Lecture 1: Signals and Signal Characterisation
In this lecture, we look at some of the more basic and frequently encountered analogue
signals, with particular reference to the sinusoid. Mathematical formulations for these
waveforms will be established and two particular parameters of interest will be defined
and calculated: the average (or DC) value and the rms (or root mean square) value. We
also give some consideration to the idea of a periodic signal being composed of a number
of harmonically related sinusoids.
Learning Outcomes:
On completing this lecture, you will be able to:



Sketch and form mathematical expressions for common periodic waveforms;
Calculate average and rms values;
Carry out power calculations based on Fourier series waveform representation.
1.1
Some Basic Signals
(i)
The Sinewave
v(ωt)
E
v(ωt) = Esin(ωt+Φ)
2π
ωt
Φ
For real world analogue signals that are periodic, sine waves play a fundamental role in
characterising the mathematical structure of such signals. Indeed, in our consideration of
typical speech signals in the last lecture we visually noted tendencies towards the
sinusoidal. As shown above, a general sinusoidal signal v is expressed by
v(t )  E sin( t   )
where E is the amplitude of the sine wave,  is the phase angle, and  is the angular
frequency expressed in rad/s. Normally, we are not all that interested in the phase of a
particular signal but rather in the phase difference between two signals. Thus we will
most usually consider the phase angle  of the primary signal to be zero.
Note that in the above formulation, the independent parameter is the angle t, where t
carries the time dependence, and the period is 2. Most often, however, we wish to
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directly represent our signal waveforms as functions of time t and the sinewave is thus
shown in the following diagram:
v(t)
v(t) = Esin(ωt)
E
0
T/2
t
T
Note that the periodic time T is related to angular frequency  by

2
 2f
T
where f is the (regular) frequency in Hz.
The sinewave is not the only periodic waveform encountered in analogue electronics. Two
others of interest are the following:
(ii) The Squarewave
squarewave
v(t)
E
0
T/2
T
2T
We can formulate the squarewave mathematically as
T
2
v(t )  E
T
t T
2
v(t )  0
0t
while the periodic nature is captured by
1-2
t
t T
v(t )  v(t  nT )
for integer n.
Essentially the squarewave is a digital signal, a regular sequence of alternating 0’s and
1’s more commonly termed a clock signal.
(iii) The Sawtooth
v(t)
sawtooth
E
T
0
2T
t
This is given mathematically by
v (t ) 
0t T
E
t
T
and again
t T
v(t )  v(t  nT )
The voltage is seen to increase linearly until it reaches E at time T and then switches
abruptly back to zero.
1.2
Average Value
Apart from amplitude, frequency, and phase, there are a number of additional signal
parameters of interest. The first of these is the average or DC value and is defined as
V DC 
1
T

T
0
v (t )dt
We now apply this to the previously specified waveforms of interest.
Sinewave
VDC 
1
T

T
0
E sin( t )dt
T
1 E

  cos(t )
T 
0
1-3

E
cos(T )  cos(0)
T
T  2
And noting that
V DC  
we get
E
1  1  0
2
As expected, the average value of the symmetrical sinewave is zero.
Squarewave
VDC
 T2

T
1 

   Edt   0dt 
T 0
T


2


T
E
 t 02
T

E T
T 2

E
2
Sawtooth
T
VDC
1 E
  tdt
T 0T
T
E 1 
 2  t2
T 2 0


E
T2 0
2
2T

E
2

While both of the latter two waveforms happen to have dc values of E/2, this is not a
general result — as will be seen in the sample problems.
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1.3
Root Mean Square (RMS)
Consider a periodic signal voltage v(t) applied to a resistor R and resulting in a current
i(t):
i(t)
R
V(t)
The instantaneous power delivered to R is
p (t )  v(t )i (t ) 
1 2
v (t )
R
The average power is thus given by
P
1
T
T
 p(t )dt
0
T
1 1 2
1 2

v (t )dt  Vrms

RT 0
R
The quantity Vrms is a measure of the average power available in the signal v(t) — it
needs the R value to specify the actual power delivered to the resistor. V 2rms is a measure
of signal strength such as is seen, for example, on a mobile phone handset. We now
calculate the Vrms for some of the signal waveforms already encountered.
Sinewave
T
2
rms
V
1
2
  E sin( t )  dt
T0

E2
2T
E2

2T

E2
2T
Again noting that
 1  cos(2t )dt
T
1


t  2 sin( 2t )

0

1
1
 

 T  2 sin( 2T )    0  2 sin( 0) 
 


T  2
and that sin( 4 )  0 , we have
1-5
V
2
rms
E2

2
Vrms 
or
1
2
E
Squarewave
It is left as an exercise to show that
Vrms 
1
2
E
We also frequently encounter the squarewave with zero DC value:
v(t)
+E
0
T/2
T
2T
t
-E
For this waveform it may readily be shown that
Vrms  E
Sawtooth
2
1 E 
   t  dt
T 0T 
T
V
2
rms
T
1 3 
3 t 

0

E2
T3

1 E2 3
T
3 T3

1 2
E
3
Each waveform has its own characteristic rms value.
1.4
Fourier Series
The relationship between any periodic waveform and the sinewave was fully developed by
the French mathematician Jean Fourier. He basically showed that a periodic function could
be written as an infinite series of harmonically related sinusoidal functions. There are a
number of equivalent ways in which the relationship may be expressed, one of which is as
follows:
Let v(t) represent a signal of period T. Then v(t) may be expanded as
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
v(t )  a 0   a k sin k 0 t   k 
k 1
a0 denotes the average value or DC component of the waveform while ak denotes the
amplitude of the k’th sinusoid and k its phase angle.
The angular frequency of the first, or fundamental, sinusoid is given by
0 
2
T
and we note that all subsequent sinusoids have a frequency which is
an integral multiple of the fundamental frequency.
The 20 component is called the second harmonic;
The 30 component is the third harmonic, etc.
Our concern is not with the mathematical problem of deriving the Fourier series for any
particular periodic signal but rather with working with the Fourier series as a given. In
particular, we are interested in how the available power in a signal distributes across the
range of frequencies; this is what is known as the spectrum of the signal. For the present
we simply note the following Fourier series expansions of waveforms already
encountered.
Squarewave
It may be shown that the squarewave of amplitude E can be expressed
v(t ) 
E 2E

2 
1
1


sin  0 t   3 sin 3 0 t   5 sin 5 0 t   
We have already determined the DC value of the squarewave to be indeed E/2. This
series expansion contains only odd harmonics.
If the DC component is removed from the waveform leaving
v(t)
+E
0
T/2
T
2T
t
-E
then the Fourier series becomes
v(t ) 
4E 
1
1

sin  0 t   sin 3 0 t   sin 5 0 t   

 
3
5

The effect of adding components to the series and producing the squarewave is illustrated
by the following diagram
1-7
K=5 (11, 49) denotes the addition of frequency components up to and including 50, ie
three sinusoids. Note that the basic outline of the squarewave quickly emerges but even
when a large number of terms are included there still remain errors at the points of
discontinuity. Basically, it would take an infinite number of sinusoids to precisely reconstruct a perfect squarewave.
Sawtooth
Again it may be shown that the Fourier series for the previously specified sawtooth
waveform is given by
v(t ) 
E E 
1
1

 sin  0 t   sin 2 0 t   sin 3 0 t   
2  
2
3

All harmonic frequencies are present in this series. Again the underlying harmonic
structure is illustrated by the following:
1-8
This composite contains all sinusoids up to and including 70.
1.5
Power Calculations
Electronic circuits tend to be frequency sensitive, ie they may react differently to
sinewaves of differing frequency. The Fourier series tool allows us to determine the
underlying sinusoidal structure of a given signal and then to calculate how the electronic
circuit effects the individual sinusoidal components, particularly with regards to the
distribution of available power across these components. The basic idea is as follows:
Consider a periodic signal v(t) for which, for convenience, the DC component is zero, ie
v(t )  a1 sin 0 t  1   a2 sin 20 t  2   a3 sin 30 t  3   
We have already seen how, given a particular waveform, the available signal power can
be calculated as the square of Vrms. This must equal the available power in the individual
sinusoids:
Ptotal  P1  P2  P3  
2
Vrms


1 2 1 2 1 2
a1  a 2  a3  
2
2
2


1 2
a1  a 22  a32  
2
Example: The human ear can perceive frequencies up to 20kHz (depending on age!). If a
zero-DC square wave is played out through a loudspeaker, what fraction of the available
signal power is heard by a listener (of normal hearing) if the squarewave (fundamental)
frequency is (a) 15kHz, and (b) 5kHz?
1-9
As previously noted, a zero-DC squarewave may be expressed
v(t ) 
4E 
1
1

sin 2f 0 t   sin 6f 0 t   sin 10f 0 t   

 
3
5

where f0 is the fundamental frequency in Hz.
We know that the total available signal power is given by
2
Vrms
 E2
(a)
since
If
f 0  15kHz , then the only component actually perceived is the f0 component
3 f 0  20kHz The perceived signal power may be expressed
2
2
1 rms
V
1  4E 
1 16 E 2 8 E 2
 
 2
 
2  
2 2

The fraction of total power that is perceived is thus
V12rms

2
Vrms
(b)
If
8E 2 1
8
 2  0.81
2
2
 E

f 0  5kHz , then the components actually perceived are the f0 and 3f0
components. The perceived signal power is now given
V12rms  V22rms 
1 16 E 2
2 2
  1 2 
1    
  3  
The fraction of the total power that is perceived is
V12rms  V22rms
2
rms
V

8  1
1
 0.9
 2  9 
Stated otherwise, 90% of the power in a squarewave is contained in the first two
harmonics.
1.6
Concluding Remarks
In this lecture we have considered the mathematical representation of a number of
frequently encountered analogue signals. Particular attention has been paid to two signal
parameters, the DC (or average) value and the rms value, the latter being a measure of
available signal power. The very important idea of regarding a periodic signal as being
composed of a number of harmonically related sinusoids has in effect introduced the
concept of signal spectrum.
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