Chapter 1.2
Characteristics of signals
• Signal - Any physical quantity that varies with time (or other independent variable)
and carries information.
– In our case currents and voltages which are time dependent; or
– Variation of pressure with depth- not time dependent
Signals
Continuous
Discrete
Smooth changes
wrt independent
variable
Changes wrt
independent
variable occur
in definite
steps
Continuous:
time
Discrete:
Step size or
quantization
time
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If what we are considering is a variation of signal amplitude with time, we can define the
following:
• Analog signal-Continuous amplitude changes within some set of limits
– Indications of a moving coil meter for example a fuel gauge
– Mercury-in-glass thermometer
• Digital signal - Discrete amplitude changes within a set of limits, for example 0-256
levels with a step size of 2 levels
– TV rotary channel selector – 13 steps, 1channel per step
– output signal from a mouse – changes in 1 pixel increments
Two other signal types frequently encountered are:
• Continuous-time signals - the signal exists for all time.
– The pressure sensors in a process plant produces continuous-time signals once the
plant is operational
• Discrete-time signals - the signal exists only at selected instances in time
nT
T 2T 3T 4T
• We can express the discrete-time signals mathematically:
x = f (nT ) n = 0,1, 2,..........N − 1, N
i.e
x = f (T ), f (2T ), f (( N − 1)T ), f ( NT )
• Discrete-time signals are produced by many computer based systems;
• Can be produced from continuous-time signals by a process called analog to digital
conversion which involves sampling and quantizing.
• They are useful since they consist of a set of values that can be stored and retrieved
for later processing.
• Many measurement systems convert continuous-time input signals to discrete-time
signals for storage and further processing- Digital storage oscilloscope for example.
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Periodic signals
• Periodic signals are an important class of signals that repeat themselves at a regular
interval called the period
Period
• Mathematically, this can be expressed as y = f(t+nT)
• The simplest periodic signal is the sinusoid which is described by either the sine or
cosine function, either of which produce a sinusoidally shaped waveform.
Sinusoids
• We can define a sinusoidal signal y(t) as:
y(t) = A sin(ωt+φ) where
A is the amplitude of the sinusoid;
ω is the angular frequency in rad/s; and
φ is the phase shift in radians (relative to some reference)
y(t)
Amplitude
π
0
2π
.
ω = 2πf
where f is the frequency of the sinusoid in Hz (or the no. of complete cycles made per
second).
∴ time for 1 cycle = 1/f seconds.
This is called the period, T, of the sinusoid.
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y(t)
Period
π
0
2π
The example shown starts at t = 0. If it starts at some other time t ≠ 0, we denote this
in the general equation by the phase shift φ.
• φ is +ve if y(t) starts before t=0
– Phase lead;
• φ is -ve if y(t) starts after t=0
– Phase lag;
y(t)
Phase lag
z(t)
π
0
2π
Phase lead
Example
v(t) = Vp sin(ωt+φ) ;
where Vp = 7V;
ω = 50 rad/s;
φ = 35° lead on reference (t = 0).
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
Solution
waveform
0.05
0.1
0.15
0.2
0.25
Reference
waveform
What is the equation for this signal and what is the frequency?
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• What is the local line frequency?
• What is the local angular frequency?
Characteristics of periodic waveforms
• Another way that we describe a periodic signal, besides its mathematical formula is by
its average value.
• Intuitively, what is the average value of the sinusoidal waveform?
• Mathematically, the average value of a waveform is given by :
FAV =
1T
f (t )dt
T ∫0
Represents the area
under the waveform
• For the sinusoid, the area under the curve is divided equally above the axis for a time
T/2 and below the axis for another time T/2, therefore the average value is 0.
• We therefore need another way of describing the average value of the periodic
waveform.
Examine the following figure:
• Passing current through a resistor, R, dissipates power. We know that since the
resistor gets warm.
• This happens with a sinusoidal current also (check out your average toaster!). So
although the average value of the sinusoid is zero, there is clearly some transfer of
energy taking place.
• The power dissipated is I2R watts
• Let us consider this situation graphically
current
i(t)
t3
t1 t2
R
i(t)
Heating
effect of
i(t) produced
in R
i2(t)R
t3
t1 t2
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t
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• Heating effect at:
t = i R;
2
1
1
t =i R ;
2
2
2
t = i R;
2
3
3
t =i R
2
n
n
• where i1denotes i(t1) etc.
• Note that at t3, the power dissipated is (-i3)(-i3)R.
Total heating effect = i R + i R + i R + .. + i R
Average heating effect =
2
2
2
2
1
2
3
n
i R + i R + i R + .. + i R
n
2
2
2
2
1
2
3
n
• Let us now pass a DC current I through R such that we generate the same heating
effect.
• We can therefore say that:
IR=
2
i R + i R + i R + .. + i R
n
2
2
2
2
1
2
3
n
 i + i + i + .. + i 
∴I = 

n

2
2
2
2
1
2
3
n
• We can conclude that there exists some DC current I, which produces the same
average heating effect as an AC current i(t), passing through the same resistor.
• We call this DC current the Root Mean Square (or rms.) value or effective value of the
AC current i(t).
RMS value of waveform
• In general the rms. value of a periodic waveform (signal) is:
T
2
1
Frms =
f
t
(
)
]dt
T ∫0 [
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For a sinusoidal signal:
v(t ) = V p sin(ωt )
Frms =
Vp
= 0.707Vp
2
Example:
v (t ) = 170 sin(377t )
What is Vrms? Does this value seem familiar?
• What is the effective ac power dissipated in a resistor R?
Effective power = I rms × Vrms
∴ Effective power = 0.707 I p × 0.707V p
= 0.5 I p × V p
Other typical periodic signals
Square wave
Digital clock
signals
Triangle wave
A/D converters
Sawtooth wave
Sweep generator in
TV or oscilloscope
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