Staffordshire University
Faculty of Computing, Engineering and Technology
August 2005
Signal Processing
Page
1
SIGNALS AND SYSTEMS
Introduction
Signals
Classification of Signals
Signal Representation
Classification of Systems
Staffordshire University
Faculty of Computing, Engineering and Technology
August 2005
Signal Processing
Page
2
Introduction
Signals
are
found in many disciplines, eg medicine, physics, optics, chemistry, music etc
In engineering
there a
re
many examples in electrical, mechanical, electronic, for example TV,
satellites, speech, communications etc
Signals arise in many forms, eg acoustic, light, pressure, flow, mechanical
Signals need to be in a form convenient for processing. It so happ
ens that electrical forms of a
signal
are convenient and hence, to some extent, the development of electronics and devices.
Signals
occur in many
different physical forms are often converted to electrical form by a
transducer, eg a microphone.
We are inte
rested in how we can represent a signal, and how we can analyse a process which
involves the signal and a system as illustrated below.
Need to characterise the signal, and the system.
Generally the signals we consider are functions of
time or frequency, x(t), or X(f), or X(
)
The signal could be one of three types:
1)
Analogue or continuous
–
time, defined for all time, denoted by x(t) or v(t)
2)
Discrete
–
time, defined only at discrete times, denoted by x(nT
S
)
(T
S
is the sampling interv
al, discussed in more detail later)
3)
Digital, binary signal representing the discrete
–
time signal, denoted eg as x[n] or
d(t).
Often, the digital signal is derived from the analogue signal by a process of analogue

to

digital (ADC) conversion.
ADC, inv
olves sampling to give a discrete time signal and encoding to give the digital signal.
System
Input signal
x
Output signal
y
System, processes
the signal
Staffordshire University
Faculty of Computing, Engineering and Technology
August 2005
Signal Processing
Page
3
A digital

to

analogue converter (DAC) converts a digital signal to analogue.
We can thus categorise signal processing as
Analogue Signal Processing (ASP) or
Digita
l Signal Processing (DSP)
An example for speech is illustrated below.
Analogue
Analogue Signal Proc. ASP
Eg Amplifier, filter
ADC
Digital Signal Proc. DSP
Eg Amplifier, Filter
DAC
Mic
x(t)
x[n]
y[n]
y(t)
y(t)
Analogue
Digital
Staffordshire University
Faculty of Computing, Engineering and Technology
August 2005
Signal Processing
Page
4
We will often refer to EQUATION, BLOCK DIAGRAM, WAVEFORM, SPECTRUM.
It is important to realise these are often different ways of rep
resenting the same information,
ie different ways of saying the same thing
Often, given one of these, the others can be deduced
Example:
Equation
v(t) = V
DC
+ Vcos
t
Can we deduce the block diagram to give v(t)
, the waveform of v(t), and the spectrum of
v(t)?
Block
Diagram
Equation
Waveform
Spectrum
Staffordshire University
Faculty of Computing, Engineering and Technology
August 2005
Signal Processing
Page
5
Block Diagram
Waveform
Spectrum
V
V
DC
Vcos
t
v
(t)
time
V
DC
Amp
Frequency
Amp
V
DC
V
= 2
f
Staffordshire University
Faculty of Computing, Engineering and Technology
August 2005
Signal Processing
Page
6
Exercise
Give an equation, block diagram and waveform which could correspond to the following
s
pectrum.
Frequency
Hz
V
3
V
5
V
f
3f
5f
Staffordshire University
Faculty of Computing, Engineering and Technology
August 2005
Signal Processing
Page
7
SIGNALS
When we say ‘Signals’, we usually mean functions representing information.
HOWEVER the term ‘Signals’ is also used to include both ‘wanted signal’, eg information,
and ‘unwanted signals’, eg noise and i
nterference.
We are mainly concerned with electrical (voltage/current) or electromagnetic (EM) signals.
EM signals include radio, optical (Optical fibre) and infra

red signals. Another form of a
signal is acoustic or sound (eg ultrasonic).
The informati
on is often a physical quantity, eg speech, image, temperature, pressure. The
physical quantity is converted to an electrical signal by a transducer, eg a microphone for
speech.
CLASSIFICATION OF SIGNALS
DETERMINISTIC AND NON

DETERMINISTIC OR RANDOM SIG
NALS
Deterministic
–
defined or predictable for all time.
Can be expressed by an explicit equation.
For example, S
1
(t) = V
1
cos
1
t, is a deterministic signal.
Non

Deterministic or Random
–
signals for which there is some uncertainty before it
ac
tually occurs.
Not defined or predictable exactly.
Noise and ‘information’ are examples of non

deterministic or random signals. Can usually
be expressed in statistical terms.
Staffordshire University
Faculty of Computing, Engineering and Technology
August 2005
Signal Processing
Page
8
PERIODIC AND NON

PERIODIC SIGNALS
A signal v(t) is periodic if there exists
a constant, T, such that v(t) = v(t + mT), for all time,
where m is integer.
T is the periodic time.
The fundamental frequency, f Hz, and periodic time T are related by
T
f
1
A non

periodic signal
is one for which there is no value of T satisfying v(t) = v(t + mT).
CONTNUOUS

TIME AND DISCRETE

TIME SIGNALS
A Continuous

Time or analogue signal
is one which is defined at every point in an interval.
t
t+T
t+2T
T
v(t)
time
Staffordshire University
Faculty of Computing, Engineering and Technology
August 2005
Signal Processing
Page
9
A Discrete

Time signal
is one whi
ch is defined only at discrete times (Instantaneous values).
Such a sequence of values arises for example from the process of sampling a continuous

time
signal.
v(nT
S
)
are the discrete time values or instantaneous values of
v(t) at the instants nT
S
, where
T
S
is the sampling interval.
SIGNAL REPRESENTATION
TIME DOMAIN AND FREQUENCY DOMAIN
Signals may be represented:
a)
In the TIME

DOMAIN, where the independent variable is time t. In the time

domain, the signal is repres
ented as a function of time, eg v(t).
This is simply the waveform, as seen on an oscilloscope.
b)
In the FREQUENCY
–
DOMAIN, where the independent variable is frequency. Ie the
signal is represented as a function of frequency, v(f).
This is the signal spe
ctrum as seen on a spectrum analyser.
Both ‘ways’ of seeing a signal are important, sometimes the spectrum is the more useful.
Consider the Signal
V cos
t
Amplitude = V, angular frequency
= 2
f, frequency = f Hz
time
0 1
2
3
4
5
6
0 T
S
2T
S
3T
S
4T
S
5T
S
6T
S
Discrete

time
value
v(nT
S
)
Staffordshire University
Faculty of Computing, Engineering and Technology
August 2005
Signal Processing
Page
10
V
t
f Hz
Frequency Hz
Signals can be represented, analysed and processed in the Time

Domain or Frequency

Domain.
Consider the product of two signals.
S
1
(t) =
V
1
cos
1
t
S
2
(t) = V
2
cos
2
t
S
OUT
A trigonometric identity is:
cos A cos B = ½ cos (A+B) + ½ cos (A

B)
ie
S
V
V
t
V
V
t
OUT
1
2
1
2
1
2
1
2
2
2
cos
cos
Sum Frequency =(f
1
+ f
2
) Hz
Difference Frequency = (f
1

f
2
) Hz
Time

Domain
Waveform
Frequency

Domain
Spectrum
v(t)
V(f)
S
1
(t), at
frequency f
1
S
2
(t)
, at
frequency f
2
S
OUT
=
S
1
(t)
S
2
(t)
= V
1
cos
1
t
.
V
2
cos
2
t
=
V
1
V
2
cos
1
t
.
cos
2
t
Staffordshire University
Faculty of Computing, Engineering and Technology
August 2005
Signal Processing
Page
11
Time

Domain Waveform
Frequency

Domain Spectrum
V
1
S
1
(t)
t
f
f
1
V
2
S
2
(t)
time
f
f
2
S
OUT
t
f
1
–
f
2
f
1
+ f
2
Sum and Difference Frequencies
Multiplication of signals is an important process in signal processing.
See handout notes for illustrations including square wave, speech and digital data.
f
Staffordshire University
Faculty of Computing, Engineering and Technology
August 2005
Signal Processing
Page
12
CLASSIFICATION OF SY
STEMS
SYSTEM
A system is a set of connected elements that perform a particular task or function.
Characterised by a block diagram and transfer function, relating output to input.
Elements are signal processing elements and include amplifiers, filters,
summers, multipliers.
The elements may be analogue signal processing (for which signals expressed as continuous

time are appropriate) or digital signal processing ( for which signals expressed as discrete

time are appropriate).
LINEAR AND NON

LINEAR SYS
TEMS
A system is linear if the ‘Principle of Superposition’ applies.
SYSTEM
SYSTEM
SYSTEM
x
1
(t)
x
2
(t)
x
1
(t) + x
2
(t)
y
1
(t)
y
2
(t
)
y
1
(t) + y
2
(t)
Input x
1
(t) gives output y
1
(t)
Input x
2
(t) gives output y
2
(t)
Applying both inputs together
The system is linear if input (x
1
(t) + x
2
(
t)) gives output (y
1
(t) + y
2
(t))
Staffordshire University
Faculty of Computing, Engineering and Technology
August 2005
Signal Processing
Page
13
TIME INVARIANT AND TIME VARYING SYSTEMS
A system is time

invariant if its characteristics do not change with time. Thus a time shift in
the i
nput will result in a corresponding time shift in the output.
Eg x(t)
y(t)
x(t +
τ
) y(t +
τ
)
CAUSAL AND NON

CAUSAL SYSTEMS
A causal system is one whose response at the output does not begin before the input function
is applied
.
Ie x(t
0
) y(t
0
+ t)
Where t> 0 represents a delay between the input and output.
Causal implies that the input function causes the output response.
Note: The above comments apply to both continuous

time and discrete

time signa
ls and
it is
o
ften useful to express a signal in a complex exponential form.
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