Experimental Methods Ahmed Rashed Aerospace Engineering Department Cairo University Monday 9th March, 2015 1 1 1 Contents Contents 1 Contents 2 2 Description 4 3 Grading 5 4 References 6 5 Assignments 7 1 8 Introduction 1 Problems 12 2 13 Static Measurements 1 Definitions 13 2 Measurement Error 16 3 Problems 22 3 24 Electronic DAQ 1 Introduction 24 2 ADC 29 2.1 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 Sig. Conditioning 33 3.1 Special Circuits\Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4 Other Comp’s 40 5 Multi-channel DAQ 41 6 Further Details 44 7 Problems 46 4 48 Dynamic Signal Analysis 1 Introduction 48 2 2 The FT 49 3 FT Properties 58 4 Convolution 63 5 Problems 66 5 Appendices 68 A Basic FT Pairs 68 6 70 Problem’s Answers 1 Part 2: Static Measurements 70 3 2 Course Description Title Experimental Methods Course Code ME 241 Instructor Assistant prof. Ahmed Mohamed Rashed Desoki Hours 3 (2 Lecture + 1 Tutorial) (1 lecture + 1 section)/week Grade 100 2 Course Aims • To understand modern engineering experimentation including experiment design, system calibration, data acquisition, analysis and presentation. • To understand how to quantify error and uncertainty in physical measurements. • To gain hands-on experience with modern instrumentation and systems-level experimentation. • To improve written and oral communication skills, to develop the ability to write engineering reports of high quality, and to improve the student’s ability to function as a member of an engineering team. 3 Course Objectives At the end of the course, the student who has mastered the course material will be able to: • Draw a concept map for a generalized measurement system that identifies the most important concepts. • Apply basic statistical methods to design experiments, to analyze, and to present the results of experiments. • Identify and describe the elements making up computer-based data acquisition systems, including alternative configurations and technologies. • Identify and describe the various types of mechanical measurements including temperature, pressure, sound, motion and position, force and torque, stress and strain, flow visualization and measurement (e.g., volume flow rate, velocity, etc.) and explain the transduction principles that underlie them. • Operate modern instrumentation systems that include mechanical and electro-optical technologies and computer-based data acquisition systems. • Work productively and effectively in an engineering team. 4 4 Course Outline(tentative) Week 1: Introduction Week 2: Generalized Measuring System Week 3: Data Acquisition, Signals, Signal Conditioning, Sampling Week 4: Lab View – Lab View Tutorial Week 5: Back ground and Introduction to thermal experiments Week 6: Background and Introduction Fluid mechanics experiments Week 7: Background and introduction to Material experiments Week 8: Background and Introduction to solid mechanics experiments Week 9: Presentation & communication skills Week 10: Accuracy, Precision, Error in Measurement, Calibration - Lab Work Week 11: Uncertainty Analysis – Exercise Week 12: Displacement and Dimensional Measurement – Lab work Week 13: Library Exercise Week 14: Oral Presentation for Selected Topic Week 15: Oral Presentation for Selected Topic Week 16: Final Exam 5 3 Grading Assessment Section work (quizzes, assignments, . . . ) 7th week Assignment/Midterm Lecture Assignments 12th week Assignment/Midterm Project Final Exam Total Grade 10 15 5 10 10 10 40 100 Announced as Grade 7th week Assignment/Midterm 30 12th week Assignment/Midterm 20 Year work Final Exam 10 40 Additionally, every absence costs you -0.5 degree 6 5 4 References Main Textbook 1. R. S. Figliola and D. E. Beasley, Theory and Design for Mechanical Measurements, John Wiley and Sons, 5th ed., 2010. Reference Books 1. National Instruments, LabView data acquisition basics manual, January 2000. 2. K. Shin and J. K. Hammond, Fundamentals of Signal Processing for Sound and Vibration Engineers, John Wiley & Sons, 2008. (Very good explanation of Signal Processing. Comprehensible for mechanical engineering students. Assisted with very interesting examples and Matlab codes. Wider and more practical than Brigham’s) 3. A. Brandt, Noise And Vibration Analysis, Wiley, 2011. (Seems very comprehensive and very good for both theoretical and experimental modal analyses.) 4. National Instruments, X series user manual 5. Yaser El-Shaer, Experimental Methods, undergraduate lecture notes, Arab Academy for Science, Technology & Maritime Transport 7 6 5 Assignments # 1 2 3 4 5 6 7 8 9 10 Assignment Name Problem (1.1); only one item, (1.3); only (f), (2.1), (2.2), (2.3), (2.4), (2.5); load cell A only Problem (2.6) Problem (3.2)-a and (3.2)-b Problem (3.1)-a & c Problem (3.2)-c to m Problems (4.1c), (4.1e) & (4.1f) Due date 2015-Feb.-23 2015-Mar.-2 2015-Mar.-9 2015-Mar.-16 8 7 Part 1 Introduction What Is Measurement? Measurement is the process of estimating a physical variable by a measuring device Measuring Instrument V C Resistance Measurand 9 Example Measuring Systems 10 70 Detector and feedback electronics 60 Laser Photodiodes Display scale Stem 20 30 40 50 Output stage Cantilever and tip Sample surface Bulb Sensor–transducer stage Atomic-force microscope; a complicated measuring system Sensor Bulb thermometer; a simple measuring system 8 Sensor and Transducer 11 Sensor is a physical element that employs some natural phenomenon to sense the variable being measured • e.g.: the cantilever beam senses the height of the surface Transducer converts the sensed information into a detectable signal that can be meaningfully recorded (mechanically, electrically, optically, numerically, . . . ) • e.g.: the laser and the light sensors (photodiodes) Detector and feedback electronics Laser Photodiodes Cantilever and tip Sample surface Atomic-force microscope; a complicated measurement system Generalized Measuring System Calibration Optional Signal conditioning stage Sensor stage Signal path Transducer stage Ampliﬁer Output stage Filter Process Control signal Control stage 12 9 Signal Domains Examples Mechanical Length, area, volume, all time derivatives such as linear/angular velocity/acceleration, mass flow, force , torque, pressure, acoustic wavelength and intensity Thermal Temperature, (specific) heat, entropy, heat flow, state of matter Electrical Voltage, current,charge, resistance, inductance, capacitance, dielectric constant, polarization, electric field, frequency, dipole moment Field intensity, flux density, magnetic moment, permeability Magnetic Radiant Intensify, phase, wavelength, polarization, reflectance, transmittance, refractive index Chemical Composition, concentration, reaction rate, pH, oxidation/reduction potential 13 Assignment • Solve problem (1.1) 14 Noise and Interference 15 6 Noise is a random variation of the value of the measured signal Signal: y (t ) = 2 + sin (2 π t) 5 4 Interference imposes undesirable trends on the measured value Signal y(t) • increases data scatter deterministic Signal + interference 3 Signal + noise 2 1 • For ex.; in electrical instruments interference comes from AC power and is seen as a sinusoidal wave superimposed onto the measured signal 0 0 .0 0 .5 1 .0 1 .5 2 .0 Time (s) Effects of noise and interference superimposed on the signal y(t) = 2 + sin (2πt) Test Standards and Codes Test standard refers to well defined test procedures, technical terminology, methods to construct test specimens or test devices, and/or methods for data reduction The goal of a test standard is to provide consistency in the conduct and reporting of a certain type of measurement between test facilities 10 For example; ASME Power Test Code 19.5 provides detailed designs and operation procedures for flow meters. See [1, problem 1.34] for details. ASTM Test Standard F558-88 provides detailed procedures for evaluating vacuum cleaner cleaning effectiveness and controls the language for product performance claims. 16 Assignment • Solve problem (1.2) 17 Static versus Dynamic Measurements Static measurement is performed when the measured quantity is not (or very slowly) changing with time • An example is beam deflection under a static load Dynamic measurements is vise versa performed • An example is plane wing vibration due to engine vibration and flutter induced by high speed flight 18 Assignment • Solve problem (1.3) 19 Measuring System Characteristics Static characteristics may include • Accuracy • Linearity • Precision • Hysteresis • Sensitivity • ... Dynamic characteristics usually describe • Transient response • Steady state response 20 11 1 Problems Important, for any research question, • you must mention your references • If the reference is electronic and accessible, save it and send me a copy 1.1. Solve problem 1.8 of [1] 1.2. Solve problem 1.53 of [1] 1.3. Solve problem 2.24 of [1] 21 12 Part 2 Static Measurements 1 Definitions Range 10 Measured values Curve fit, y =f(x) Output value, y (units) 8 Eo y =f(x) 6 4 K (x1) = dy dx x = x1 2 Ei 0 0 1 x1 2 3 4 5 Input value, x (units) Measurand range (or operating range or full-scale range or span) It is the range of the input variable, x, ri ≡ xmax − xmin (2.1) Full Scale Output (FSO) range It is the range of the output variable, y, ro ≡ ymax − ymin (2.2) 22 Assignment • Solve problem (2.1) 23 Calibration 10 Measured values Curve fit, y =f(x) Output value, y (units) 8 y =f(x) 6 4 K (x1) = dy dx x = x1 2 0 0 1 2 x1 3 Input value, x (units) 13 4 5 A calibration applies a known input value to a measurement system for the purpose of observing its output value • It establishes the relationship between the input and output values Static calibration implies that the values of the variables involved remain constant with time • In static calibrations, only the magnitudes of the known input and the measured output are important Dynamic calibration implies that variables are time (or space) dependent • A dynamic calibration measures the measurement system response due to an input of known dynamic behavior • Usually, such calibrations involve applying either a sinusoidal signal or a step change as the known input signal 24 Static Sensitivity • Simply can be called Sensitivity – Also called Gain • It is the change of an instrument/transducer output per unit change in the measured quantity. That is dy (2.3) K(x) ≡ dx 10 Measured values Curve fit, y =f(x) Output value, y (units) 8 y =f(x) 6 4 K (x1) = dy dx x = x1 2 0 0 1 2 x1 3 4 5 Input value, x (units) – A more sensitive instrument’s reading changes significantly in response to smaller change in the measured quantity – If the output is a linear function of the input, K(x) = constant =K (2.4) 25 14 Resolution • It is quantified by the smallest scale increment of the output readout indicator 26 15 Measurement Error Actual data trend (2.5) Output value Downscale Hysteresis Best linear curve fit Upscale where y is the measured value and ytrue is the true value Input value Input value (a) Hysteresis error (b) Linearity error Maximum for typical device Output value Nominal curve for typical device K Minimum for typical device Typical shift (high) Output value e ≡ y − ytrue e e(%) ≡ × 100 ytrue e × 100 e(% FSO) ≡ ro Output value 2 Nominal Typical shift (low) Input value (d) Zero shift (null) error Output value Input value (c) Sensitivity error Probable (±2s x) data scatter band on successive measurements Input value (e) Repeatability error • A primary objective in designing and executing an experiment is to minimize the measurement error • Usually, the true value, ytrue , is rarely known exactly, e(%) 27 Types of Error Errors are usually classified as • Systematic error – Bias/Accuracy error • Random error – Precision error • Other (illegitimate) errors 28 16 Uncertainty • Since the true value is rarely known exactly, the error is not known exactly • Uncertainty is a numerical estimate of the possible range of error in a measurement • Based on available information, the operator might feel confident that the error is within certain bounds – e.g.: a plus or minus range of the indicated reading – This is the assigned uncertainty – For example, from the figure below, we might assign an estimate to the random error based on the data scatter of the measurements Typical pressure transducer manufacturer’s Actual data trend specifications Output value Output value Downscale Hysteresis Operation Best linear curve fit Upscale Input value Input value (a) Hysteresis error (b) Linearity error Maximum for typical device K Minimum for typical device Excitation ±15V DC Output range 0−5 V Temperature range 0 − 50◦ C Performance Errors Nominal Typical shift (low) Input value Input value (c) Sensitivity error (d) Zero shift (null) error Output value 0 − 1000 cm H2 O Typical shift (high) Output value Output value Nominal curve for typical device Input range 1 Linearity 0.5% FSO Hysteresis Less than ±0.15% FSO Sensitivity 0.25% of reading Thermal sensitivity ±0.02%/◦ C of reading Thermal zero drift ±0.02%/◦ C FSO Probable (±2s x) data scatter band on successive measurements Input value (e) Repeatability error 29 Overall Uncertainty q u = u21 + u22 + u23 + · · · (2.6) • More details are in [1, ch. 5] 30 17 Assignment • Solve problem (2.2) • Solve problem (2.3) 31 Precision (Random Error) Sensor reading Acuracy (Bias Error) Precision (Random error) Frequency of readings Measurement trial Precision is the difference between (or scattering of) measured values during repeated measurements of the same quantity. • Calculated through statistical analysis on the results of several experiments Precision(% FSO) ≡ max (|y − ȳ|) × 100 ro (2.7) • e.g.: 100kPa ± 1%. • represents the ability to reproduce a reading (not necessarily correct) – also called reproducibility or repeatability 32 Accuracy (Bias Error) Sensor reading Acuracy (Bias Error) Precision (Random error) Frequency of readings Measurement trial Accuracy is the difference between the true value and the average measured value 1 In fact, this is uncertainty. It is customary to call uncertainty as error. 18 • calculated through statistical analysis on the results of several experiments Accuracy(% FSO) ≡ |ȳ − ytrue | × 100 ro (2.8) 33 Accuracy vs Precision Accurate Precise Not accurate Precise Not accurate Not precise 34 Example 2.1. Sensor reading Acuracy (Bias Error) Precision (Random error) Frequency of readings Measurement trial • Four readings of 100 V yields 104, 102, 105 and 105 V; – ytrue = 100 V – ȳ = 104 V – Precision = max (|y − ȳ|) = max (ymax − ȳ, ȳ − ymin ) = 2 V – Accuracy = |ȳ − ytrue | = 4 V 35 19 Assignment • Solve problem (2.4) • Solve problem (2.5) 36 Hysteresis maximum hysteresis upscale output (%FSO) 100 downscale 0 0 100 measurand (% range) • It is the difference in a test output value, y, for the same input value x, when x is sequentially increased and then decreased; or vice versa eh (% FSO) = yupscale − ydownscale × 100 ro (2.9) • Usually the maximum hysteresis is reported in a transducer data sheet 37 Sensitivity and Zero Shift Errors Sensitivity error 100 output (%FSO) Nominal response 0 0 100 measurand (% range) Zero shﬁt error 20 Zero Shift error is the change in the output level of a sensor or instrument for zero input Sensitivity error is the change in the sensitivity of a sensor or instrument • Possible reasons are aging, temperature variation, sensor damage, operating conditions change, . . . 38 Assignment • Solve problem (2.6) 39 21 3 Problems 2.1. Solve problem 1.9 of [1] 2.2. Determine the overall uncertainty of the following transducer Typical pressure transducer manufacturer’s specifications Operation Input range 0 − 1000 cm H2 O Excitation ±15V DC Output range 0−5 V Temperature range 0 − 50◦ C Performance Error Linearity 0.5% FSO Hysteresis Less than ±0.15% FSO Sensitivity 0.25% of reading Thermal sensitivity ±0.02%/◦ C of reading Thermal zero drift ±0.02%/◦ C FSO 2.3. Solve problem 1.23 of [1] 2.4. Multiple Choice Question (a) The difference between the instrument’s reported values during repeated measurements of the same quantities is known as i. Accuracy ii. Precision iii. Resolution iv. Sensitivity (b) The difference between the measured and the true value is known as i. Accuracy ii. Resolution iii. Precision iv. Sensitivity (c) The smallest increment of change in the measured value that can be determined from the instrument’s readout scale is known as i. Accuracy ii. Precision iii. Resolution iv. Sensitivity (d) The change of an instrument or transducer’s output per unit change in the measured quantity is known as i. Accuracy ii. Precision iii. Resolution iv. Sensitivity (e) Errors that occur the same way each time a measurement is made known as 22 i. Bias error ii. Computational error iii. Precision error iv. Random errors (f) Error that are different for each successive measurement but have an average value of zero are know as i. Bias error ii. Computational error iii. Precision error iv. Random errors 2.5. Three load cells are tested for repeatability. The same 50 kg weight is placed on each load cell 10 times. The load cells’ output along with their minimum, maximum and mean are given in the following table. The expected (true) output is 10 mV. The maximum weight the load cell can measure is 100 kg. • Calculate the precision and accuracy of each of the three load cells. Trail no. 1 2 3 4 5 6 7 8 9 10 Maximum Average Minimum A 10.02 10.96 11.20 9.39 10.50 10.94 9.02 9.47 10.08 9.32 11.20 10.09 9.02 Load cell output (mV) B 11.50 11.53 11.52 11.47 11.42 11.51 11.58 11.50 11.43 11.48 11.58 11.49 11.42 C 10.00 10.03 10.02 9.93 9.92 10.01 10.08 10.00 9.97 9.98 10.08 9.99 9.92 2.6. Solve problem 1.12 of [1] 40 23 Part 3 Electronic Data Acquisition (DAQ) 1 Introduction A data acquisition (DAQ) system is the portion of a measurement system that quantifies and stores data Examples • Experimenter – reads a transducer dial – records the measurement • Electronic DAQ system – automates data quantification and storage We focus herein on electronic DAQ systems 41 Electronic DAQ system Physical variable Voltage Current Power Force etc Proportional Voltage ADC Transducer Signal conditioning DAQ system 24 Digital form Storage Sample DAQ’s • Sound card of a PC • National Instruments products – e.g., USB line, PXI line, compact DAQ line, compact Rio line, PCI line, PCI express line, . . . • Brüel & Kjær products – e.g., Brüel & Kjær Multichannel Analysis System Type 3550 Multichannel Data Acquisition Interface 7520 25 kHz Zoom Processor 3156 100 kHz/Multichannel Zoom Processor 3157 Dual Channel Signal Analyzer Type 2032 Signal Analyzer Unit 2035 Dual-channel Analysis Software 7649 B ^ ¨ ESC CTRL ! 1 ¤ 2 TAB Q ALPHA LOCK A > < ↑ # 3 W $ 4 E S Z % 5 R D X ^ 6 Y G B M 4/8-channel Analysis Software 7674 CLR : ; L > . LINE FEED ) ] ( [ P L < , ? - — _ ) 0 K J N O I U H V ( 9 * 8 & 7 T F C ? / ' ` SHIFTS ← ↓ DEL RETURN REP → ↓ Keyboard NP 0028 Preamp. Input Acc. Input Charge Pol. Voltage Direct Input 1MΩ / / 100pF 200V Signal Analysis Software 7671 Multichannel Analysis Software 7640 Preamp. Input Acc. Input Charge Pol. Voltage Direct Input 1MΩ / / 100pF 4–channel Input Module Type 3023 Input 1 Sampling Input/Output Input 4 Input 3 Input 2 Vibration Diagnostic Software 7672 200V 28V Tracking Analysis Software 7670 K Bruël & Kjær SHIFTS Extended Analysis Software 7639 Trigger Input Intensity Probe/ Remote Control Sampling Input/Output Trigger Input Signal Generator Output 28V 0V 0V Channel A ! Floating / Signal Ground 25 kHz Input Module 3019 Channel B ! ! B K 7/6-'89 Floating / Signal Ground ! Brüel & Kjær ! 4-channel Input Module 3023 100 kHz Input Module 3020 Multichannel Data Acquisition Unit 2816 Unit 1 B Unit 2 Direct Input 1MΩ / / 100pF Active Direct Input 1MΩ / / 100pF Active Direct Input 1MΩ / / 100pF Active Active Active Sampling Module 3018 Blank Panel FA 1009 Active K 7/6-'89 Direct Input 1MΩ Ground / / 100pF Active Signal Brüel & Kjær Signal Ground Sampling Input Delay Adj. ! Signal Ground ! Direct Input 1MΩ / / 100pF Active Floating Trigger Input On B Off Input Module 3015 Acc. Input 200V 28V 0VCharge Pol. Voltage Input Module Preamp. Input B Preamp. Input B K 7/6-'89 7/6-'89 Brüel & Kjær Signal Ground Signal Ground ! B 7/6-'89 Intensity Probe/ Remote Control 1 Intensity Probe/ Remote Control ! Floating Floating Generator Module Generator Module 3107 3107 Acc. Input Charge Acc. Input Charge 1 3015 Input Module 3015 Input Module 3016 Sound Intensity Module 3017 200V 28V 0V Pol. Voltage 200V 28V 0V Pol. Voltage 200V 28V 0V Pol. Voltage 200V 28V 0V Pol. Voltage Preamp. Input Preamp. Input Signal Generator Output Preamp. Input K Brüel & Kjær Active Floating Signal Ground ! Floating Acc. Input Charge Input Module 3015 200V 28V 0V Pol. Voltage K 7/6-'89 Acc. Input Charge Input Module 3015 200V 28V 0V Pol. Voltage Brüel & Kjær Signal Ground Floating Signal Ground Power ! Floating ! Acc. Input Charge Sampling Output Signal Analyzer Interface Module Type 7521 Direct Input 1MΩ / / 100pF Active ! Floating Multichannel Data Acquisition Unit Type 2816 B K 7/6-'89 Brüel & Kjær B B K 7/6-'89 & Kjær K Brüel & Kjær Preamp. Input 7/6-'89 Brüel & Kjær B K 7/6-'89 Brüel & Kjær Signal Generator Output B Sound Intensity Module 3017 200V 28V 0V Pol. Voltage 2 2 K 7/6-'89 Intensity Probe/ Remote Control B 7/6-'89 Brüel & Kjær K Brüel & Kjær B 7/6-'89 K Brüel & Kjær Intensity Probe/ Remote Control B K K 7/6-'89 Brüel & Kjær Brüel & Kjær Signal Analyzer Interface Module 7521 Sound Intensity Module 3017 25 kHz or 100 kHz Input Modules 3015 or 3016 Generator Module 3107 Unit 1 B Unit 2 Direct Input 1MΩ / / 100pF Active Direct Input 1MΩ / / 100pF Active Direct Input 1MΩ / / 100pF Active Active Signal Ground Sampling Input Delay Adj. ! Direct Input 1MΩ Ground / / 100pF Active Signal Signal Ground ! Direct Input 1MΩ / / 100pF Active Floating Floating Acc. Input Charge Signal Ground ! Input Module 3015 Input Module 3015 Input Module 3015 200V 28V 0V Pol. Voltage 200V 28V 0V Pol. Voltage Acc. Input 200V 28V 0VCharge Pol. Voltage Input Module Signal Ground Signal Ground ! Signal Ground Active Active Floating Noise Floating Floating 500 Hz Floating Active Input Module 3016 200V 28V 0V Pol. Voltage Active Floating Acc. Input Charge Input Module 3015 200V 28V 0V Pol. Voltage Σ Σ Floating Generator Module Generator Module 3107 3107 Acc. Input Charge 3015 200V 28V 0V Pol. Voltage Noise Active Floating ! Floating Acc. Input Charge ! Floating Signal Ground ! Acc. Input Charge Direct Input 1MΩ / / 100pF Active ! Floating Sampling Output Signal Analyzer Interface Module Type 7521 Signal Output Module Type 5955 Active Signal Output Module Type 5955 K 7/6-'89 Brüel & Kjær Multichannel Data Acquisition Unit Type 2816 Floating / Signal Ground Generator & Sampling Module 3106 3 ms Delay x0,5 x 0.5 Σ Ch.B Gen. Active Σ Floating Noise Active Power On Off Trigger Input B 7/6-'89 K Brüel & Kjær Preamp. Input B 7/6-'89 K Brüel & Kjær Preamp. Input B 7/6-'89 K Brüel & Kjær B 7/6-'89 B K 7/6-'89 Brüel & Kjær K Brüel & Kjær Preamp. Input B 7/6-'89 & Kjær Floating Signal Generator Output Preamp. Input K Brüel & Kjær Preamp. Input B 7/6-'89 Active Signal Generator Output B 7/6-'89 Ch. A Floating B K 7/6-'89 Brüel & Kjær Preamp. Input K Brüel & Kjær B 7/6-'89 K Brüel & Kjær B 7/6-'89 ZZ 0220 Simulator Unit Simulator Unit ZZ 0220 K Brüel & Kjær Floating K Brüel & & Kjær Signal Output Module 5955 910704/3e Hardware and software components of M ultichannel Analysis System Type 3550 • ... 42 25 DAQ System Components Digital Inputs Process feedback Analog Transducers Signal Conditioning Display Multiplexer Buffer Storage ADC Clock External Computer Controller Trigger Multiplexed DAQ The main DAQ components are: 1. Analog to Digital Converter (ADC) 2. Signal Conditioning (optional) • Amplifier, Filters, . . . 3. Other Components • Clock, Trigger, . . . 43 Single-Ended/Floating Signals + _ + _ Vs Ground Vs Ground Single ended (grounded) signal Floating signal 44 Floating (Differential) signal Single-ended (Grounded) signal • referenced to a system ground • Examples are devices that plug into a building ground through wall outlets, such as: • not connected to an absolute reference • Examples include: – batteries – signal generators and – battery-powered sources – power supplies – thermocouples – transformers – isolation amplifiers 26 Single-ended/Differential Measurement Systems Multiplexer Channel 1 + Transducer 1 + CH1 Hi – ADC Channel N + Transducer N Vm + CH N Hi – GND GND Single-ended DAQ system Multiplexer 1 Channel 1 + Transducer 1 + CH1 Hi – + CH N/2 Hi ADC Channel N/2 Transducer N/2 + - CH1 Lo Vm – - CH N/2 Lo Multiplexer 2 GND Differential DAQ system Single-ended measurement system Differential measurement system can be used when all input signals meet • Preferred way to measure low-level the following criteria signals, because it rejects: • Input signals are high level (greater than 1 V as a rule of thumb) – ground loop-induced errors • Signal cabling is short and travels through a noise-free environment or is properly shielded • All input signals can share a common reference signal at the source 27 – the noise picked up in the environment (to a certain degree) • By using twisted pairs, noise effects are greatly reduced • One differential channel consumes two single ended channels • More details are in – “Ground Loops and Returns” (http://www.ni.com/white-paper/3394/en/) ∗ Chapter 4 of [4] 45 The Oscilloscope It is a very useful diagnostic tool • It is used to measure and display voltage magnitude versus time for dynamic signals over a wide range of frequencies extending commonly to megahertz (or even gigahertz) • It can display the signal magnitude, frequency, distortion, and DC and AC components – Typical oscilloscopes can also display several more signals (e.g. x(t) and y(t)), perform addition and subtraction (x(t) ± y(t))and other features such as fast Fourier transform (FFT) • This visual display can help the user to detect noise/interference superimposed on a signal, something nonvisual metering devices cannot do • The LabView interactive program “Oscilloscope.vi” with the software accompanying reference [1] – It features a basic two-channel oscilloscope. The user can vary channel displayed and the time sweep and gain using two built-in signals (sine wave and square wave) and an active signal trigger 46 28 Signal Signal The Analog to Digital Converter (ADC) Signal 2 Time Quantization levels Time Analog signal Time Discrete signal Digital signal • It converts analog voltage into a binary number through a process called quantization • The ADC has both an analog side and a digital side. 47 The Analog side ADC • It is specified in terms of the full-scale voltage range, EFSR ADC defines the voltage range over which the ADC operates – The EFSR ADC Example 3.1. An ADC with EFSR = 10 V can accept voltages in the range 0 ∼ 10 V (unipolar) or −5 ∼ 5 V (bipolar), depending on the ADC setting 48 The Digital side • It is specified in terms of the number of bits of its register • An M -bit ADC outputs an M -bit binary number – It can represent 2M different numbers Example 3.2. • An 3-bit ADC represents analog voltages with 23 = 8 different values Amplitude (volts) • A 16-bit ADC represents analog voltages with 216 = 65, 536 different values 10.00 8.75 7.50 6.25 5.00 3.75 2.50 1.25 0 0 16-Bit Versus 3-Bit Resolution (5 kHz Sine Wave) 111 16-bit 110 101 100 011 010 3-bit 001 000 50 100 Time (µs) 150 200 49 29 2.1 ADC Characteristics Amplitude (volts) Resolution 16-Bit Versus 3-Bit Resolution (5 kHz Sine Wave) 10.00 8.75 7.50 6.25 5.00 3.75 2.50 1.25 0 0 111 16-bit 110 101 100 011 010 3-bit 001 000 50 100 Time (µs) 150 200 • It is defined in terms of the smallest voltage change that causes a change of the least significant bit – Same as was defined earlier in section 1 Q ADC ADC EFSR = M 2 (3.1) Example 3.3. For a 3-bit ADC • setting the range to 0 ∼ 10 V yields QADC = ADC 10 EFSR = 3 = 1.25 V M 2 2 (3.2) • setting the range to –10 ∼ 10 V yields QADC = ADC EFSR 20 = 3 = 2.5 V M 2 2 (3.3) The figure below shows a 0 ∼ 10 V sine signal digitized by both settings 10.00 8.75 7.50 6.25 5.00 3.75 2.50 1.25 0 0 Range = 20 V [ -10 to 10 V ] 111 Amplitude (vo lts ) Amplitude (vo lts ) Range = 10 V [ 0 to 10 V ] 110 101 100 011 010 001 000 50 100 150 Time (µs ) 200 10.00 7.50 5.00 2.50 0 -2.50 -5.00 -7.50 -10.00 0 111 110 101 100 011 010 001 000 50 100 150 Time (µs ) 200 50 30 Clipping ADC • Signals should use as much as possible of the EFSR range, but without overloading – Otherwise, clipping will occur 51 Amplitude (volts) Dynamic Range 10.00 8.75 7.50 6.25 5.00 3.75 2.50 1.25 0 0 16-Bit Versus 3-Bit Resolution (5 kHz Sine Wave) 111 16-bit 110 101 100 011 010 3-bit 001 000 50 100 Time (µs) 150 200 • It is the ratio between the largest and smallest number that can be represented by a measuring system – calculated in dB • For an M -bit ADC, the dynamic range is calculated as ADC EFSR ADC D (dB) ≡ 20 log10 QADC ≡ 20 log10 2M ≈ 6M (3.4) • More details are in [2; section 5.3] 52 Practical DAQ System Dynamic Range • A DAQ system includes several components other than the ADC – Hence, the useful dynamic range of the whole system is usually less than that of the ADC 31 Example 3.4. For a typical 16-bit B&K DAQ system, • DADC = 96 ≈ 100 dB • but DSYS ≈ 80 dB for the whole system Example 3.5. For a modern 24-bit B&K DAQ systems, • DADC = 6 × 24 = 144 dB • but DSYS ≈ 100 ∼ 110 dB for the whole system 53 Quantization Error • If the input voltage falls between two adjacent output codes, it is erroneously digitized • This error is referred to as the quantization error eQ Amplitude (volts) • It behaves as noise imposed on the digital signal 10.00 8.75 7.50 6.25 5.00 3.75 2.50 1.25 0 0 16-Bit Versus 3-Bit Resolution (5 kHz Sine Wave) 111 16-bit 110 101 100 011 010 3-bit 001 000 50 100 Time (µs) 150 200 54 32 Demonstration • “Chapter 7 > Bits of Resolution” in the Mechanical Measurements LabView demo companion of [1] • “Chapter 7 > Digitizing for Spectral Analysis” in the Mechanical Measurements LabView demo companion of [1] 55 Assignment • Solve problem (3.2)-a and b 56 3 Signal Conditioning Signal Conditioning AC Analog transducer DC Special circuit Transducer specific signal conditioning AC/DC- Amplification/ Coupling Attenuation ADC Analog filters Sampling • Transducer outputs must be ‘conditioned ’ to accommodate cabling, environmental considerations, features of the recording instrumentation, . . . Conditioning usually includes: • amplification, • filtering, • enhance power supplies and cabling – For example, some transducers, such as strain gauges, require power supply ∗ For this case, the signal conditioner must provide stabilized the power supply with little ripple, low noise, . . . • ... 57 33 Amplifier/Attenuator • Usually sensor output is a very small signal – For best digitization, the sensor output has to be amplified/attenuated to fill ADC the EFSR ∗ This is accomplished using both amplifier and offset circuit Range = 20 V [ -10 to 10 V ] 111 Amplitude (vo lts ) Amplitude (vo lts ) Range = 10 V [ 0 to 10 V ] 10.00 8.75 7.50 6.25 5.00 3.75 2.50 1.25 0 0 110 101 100 011 010 001 000 50 100 150 Time (µs ) 200 10.00 7.50 5.00 2.50 0 -2.50 -5.00 -7.50 -10.00 0 111 110 101 100 011 010 001 000 50 100 150 Time (µs ) 200 • Take care; if you amplified the signal too much, clipping will occur DAQ system Analog transducer G ADC Amplification/ Attenuation • Using low noise amplifier/attenuator with gain G, the usable range of the DAQ system is expanded/contracted as sys EFSR = ADC EFSR G (3.5) – By using selectable variable amplifier gains, the ADC can be adjusted to wider/smaller ranges for optimum signal acquisition 58 34 ADC Example 3.6. • A typical 12 bit DAQ system has EFSR = 10 V. The ADC resolution is thus calculated as ADC EFSR ADC Q = M = 2.44 mV 2 • Using low noise amplifier with adjustable gains from G = 0.5 ∼ 1000; – when set at maximum gain G = 1000, the input voltage range contracts to sys EFSR ADC EFSR = = 0.01 V 1000 and the resolution becomes Qsys = sys ADC EFSR QADC EFSR = 2.44 µV = = 2M G 2M G – when set at minimum gain (maximum attenuation) G = 0.5, the input voltage range expands to E ADC sys EFSR = FSR = 20 V 0.5 and the resolution becomes Qsys = QADC = 4.88 mV G 59 More Information • Analog amplifiers; Typical op amp circuits are in [1; sec. 6.6] • Basic amplifier for digital DAQ is in [1; sec. 7.14] 60 Offset Circuit 10.00 8.75 7.50 6.25 5.00 3.75 2.50 1.25 0 0 Range = 20 V [ -10 to 10 V ] 111 Amplitude (vo lts ) Amplitude (vo lts ) Range = 10 V [ 0 to 10 V ] 110 101 100 011 010 001 000 50 100 150 Time (µs ) 200 10.00 7.50 5.00 2.50 0 -2.50 -5.00 -7.50 -10.00 0 111 110 101 100 011 010 001 000 50 100 150 Time (µs ) 200 • For best digitization, the sensor output has to be expanded/contracted to fill the ADC EFSR range – This is accomplished using both amplifier and offset circuit • Usually the offset circuit is included in the ADC amplifier • Typical offset circuit is in [1; Fig. 7.16] 61 35 More Information • Check the “Calibration Circuitry” section in NI user manual of X series or M series and learn about – EEPROM installed in Integrated Electronics Piezo Electric (IEPE) sensors and – device self calibration • More information about IEPE are in [3, sec. 7.3] 62 Filters • Analog filters are used to control the frequency content of the signal being sampled. Examples include: – low-pass filter to remove high frequency noise – anti-alias filter which is a low pass filter that removes frequencies higher than fs /2 More details are in [1; sec. 6.8] 63 Anti-Aliasing Filter folding Frequency spectrum of anti-aliasing filter Usable range (Passband) Aliasing 0 dB -40 dB -80 dB • This must be an analog filter, since it appears before the ADC • Not all DAQ boards contain analog anti-aliasing filter, so be careful when you buy a DAQ board • In early equipment for noise and vibration analysis, filters with oversampling factor β ≈ 1.28 (fc = fsβ/2 ≈ 0.39 fs ) was established • In more recent systems utilizing sigma–delta ADCs, this factor is somewhat smaller, say down to β ≈ 1.1 (fc ≈ 0.45 fs ) • More information are in [1; section 6.8] 64 36 AC Coupling Circuit DC coupling AC coupling 35 2 .5 Fluctuating voltage 30 20 15 y(t) – y y(t) Voltage 25 10 5 0 0 .5 1 1 .5 2 Time (s) 2 .5 3 3 .5 2 1 .5 1 0 .5 0 –0 .5 –1 –1 .5 –2 –2 .5 0 0 .5 Signal prior to subtracting DC offset 1 1 .5 2 Time (s) 2 .5 3 3 .5 Fluctuating component of signal AC coupling removes the DC component from the signal • desired when the AC component we want to analyze is relatively small in comparison to a superimposed DC component • Most dynamic signals should preferably be acquired with AC coupling, since we are in most cases only interested in the dynamic part of the signal • AC coupling can be implemented by passing the signal through a very low frequency high-pass filter 65 More Information • http://www.ni.com/white-paper/8734/en/ 66 3.1 Special Circuits\Modules Signal Conditioning AC Analog transducer Special circuit Transducer specific signal conditioning DC AC/DC- Amplification/ Coupling Attenuation ADC Analog filters Sampling • Several modules exist for different types of transducers • They convert the transducer output to a form suitable for ADC (usually a volt) 67 37 Resistance bridge (Wheatstone bridge) Gauge 3 S3 R Amplifier + R E supply – R Ei Gauge 2 Gauge 4 S2 S4 G Channel 1 M u l t i p l e Channel 2 x e r A/D P C b u s I n t e r f a c e Gauge 1 • Used to measure small resistance changes as in strain gauges • This module allows direct interface of strain gauges or other resistance-based sensors 68 Charge Amplifier • Piezoelectric transducers produce charge2 rather than voltage Coulomb / sec – Also charge signals are sensitive to noise/disturbances – Hence, the measured signal must be converted to a proportional voltage suitable for a normal DAQ system • The conversion is accomplished using a charge amplifier , see [3; Figure 7.2] – Charge amplifiers are relatively expensive • More information are in – Triboelectric (static electricity) effect; [3; sec. 7.2] and [2; sec. 5.4] – Mark Serridge and Torben R. Licht, Piezoelectric Accelerometers and Vibration Preamplifiers, Theory and Application Handbook, Brüel & Kjær documentation 69 2 1 A = 1 C/1 s 38 Thermocouple module Thermocouple wires Copper connecting wires Cu A 1 Cu B A 2 1′ Ice bath (reference junction) B A N 2′ Measuring junctions A Potentiometer Measuring junctions B N' Ice bath (reference junction) Thermocouples arranged to sense temperature differences. (From Benedict, R. P. Fundamentals of Temperature, Pressure and Flow Measurements, 3rd ed. Copyright 1984 by John Wiley and Sons, New York.) • It allows for cold junction compensation for signal linearization for reasonably accurate (down to 0.5 C) temperature measurements • More details are in [1; p. 347] 70 Shunt Resistor • Used to measure current • It converts current signals into proportional voltage signals • Typical shunt resistor circuit is in [1; Fig. 7.15] 71 Assignment • Solve problem (3.1) 72 Demonstration • “Chapter 2 > Sound in from Microphone” in the Mechanical Measurements LabView demo companion of [1] 73 More Information • Field Wiring and Noise Considerations for Analog Signals (go to ni.com/info and enter the Info Code “analogwiring”) 74 39 4 Other Components Multiplexer Digital Inputs Process feedback Analog Transducers Signal Conditioning Display Multiplexer Buffer Storage ADC Clock External Computer Controller Trigger Multiplexed DAQ • It is used to switch between connections when multiple input signal lines are connected by a common throughput line to a single ADC 75 Clock Single channel sampling Channel 1 Single channel sample clock • More information are in http://zone.ni.com/reference/en-XX/help/370466AA-01/ mxcncpts/clocks/#GUID-8A10ED1D-A7A9-42EA-8987-2F8225D7500D 76 Trigger 77 40 Digital to Analog Converter (DAC) • It converts digital numbers into analog voltages, which might be used – for process control – to activate a device – to drive a sensor positioning motor – ... 78 Digital Input/Output Digital I/O signals may be used as single state (HIGH or LOW; 5 V or 0 V) • might be used to operate a switch or relay, light a led, signal an alarm, . . . series of pulses of HIGH/LOW states • Pulse stepping – used to drive stepper motors and servos – sends a predetermined number of pulses in a series – counting pulses that occur over a specified period of time enables frequency determination (number of pulses/unit time) and counting/timing applications 79 5 Multi-channel DAQ Single, Aggregate and Multiplexed Sample Rates Digital Inputs Process feedback Analog Transducers Signal Conditioning Display Multiplexer Buffer Storage ADC Clock Controller Trigger Multiplexed DAQ 41 External Computer • A multiplexed DAQ device may have several sampling rates – Single-channel sampling rate, fssingle ∗ It is the fastest sampling rate attainable from a single channel Single channel sampling Channel 1 Single channel sample clock – Aggregate sampling rate, fsA ∗ It is the fastest sampling rate attainable from all the multiplexed channels ∗ Due to the multiplexer hardware limitations fsA < fssingle Multiplexed sampling of 3 channels on a 4 channel DAQ device Channel 1 Channel 2 Channel 3 Sample clock Convert clock – Multiplexed sampling rate, fsMux ∗ It is the fastest sampling rate attainable from each multiplexed channel ∗ For a K channel DAQ device, 1 fsMux ≥K· 1 fsA ⇐⇒ fsMux ≤ fsA K Example 3.7. The NI 6351 has 16 analog inputs, fssingle = 1.25 MS/s and fsA = 1 MS/s. • Thus it can sample – one channel at fssingle = 1.25 MS/s 42 – 8 channels at fsMux = fsA 8 – 16 channels at fsMux = = 0.125 MS/s each fsA 16 = 0.0625 MS/s each • Check http://www.ni.com/white-paper/9376/en/ for typical examples 80 Multi-channel Sampling Multi-channel DAQ devices use either multiplexed or simultaneous sampling; • Simultaneous sampling devices have an ADC for each analog input channel Digital Inputs Process feedback Analog Transducers Signal Conditioning ADC Buffer ADC Buffer Buffer ADC Display Storage Buffer Clock ADC External Computer Controller Trigger Simultaneous DAQ Simultaneous Sampling Channel 1 Channel 2 Channel 3 Single channel sample clock – It can sample all channels at the same time (simultaneous or synchronous sampling) 43 • Multiplexed sampling devices have a single ADC for all analog input channels Digital Inputs Process feedback Analog Transducers Signal Conditioning Display Multiplexer Buffer Storage ADC Clock External Computer Controller Trigger Multiplexed DAQ Multiplexed sampling of 3 channels on a 4 channel DAQ device Channel 1 Channel 2 Channel 3 Sample clock Convert clock 81 6 Further Details Sample DAQ Systems Sampling clock ADC Signal conditioning Multichannel sampling Basic none single none multiplexed 44 Dedicated exist single none multiplexed Highly Dedicated exist multiple exist simultaneous Basic • PIC Dedicated • Sound card • Arduino • MBed – DC ??? coupling • LabJack • ... – No anti aliasing filter Highly Dedicated • Brüel & Kjær Multichannel Analysis System Type 3550 • Many National Instruments products as well • ... – No AC coupling • NI USB-6251 – No anti aliasing filter – No AC coupling • ... 82 Hardware Specifications Check [3; sec. 11.2.4] for further topics as: • Cross channel match • Cross-channel talk 83 Build or buy DAQ System • Of course it is possible to build a system based on general ADC components – but once all necessary parts are put together, the price will likely be higher than using dedicated hardware – For example, a good system for noise and vibration applications, requires very expensive components such as: ∗ highly accurate anti-aliasing filters ∗ matched between the channels, and ∗ electronics with relatively low noise floor 84 45 7 Problems 3.1. Study the specifications of the following multi-channel DAQ devices and describe sys (i) channels input range (EFSR ) (ii) maximum number of analog input channels (iii) single-ended or floating input channels (iv) AC or DC coupling (v) sampling rate/channel (vi) aggregate (all channels) analog input throughput (vii) whether there is anti-aliasing filter, or not • In NI documents, the cutoff frequency of the low-pass filter is also called the “small signal bandwidth” (viii) available other signal conditioning modules; if any (ix) ADC buffer size (x) and whether they have simultaneous channel sampling, or not (a) (b) (c) (d) B&K input module type 3015 NI-USB-6251-BNC NI USB 6009 NI 9237 Strain gauge module (e) NI 9234 dynamic signal acquisition module (f) NI myRIO-1900 3.2. A typical DAQ system has a 16-bit ADC and a low noise amplifier. The ADC range is −1 ∼ 1 V. The amplifier has adjustable gains G = 0.1, 0.2, 0.5, 1, 10, 100 & 1000. Calculate: (a) Range of the ADC (b) Resolution of the ADC (c) Maximum range of the DAQ system (d) Minimum rang of the DAQ system (e) Maximum resolution of the DAQ system (f) Minimum resolution of the DAQ system (g) Determine the best amplifier gain to use for measuring a bipolar −10 ∼ 10 V signal (h) Do you need any additional circuit to improve the measurement of this signal? (i) If yes, what is this circuit? Explain how/why it improves the measurement. (j) Determine the best amplifier gain to use for measuring a unipolar 0 ∼ 10 V signal (k) Do you need any additional circuit to improve the measurement of this signal? (l) If yes, what is this circuit? Explain how/why it improves the measurement. 46 (m) Determine the best amplifier gain to use for measuring a bipolar −10 ∼ 10 mV signal 3.3. Create 1 s wav file with the sound of the telephone tone number 7 (a) Choose a suitable sampling frequency (b) Calculate the bit rate of your file (c) Plot one period of both the continuous and discretized sound signal (d) Calculate the size of your file (e) How can you reduce the file size? i. Explain the advantages/drawbacks of your answer (f) If you read your wav file, will there be any difference as compared to the original data? (g) Plot one period of both the original continuous and read sound signal (h) Explain the difference if any Hints • Tone information are in http://en.wikipedia.org/wiki/Telephone_keypad • “wavwrite” Matlab function can create wav files • “wavread” Matlab function can read wav files 3.4. Plot a schematic chart showing the internal modules inside of sound card of your PC. 85 47 Part 4 Dynamic Signal Analysis 1 Introduction Signal Properties Amplitude PeakPeak Peak RMS Average Time 86 Signal Types Stationary Properties do not vary with time Non-stationary Properties vary with time Deterministic Random Transient Predictable Described using its statistical properties Starts and stops within the analysis time Continuous • Example real signals are in [2; sec. 1.2] 87 48 2 The Fourier Transform The Fourier Transform of a Signal The Fourier Integral Z ∞ x(t) e−jωt dt ∀ω ,: ω ∈ R (4.1) x(t) e−j2πf t dt ∀f ,: f ∈ R (4.2) X(ω) = −∞ or Z ∞ X(f ) = −∞ where ω = 2πf • Signals, x(t), are a function of time (usually real data) • Frequency (Fourier) spectrum, X(f ), is an equivalent representation of the signal – Complex domain (Real, imaginary) or (Magnitude, phase) The Inverse Fourier Transform Z ∞ x(t) = X(f )ej2πf t df ∀t ,: t ∈ R (4.3) −∞ Fourier Transform Pair F x(t) ⇐⇒ X(f ) 88 49 Vi ol et Physical Meaning of The Fourier Transform ue Bl White light n ee Gr Yellow Spectrum ge Oran Red Prism Separation of white light into its color spectrum. Color corresponds to a particular frequency or wavelength; light intensity corresponds to varying amplitudes • Any signal can be represented as a summation of harmonic (sinusoidal) signals 89 50 Demonstrations 51 Gibb’s phenomenon • Square wave 52 • Sawtooth wave 53 Matlab Illustration • Study example 3.1 of [2] Wolframe demonstrations (http://demonstrations.wolfram.com) • Approximation of Discontinuous Functions by Fourier Series 90 Frequency Spectrum of a Signal Input Output x(t) y(t) FFT IFT INPUT SPECTRUM OUTPUT SPECTRUM • Frequency spectrum of a signal can highlight characteristics of the signal – numerous applications • Transformation is done using Fourier Transform – Frequency Spectrum ≡ Fourier Spectrum≡ Spectrum 91 54 Typical Applications of Fourier Analysis[??] Signal Processing Numerical Methods Applied Mechanics • matched filters • high-speed interpolation • structural dynamics • deconvolution • conjugate gradient method • real-time spectral analysis • aircraft wing-flutter suppression • boundary value problems • cepstrum analysis • Riccati and Dirichlet equations • machinery dynamics diagnostics • coherence function estimation • Rayleigh’s integral • nuclear power plant modeling • speech synthesis and recognition • Wiener-Hopf integral equation • vibration analysis • random process generation • diffusion equation • transfer function estimation • numerical integration • echo/reverbation removal • Karhunen-Loeve transform • elliptic differential equations Electromagnetics • chromatography • microscopy • spectroscopy • x-ray diffraction • micros trip line propagation Sonics And Acoustics • conducting bodies scattering • acoustic imaging • antenna radiation patterns • passive sonar • dielectric substrate capacitance • ultrasonic transducers • phased-array antenna analysis Instrumentation • electrochronography Radar • array processing • time-domain reflectometry • architecture acoustic measurement • waveguide analysis • music synthesis • network analysis • cross-section measurement • moving target indicator • synthetic aperture • doppler processor • pulse compression • clutter rejection Communications Biomedical Engineering • diagnosis of airways obstruction • muscle fatigue monitoring • assessing heart valve damage • tissue structure characterization Miscellaneous • systems analysis • transmultiplexers • demodulators • speech scrambler system • multichannel filtering • M -ary signaling • gastric disturbances investigation • signal detection • cardiac patients diagnosis • high-speed digital filters • ECG data compression • voice coding systems • artery dynamics investigation • video bandwidth compression • magnetotellurics • metallurgy • electrical power systems • image restoration • nonlinear system analysis • geophysics • GaAs FET transient response • integrated circuit modeling • quality control 92 Signal Processing is a Tool • In this course, Signal Processing is no more than a tool used to extract the model properties 55 – Understanding the physical meaning of the FT and its properties is the goal – Theoretical derivation of the FT and its properties is assumed trivial ∗ All details can be found for example in [2; sec. 4.3] or any other signal processing textbook – Check sample FT pairs in appendix A 93 56 Plotting FT pairs 94 57 3 Fourier Transform Properties F x(t) ⇐⇒ X(f ) F a x(t) ⇐⇒ a X(f ) F x(t) + y(t) ⇐⇒ X(f ) + Y (f ) F X(t) ⇐⇒ x(−f ) f 1 F X x(kt) ⇐⇒ |k| k 1 t F x ⇐⇒ X(k f ) |k| k F x(t − t0 ) ⇐⇒ X(f ) e−j2πf t0 F x(t) ej2πf0 t ⇐⇒ X(f − f0 ) 1 F [X(f + f0 ) + X(f − f0 )] x(t) cos (2πf0 t) ⇐⇒ 2 F x(t) = xEven (t) ⇐⇒ X(f ) = XReal (f ) F x(t) = xOdd (t) ⇐⇒ X(f ) = j Ximaginary (f ) F x(t) = xReal (t) ⇐⇒ X(f ) = XEven (f ) + j XOdd (f ) 58 (4.4) (4.5) (4.6) (4.7) (4.8) (4.9) (4.10) (4.11) (4.12) (4.13) (4.14) (4.15) Linearity; equation (4.6) Symmetry; equation (4.7) Scale F x(t) + y(t) ⇐⇒ X(f ) + Y (f ) Scale Dt & Df = 2.5 mm T & fs = 20mm F Dt & Df = 2.5 mm X(t) ⇐⇒T x(−f ) & fs = 20mm 59 FT of even function; equation (4.13) FT of odd function; equation (4.14) F F x(t) = xEven (t) ⇐⇒ X(f ) = XReal (f ) x(t) = xOdd (t) ⇐⇒ X(f ) = j Ximaginary (f ) 60 FT of real function; equation (4.15) F x(t) = xReal (t) ⇐⇒ X(f ) = XEven (f ) + j XOdd (f ) Time shift; equation (4.10) F x(t − t0 ) ⇐⇒ X(f ) e−j2πf t0 61 95 Demonstrations Wolframe demonstrations (http://demonstrations.wolfram.com) • Rectangular Pulse and Its Fourier Transform 96 Logarithmic Scale 5 x(t) (volts) 3 1 –1 –3 –5 0 0.005 0.01 0.015 0.02 t (seconds) 0.025 0.03 0.035 0.04 (a) Time history |X( f )| (linear scale, volts / Hz) 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 1.5 1.6 1.7 1.8 1.9 2 Frequency (kHz) (b) Frequency components (linear scale) |X( f )| (log scale, volts /Hz) 100 10–2 10–4 10–6 10–8 10–10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 Frequency (kHz) (c) Frequency components (log scale) Measured telephone tone (No. 8) signal considered as periodic 62 • More information are in [3; Appendix B] 97 Assignment • Solve problems (4.1); 4.1a to 4.1e 98 4 Convolution Z ∞ x(τ ) y(t − τ ) dτ x(t) ∗ y(t) ≡ (4.16) −∞ Corollary 4.1. x(t) ∗ y(t) = y(t) ∗ x(t) (4.17) Important for t − tstart t • understanding the Discrete Fourier Transform 1 0 T x(τ ) y(τ ) y(t − τ ) (x ∗ y)(t) • dealing with Linear Time-Invariant (LTI) systems [2; sec. 4.7] 0.5 99 0 −8 −6 −4 −2 Graphical evaluation t − tstart t 1 0.5 0 2 4 6 8 τ t 0 T 1 x(τ ) y(τ ) y(t − τ ) (x ∗ y)(t) 1 0.5 0.5 x(τ ) t y(τ y(τ )) y(t − x(τ ) τ) (x ∗−y)(t) x(t τ) t − tstart t − t0start T 0 T (y ∗ x)(t) 0 0 −8 −6 −4 −2 0 2 4 6 8 τ t 1 1 0.5 0.5 x(τ ) t y(τ y(τ )) y(t x(τ − ) τ) (x x(t∗−y)(t) τ) t − tstart t − t0start T 0 T 0 −8 −6 −4 −2 0 2 4 6 8 −8 −6 −4 −2 τ0 τ 2 4 6 8 t 1 1 0.5 0.5 (y ∗ x)(t) x(τ ) y(τ y(τ )) y(t − x(τ ) τ) (x ∗−y)(t) x(t τ) t t − tstart t − t0start T 0 T (y ∗ x)(t) 0 0 0 −8 −6 −4 −2 0 2 4 6 8 −8 −6 −4 −2 τ0 τ 2 4 6 8 t 1 1 0.5 0.5 x(τ ) y(τ y(τ )) y(t − x(τ ) τ) (x x(t∗−y)(t) τ) t t − tstart t − t0start T 0 T 0 −8 −6 −4 −2 0 2 4 6 8 −8 −6 −4 −2 τ0 τ 2 4 6 8 t 1 1 0.5 0.5 (y ∗ x)(t) t x(τ ) y(τ y(τ )) y(t − x(τ ) τ) (x x(t∗−y)(t) τ) t − tstart t − t0start T 0 T (y ∗ x)(t) 0 0 0 −8 −6 −4 −2 0 2 4 6 8 −8 −6 −4 −2 τ0 τ 2 4 6 8 t 1 1 0.5 0.5 t x(τ ) y(τ y(τ )) y(t − x(τ ) τ) (x ∗−y)(t) x(t τ) t − tstart t − t0start T 0 T −4 −2 0 2 4 6 8 −6 −4 −2 τ0 τ 2 4 6 8 t − tstart t 0.5 (y ∗ x)(t) 0 0 T y(τ ) x(τ ) x(t − τ ) (y ∗ x)(t) 0 −6 −4 −2 0 2 4 6 8 −8 −6 −4 −2 τ0 τ 2 4 6 8 t − tstart t 0.5 −6 −8 1 0 −8 1 0 −8 y(τ ) x(τ ) x(t − τ ) (y ∗ x)(t) 0 63 T −8 −6 −4 −2 0 τ 2 4 6 8 0.5 y(τ ) y(t − τ ) (x ∗ y)(t) 0 −8 −6 −4 −2 0 2 4 6 8 τ t − tstart t 1 0.5 0 t T x(τ ) y(τ ) y(t − τ ) (x ∗ y)(t) 1 1 0.5 0.5 0 0 −8 −8 0 −8 −6 −4 −2 0 2 4 6 8 τ t 1 1 0.5 0.5 0 0 −8 −8 t x(τ ) y(τ )) y(τ x(τ − ) τ) y(t x(t∗−y)(t) τ) (x (y ∗ x)(t) −6 −6 t − tstart t − tstart 0 0 −2 −2 0 τ0 2 2 4 4 6 6 x(τ ) y(τ )) y(τ x(τ ) τ) y(t − x(t∗−y)(t) τ) (x (y ∗ x)(t) −6 −6 −4 −4 −2 −2 1 1 0.5 0.5 0 0 −8 −8 t x(τ ) y(τ )) y(τ x(τ ) τ) y(t − x(t∗−y)(t) τ) (x (y ∗ x)(t) −6 −6 t 0 0 1 1 0 0 −8 −8 8 8 −2 −2 0 τ0 τ 2 2 4 4 6 6 t − tstart t − tstart 1 1 0.5 0.5 0 0 −8 −8 0 t t 0 x(τ ) y(τ )) y(τ x(τ − ) τ) y(t x(t∗−y)(t) τ) (x (y ∗ x)(t) −6 −6 −2 −2 0 τ0 τ t − tstart t − tstart 1 1 0.5 0.5 0 0 −8 −8 0 0 t x(τ ) y(τ )) y(τ x(τ − ) τ) y(t x(t τ) (x ∗−y)(t) (y ∗ x)(t) −6 −6 1 1 0 0 −8 −8 8 8 x(τ ) y(τ )) y(τ x(τ − ) τ) y(t x(t∗−y)(t) τ) (x (y ∗ x)(t) 2 2 4 4 6 6 t 1 1 0 0 −8 −8 8 8 −2 −2 0 τ0 2 2 4 4 6 6 −6 −6 −4 −4 −2 −2 1 1 0.5 0.5 0 0 −8 −8 0 0 x(τ ) y(τ )) y(τ x(τ − ) τ) y(t x(t∗−y)(t) τ) (x (y ∗ x)(t) t −6 −6 −4 −4 −2 −2 0 τ0 τ 0 τ0 τ 0 0 t x(τ ) y(τ )) y(τ x(τ − ) τ) y(t x(t∗−y)(t) τ) (x (y ∗ x)(t) −6 −6 −4 −4 −2 −2 1 1 0 0 −8 −8 8 8 x(τ ) y(τ )) y(τ x(τ ) τ) y(t − x(t∗−y)(t) τ) (x (y ∗ x)(t) −6 −6 −4 −4 −2 −2 0 τ0 τ 2 2 4 4 6 6 8 8 T T 2 2 4 4 6 6 t t 0 2 2 8 8 T T 2 2 0 0 τ0 τ 8 8 4 4 6 6 t 8 8 T T 4 4 6 6 8 8 t − tstart T T t 6 6 0 t t 0 x(τ ) y(τ )) y(τ x(τ − ) τ) y(t x(t τ) (x ∗−y)(t) (y ∗ x)(t) τ t − tstart t − tstart 4 4 T T t − tstart t − tstart T T 0.5 0.5 −4 −4 2 2 t 0 0 t − tstart t − tstart T T 0.5 0.5 −4 −4 0 T T t − tstart t − tstart T T 0.5 0.5 −4 −4 0 0 τ0 τ t τ t − tstart t − tstart t − tstart t − tstart t − tstart t − tstart T T 0.5 0.5 −4 −4 t 0 1 0.5 t T y(τ ) x(τ ) x(t − τ ) (y ∗ x)(t) 0 −6 −6 −4 −4 −2 −2 0 τ0 2 2 4 4 6 6 8 8 τ −8 −6 −4 −2 0 2 4 6 8 τ 100 t − tstart 0 1 0.5 t T y(τ ) x(τ ) x(t − τ ) (y ∗ x)(t) 0 −8 −6 −4 −2 0 2 4 6 8 τ 64 Convolving with impulse function 101 Convolution Theorems Theorems Time convolution F x(t) ∗ y(t) ⇐⇒ X(f ) Y (f ) (4.18) Frequency convolution F x(t) y(t) ⇐⇒ X(f ) ∗ Y (f ) (4.19) Continuous LTI system 102 Assignment • Solve problems (4.1); 4.1f to ?? 103 65 5 Problems Important: For any FT pair you sketch in the following questions, make sure to: • clearly write all the horizontal and vertical axes tick values on both time-domain and frequency-domain curves; whenever possible. • use the and symbols whenever applicable. • If a frequency domain curve is real, write the vertical axis label similar to “X(f ); X(f ) is real”. Otherwise, write the vertical axis label similar to “|X(f )|”. 4.1. Using the FT pairs in appendix A and the FT properties of section (3), derive a mathematical expression and sketch (magnitude of the FT is enough) the (a) FT of y(t) = A sin (2πf0 t) (b) FT of x(t) = A cos (2πf0 t + φ) + B (c) FT of x(t) = A sin (2πf0 t + φ) + B (d) the IFT of the low pass filter ??? (e) IFT of ??? 66 (f) IFT of band pass filter, where fc = 4fmax ??? (g) FT of F ⇐⇒ ??? 104 67 Part 5 Appendices A Basic Fourier Scale Scale Scale Dt & Df = 2.5 mm Scale Dt = Dt &fsDf Df = 2.5 2.5 mm mm = 20mm T && Dt &f Df = 2.5 mm = 20mm T & Transform TTPairs & fss = 20mm & fs = 20mm 68 105 69 Part 6 Problem’s Answers 1 Part 2: Static Measurements Problem (2.5) • Load cells’ output is plotted in the following charts. Load cell A Load cell B Load cell C •Accurate and repeatable • Not accurate but repeatable • Seem to accurate but not usable Max. 11.6 11.6 x 11.0 x 10.6 Output (mV) Output (mV) 10.8 x 10.4 Ave. x x 10.0 x x 11.2 x 10.2 x x x x x x x 11.4 Min. 11.2 11.0 11.0 10.8 10.8 10.6 Output (mV) 11.2 x 11.4 Max. 11.4 11.6 10.4 10.2 10.0 10.6 10.4 9.8 9.6 9.6 9.6 9.4 9.4 9.2 x x x 9.2 Min 9.2 9.0 x 9.0 0 1 2 3 4 5 6 7 8 9 10 x 10.0 9.8 9.4 Max. 10.2 x x x x x x x x x 8 9 10 9.8 0 1 2 3 4 5 6 7 8 9 10 Min. 9.0 0 1 Trial no. Trial no. 2 3 4 5 6 7 Trial no. • Inputs of the problem are summarized as – x = 50 kg & ytrue = 10 mV – xmax = 100 kg • The load cell nominal sensitivity is calculated as S= 10 = 0.2 mV/kg 50 and the FSO is calculated as ro = S xmax = 20 mV (Of course it is understood that xmin = 0 kg and ymin = 0 mV. Hence ro = ymax − ymin = ymax = S xmax ) • Thus the accuracy and precision are calculated as |ȳ − ytrue | max (|y − ȳ|) = max (ymax − ȳ, ȳ − ymin ) 70 Load cell ymin ȳ ymax Accuracy = |ȳ − ytrue | Accuracy(%FSO) ymax − ȳ ȳ − ymin Precision= max (|y − ȳ|) Precision (%FSO) A 9.02 10.09 11.2 0.09 mV 0.45% 1.11 1.07 1.11 mV 5.55% B 11.42 11.49 11.58 1.49 mV 7.45% 0.09 0.07 0.09 mV 0.45% C 9.92 9.99 10.08 0.01 mV 0.05% 0.09 0.07 0.09 mV 0.45% Conclusion A transducer or sensor that is repeatable but not overly accurate may still be quite usable in a measurement or control application. As long as the transducer or sensor is repeatable, you will get consistent results. We may correct this inaccuracy by the recalibration this transducer or sensor 71 106