Signals, Classification of Signals

1. ELECTRICAL DEPARTMENT
Topic :Signals,Classification of
signals,Basic signals..
Subject: Signal & System-2141005

2. Signals Concepts
Specific objectives are
What is signal?
Different types of signals
1.Analog &Digital Signal
2.Periodic & Aperiodic signal
3.Power & Energy signal etc
Representing signals in Matlab & Simulink

3. What is signal?
In electrical engineering, the fundamental quantity of
representing some information is called a signal. It does not
matter what the information is i-e: Analog or digital
information. In mathematics, a signal is a function that
conveys some information. In fact any quantity measurable
through time over space or any higher dimension can be taken
as a signal. A signal could be of any dimension and could be of
any form.

4. Analog and Digital Signal
Analog signal is a continuous signal for which the time varying
feature of the signal is a representation of some other time varying
quantity.
Digital Signal is a signal that represents a sequence of discrete
values.
A logic signal is a digital signal with only two possible values, and

5. Periodic Signals
An important class of signals is the class of periodic
signals. A periodic signal is a continuous time signal
x(t), that has the property
where T>0, for all t.
Examples:
cos(t+2π) = cos(t)
sin(t+2π) = sin(t)
Are both periodic with period 2π
for a signal to be periodic, the relationship must hold
for all t.
)()( Ttxtx +=
2π

6. Aperiodic signal
Aperiodic signal: An aperiodic function never
repeats, although technically an aperiodic function
can be considered like a periodic function with an
infinite period.
Examples: Sound signal, noise signal etc.

7. Continuous & Discrete Signals
Continuous-Time Signals
Most signals in the real world are
continuous time, as the scale is
infinitesimally fine.
E.g. voltage, velocity,
Denote by x(t), where the time
interval may be bounded (finite) or
infinite
Discrete-Time Signals
Some real world and many digital
signals are discrete time, as they are
sampled
E.g. pixels, daily stock price (anything
that a digital computer processes)
Denote by x[n], where n is an integer
value that varies discretely
Sampled continuous signal
x[n] =x(nk)
x(t)
t
x[n]
n

8. Discrete Unit Impulse and Step Signals
The discrete unit impulse signal is defined:
Useful as a basis for analyzing other signals
The discrete unit step signal is defined:
Note that the unit impulse is the first
difference (derivative) of the step signal
Similarly, the unit step is the running sum
(integral) of the unit impulse.



=
≠
==
01
00
][][
n
n
nnx δ



≥
<
==
01
00
][][
n
n
nunx
]1[][][ −−= nununδ

9. Continuous Unit Impulse and Step Signals
The continuous unit impulse signal is
defined:
Note that it is discontinuous at t=0
The arrow is used to denote area, rather than
actual value
Again, useful for an infinite basis
The continuous unit step signal is defined:



=∞
≠
==
0
00
)()(
t
t
ttx δ
∫ ∞−
==
t
dtutx ττδ )()()(



>
<
==
01
00
)()(
t
t
tutx

10. Electrical Signal Energy & Power
It is often useful to characterise signals by measures
such as energy and power
For example, the instantaneous power of a resistor is:
and the total energy expanded over the interval [t1, t2]
is:
and the average energy is:
)(
1
)()()( 2
tv
R
titvtp ==
∫∫ =
2
1
2
1
)(
1
)( 2
t
t
t
t
dttv
R
dttp
∫∫ −
=
−
2
1
2
1
)(
11
)(
1 2
1212
t
t
t
t
dttv
Rtt
dttp
tt

11. Odd and Even SignalsAn even signal is identical to its time reversed signal, i.e. it
can be reflected in the origin and is equal to the original:
Examples:
x(t) = cos(t)
x(t) = c
An odd signal is identical to its negated, time reversed signal,
i.e. it is equal to the negative reflected signal
Examples:
x(t) = sin(t)
x(t) = t
This is important because any signal can be expressed as the
sum of an odd signal and an even signal.
)()( txtx =−
)()( txtx −=−

12. Time Shift Signal
A central concept in signal analysis is the transformation of one signal
into another signal. Of particular interest are simple
transformations that involve a transformation of the time axis only.
A linear time shift signal transformation is given by:
where b represents a signal offset from 0, and the a parameter
represents a signal stretching if |a|>1, compression if 0<|a|<1 and a
reflection if a<0.
)()( batxty +=

13. Exponential and Sinusoidal Signals
Exponential and sinusoidal signals are characteristic of real-world
signals and also from a basis (a building block) for other signals.
A generic complex exponential signal is of the form:
where C and a are, in general, complex numbers. Lets investigate
some special cases of this signal
Real exponential signals
at
Cetx =)(
0
0
>
>
C
a
0
0
>
<
C
a
Exponential growth Exponential decay

14. Periodic Complex Exponential &
Sinusoidal SignalsConsider when a is purely imaginary:
By Euler’s relationship, this can be expressed as:
This is a periodic signals because:
when T=2π/ω0
A closely related signal is the sinusoidal signal:
We can always use:
tj
Cetx 0
)( ω
=
tjte tj
00 sincos0
ωωω
+=
tj
Ttj
etjt
TtjTte
0
0
00
00
)(
sincos
)(sin)(cos
ω
ω
ωω
ωω
=+=
+++=+
( )φω += ttx 0cos)(
00 2 fπω =
( ) ( )
( ) ( ))(
0
)(
0
0
0
sin
cos
φω
φω
φω
φω
+
+
ℑ=+
ℜ=+
tj
tj
eAtA
eAtA
T0 = 2π/ω0
= π
cos(1)
T0 is the fundamental
time period
ω0 is the fundamental
frequency

15. Exponential & Sinusoidal Signal
Periodic signals, in particular complex periodic
and sinusoidal signals, have infinite total
energy but finite average power.
Consider energy over one period:
Therefore:
Average power:
Useful to consider harmonic signals
Terminology is consistent with its use in music,
where each frequency is an integer multiple of
a fundamental frequency
0
0
0
2
0
0
0
1 Tdt
dteE
T
T
tj
period
==
=
∫
∫
ω
1
1
0
== periodperiod E
T
P
∞=∞E

16. General Complex Exponential Signals
So far, considered the real and periodic complex exponential
Now consider when C can be complex. Let us express C is polar form and
a in rectangular form:
So
Using Euler’s relation
These are damped sinusoids
0ω
φ
jra
eCC j
+=
=
tjrttjrjat
eeCeeCCe )()( 00 φωωφ ++
==
))sin(())cos(( 00
)( 0
teCjteCeeCCe rtrttjrjat
φωφωωφ
+++== +

17. Generic Signal Energy and Power
Total energy of a continuous signal x(t) over [t1, t2] is:
where |.| denote the magnitude of the (complex) number.
Similarly for a discrete time signal x[n] over [n1, n2]:
By dividing the quantities by (t2-t1) and (n2-n1+1), respectively,
gives the average power, P
Note that these are similar to the electrical analogies
(voltage), but they are different, both value and dimension.
∫=
2
1
2
)(
t
t
dttxE
∑ =
= 2
1
2
][
n
nn
nxE

18. Energy and Power over Infinite
Time
For many signals, we’re interested in examining the power and energy
over an infinite time interval (-∞, ∞). These quantities are therefore
defined by:
If the sums or integrals do not converge, the energy of such a signal is
infinite
Two important (sub)classes of signals
1. Finite total energy (and therefore zero average power)
2. Finite average power (and therefore infinite total energy)
Signal analysis over infinite time, all depends on the “tails” (limiting
behaviour)
∫∫
∞
∞−−
∞→∞ == dttxdttxE
T
T
T
22
)()(lim
∑∑
∞
−∞=−=∞→∞ == n
N
NnN nxnxE
22
][][lim
∫−
∞→∞ =
T
T
T dttx
T
P
2
)(
2
1
lim
∑ −=∞→∞
+
=
N
NnN nx
N
P
2
][
12
1
lim

19. Introduction to Matlab
Simulink is a package that runs inside the Matlab environment.
Matlab (Matrix Laboratory) is a dynamic, interpreted, environment
for matrix/vector analysis
User can build programs (in .m files or at command line) C/Java-like
syntax
Ideal environment for programming and analysing discrete (indexed)
signals and systems

20. Basic Matlab Operations
>> % This is a comment, it starts with a “%”
>> y = 5*3 + 2^2; % simple arithmetic
>> x = [1 2 4 5 6]; % create the vector “x”
>> x1 = x.^2; % square each element in x
>> E = sum(abs(x).^2); % Calculate signal energy
>> P = E/length(x); % Calculate av signal power
>> x2 = x(1:3); % Select first 3 elements in x
>> z = 1+i; % Create a complex number
>> a = real(z); % Pick off real part
>> b = imag(z); % Pick off imaginary part
>> plot(x); % Plot the vector as a signal
>> t = 0:0.1:100; % Generate sampled time
>> x3=exp(-t).*cos(t); % Generate a discrete signal
>> plot(t, x3, ‘x’); % Plot points

21. Example: Generate and View a Signal
Copy “sine wave” source and
“scope” sink onto a new
Simulink work space and
connect.
Set sine wave parameters
modify to 2 rad/sec
Run the simulation:
Simulation - Start
Open the scope and leave open
while you change parameters
(sin or simulation
parameters) and re-run

22. Summary
This presentation has looked at signals:
Power and energy
Signal transformations
Time shift
Periodic
Even and odd signals
Exponential and sinusoidal signals
Unit impulse and step functions
Matlab and Simulink are complementary environments for
producing and analysing continuous and discrete signals.
This will require some effort to learn the programming
syntax and style!