Download RMS, power, dB, PDF. Analogue-to

Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
Retrieval of experimental data files.
Signal types.
Signal characteristics:
RMS, power, dB, PDF.
Analogue-to-Digital Conversion (ADC).
Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
Outline
• Retrieval of experimental data files
• Power in signal analysis
• Root Mean Squares (RMS)
• dB scale
• Dynamic range
• Specification of ADC
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Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
Retrieval of experimental data files
• Text (ASCII) files
• Binary files, e.g.
–
–
–
–
*.mat (Matlab native compressed)
*.wav files
*.daq files (Matlab data acquisition)
other binary files. e.g. ‘float32’
Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
Reading *.wav file in Matlab
[fName, fPath] = uigetfile('*.wav',…
'Select *.wav file');
[g,sFreq] = audioread([fPath fName]);
%or wavread ([fPath fName]);
dt = 1 / sFreq; %sampling interval
nPts = length(g);
time = (0:nPts-1)’ * dt;
plot(time*1000,g(:,1)) %2 cols if stereo
xlabel('Time, ms')
ylabel('Intantaneous sound pressure, -')
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Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
Example of reading a binary file
• Each data point is stored as 4 byte
floating point number:
%open the file for reading
fid = fopen(([fPath fName], ’r’);
%read the whole contents
g = fread(fid, ’float32’);
Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
Reading an ASCII file
%Example:
%A single column data file
%first number is the sample frequency:
[fName, fPath] = uigetfile('*.txt',…
'Select data file’);
g = load([fPath, fName], ’-ascii’);
sFreq = g(1); %first data point is sample freq.
g(1) = []; %eliminate the first point
dt = 1/sFreq; %sample interval
nPts = length(g);
time = (0 : nPts – 1)’ * dt;
%or time = linspace(0, (nPts-1) * dt, nPts);
plot(time, g)
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Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
Signal types
• Stationary (average properties don’t
vary with time)
– Deterministic
• Instantaneous value is predictable at all
points in time
– Random
• Only statistical properties are predictable
• Spectrum is continuous
• Non-stationary
– Continuous - eg speech
– Transient – eg shock
Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
wav
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Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
Signal characteristics ?
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0
2
4
6
8
10
Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
Power of a signal
X’er
Signal
Conditioning
(amplifier)
V2
Power  Vi 
R
V
Units 
sensitivity
V  Units  sensitivity
i
i=V/R
R
Voltage
across a
resistor
(recording
device)
V2
Units  sensitivity 2

R
R
sensitivity 2
Power 
Units2
R
Power  Units2
Power 
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Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
Instantaneous power
• In signal analysis the instantaneous
sample squared is referred to as
‘POWER”
Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
Mean power of vibration and Root Mean
Squares (RMS)
• Mean power of a sampled vibration
signal (mean squares)
1
Mean power 
N
N
 gi
2
i 1
• Mean level that
produces the same
power as the signal
0.4
RMS
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
RMS 
1
N
N
 gi
-0.4
2
0
i 1
or Root Mean Squares (RMS)
2
4
6
8
10
1 N
 g i  g 2
N i 1
Standard deviation
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Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
RMS of a signal stored in vector g
• rms=sqrt(mean(g.^2))
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0
2
4
6
8
10
Average Power  RMS 2
RMS  Average Power
Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
1
Mean power of harmonic
signal
A cos x
over the period of 2
(analytical solution)
0.5
0
-0.5
-1
1
2
2

A 2 cos2x dx 
0
RMS of a
harmonic
signal
0
1
2
3
4
5
6
7
1
A 1
A
 2
 x  sin 2x  
2  2
4
2
0
2
RMS 
2
A2
A

 0.707 A
2
2
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Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
dB scale (decibel)
• dB scale is a relative logarithmic
scale
dB  10 log
Power
Powerref
Since Power  RMS 2
 RMS
dB  10 log
 RMS ref



2
 20 log
RMS
RMSref
• 0 dB corresponds to ?
the reference level
Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
dB scale
• In air acoustics in which the sound pressure
level is measured, the reference for the dB
scale is the average lowest threshold of
audibility, by convention taken as 20 Pa
(2 x 10-5 Pa )
• This 20 Pa is the RMS of the reference
signal
• In other applications, however, the dB
scale is used to compare the levels of two
signals and the choice of reference level is
arbitrary
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Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
dB - example
• The level increases from the reference level
(0 dB) to 40 dB in increments of 10 dB. What is
the corresponding factor by which the RMS
level and power increase?
dB  20 log
RMS  10
P  10
dB
10
RMS
RMS ref
dB
20
RMS ref
Pref
dB
0dB
10dB
20dB
30dB
40dB
RMS
1
3.16
10
31.6
100
Power
1
10
100
1000
10000
Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
Dynamic range of ADC
• Dynamic range (in dB), ref.=1
– bi-polar ADC
Dynamic range = 20 log (2^resolution/2)
Eg. 20 log (2^12/2) = 20 log 4096/2 = 66 dB
20 log (2^16/2) = 20 log 65536/2 = 90 dB
– uni-polar ADC
Dynamic range = 20 log (2^resolution)
Eg. 20 log (2^12) = 20 log 4096= 72 dB
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Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
Probability Density
Units
random-gaussian
‘pdfExample.m’
Probability Density Function
4
4
3
3
2
2
1
1
0
0
-1
-1
-2
-2
-3
-3
-4
-4
0
5
10
15
0
0.05
0.1
0.15
0.2
Time (s)
• Normally distributed (Gaussian) random signal
• White noise
Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
Probability Density Function - PDF
random-gaussian
Probability Density Function
3
2
1
0
i
p  p / length( g ) / diff (bins (1 : 2));
plot (bins, p ) %or bar (bins, p)
xlabel (' Signal g (units)' )
0
-1
 t
T binWidth
In MATLAB :
[ p, bins ]  hist ( g , nBins );
2
1
Units
pdf (bin) 
-1
t1 t2
ylabel (' PDF (1/unit ' )
-2
-2
-3
-3
5
10
Time (s)
15
0
0.1
0.2
0.3
0.4
-3
x 10
• PDF estimate – the fraction of total time the signal is within
a particular bin, normalised (divided) by the bin width.
• The same as the number of points of data within a bin
divided by the bin width
• Histogram of frequency of occurrence
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Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
Exercise
• Calculate the 3-phase mains voltage in
Australia.
Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
ADC parameters
• Analogue-to-Digital Converter
• Sample rate or frequency (Hz)
• Sampling interval (s, ms, s)
Sample interval 
1
Sample frequency
• Resolution – the ability of ADC to
distinguish the voltage
• Gain
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Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
Resolution of ADC
• Expressed in bits, eg. 12 bit
resolution
212  4096 states
from 0 to 4095
• 16 bit resolution
216  65536 states
from 0 to 65535
Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
Gain
• Internal amplification in the ADC
• Input voltage is amplified the ‘gain’ times
and supplied to the AD converter
• Why bother?
• Gain is used to improve the resolution of
ADC conversion for lower voltage signals
• Example
– Voltage of 2V measured with the gain of 4
would appear to the ADC as 8V
• Low level ADC boards with the high gain
(eg 500) are used for low sensitivity
transducers such as thermocouples (eg
full range +/-10V/500 = 20mV)
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Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
ADC - Voltage range (span)
• Bi-polar, eg. +/-10V
• Uni-polar, eg. 0-5V
• The range is divided into the 2resolution
number of states, eg 212 = 4096
• Bi-polar +/-10V, 12 bit resolution
Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
Offset binary coded ADC
• The lowest voltage of the range is
mapped to 0 by the ADC
• The highest voltage is mapped to
2bit_ resolution
10V
4096 = 212
0V
2048
-10V
Voltage
0
Digital values
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Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
Example – offset binary coding
• A bi-polar ADC with the voltage range of
+/-10V, 12 bit resolution and the gain of
one returns a digital value (DV) of 2500.
What is the voltage?
• Voltage span is 20 V (from –10V to 10V)
• Number of states is 4096 (212)
• -10V correspond to the DV of zero
• 10V correspond to the DV of 4096
• 0V corresponds to 2048 (4096/2)
DV  2048
10
2048
2500  2048
Voltage 
10  2.207V
2048
Voltage 
Dr Michael Sek – Signal characteristics: rms, power, dB; ADC
Conversion of DV to voltage
• Various forms of conversion
DV  2048
10
or
2048
Specific cases
20
Voltage 
DV  10
or
4096
Span
Voltage  resolution DV  lowest voltage
2
 Span DV  lowest voltage
 resolution

2


Voltage 
gain
Voltage 
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