Biomedical Instrumentation Signals and Noise Chapter 5 in

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

Types of Signals l Signals can be represented in time or frequency domain

Types of Time Domain Signals l l l 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: l l l 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 l 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 l Definition • Fourier = • Inverse Fourier =

Eg. v = Vm sin(2ωt) l l l 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 Time (sec) 1 sec Sine Wave in time domain f(t) = sin(2 3 t) 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

Time vs Frequency Relationship l l l 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 l l l Spectrum • Absolute bandwidth • width of spectrum Effective bandwidth • • • l 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 l 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 l 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 l l 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

l Two types of error with digitalization. • Sampling Error • Quantization Error

Sampling Rate l Sample Rate must follow Nyquist’s theorem. • Sample rate must be at least 2 times the maximum frequency.

Quantization Error l 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 12 are at least 12 bits or 2 = 4096 levels • The “staircase” effect is call the quantization noise or digitization noise

Quantization Noise l 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 > 2 Fm l 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+ Fm -Fm 0 Fm Fs-Fm Fs Fs+ Fm

Spectral Information: Sampling when Fs < 2 Fm l Aliasing occurs when Fs< 2 Fm where you begin to see overlapping in frequency domain. -Fm 0 Fm

l Problem: if you try to filter the signal you will not get the original signal • Solution use a LPF with a cutoff frequency to • l pass only maximum frequencies in waveform Fm not Fs Set sampling Frequency Fs >=2 Fm Shows how very fast sampled frequency if sampled incorrectly can be a slower frequency signal

Noise l Every electronic component has noise • thermal noise • shot noise • distribution noise (or partition noise)

Thermal Noise l l l 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 l thermal noise is caused by thermal motion of the charge carriers; as a result the random electromotive force appears between the ends of resistor;

Johnson Noise, or Thermal Agitation Noise l l 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 • T = temperature in Kelvin • R = resistance in ohms (Ώ) • B = Bandwidth in Hertz (Hz) -23 Joules/ Kelvin

Eg. of Thermal Noise • Given R = 1 Kohm • Given B = 2 KHz to 3 KHz = 1 KHz • Assume: T = 290 K (room Temperature) • Vn 2 = 4 KTRB units V 2 • Vn 2= (4) (1. 38 x 10 – 23 J/K) (290 K) (1 Kohm) • • V (1 KHz) = 1. 6 x 10 -14 V 2 – 7 V = 0. 126 u. V n = 1. 26 x 10

Eg of Thermal Noise • V = 4 (R/1 Kohm) ½ units n. V/(Hz)1/2 • Given R = 1 MW find noise • V = 4 (1 x 106 / 1 x 103) ½ units n. V/ (Hz) ½ n • n = 126 n. V/ (Hz) ½ • Given BW = 1000 Hz find Vn with units of V • Vn = 126 n. V/ (Hz) ½ * (1000 Hz)1/2 = 400 n. V = 0. 4 u. V

Shot noise l 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 l Shot Noise: noise from DC current flowing in any conductor l • • • where In = noise current (amps) q = elementary electric charge = 1. 6 x 10 -19 Coulombs I = Current (amp) B = Bandwidth in Hertz (Hz)

Eg: Shot Noise l l l Given I = 10 m. A Given B = 100 Hz to 1200 Hz = 1100 Hz In 2= 2 q I B = = 2 (1. 6 x 10 – 19 Coulomb) ( 10 X 10 – 3 A)(1100 Hz) = 3. 52 x 10 – 18 A 2 In = (3. 52 x 10– 18 A 2) ½ = 1. 88 n. A

Noise cont l Flicker Noise also known as Pink Noise or 1/f noise is the lower frequency < 1000 Hz 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 k. Hz the level of this noise can be some orders higher then thermal noise.

distribution noise l 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 l 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 l Noise Power Pn • Pn = k. TB, where • Pn =noise power in watts • k = Boltzman’s constant • Boltzman’s constant = 1. 38 x 10 -23 Joules/ Kelvin • T = temperature in Kelvin • B = Bandwidth in Hertz (Hz)

Internal and External Noise l l l Internal Noise External Noise Total Noise Calculation

Internal Noise l Internal Noise: Caused by thermal currents in semiconductor material resistances and is the difference between output noise level and input noise level

External Noise l External Noise: Noise produced by signal sources also called source noise; cause by thermal agitation currents in signal source

External Noise l Total Noise Calculation = square root of sum of squares Vne = (Vn 2+(In. Rs)2) ½ necessary because otherwise positive and negative noise would cancel and mathematically show less noise that what is actually present

Noise Factor l Noise Factor = ratio of noise from real resistance to thermal noise of an ideal resistor

Noise Factor l Fn = Pno/Pni evaluated at T = 290 o. K (room temperature) where • Pno = noise power output and • Pni = noise power input

Noise Factor l Pni =k. TBG where • G = Gain; • T = Standard Room temperature = 290 o. K • K = Boltzmann’s Constant = 1. 38 x 10 -23 J/o. K • B = Bandwidth (Hz)

Noise Factor l Pno = k. TBG + ΔN where • ΔN = noise added to system by network or amplifier

Noise Figure l l Noise Figure : Measure of how close is an amplifier to an ideal amplifier NF = 10 log (Fn) where • NF = Noise Figure (d. B) • Fn = noise factor (previous slide)

Noise Figure l 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

l Example: Given a 2 stage amplifier where A 1 has a gain of 10 and a noise factor of 12 and A 2 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 Vin + Σ E + G 1 β Vo

Use of Feedback to reduce Noise Vn = Noise Vin + Σ + V 1 B Vo V 1 G 1 + G 1 Σ Β V 2 G 2 V 2 G 2 Vo

Use of Feedback to reduce Noise Vn = Noise Vin + Σ + V 1 B Vo V 1 G 1 + G 1 Σ Β V 2 G 2 V 2 G 2 Vo

Use of Feedback to reduce Noise Derivation: Vn = Noise Vin + Σ + V 1 B Vo V 1 G 1 + G 1 Σ Β Thus Vn is reduced by Gain G 1 Note Book forgot V in equation 5 -35 V 2 G 2 V 2 G 2 Vo

Noise Reduction by Signal Averaging l l Un processed SNR Sn =20 log (Vin/Vn) Processed SNR Ave Sn = 20 log (Vin/Vn/ N 1/2) • Where • SNR Sn = unprocessed SNR • SNR Ave Sn = time averaged SNR • N = # repetitions of signals • Vin = Voltage of Signal • Vn = Voltage of Noise l Processing Gain = Ave Sn – Sn in d. B

Noise Reduction by Signal Averaging l Ex: EEG signal of 5 u. V with 100 u. V of random noise • Find the unprocessed SNR, processed SNR with 1000 repetitions and the processing Gain

Noise Reduction by Signal Averaging l Unprocessed SNR l Processing SNR • Sn = 20 log (Vin/Vn) = 20 log (5 u. V/100 u. V) = -26 d. B • Ave Sn = 20 log (Vin/Vn/N 1/2) = 20 log (5 u/100 u / (1000)1/2) = 4 d. B l Processing gain = 4 – (- 26) = 30 d. B

Review l Types of Signals (Static, Quasi Static, Periodic, Repetitive, Single-Event Transient, Quasi Transient) l Time vs Frequency • • • l Fourier Bandwidth Alaising Sampled signals: Quantization, Sampling and Aliasing

Review l Noise: Johnson, Shot, Friis Noise l Noise Factor vs Noise Figure l 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 l l 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