MECH 373
Instrumentation and Measurements
Lecture 9
(Course Website: Access from your “My Concordia” portal)
Discrete Sampling & Analysis of Time-Varying
Signals (Chapter 5)
• Sampling-Rate Theorem
Sampling
Nyquist frequency
Aliasing
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
1
Sampling and Recording
When perform a measurement … a transducer converts the
measurand into an electrical signal … and this signal is
“sampled” using a digital computer
We normally record a continuous signal y(t) by a set of
samples ys(t) at discrete intervals of time t.
y(t)
yS(t)
t
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
t
2
Sampling Frequency
t
yS(t)
t
The number of samples recorded each second is
defined as the sampling frequency, fS.
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
3
Resemble Sampling Data
Original signal
Sampled signal
If a signal is sampled and recorded relative rapidly, the
sampled data will closely resemble the original signal.
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
4
Under Sampling of Test Data
Original signal
Sampled data
If we sampled too slowly, a recorded data will present a
distortion from the original signal. Such distortion will
introduce some measurement errors.
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
5
Sampling and Hold
Almost any analog to digital converter will have some
form of voltage “hold” before sampling. A sampling and
hold unit is used to hold each sample value until the next
pulse occurs. The sampling and hold unit is necessary
because the A/D converter requires a finite amount of
time.
y(t)
yS(t)
t
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
t
6
A 10 Hz Sine Wave Signal
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
7
Sampling Rate ~ 5 Hz
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
8
Sampling Rate ~ 11 Hz
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
9
Sampling Rate ~ 18 Hz
At a sampling frequency of 18 Hz, the reconstructed sine wave appears to
be of 8 Hz sine wave. That is, the frequency of the sine wave reconstructed
from the sampled data is still different from that of the original signal.
These incorrect frequencies that appear in the output data range are known
as alias frequencies or aliases. The alias frequencies are false frequencies
that appear in the output data, that are simply artifacts of the sampling
process, and that do not (in any matter) occur in the original data.
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
10
Sampling Rate ~ 20.1 Hz
• Now, consider the reconstruction of the original signal based on the
sampling rate of 20.1 Hz.
• The above figure shows the signal reconstructed from the data sampled at
a frequency of 20.1 Hz. The reconstructed wave has a frequency of 10 Hz,
which is the same frequency as the original signal. However, the amplitude
of the reconstructed wave is lower than the original one.
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
11
Nyquist Sampling Theorem
A continuous signal can be represented by, and
reconstituted from, a set of sample values
providing that the number of samples per
second is at least twice the highest frequency
presented in the signal.
f max is the signal frequency (or the maximum
signal frequency if there is more than one
frequency in the signal)
f S is the sampling rate
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
12
Nyquist Sampling Theorem
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
13
Nyquist Sampling Theorem
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
14
Discussion Question
Nyquist sampling theorem tells us that the sampling
frequency should be at least twice the highest
frequency presented in the signal to be able to
resemble the pattern of the original signal.
Use your common sense knowledge, to determine what
is the proper sampling frequency if you are requested
to design a vibration monitoring system for a passenger
car. Assume that the natural frequency of the car
system is 10 Hz, the driving stimulated vibration on
driver’s seat is 120 Hz, and the measurement system
natural frequency is 10 kHz.
What is the sampling frequency you will select?
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
15
Nyquist Frequency and Aliasing (1)
High frequency signal to be sampled by a low sampling rate may cause
to “fold” the sampled data into a false lower frequency signal. This
phenomena is known as aliasing.
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
16
Aliasing Example
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
17
Higher Frequency Aliases
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
18
Nyquist Frequency and Aliasing (2)
Definitions:
Sampling Time, T:
Total measuring time of a signal
Sampling Interval t:
Time between two samples
Sample Rate
1/t, the number of samples per second
:
T = N t = N/fs [sec]
fs
1
 fs 
2
t
Nyquist Frequency, Fnyq:
Maximum frequency that
can be captured by a sample interval, t
Resolution Bandwidth:
Minimum frequency that can be represented by a sample
f 
fNyq 
1
f
 s
Nt N
• The faster you sample, the higher frequency you can represent
• The longer you sample, the smaller the frequency represented
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
19
Frequency Resolution
• Duration of the sample T = N t = N/fs [sec]
• Minimum frequency that can be resolved is function of
sample length …. “bandwidth resolution”
1
fs
• Must capture a full period of the frequency f 

Nt N
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
20
Alias Frequency
A simple method to estimate alias frequencies is by using the folding diagram
shown below.
fm/ fN and
where, fN is the Nyquist (folding) frequency equal to half the sampling frequency fS.
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
21
Aliasing Formulas
fa 
fN 
Alias frequency
Folding frequency (Nyquist frequency)
f N  fs / 2
fa  ( fa / f N ) f N
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
22
Question 2
Given that sampling frequency equals 250Hz and the
ratio of folding frequency to alias frequency equals
0.5, find the alias frequency.
Solution:
f s  250
f a / f N  0.5
f N  250 / 2  125
f a  ( f a / f N ) f N  0.5 125  62.5Hz
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
23
Alias Frequency
fa = 100 Hz
Lecture 9
Lecture Notes on MECH 373 – Instrumentation and Measurements
24