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MAE014 Computer and Instruments

1.1 Introduction

Example 1: Sounds of the ocean

Suppose we want to investigate the sounds that travel through the ocean.

A microphone is placed in the water and the resulting electronic signal amplified to a reasonablelevel. An analog low-pass filter is then used to remove all frequencies above 80 hertz, so that the signalcan be digitized at 160 samples per second.After acquiring and storing several thousand samples, what next?

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Example 2: Electronic scales

Use a weighing device called a load cell That is an aluminium alloy beam, eliminates the need forsprings, cogs, or other moving parts which can wear, break, or cause inaccuracy in mechanicalscales. A strain gauge is bonded on the load cell. The strain gauge consists of a small piece of metal foilwhich detects any bending of the beam. When a load is placed on the platform, it causes the load cell to bend very slightly. This, in turn,causes a change in strain, which triggers a change in the electrical resistance of the strain gauge. As the resistance changes, so does the output voltage from the strain gauge. In short, the change

in voltage across the strain gauge is proportional to the load on the platform.

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1.2.1 The sensor

Sensor is a physical element that employs some natural phenomenon by which it senses thevariable being measured.

1.2.2 The transducer

Transducer converts this sensed information into a detectable signal, which might be electrical,mechanical, optical, or otherwise. The goal is to convert the sensed information into a form that canbe easily quantified.

1.2.3 Signal conditioning

The signal conditioning equipment takes the transducer signal and modifies it to a desiredmagnitude.

1.2.4 The output stage

The output stage indicates or records the value measured.

1.2.5 The feedback feedback control stage

The control stage contains a controller that interprets the measured signal and makes a decisionregarding the control of the process.

1.3 Components of a measurement system

1.3.1 Variables

Variables are entities that influence the test.

A variable that can be changed independently of other variables is known as an independentvariable. A variable that is affected by changes in one or more other variables is known as a dependentvariable. Variables that are not or cannot be controlled during measurement, but that affect the value of thevariable measuredare called extraneous variables.

1.3.2 Noise

Noise is a random variation of the value of the measured signal as a consequence of the variation ofthe variables increases data scatter.

1.3.3 Interference

Interference imposes undesirable deterministic trends on the measured value. Any different from itstrue behavior is interference.

1.3.4 Parameters

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A parameter is defined as a functional grouping of variables. A parameter that has an effect on the behavior of the measured variable is called a controlparameter. Repeated measurements made during any single test run or on a single batch are calledrepetitions. Repetition helps to quantify the variation in a measured variable as it occurs during anyone test or batch while the operating conditions are held under nominal control.

An independent duplication of a set of measurements using similar operating conditions is referredto as a replication.Replication allows for quantifying the variation in a measured variable as it occurs between differenttests, each having the same nominal values of operating conditions.

1.3.5 Calibration

A calibration applies a known input value to a measurement system for the purpose of observingthe system output value, establishing the relationship between the input and output values. The known value used for the calibration is called the standard. The most common type of calibration is known as a static calibration. The term static implies thatthe values of thevariables involved remain constant, that is, they do not vary with time or space. A dynamic calibration determines the relationship between an input of known dynamic behaviorand the measurement system output.

1.3.6 Random and systematic errors

Random error is a measure of the random variation found during repeated measurements of avariable. A system that repeatedly indicates the same wrong value upon repeated application of a particularinput would be considered to have small random error contributions regardless of its accuracy. The repeatability of a measurement system refers its ability to indicate the same value on repeatedmeasurements for aspecific value of input. The term precision is sometimes used as a measure of the repeatability of a measurement orsystem. A higher precision infers a lower random error, better repeatability, or less variation betweenmeasurements.

The portion of the absolute error that remains constant on repeated measurements is called thesystematic error. Withsystematic error, there is an offset or bias from the true value that cannot be discerned fromrepeated measurements.The slope of static calibration curve yields the static sensitivity sensitivity sensitivity sensitivity of themeasurement system.

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Theaccuracyof a measurement system refers to its ability to indicate a true value exactly.Accuracy is related to absolute error.Absolute error, $\epsilon$, is defined as the difference between the true value applied to ameasurement system and the indicated value of the system:

Theprecision erroris a measure of the random variation found during repeated measurements. Thebias erroris the difference between the average value and the true value.

1.4 Base dimensions and their units

Mass

Pound (mass) - Kg: 1lbm = 0.4535924Kg

Length

1ft = 0.3048m1in = 0.0254m

Temperature

CelsiusKelvin

Fahrenheit - Rankine

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FahrenheitCelsius

Force (Newtons law)

where is a proportionality constant.

Energy

Joule:

Power

Watt:

Stress and pressure

Pascal:

2.1 Introduction to signals

2.1.1 Classification of waveforms

Analog describes a signal that is continuous in time. Discrete time signal is available only at discrete points in time. Digital signal exists at discrete values in time and the magnitude is also discrete that is determinedby a processknown as quantization at each discrete point in time. The quantization assigns a single number torepresent a range ofmagnitudes of a continuous signal. Analog-to-digital (A/D) converter is a solid-state device that converts an analog signal to a binarynumber systemrepresentation.

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2.1.2 Classification of signals

A static signal does not vary with time.

A dynamic signal is defined as a time-dependent signal.

1. A deterministic signalvaries in time in a predictable manner, such as a sine wave, a stepfunction, or a rampfunction.

Adynamic signalis defined as a time-dependent signal.

2. Aperiodicis the term used to describe deterministic signals that do not repeat at regular intervals,such as a stepfunction:

Anondeterministic signalthat has no discernible pattern of repetition. A nondeterministic signal

cannot be prescribed before it occurs, although certain characteristics of the signal may be known inadvance.

2.2 Signal analysis

2.2.1 Root-Mean-Square value

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The average or mean value of the signal is

The power dissipated in a resistor due to the flow of a current is

The total electrical energy dissipated in the resistor from to is

Finding the magnitude of a constant effective current , that would produce the same total

energy dissipation in the resistor as the time-varying current over the time

period :

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8/11/2019 MAE014 Computer and Instruments

10/23

Similarly, the root-mean-square(RMS) value of any continuous analog variable over the

time is expressed as

2.2.2 Discrete-time signals

A time-dependent analog signal can be represented by values measured over thetime period from to

where

The mean and RMS value

Example 2.1

Suppose the current passing through a resistor can be described by

Calculate the mean and RMS values of the current over a time from 0 to ,

with and .

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8/11/2019 MAE014 Computer and Instruments

11/23

Solution

Hence

Example 2.2

The following values are obtained by sampling a time-varying signal once every 0.1s

http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{array}{ll}%20t_{f}%20=%20/pi%20&%20/rightarrow%20I_{RMS}%20=%20/sqrt{50}%20//%20t_{f}%20=%202/pi%20&%20/rightarrow%20I_{RMS}%20=%20/sqrt{50}/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20I_{RMS}%20=%20/sqrt{/displaystyle%20/frac{100}{t_{f}}%20/left.%20/left(%20/displaystyle%20/frac{t}{2}%20-%20/frac{1}{2}%20cos%20t%20sin%20t%20/right)%20/right|_{0}^{t_{f}}%20}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20I_{RMS}%20=%20/sqrt{/displaystyle%20/frac{1}{t_{f}}%20/displaystyle/int_{0}^{t_{f}}%20i^{2}(t)%20dt%20}%20=%20/sqrt{/displaystyle%20/frac{1}{t_{f}}%20/displaystyle/int_{0}^{t_{f}}%20100%20sin^{2}%20t%20dt%20}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{array}{ll}%20t_{f}%20=%20/pi%20&%20/rightarrow%20/overline{I}%20=%2020/%20/pi%20//%20t_{f}%20=%202/pi%20&%20/rightarrow%20/overline{I}%20=%200%20/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/overline{I}%20=%20/displaystyle%20/frac{/displaystyle/int_{0}^{t_{f}}%20i(t)dt}{/displaystyle/int_{0}^{t_{f}}%20dt}%20=%20/displaystyle%20/frac{/displaystyle/int_{0}^{t_{f}}%2010%20sin%20t%20dt}{t_{f}}%20=%20/frac{1}{t_{f}}%20/left[%20-10%20cos%20t%20/right]_{0}^{t_{f}}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{array}{ll}%20t_{f}%20=%20/pi%20&%20/rightarrow%20I_{RMS}%20=%20/sqrt{50}%20//%20t_{f}%20=%202/pi%20&%20/rightarrow%20I_{RMS}%20=%20/sqrt{50}/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20I_{RMS}%20=%20/sqrt{/displaystyle%20/frac{100}{t_{f}}%20/left.%20/left(%20/displaystyle%20/frac{t}{2}%20-%20/frac{1}{2}%20cos%20t%20sin%20t%20/right)%20/right|_{0}^{t_{f}}%20}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20I_{RMS}%20=%20/sqrt{/displaystyle%20/frac{1}{t_{f}}%20/displaystyle/int_{0}^{t_{f}}%20i^{2}(t)%20dt%20}%20=%20/sqrt{/displaystyle%20/frac{1}{t_{f}}%20/displaystyle/int_{0}^{t_{f}}%20100%20sin^{2}%20t%20dt%20}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{array}{ll}%20t_{f}%20=%20/pi%20&%20/rightarrow%20/overline{I}%20=%2020/%20/pi%20//%20t_{f}%20=%202/pi%20&%20/rightarrow%20/overline{I}%20=%200%20/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/overline{I}%20=%20/displaystyle%20/frac{/displaystyle/int_{0}^{t_{f}}%20i(t)dt}{/displaystyle/int_{0}^{t_{f}}%20dt}%20=%20/displaystyle%20/frac{/displaystyle/int_{0}^{t_{f}}%2010%20sin%20t%20dt}{t_{f}}%20=%20/frac{1}{t_{f}}%20/left[%20-10%20cos%20t%20/right]_{0}^{t_{f}}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{array}{ll}%20t_{f}%20=%20/pi%20&%20/rightarrow%20I_{RMS}%20=%20/sqrt{50}%20//%20t_{f}%20=%202/pi%20&%20/rightarrow%20I_{RMS}%20=%20/sqrt{50}/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20I_{RMS}%20=%20/sqrt{/displaystyle%20/frac{100}{t_{f}}%20/left.%20/left(%20/displaystyle%20/frac{t}{2}%20-%20/frac{1}{2}%20cos%20t%20sin%20t%20/right)%20/right|_{0}^{t_{f}}%20}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20I_{RMS}%20=%20/sqrt{/displaystyle%20/frac{1}{t_{f}}%20/displaystyle/int_{0}^{t_{f}}%20i^{2}(t)%20dt%20}%20=%20/sqrt{/displaystyle%20/frac{1}{t_{f}}%20/displaystyle/int_{0}^{t_{f}}%20100%20sin^{2}%20t%20dt%20}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{array}{ll}%20t_{f}%20=%20/pi%20&%20/rightarrow%20/overline{I}%20=%2020/%20/pi%20//%20t_{f}%20=%202/pi%20&%20/rightarrow%20/overline{I}%20=%200%20/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/overline{I}%20=%20/displaystyle%20/frac{/displaystyle/int_{0}^{t_{f}}%20i(t)dt}{/displaystyle/int_{0}^{t_{f}}%20dt}%20=%20/displaystyle%20/frac{/displaystyle/int_{0}^{t_{f}}%2010%20sin%20t%20dt}{t_{f}}%20=%20/frac{1}{t_{f}}%20/left[%20-10%20cos%20t%20/right]_{0}^{t_{f}}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{array}{ll}%20t_{f}%20=%20/pi%20&%20/rightarrow%20I_{RMS}%20=%20/sqrt{50}%20//%20t_{f}%20=%202/pi%20&%20/rightarrow%20I_{RMS}%20=%20/sqrt{50}/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20I_{RMS}%20=%20/sqrt{/displaystyle%20/frac{100}{t_{f}}%20/left.%20/left(%20/displaystyle%20/frac{t}{2}%20-%20/frac{1}{2}%20cos%20t%20sin%20t%20/right)%20/right|_{0}^{t_{f}}%20}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20I_{RMS}%20=%20/sqrt{/displaystyle%20/frac{1}{t_{f}}%20/displaystyle/int_{0}^{t_{f}}%20i^{2}(t)%20dt%20}%20=%20/sqrt{/displaystyle%20/frac{1}{t_{f}}%20/displaystyle/int_{0}^{t_{f}}%20100%20sin^{2}%20t%20dt%20}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{array}{ll}%20t_{f}%20=%20/pi%20&%20/rightarrow%20/overline{I}%20=%2020/%20/pi%20//%20t_{f}%20=%202/pi%20&%20/rightarrow%20/overline{I}%20=%200%20/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/overline{I}%20=%20/displaystyle%20/frac{/displaystyle/int_{0}^{t_{f}}%20i(t)dt}{/displaystyle/int_{0}^{t_{f}}%20dt}%20=%20/displaystyle%20/frac{/displaystyle/int_{0}^{t_{f}}%2010%20sin%20t%20dt}{t_{f}}%20=%20/frac{1}{t_{f}}%20/left[%20-10%20cos%20t%20/right]_{0}^{t_{f}}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{array}{ll}%20t_{f}%20=%20/pi%20&%20/rightarrow%20I_{RMS}%20=%20/sqrt{50}%20//%20t_{f}%20=%202/pi%20&%20/rightarrow%20I_{RMS}%20=%20/sqrt{50}/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20I_{RMS}%20=%20/sqrt{/displaystyle%20/frac{100}{t_{f}}%20/left.%20/left(%20/displaystyle%20/frac{t}{2}%20-%20/frac{1}{2}%20cos%20t%20sin%20t%20/right)%20/right|_{0}^{t_{f}}%20}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20I_{RMS}%20=%20/sqrt{/displaystyle%20/frac{1}{t_{f}}%20/displaystyle/int_{0}^{t_{f}}%20i^{2}(t)%20dt%20}%20=%20/sqrt{/displaystyle%20/frac{1}{t_{f}}%20/displaystyle/int_{0}^{t_{f}}%20100%20sin^{2}%20t%20dt%20}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{array}{ll}%20t_{f}%20=%20/pi%20&%20/rightarrow%20/overline{I}%20=%2020/%20/pi%20//%20t_{f}%20=%202/pi%20&%20/rightarrow%20/overline{I}%20=%200%20/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/overline{I}%20=%20/displaystyle%20/frac{/displaystyle/int_{0}^{t_{f}}%20i(t)dt}{/displaystyle/int_{0}^{t_{f}}%20dt}%20=%20/displaystyle%20/frac{/displaystyle/int_{0}^{t_{f}}%2010%20sin%20t%20dt}{t_{f}}%20=%20/frac{1}{t_{f}}%20/left[%20-10%20cos%20t%20/right]_{0}^{t_{f}}

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Calculate the mean and RMS values for this discrete data.

Solution

The mean value

Calculating

The RMS value

http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{array}{rl}y_{RMS}%20&=%20/sqrt{%20246.3915%20}%20=%2015.6968/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{array}{rl}/displaystyle%20/frac{1}{N}%20/sum_{k=0}^{N-1}%20y^{2}_{k}%20&=%20/displaystyle%20/frac{1}{11}%20/left(%200%20+%20235.3156%20+%20442.2609%20+%20442.2609%20+%20235.3156%20/right.%20//%20&%20/left.%20+%200%20+%20235.3156%20+%20442.2609%20+%20442.2609%20+%20235.3156%20+%200%20/right)%20//&%20=%20/displaystyle%20/frac{1}{11}%20(1769.0436%20+%20941.2624)%20=%20/displaystyle%20/frac{1}{11}%202710.306%20=%20246.3915/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=/begin{array}{rl}/overline{y}%20&=%20/displaystyle%20/frac{1}{N}%20/sum_{k=0}^{N-1}%20y_{k}%20//%20&=%20/displaystyle%20/frac{1}{11}%20(0%20+%2015.34%20+%2021.03%20+%2021.03%20+%2015.34%20//%20&%20+%200%20-%2015.34%20-%2021.03%20-%2021.03%20-%2015.34%20+%200)%20=%200/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{tabular}{|c|c|c|c|}%20/hline/mathbf{t}%20&%20/mathbf{y(t)}%20&%20/mathbf{t}%20&%20/mathbf{y(t)}%20//%20/hline0%20&%200%20&%200.6%20&%20-15.34%20//%200.1%20&%2015.34%20&%200.7%20&%20-21.03%20//%200.2%20&%2021.03%20&%200.8%20&%20-21.03%20//%200.3%20&%2021.03%20&%200.9%20&%20-15.34%20//0.4%20&%2015.34%20&%201.0%20&%200%20//%200.5%20&%200%20&%20&%20//%20/hline%20/end{tabular}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{array}{rl}y_{RMS}%20&=%20/sqrt{%20246.3915%20}%20=%2015.6968/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{array}{rl}/displaystyle%20/frac{1}{N}%20/sum_{k=0}^{N-1}%20y^{2}_{k}%20&=%20/displaystyle%20/frac{1}{11}%20/left(%200%20+%20235.3156%20+%20442.2609%20+%20442.2609%20+%20235.3156%20/right.%20//%20&%20/left.%20+%200%20+%20235.3156%20+%20442.2609%20+%20442.2609%20+%20235.3156%20+%200%20/right)%20//&%20=%20/displaystyle%20/frac{1}{11}%20(1769.0436%20+%20941.2624)%20=%20/displaystyle%20/frac{1}{11}%202710.306%20=%20246.3915/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=/begin{array}{rl}/overline{y}%20&=%20/displaystyle%20/frac{1}{N}%20/sum_{k=0}^{N-1}%20y_{k}%20//%20&=%20/displaystyle%20/frac{1}{11}%20(0%20+%2015.34%20+%2021.03%20+%2021.03%20+%2015.34%20//%20&%20+%200%20-%2015.34%20-%2021.03%20-%2021.03%20-%2015.34%20+%200)%20=%200/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{tabular}{|c|c|c|c|}%20/hline/mathbf{t}%20&%20/mathbf{y(t)}%20&%20/mathbf{t}%20&%20/mathbf{y(t)}%20//%20/hline0%20&%200%20&%200.6%20&%20-15.34%20//%200.1%20&%2015.34%20&%200.7%20&%20-21.03%20//%200.2%20&%2021.03%20&%200.8%20&%20-21.03%20//%200.3%20&%2021.03%20&%200.9%20&%20-15.34%20//0.4%20&%2015.34%20&%201.0%20&%200%20//%200.5%20&%200%20&%20&%20//%20/hline%20/end{tabular}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{array}{rl}y_{RMS}%20&=%20/sqrt{%20246.3915%20}%20=%2015.6968/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{array}{rl}/displaystyle%20/frac{1}{N}%20/sum_{k=0}^{N-1}%20y^{2}_{k}%20&=%20/displaystyle%20/frac{1}{11}%20/left(%200%20+%20235.3156%20+%20442.2609%20+%20442.2609%20+%20235.3156%20/right.%20//%20&%20/left.%20+%200%20+%20235.3156%20+%20442.2609%20+%20442.2609%20+%20235.3156%20+%200%20/right)%20//&%20=%20/displaystyle%20/frac{1}{11}%20(1769.0436%20+%20941.2624)%20=%20/displaystyle%20/frac{1}{11}%202710.306%20=%20246.3915/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=/begin{array}{rl}/overline{y}%20&=%20/displaystyle%20/frac{1}{N}%20/sum_{k=0}^{N-1}%20y_{k}%20//%20&=%20/displaystyle%20/frac{1}{11}%20(0%20+%2015.34%20+%2021.03%20+%2021.03%20+%2015.34%20//%20&%20+%200%20-%2015.34%20-%2021.03%20-%2021.03%20-%2015.34%20+%200)%20=%200/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{tabular}{|c|c|c|c|}%20/hline/mathbf{t}%20&%20/mathbf{y(t)}%20&%20/mathbf{t}%20&%20/mathbf{y(t)}%20//%20/hline0%20&%200%20&%200.6%20&%20-15.34%20//%200.1%20&%2015.34%20&%200.7%20&%20-21.03%20//%200.2%20&%2021.03%20&%200.8%20&%20-21.03%20//%200.3%20&%2021.03%20&%200.9%20&%20-15.34%20//0.4%20&%2015.34%20&%201.0%20&%200%20//%200.5%20&%200%20&%20&%20//%20/hline%20/end{tabular}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{array}{rl}y_{RMS}%20&=%20/sqrt{%20246.3915%20}%20=%2015.6968/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{array}{rl}/displaystyle%20/frac{1}{N}%20/sum_{k=0}^{N-1}%20y^{2}_{k}%20&=%20/displaystyle%20/frac{1}{11}%20/left(%200%20+%20235.3156%20+%20442.2609%20+%20442.2609%20+%20235.3156%20/right.%20//%20&%20/left.%20+%200%20+%20235.3156%20+%20442.2609%20+%20442.2609%20+%20235.3156%20+%200%20/right)%20//&%20=%20/displaystyle%20/frac{1}{11}%20(1769.0436%20+%20941.2624)%20=%20/displaystyle%20/frac{1}{11}%202710.306%20=%20246.3915/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=/begin{array}{rl}/overline{y}%20&=%20/displaystyle%20/frac{1}{N}%20/sum_{k=0}^{N-1}%20y_{k}%20//%20&=%20/displaystyle%20/frac{1}{11}%20(0%20+%2015.34%20+%2021.03%20+%2021.03%20+%2015.34%20//%20&%20+%200%20-%2015.34%20-%2021.03%20-%2021.03%20-%2015.34%20+%200)%20=%200/end{array}http://elearning.tnut.edu.vn/filter/tex/displaytex.php?texexp=%20/begin{tabular}{|c|c|c|c|}%20/hline/mathbf{t}%20&%20/mathbf{y(t)}%20&%20/mathbf{t}%20&%20/mathbf{y(t)}%20//%20/hline0%20&%200%20&%200.6%20&%20-15.34%20//%200.1%20&%2015.34%20&%200.7%20&%20-21.03%20//%200.2%20&%2021.03%20&%200.8%20&%20-21.03%20//%200.3%20&%2021.03%20&%200.9%20&%20-15.34%20//0.4%20&%2015.34%20&%201.0%20&%200%20//%200.5%20&%200%20&%20&%20//%20/hline%20/end{tabular}

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2.3 Signal amplitude and frequency

2.3.1 Periodic signals

Consider a spring-mass system

The spring force , is the spring constant.

is the displacement. is the mass.

By Newtons second law

The general form of the solution is

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Where , , .

2.3.2 Frequency analysis

A complex signal can be represented by the addition of a number of simpler periodic functions.

Example: The separation of white light through a prism.

Any

complex signal can be thought of as made up of sines and cosines of differing periods andamplitudes, which are added together in an infinite trigonometric series Fourier series.

A function is called a periodic function if there is some positive number such that

If both and have period , then

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8/11/2019 MAE014 Computer and Instruments

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also has period of .

A periodic function with a period is to be represented by

with known, the coefficients and are to be determined.

Taking integration from to :

Note that

Hence

Multiplying by and integrating from to we get

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