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Signals and Measuring - A Big Picture View
Transcript of Signals and Measuring - A Big Picture View
Values of the input and output to which the measurement system will respond properly difference between the upper and lower values of the range dO/dI
slope of the ideal straight line (K) intercept of the best-fit straight line with the output axis
"b"in the ideal case difference in output values for increasing and decreasing inputs difference between actual and ideal straight-line behavior
N(I_l) = O(I_l) - (KI_l + b) Resolution largest change in input that can occur without a detectable change in output Big Idea: click on circles and frames to learn more; press the down arrow key to zoom back out. organized by course sequence: Question:How do they relate? Data Representation Number Systems ADC Converstion Sampling & Aliasing Big Idea: Digital representation is 0's and 1's, analog is our typical base 10 system. Digital is less expensive, easy to manipulate, can be transmitted through a noisy system; however it is prone to sampling error, clipping, aliasing etc.
Analog tends to be more accurate, however is much more susceptible to environmental noise causing disruption and distortion. Binary: "n" bits in binary represents 2*n integers. Example: 149 to Binary (click to zoom) main idea:
a binary number is
just a normal number
converted to the sum
of 2's powers. Least Significant Bit (LSB) Most Significant Bit: Does this number determine whether it's an even or odd number in base 10? If yes, you have your LSB! greatest value For sign convention, the MSB is the sign bit. 0 means positive, 1 means negative. Converts analog voltages to binary (ie digital) integers. clipping notice that in the 3 bit example it is divided into 2^3 - 1 intervals (7 intervals) quantization coding quantization error = Q/2;
smallest change in voltage measurable is equal to quantization interval. aperture time = time needed to convert voltage to a binary code, signal can change in this time (nothing is real time!) Hokay, so:
We know there is a time delay involved in signal processing; what happens because of that? the signal changes in the time taken to measure it if you don't sample fast enough, it will seem like you're sampling a different signal than you are! engineering challenge:
you want all of the information, as accurate as possible, as fast as possible, using as little effort as possible. Sample Statistics Confidence Interval Linear Regression Propagation
of Error Accuracy + Precision (click to zoom) accurate, not precise accurate, precise not accurate, not precise not accurate, precise probability distribution function
probability density function describes probability distribution of all possible measures of x (relative likelihood of x occuring) describes probability that some number X is between certain values. Integral of the probability distribution function. Analysis
of a Distribution Common Distributions: Gaussian (function of mean and standard deviation)
Uniform (function of lower and upper limits... is either one value or nothing) average or expected value typical deviation of values from the mean Gaussian Uniform If you sample a room full of 20 people and ask them something, and then you sample a different room full of 20 people, your mean values are likely to be more different than if you sampled two rooms of 2000 people. Which leads to the question "How confident can I be in my mean value?" Look to the mean and standard deviation of the means! When we measure something,
we want it to be accurate and
precise. Understanding how
precise and accurate measurements are
requires statistical analysis. It's not enough just to
know a measurement--
we also need to know
how likely that
to be correct. Given some inputs and outputs, what is
the relationship between them? The least squares method is a method of evaluating the relationship between a set number of input-output pairs. The correlation coefficient, R, is a measure of how well a model fits the data. The closer to 1 R is, the closer the model fits the data. if a function y is dependent on a set number of x variables, the error will be proportional to the error in each of the variables. Engineering Implications:
Optimization... which variable contributes most to overall error? How can cost and error minimization be optimized? Time
Domain Analysis of a signal with respect to time (for example, an oscilloscope typically outputs what a signal looks like as a function of time) 1st Order: 2nd Order: Time Constant and Static Sensitivity are evaluated Time Constant, Damping Ratio, Natural Frequency and Static Sensitivity are evaluated. = 0: undamped
< 1: underdamped
= 1: critically damped
>1: overdamped Analyzing a signal with respect to frequency, rather than time. Gain Phase Angle in bode plot, -20n dB/decade rolloff, for an nth order system. in bode plot, -90n degrees as frequency approaches infinity, for an nth order system. Time Domain: How does the signal vary over
Frequency domain: How does
the signal vary as a function
of frequency? Frequency
Domain Time Domain 1st Order: 63.2% Method Log Linearization Method Find K: 2nd Order: Given a response plot, how can we determine key characteristics of the function? Damped Natural Frequency Damping Ratio Overview 1st Order 2nd Order In a 1st order system, you're looking for the time constant and static sensitivity. In a 2nd order system you're trying to find the damping ratio, natural frequency, and static sensitivity. Both time domain and frequency domain information can be used to obtain this information. Filters Loading Op-Amps Filters allow you to attenuate components of a signal that you don't want. High Pass:
Attenuate Low Frequency Signals Low Pass:
Attenuate High Frequency Signals Cascading filters allows us to filter frequencies within or outside a given range (band or notch pass filters) *Examples shown are passive filters When filters are cascaded, the impedance of each filter affects the neighboring filters, producing what is known as a loading effect. The graphic below details the analysis of loading effects. click to enlarge Op-Amps are high input impedance,high gain devices. They are active devices that draw negligible current and make the voltage difference between two inputs equal to zero. They are used in filtering systems. Fourier Series Amplitude and Phase Spectra Main Idea:
Any periodic function x(t) can be represented by an infinite series of sine and consine functions. average value of signal (DC Offset) 0 for odd functions (x(-t) = -x(t)) 0 for even functions (x(-t) = x(t)) Using fourier series we can translate functions into spectra to be analyzed. Engineering implication? Given spectrum information we can calculate fourier series and understand the signal. Periodic signals can be analyzed using fourier transforms. A fourier transform is used to translate a signal between the time and frequency domain. Say you have a musical chord... the Fourier transform of that chord is a mathematical representation of the notes that make up the chord. Fourier transforms are a compact representation of a signal Bridge Circuit and Strain Gages Electrical impedance is directly related to the thickness of the wire the current passes through. Strain gages use this property to detect strain. A bridge circuit produces a voltage output proportional to a change in impedance rather than the absolute value of the impedance. The general idea here is that electrical properties can be used to measure physical changes. strain eq. Variable Impedance Devices measure of opposition to current Variable impedance devices are often used as displacement sensors. They include potentiometers, variable capacitance displacement sensors, and variable inductance displacement sensors. Output voltage is typically a function of some distance. Carrier Systems Demodulation Amplitude Modulation carrier signal carrier amplitude carrier frequency (rad/s) carrier phase at t = 0 s Most of the time, signals are transmitted directly through wires. Carrier Systems modulate signals using a high frequency voltage source modulated by a low frequency transducer signal. Benefit? Information can be transmitted over large distances without wires! Also, smaller devices are possible. youtube example As can be seen in the visual to the right, the output of amplitude modulation is just the product of the original signal and the carrier signal. Note that the carrier signal is high frequency and the incoming signal is a lower frequency. When we demodulate a signal we are trying to obtain the envelope of the modulated signal. This can be achieved through multiplication by the carrier signal and passing through a low pass filter (remember how the original signal was low frequency?) Signal to Noise Ratio Power Spectral Density Noise Reduction An unsteady component of a signal which causes the instantaneous value of the signal to vary from the true value. Looks like a bunch of humps superimposed on the desired signal. Generally random in nature. Can result from internal and external sources. Generally a signal can be thought of the sum of the true signal and the noise:
x(t) = s(t) + n(t) where s is the true signal and n is noise. The SNR tells us how "noisy" a signal is. SNR =
signal power / noise power
10log(signal power / noise power) (dB)
20log(signal RMS / noise RMS) (dB) signal power and noise power are given by squaring s(t) and n(t) noisiness of a system is the ratio of input and output SNR's... a ratio of ratios! the power spectral density is the distribution of the average signal power across frequency... essentially measuring the power of the noise per unit frequency. click to zoom Know your enemy... before you can reduce noise, know where it comes from. Johnson Noise: Temperature Induced phi = 4kRT Plan of attack? Reduce Temperature. *roughly Gaussian Shot Noise: Electron Quantization Induced phi = 2IqR^2 Plan of attack? Reduce system bandwidth Flicker Noise: Low Frequency Noise phi = material constant / freq. Plan of attack? Avoid DC Measurements for small signals because it affects lower frequency signals more Filtering, averaging repeated measurements of a signal, and differential amplifiers can help combat noise Calibration of the sensing device is the foundation of measurement. If inputs don't correspond in any logical way to an output, the data isn't useful. damping ratio can be found using amplification, half power, and slope of phase angle methods. Eo = (-Zf/Zi) * Ei *Apply KCL!