = noise Voltage (V) k = Boltzman’s constant Boltzman’s constant = 1.38 x Joules/  Kelvin T = temperature in  Kelvin R = resistance in ohms (Ώ) B = Bandwidth in Hertz (Hz)

Eg. of Thermal Noise Given R = 1Kohm Given B = 2 KHz to 3 KHz = 1 KHz Assume: T = 290K (room Temperature) V n 2 = 4KTRB units V 2 V n 2 = (4) (1.38 x 10 –23 J/K) (290K) (1 Kohm) (1KHz) = 1.6 x V 2 V n = 1.26 x10 –7 V = uV

Eg of Thermal Noise V n = 4 (R/1Kohm) ½ units nV/(Hz) 1/2 Given R = 1 M  find noise V n = 4 (1 x 10 6 / 1x 10 3 ) ½ units nV/ (Hz) ½ = 126 nV/ (Hz) ½ Given BW = 1000 Hz find V n with units of V V n = 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 I n = noise current (amps) q = elementary electric charge = 1.6 x Coulombs I = Current (amp) B = Bandwidth in Hertz (Hz)

Eg: Shot Noise Given I = 10 mA Given B = 100 Hz to 1200 Hz = 1100 Hz I n 2 = 2q I B = = 2 (1.6 x 10 –19 Coulomb) ( 10 X10 –3 A)(1100 Hz) = 3.52 x10 –18 A 2 In = (3.52 x10 –18 A 2 ) 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 P n P n = kTB, where P n =noise power in watts k = Boltzman’s constant Boltzman’s constant = 1.38 x Joules/  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 = (Vn 2 +(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

F n = P no /P ni evaluated at T = 290 o K (room temperature) where P no = noise power output and P ni = noise power input Noise Factor

P ni =kTBG where G = Gain; T = Standard Room temperature = 290 o K K = Boltzmann’s Constant = 1.38 x J/ o K B = Bandwidth (Hz) Noise Factor

P no = kTBG + ΔN where ΔN = noise added to system by network or amplifier 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) F n = 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 F N = total noise F n = 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. 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 G1 β Σ + +Vin E Vo

Use of Feedback to reduce Noise G1G2Σ Σ Β Vin VoV1 V1G1 Vn = Noise V2V2G2 B Vo + + +

Use of Feedback to reduce Noise G1G2Σ Σ Β Vin VoV1 V1G1 Vn = Noise V2V2G2 B Vo + + +

Use of Feedback to reduce Noise Thus Vn is reduced by Gain G1 Note Book forgot V in equation 5-35 G1G2Σ Σ Β Vin VoV1 V1G1 Vn = Noise V2 V2G2 B Vo Derivation:

Un processed SNR S n =20 log (V in /V n ) Processed SNR Ave S n = 20 log (V in /V n / N 1/2 ) Where SNR S n = unprocessed SNR SNR Ave S n = time averaged SNR N = # repetitions of signals V in = Voltage of Signal V n = Voltage of Noise Processing Gain = Ave S n – S n 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/N 1/2 ) = 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 1 st 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