Lecture 2: Measurement and Lecture 2: Measurement and InstrumentationInstrumentation

Time vs. Frequency DomainTime vs. Frequency Domain

• Different ways of looking at a problem– Interchangeable: no information is lost in changing

from one domain to another– Benefits from changing perspective: the solution to

difficult problems can often become quite clear in the other domain

Time Domain AnalysisTime Domain Analysis

• The traditional way of observing signals is to view them in the time domain. Record in the time domain typically describes the variation of system output or system parameter over time.

Record from SeismographRecord from Seismograph

• This record (of displacement, also called seismogram) displays 24 hours of data, beginning approximately 8 hours before the mainshock, recorded by the BDSN station BKS at a distance of 525 km.

Frequency Domain AnalysisFrequency Domain Analysis

• It was shown over one hundred years ago by Jean B. Fourier that any waveform that exists in the real world can be generated by adding up sine waves.

where the Fourier coefficients of f(x) can be calculated using the so-called Euler formulas

10 )sincos()(f

nnn nxbnxaax

dxxfa )(2

10

nxdxxfbn sin)(1

nxdxxfan cos)(1

Frequency Component of a SignalFrequency Component of a Signal

• This frequency domain representation of our signal is called the spectrum of the signal, in which every sine wave from the signal appears as a vertical line. Its height represents its amplitude and its position represents its frequency.

• Each sine wave line of the spectrum is called a component of the total signal.

Spectrum of Various Signal TypesSpectrum of Various Signal Types

• Figures on the right show a few common signals in both the time and frequency domains.

Laplace TransformationLaplace Transformation

Measurement SystemMeasurement System

• Components– Transducer: convert a physical quantity into a time-

varying electrical signal, i.e. analog signal– Signal conditioner: modify/enhance the analog

signal (profiltering, amplification, etc)– Analog-to-digital converter: convert analog signal

into digital format– Digital signal processing– Recorder: display or data storage

Actual Input

Measurement System

Recorder output

Transfer FunctionTransfer Function

• Block diagram and transfer function provide an efficient way to describe a dynamic system

• Transfer Function – Used to describe linear time-invariant systems

– Can be expressed using the Laplace transform of the ratio of the output and input variables of the system

– Can be experimentally determined by curve-fitting• Frequency response (sinusoidal input)

• Impulse response (short-duration pulse input)

Input, r(t) Output, y(t)System

Frequency Response of a SystemFrequency Response of a System

)(

)(

)]([

)]([)(

sR

sY

tr

tysG

L

L

Characteristics of a Good Measurement Characteristics of a Good Measurement SystemSystem

• Amplitude Linearity

• Adequate Bandwidth

• Phase Linearity

)2sin()( ftAty

A: Amplitude

f: frequency

phase

Amplitude LinearityAmplitude Linearity

)0()()0()( ininouout VtVtVtV

Factors impact linearity • Limited range of input amplitude• Bandwidth

: proportion constant

Frequency BandwidthFrequency Bandwidth

• Requirement: a measurement system should replica all frequency components

• Unit-Decibel scale:

• Bode plot: frequency response curve of a system, i.e. a plot of amplitude ratio Aout/Ain, vs. the input frequency

in

out

A

AdB 10log20

Bode PlotBode Plot

fL fc fH Frequency

Am

plitu

de (

dB)

BandwidthBandwidth

• Bandwidth: the range of frequencies where the input of the system is not attenuated by morn than -3dB

• Low & high cutoff frequency:• Bandwidth: • Questions:

– what happen if a square wave is measured with a limited BW system

– How to determine the BW of a measurement system

LH ffBW