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EEE 226: Signals and Systems
Lecture Notes # 1
Dr. Aykut Hocann
Department of Electrical and Electronics Engineering
Eastern Mediterranean University
March 11, 2002
Chapter 1 of the textbook.
1 Continuous and Discrete-Time Signals
Signals are represented mathematically as functions of one or
more inde-pendent variables. In this course, we will consider
signals of one indepen-dent variable.
Continuous-Time (CT) signals are defined for a continuous
independentvariable. Discrete-Time (DT) signals are defined at
discrete times.
Examples of CT: speech signal, atmospheric pressure. Examples of
DT:weekly stock market index, number of goals scored in one
season.
2 Signal Energy and Power
The instantaneous power for given instantaneous voltage v(t) and
current i(t)across a resistor R is given by
p(t) = v(t)i(t) =1
Rv2(t). (1)
The total energy expended over the time interval t1 t t2 is
t2
t1
p(t)dt =
t2
t1
1R
v2(t)dt. (2)
The average power over the time interval is
1
t2 t1
t2
t1
p(t)dt =1
t2 t1
t2
t1
1
Rv2(t)dt. (3)
For complex CT signals which are also defined over t1 t t2
is
t2
t1
|x(t)|2dt (4)
1
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and similarly for DT signals defined over n1 n n2
n=n2
n=n1 |x[n]|
2
(5)
When we consider average energy over the infinite time interval,
i.e., for t + and n + the E is given by for CT:
E = limT
T
T
|x(t)|2dt (6)
and for DT:
E = limN
NN
|x[n]]|2 (7)
The average Power over infinite time P are as follows:
P = limT
1
2T
T
T
|x(t)|2dt (8)
and for DT:
E = limN
1
2N + 1
NN
|x[n]]|2 (9)
It is important to note that finite total energy signals E <
must have zeroaverage power P = 0 since
P = limT
E
2T = 0. (10)
Similarly, if a signal has a nonzero finite average power P then
the signal musthave infinite total energy E = .
3 Transformations of the Independent Variable
Given x(t + ), depending on the values of and we have
time shift
time reversal
time scaling
We will investigate x(t + ) given x(t) for different values of
and .
if || < 1, then linearly stretched signal
if || > 1, then linearly compressed signal
if < 0, then reversed in time
if > 0, then time advance (the signal shifts left)
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if < 0, then time delay (the signal shifts right)
An easy way to find the signal as a result of the transformation
of the indepen-
dent variable, is to transform each point of the original
signal:
t + = t =
(11)
As it can be seen from the previous equation, it is important to
shift first andthen compress/stretch.
