Transcript Slide 1

Lecture 2: Measurement and
Instrumentation
Time 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 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 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 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.

f ( x)  a0   (an cosnx  bn sin nx)
n 1
where the Fourier coefficients of f(x) can be calculated using the
so-called Euler formulas
1 
a0 
 f ( x)dx
2
an 
bn 
1

1


 f ( x) cos nxdx

 f ( x) sin nxdx
Frequency 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 Types
• Figures on the
right show a few
common signals
in both the time
and frequency
domains.
Laplace Transformation
Measurement System
• Components
– Transducer: convert a physical quantity into a timevarying 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 Function
• Block diagram and transfer function provide an
efficient way to describe a dynamic system
Input, r(t)
Output, y(t)
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)
Frequency Response of a System
L[ y (t )] Y ( s)
G( s) 

L[r (t )] R( s)
Characteristics of a Good Measurement
System
y(t )  A sin(2ft   )
A: Amplitude
f: frequency
• Amplitude Linearity
: phase
• Adequate Bandwidth
• Phase Linearity
Amplitude Linearity
Vout(t )  Vout (0)   Vin (t )  Vin (0)
: proportion constant
Factors impact linearity
• Limited range of input amplitude
• Bandwidth
Frequency Bandwidth
• Requirement: a measurement system should
replica all frequency components
 Aout 

• Unit-Decibel scale: dB  20 log10 
 Ain 
• Bode plot: frequency response curve of a
system, i.e. a plot of amplitude ratio Aout/Ain, vs.
the input frequency
Amplitude (dB)
Bode Plot
fL
fc
fH
Frequency
Bandwidth
• Bandwidth: the range of frequencies where the
input of the system is not attenuated by morn
than -3dB
• Low & high cutoff frequency:
• Bandwidth: BW  f H  f L
• Questions:
– what happen if a square wave is measured with a
limited BW system
– How to determine the BW of a measurement system