Linear Circuit Analysis (EE-101) : Electric Signals

Linear Circuit Analysis
(EE-101)
Electric signals
Signals
Concepts
Specific objectives are
 What is signal?
 Differenttypes of signals
1.Analog &Digital Signal
2.Periodic & Aperiodic
signal 3.Power & Energy
signal
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.
Analog and Digital
 Analog Signal
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
Periodic
Signals
An important class of signals is the class of periodic
signals. A periodic signal is a continuous time
x(t), that has the
signal 2
property
x(t)  x(t  T )
where T>0, for all t.
Examples:
cos(t+2) =
cos(t) sin(t+2)
= sin(t)
Are both periodic
with period 2
for a signal to be periodic, the relationship must hold

for all t.
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.

Continuous & Discrete
MostSignals
Continuous-Time Signals
signals in the real world are
continuous time, as the scale x(t)
is infinitesimally fine.
E.g. voltage, velocity,
Denote by x(t), where the time
interval may be bounded (finite) t
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 x[n]
processes)
Denote by x[n], where n is an integer
value that varies discretely
Sampled continuous signal
n
x[n] =x(nk)
Discrete Unit Impulse and Step
The discrete unit impulse signal is
Signals
defined:
x[n]   [n]  n  0
0 1 n 
Useful as a basis for analyzing other
0
signals
The discrete unit step signal is
defined:
0
x[n]  u[n]   n  0
1 n 
Note that the unit 0 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.
Continuous Unit Impulse and Step
TheSignals
continuous unit impulse signal
is defined:
t0

x(t)   (t)  t0

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:x(t)  u(t)
t  ( )
 d
t0
0
x(t)  u(t)  
1 t  0
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:
1
p(t)  v(t)i(t)  v 2 (t)
R
and the total energy expanded over the interval
[t1, t2] t t
t  t R
2
p(t)dt 1 v 2 (t)dt
2
is:
1 1
and the average energy

is: 1 t 1 t
 p(t)dt  
2 2 2
v (t)dt
t2  t1 t1 1 t1
t 2  t1
Odd and Even
An even signal is identical to its time reversed signal, i.e. it
Signals
can be reflected in the origin and is equal to the
original:
x(t)  x(t)
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
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:
y(t)  x(at  b)
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.
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:
x(t)  Ceat
where C and a are, in general, complex numbers. Lets
investigate some special cases of this signal
Real exponential signals
Exponential growth Exponential decay
a0 a0
C0 C0
Periodic Complex Expo n en t
Sinusoidal
ial &
Consider when a is purely imaginary:
Signals x(t)  Ce j0t
By Euler’s relationship, this can be expressed : cos()

as
e j0t  cos0t  j sin
0t signals because:
This is a periodic
e j (t T )  cos (t  T )  j sin  (t  T
0
) j t
when T=2/0  cos00t  j sin 0t  e 0
0
A closely related signal is the sinusoidal
T0 = 0
signal:
2/=

We can always use: 0  2f0 T0 is the fundamental
x(t)  cos0 t 
time period
 0 is the fundamental
A cos0 t    Aej() t 0
frequency
Asin t    

0  j ( t 0
 ) 
Exponential & Sinusoidal
Periodic signals, in particular complex
Signal
periodic and sinusoidal signals, have
infinite total energy but finite average
power.
Consider energy over one period:
E period  T e j t 2 dt
0
0
0
T0
  1dt  0
0
Therefore: T
E  
Average power:
1
Pperiod  T Eperiod 
0
Useful to 1consider harmonic signals
Terminology is consistent with its use in music,

where each frequency is an integer multiple of
a fundamental frequency
Generic Signal Energy and
Power
Total energy of a continuous signal x(t) over [t , t ]
t2
1 2
is: E   x(t) dt
2
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] 2
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.
Energy and Power over Infin
Tim
ite
For many signals, we’re interested in examining the power and energy
e over an infinite time interval (-∞, ∞). These quantities are therefore
defined by: T
E  T  x(t) 2 dt 
 x(t) 2
lim 
T
dt 
1 T
P  limT  
2
x(t) dt
2T T
If the sums or integrals do not converge, the energy of such a signal is

infinite
N  n

E  N
x[n] 2
 n
x[n]
2
lim

N
n x[n] 2
N
P 
lim N
1
Two important (sub)classes of signals
1. Finite total energy (and therefore zero average power)
N 
2N 1
2. Finite average power (and therefore infinite total energy)
Signal analysis over infinite time, all depends on the “tails” (limiting
behaviour)
Thanks!

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Linear Circuit Analysis

 Digital Signal is a signal that represents a sequence of

for a signal to be periodic, the relationship must hold

Examples: Sound signal, noise signal etc.

The continuous unit step signal is

and the average energy

By Euler’s relationship, this can be expressed : cos()

Terminology is consistent with its use in music,

where |.| denote the magnitude of the (complex) number.

By dividing the quantities by (t2-t1) and (n2-n1+1),

Note that these are similar to the electrical analogies

If the sums or integrals do not converge, the energy of such a signal is

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