SIGNALS AND SYSTEMS

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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.