2. 1. Introduction
Measurement systems are designed to convert physical quantities (like temperature, pressure, or voltage) into
readable data, often using sensors or transducers. However, these systems are rarely perfect and are subject to
various forms of interference or degradation, most notably noise.
2. Signal Definition
A signal refers to any time-varying physical quantity that conveys information. In the context of a
measurement system, the signal is the quantity that is being measured, for example:
Voltage in an electrical circuit.
Temperature in a temperature sensor.
Pressure in a pressure sensor.
The ideal signal is assumed to be:
Noise-free.
Accurate.
Representing the true value of the measured parameter.
However, in practice, signals are always accompanied by noise, which can distort the true value.
3. The signal in a measurement system represents the quantity we want to measure. Here are more examples
of signals in various systems:
Temperature Measurement: In a thermometer or thermocouple, the signal is the temperature of the
object being measured. For instance, a thermocouple generates a small voltage that varies with
temperature (Seebeck effect).
Pressure Measurement: In a barometer or a pressure sensor, the signal is the pressure being measured.
This could be the change in capacitance or resistance that corresponds to different levels of pressure.
Flow Measurement: In a flowmeter, the signal could be the change in frequency or voltage that
corresponds to the rate of fluid flow through a pipe. An example is an electromagnetic flowmeter,
where the flow of conductive fluid generates a voltage.
Sound Measurement: A microphone converts sound (pressure variations in air) into an electrical
signal, representing the intensity and frequency of the sound. The signal in this case is the variation in
air pressure caused by sound waves.
4. 3. Noise in Measurement Systems
Noise refers to any unwanted random or deterministic fluctuation that contaminates the true signal. Noise can
come from various sources and is typically random in nature. It can result from:
Electrical noise in the components.
Environmental factors like temperature changes or electromagnetic interference (EMI).
Mechanical vibrations or fluctuations.
Types of Noise:
Thermal (Johnson-Nyquist) Noise: Generated by random thermal motion of charge carriers (electrons) in
conductors.
Shot Noise: Results from discrete nature of charge (electrons) passing through a junction (e.g., in diodes).
Flicker Noise (1/f noise): Noise that becomes more significant at lower frequencies, typically found in
resistors and transistors.
Quantization Noise: Arises when continuous signals are sampled and quantized into discrete values.
5. Noise is any unwanted signal that interferes with the accurate measurement of the true signal. Here are some
practical examples of noise sources:
Thermal Noise in Electronics: When you are measuring voltage in an electronic circuit, components like
resistors can generate thermal noise due to random motion of electrons in the material. For example, when
measuring small voltages in precision circuits, thermal noise can obscure the true signal.
Electromagnetic Interference (EMI): If you're using a measurement system near an industrial motor or a
high-voltage power line, the system may pick up electromagnetic interference. For instance, a multimeter
measuring small voltage may experience fluctuations in its readings due to EMI from nearby electronic
devices.
Mechanical Noise: Suppose you are using a vibration sensor (like an accelerometer) in a factory floor
environment. The machine's motors or external machinery can generate vibrations that interfere with the
sensor's ability to accurately measure vibrations from the system you intend to monitor.
Quantization Noise: In a digital system, when you convert an analog signal to a digital signal through an
Analog-to-Digital Converter (ADC), you can experience quantization noise. For instance, if your ADC has
limited resolution (e.g., 8 bits), the continuous signal is mapped to discrete levels, introducing small errors or
noise into the digital representation.
Flicker Noise (1/f Noise): In low-frequency sensors (e.g., thermistors or strain gauges), you may observe flicker
noise, which dominates at lower frequencies. For example, if you're measuring low-frequency vibrations or
small temperature changes, you might see an increase in noise as the frequency of the signal decreases.
6. The dynamic response of measurement systems to step, sine wave and square wave input
signals. These signals are examples of deterministic signals: a deterministic signal is one whose
value at any future time can be exactly predicted. Thus if we record these signals for an
observation period TO (Figure 6.1), the future behavior of the signal, once the observation period
is over, is known exactly.
7. Statistical representation of random signals
These observed statistical quantities will provide a good estimate of the future behaviour of the signal,
once the observation period is over, provided:
(a) TO is sufficiently long, i.e. N is sufficiently large;
(b) the signal is stationary, i.e. long-term statistical quantities do not change with time.
10. Probability density function p( y)
This is a function of signal value y and is a measure of the probability that the signal will have a certain range of
values. Figure 6.3 shows the set of sample values yi and the y axis divided into m sections each of width Δy.
11. A random signal for several observation periods, each of length TO (Figure 6.4), the waveform will be different for
each period. However, the average signal power will be approximately the same for each observation period. This
means that signal power is a stationary quantity which can be used to quantify random signals. In Section 4.3 we
saw that a periodic signal can be expressed as a Fourier series, i.e. a sum of sine and cosine waves with frequencies
which are harmonics of the fundamental frequency.
Figure 6.4 Power spectrum and power spectral
12. The cumulative probability Cj is the total probability that the signal will occur in the first j sections and is given by:
This is the cumulative probability distribution function (c.d.f.) P( y), which is defined by:
14. Figure 6.6 Relationships between power spectrum and autocorrelation function for periodic and
random signals.
15. Effects of noise and interference on measurement circuits
Figure 6.7 Effects of interference on measurement circuit:
(a) Voltage transmission – series mode interference
This means that with a voltage
transmission system all of VSM is
across the load; this affects the
next element in the system and
possibly results in a system
measurement error. We define
signal-to-noise or signal to
interference ratio S/N in decibels
by:
16. Figure 6.7 Effects of interference on measurement circuit:
(c) Voltage transmission – common mode interference.
18. External noise and interference sources
The most common sources of external interference are nearby a.c. power circuits which usually
operate at 240 V, 50 Hz. These can produce corresponding sinusoidal interference signals in the measurement
circuit, referred to as mains pick-up orhum. Power distribution lines and heavy rotating machines such as
turbines and
generators can cause serious interference.
D.C. power circuits are less likely to cause interference because d.c. voltages are not coupled
capacitively and inductively to the measurement circuit.
However, switching often occurs in both a.c. and d.c. power circuits when equipment such as motors
and turbines is being taken off line or brought back on line. This causes sudden large changes in power, i.e. steps
and pulses, which can produce corresponding transients in the measurement circuit.
The air in the vicinity of high voltage power circuits can become ionised and a corona discharge
results. Corona discharge from d.c. circuits can result in random noise in the measurement circuit and that from
a.c. circuits results in sinusoidal interference at the power frequency or its second harmonic.
Fluorescent lighting is another common interference source; arcing occurs twice per cycle so that most
of the interference is at twice the power frequency.
Radio-frequency transmitters, welding equipment and electric arc furnaces can produce r.f.
interference at frequencies of several MHz.
20. Figure 6.7 Effects of interference on measurement circuit:
(b) Current transmission – series mode interference
Editor's Notes
#7:random signal obtained during an observation period TO. Since the signal is random we cannot write down a continuous algebraic equation y(t) for the signal voltage y at time t. We can, however, write down the values y1 to yN of N samples taken at equal intervals ΔT during TO. The first sample y1 is taken at t = ΔT, the second y2 is taken at t = 2ΔT, and the ith yi is taken
at t = iΔT, where i = 1, . . . , N. The sampling interval ΔT = TO /N must satisfy the Nyquist sampling theorem, which is explained in Section 10.1. We can now use these samples to calculate statistical quantities for the observed section of the signal.