Introduction to Signals
Hany Ferdinando
Dept. of Electrical Engineering
Petra Christian University
General Overview
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This gives students an introduction
about signal and the notation used in
this course
It discusses time signals with
elementary operation on and among
signals
Students learn the generalized signals
Introduction to Signals - Hany Ferdinando
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Definition
Signal is a phenomenon that
represents information
Since any signals always is one of a collection
of several of many possible signals, signals
may mathematically be represented as
elements of a set, called signal set
Introduction to Signals - Hany Ferdinando
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Example…
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Human vocal mechanism produces
speech by creating fluctuation in
acoustic pressure
Monochromatic picture consists of
variation patterns in brightness
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Signals
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The signals we are interested in are
functions of a variable (usually time)
Domain of a signal is a subset T of the
real line and is called signal axis
The signal may take values in any set
A, called the signal range
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Signals
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x: T  A means a signal with signal
axis T and signal range A
It can be written also as x: AT
if T is in time, then the signal is called
time signal, and the signal axis is
called time axis
Another signal is frequency signal
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Various kinds of signals
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Discrete- and continuous-time signals
Finite- and infinite-time signals
Periodic and Harmonic signals
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Discrete-time Signal
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Time axis T is discrete if it consists of a
finite or countable set of time instant
Discrete-time signals is written as x(n),
n is integer
Example:
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
sampled signal from ADC
The number of product in one hour
production
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Continuous-time Signal
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Time axis T is continuous is if it
consists of an interval of R (real
number)
Continuous-time signal is written as
x(t), t is real number
Example:
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
x(t) is electric signal in a circuit
Temperature measurement in a room
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Finite-, Semi-infinite-, Infinite-time
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Finite-time axis: if a time axis is
contained in a finite interval, it is called
a finite axis
Semi-infinite-time axis: if a time axis is
bounded from the left, it is called right
semi-infinite time axis, and vice versa
Infinite-time axis: it is neither bounded
from the left nor from the right
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The Unit Impulse
d(n) =
1, for n = 0
nT
0, otherwise
T is discrete-time axis
1
-2
0
2
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The Unit Step Signal
u(n) =
1, for n ≥ 0
…
nT
0, for n < 0
n=0
u(t) =
1, for t > 0
t T
0, for t < 0
t =0
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Remark!!
δ(n)  u(n)  u(n  1)
and

u(n)   δ(n  m)
m 0
This is only for discrete-time signal!!
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The Unit Ramp Signal
ramp(n) =
n, for n ≥ 0
nT
0, for n < 0
ramp(t) =
t, for t ≥ 0
t T
0, for t < 0
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Rectangular & Triangular Pulse
rect(t) =
1, for -0.5 ≤ t ≤ 0.5
t R
0, otherwise
trian(t) =
1-|t|, for |t| < 1
0, otherwise
Introduction to Signals - Hany Ferdinando
t R
15
Complex Exponential Signal
aat
4
4
3.5
3.5
3
3
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0
-2
-1.5
-1
a>1
-0.5
0
0.5
1
1.5
2
0<a<1
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Complex Exponential Signal
aan
4
4
3.5
3.5
3
3
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0
-2
-1.5
-1
a>1
-0.5
0
0.5
1
1.5
2
0<a<1
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Complex Exponential Signal
aan
a < -1
-1 < a < 0
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Periodic Signals
Definition: a signal that repeats itself indefinitely
The length of time after which the signal starts
repeating itself is called period
x(t + T) = x(t) for all t  T
T is the period
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Harmonic Signals
x(t) = ejwt or x(n) = ejWn
w (and W) = 2pf is called angular frequency of the
harmonic
The signal of the form x(t) = a ejwt
a is called the complex amplitude of the
harmonic signal
If T is the period, then ejwT = 1
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Periodicity of harmonic signal
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Periodicity of continuous-time harmonic
signal is simple. it is formulated as T = 1/|f|
Periodicity of discrete-time harmonic signal
is a little more complicated. The reason is
sampling two continuous-time signal with
different freq. may result in the same
discrete-time signal. This phenomenon is
called aliasing. See next figure…
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Periodicity of harmonic signal
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Periodicity of harmonic signal
e
j ( W o  2p ) n
e
jW o n
e
j 2pn
e
jW o n
1
Discrete-time signal with Wo varies from 0 to 2p will
show this ‘strange’ situation. Try it using Matlab!!
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Periodicity of harmonic signal
In order for the signal complex exponential to
be periodic with period N > 0,
e
jW o ( n  N )
e jW o N  1
e
jW o
WoN must be multiple of 2p
Ω o N  2π m
Ωo m

2π N
Period
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Compare…
sin(2*pi*n)
1
0.5
0
-0.5
-1
0
10
20
30
40
50
60
70
40
50
60
70
sin(4*n)
1
0.5
0
-0.5
-1
0
10
20
30
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Elementary Operations
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Signal range transformation
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Amplification or attenuation
Quantization
Time transformation
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Time expansion, time compression and
time reversal
Time translation
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Amplification and Attenuation
y(t) = a x(t)
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If |a| > 1 then it is an amplification
If |a| < 1 then it is an attenuation
The time axis remains the same while the
range is changed!
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Quantization
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This is an important range
transformation
This transformation is needed when
signals are processed by a digital
computer or other digital equipment
The Analog to Digital Converter (ADC)
does this operation
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Expansion, compression and reversal
y(t) = x(at)
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If 0 < a < 1 then it is time expansion
If a > 1 then it is time compression
if a = -1 then it is time reversal or
reflection
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Translation
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y(t) = x (t – q) shift x(t) to right (delay)
by q
y(t) = x (t + q) shift x(t) to left by q
What about
y(t) = x(-t – q) and y(t) = x(-t + q)
???
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Translation and Reflection
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We can get x(t-q) from x(-(-t+q))
We can get x(-t-q) from x(-(t+q))
We can get x(t+q) from x(-(-t-q))
We can get x(-t+q) from x(-(t-q))
Check these
statements!!
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Next…
An introduction to signals is discussed! Students
should do some assignment either from the lecturer or
from the books.
For the next meeting, please prepare yourself by
reading chapter about system.
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Signals and System by Alan V.
Oppenheim, chapter 2, p 35-45
Signals and Linear Systems by Robert A.
Gabel, chapter 2, p 23-37, chapter 3, p
121-127
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