Term 332
EE3010: Signals and Systems Analysis
3. Introduction to Signal and Systems
Dr. Mujahed Al-Dhaifallah
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Lecture Objectives
General properties of signals
 Energy and power for continuous &
discrete-time signals
 Signal transformations
 Specific signal types
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Reading List/Resources
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Essential
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AV Oppenheim, AS Willsky: Signals and
Systems, 2nd Edition, Prentice Hall, 1997
Sections 1.1-1.4
Recommended

S. Haykin and V. Veen, Signals and
Systems, 2005.
Sections 1.4-1.9
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Last Lecture Material
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What is a Signal?
Examples of signal.
How is a Signal Represented?
Properties of a System
What is a System?
Examples of systems.
How is a System Represented?
Properties of a System
How Are Signal & Systems Related (i), (ii), (iii)
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Signals and Systems
Signals are variables that carry
information.
 Systems process input signals to
produce output signals.
 Today: Signals, next lecture: Systems.

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Examples of signals
Electrical signals --- voltages and
currents in a circuit
 Acoustic signals --- audio or speech
signals (analog or digital)
 Video signals --- intensity variations in
an image (e.g. a CAT scan)
 Biological signals --- sequence of bases
in a gene
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Signal Classification
Type of Independent Variable

Time is often the independent variable.
Example: the electrical activity of the
heart recorded with chest electrodes ––
the electrocardiogram (ECG or EKG).
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The variables can also be spatial
Eg. Cervical MRI
In this example, the
signal is the intensity
as a function of the
spatial variables x and
y.
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Independent Variable Dimensionality

An independent variable can be 1-D (t in
the ECG) or 2-D (x, y in an image).
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Continuous-time (CT) Signals

Most of the signals in the physical world
are CT signals, since the time scale is
infinitesimally fine, so are the spatial
scales. E.g. voltage & current, pressure,
temperature, velocity, etc.
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Discrete-time (DT) Signals
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x[n], n — integer, time varies discretely
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Examples of DT signals in nature:
— DNA base sequence
 — Population of the nth generation of
certain species
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Transformations of the
independent Variable
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:
y (t )  x(t  b)

Time reversal
y (t )  x(t )
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Transformations of the
independent Variable
Time scaling
represents a signal stretching if 0<|a|<1,
compression if |a|>1
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Periodic Signals
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An important class of signals is the class of periodic
signals. A periodic signal is a continuous time signal
2p
x(t), that has the property
x(t )  x(t  T )

where T>0, for all t.
Examples:
cos(t+2p) = cos(t)
sin(t+2p) = sin(t)
Are both periodic with period 2p
The fundamental period is the smallest t>0 for which
x(t )  x(t  T )
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Odd and Even Signals
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An even signal is identical to its time reversed signal,
i.e. it can be reflected in the origin and is equal to the
original x(t )  x(t ) or x[n] = x[−n]
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Examples:
x(t) = cos(t)
x(t) = c
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Odd and Even Signals
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An odd signal is identical to its negated, time reversed
signal, i.e. it is equal to the negative reflected signal
x(t )   x(t )
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or x[n] = − x[−n]
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.
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Odd and Even Signals
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Exponential and Sinusoidal
Signals
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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:
x(t )  Ceat
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
where C and a are, in general, complex numbers. Lets
investigate some special cases of this signal
Exponential decay
Real exponential signals
a0
C 0
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Exponential growth
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a0
C 0
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Right- and Left-Sided Signals
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Bounded and Unbounded
Signals
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Periodic Complex Exponential &
Sinusoidal Signals
Consider when a is purely imaginary:
x(t )  Ce jw0t
By Euler’s relationship, this can be expressed
as:
e jw0t  cos w0t  j sin w0t
cos(1)
This is a periodic signals because:
e jw0 (t T )  cos w0 (t  T )  j sin w0 (t  T )
 cos w0t  j sin w0t  e jw0t
when T=2p/w0
A closely related signal is the sinusoidal
signal:
x(t )  cosw0t   
w0  2pf 0
We can always use:
A cosw0t     A e j (w0t  )

A sin w t     Ae
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j (w0t  )


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T0 = 2p/w0
=p
T0 is the fundamental
time period
w0 is the fundamental
frequency
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Exponential & Sinusoidal Signal
Properties
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Periodic signals, in particular
complex periodic and sinusoidal
signals, have infinite total energy
but finite average power.
Consider energy over one period:
T0
E period   e
jw 0 t 2
0
dt
T0
  1dt  T0
0

Therefore:
E  

Average power:Pperiod  1 E period  1
T0
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General Complex Exponential
Signals
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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:
C  C e j
a  r  jw 0

So

Using Euler’s relation
Ceat  C e j e( r  jw0 )t  C ert e j (w0  )t
Ceat  C e j e( r  jw0 )t  C ert cos((w0   )t )  j C ert sin(( w0   )t )
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“Electrical” Signal Energy & Power
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It is often useful to characterise signals by measures
such as energy and power
For example, the instantaneous power of a resistor is:
1
p (t )  v(t )i (t )  v 2 (t )
R
and the total energy expanded over the interval [t1, t2]
t
t 1
is:
2
p
(
t
)
dt

v
t
t R (t )dt
2
2
1

1
and the average energy is:
1 t
1 t 1 2
p(t )dt 
v (t )dt


t
t
t 2  t1
t 2  t1 R
How are these concepts defined for any continuous or
discrete time signal?
2
2
1

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Generic Signal Energy and Power
Total energy of a continuous signal x(t) over [t1, t2] is:
t2
2
E   x(t ) dt
t1
where |.| denote the magnitude of the (complex) number.
Similarly for a discrete time signal x[n] over [n1, n2]:
E  n n x[n]
n2
2
1
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.
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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:
T

E  lim T   x(t ) dt   x(t ) dt
2
T
2

E  lim N  n N x[n]  n x[n]
N
2

2
If the sums or integrals do not converge, the energy of
such a signal is infinite
1 T
2
x
(
t
)
dt


T
2T
1
N
2
x
[
n
]

2 N  1 n N
P  lim T 
P  lim N 
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Classes of Signals
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Two important (sub)classes of signals
1. Finite total energy (and therefore zero
average power). f (t )  1 0  t  1
0 otherwise
2. Finite average power (and therefore
infinite total energy). x[n]=4
3. Neither average power or total energy
are finite. x(t)=t
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Discrete Unit Impulse and Step
Signals

The discrete unit impulse signal is defined:
0 n  0
x[n]   [n]  
1 n  0

Useful as a basis for analyzing other signals
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The discrete unit step signal is defined:
0 n  0
x[n]  u[n]  
1 n  0
 Note that the unit impulse is the first difference
(derivative) of the step signal
 [n]  u[n]  u[n  1]

Similarly, the unit step is the running sum
(integral) of the unit impulse.
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Continuous Unit Impulse and Step
Signals
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The continuous unit impulse signal is defined:
0 t  0
x(t )   (t )  
 t  0

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:
t
x(t )  u(t )    ( )d

0 t  0
x(t )  u (t )  
1 t  0
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