This lecture discusses important continuous time signals including energy/power signals, the unit impulse function, and complex exponentials. It defines energy signals as those with finite energy, and power signals as those with finite power. The unit impulse function δ(t) is introduced as a limit of narrow pulses with area 1, and it has the sifting property that ∫x(t)δ(t)dt = x(0). The complex exponential ejwt is a periodic signal with constant magnitude of 1, and it relates angular frequency ω to frequency f as ω = 2πf.

1. Principles of Signals and Systems
Prof. Aditya K. Jagannatham
Department of Electrical Engineering
Indian Institute of Technology, Kanpur
Lecture - 03
Energy/ Power Signals, Unit Impulse Function, Complex Exponential
Keywords: Energy and Power Signals, Unit Impulse Function, Complex Exponential
Hello welcome to another module in this massive open online course. So let us continue
our discussion on the classification of signals, so let us look at energy and power signals.
(Refer Slide Time: 00:24)
The energy of a continuous signal x(t) is defined as
2
( )E x t dt


  . The energy of a
discrete time signal is
2
( )
n
E x n


  . Now a signal is an energy signal if the energy is
finite.

2. (Refer Slide Time: 01:48)
If 0 E  , this implies that it has a finite energy and hence it is an energy signal. For
instance, a good example of an energy signal is ( )t
e u t
or ( )n
e u n
and these are energy
signals that is signals whose energies are finite.
(Refer Slide Time: 02:44)
And we also have the notion of power signals. The power of a signal,
2
2
2
1
lim ( )
T
T
T
P x t dt
T

  and the same thing can be defined for a discrete time signal that is

3. 21
lim ( )
2 1
N
N
n N
P x n
N



 . If the power of the signal is finite that is 0 P   , then it is
known as a power signal. An example of a power signal is the sinusoid, sin(2 )ft .
(Refer Slide Time: 05:03)
So we have covered most of the major classes of signals and now let us look at some
important continuous and discrete time signals which occur very frequently in
applications and in the analysis of signals and systems. The first important signal you
must be very familiar with is the unit step signal.
(Refer Slide Time: 06:33)

4. The unit step signal is defined as 1 0( )
0 0
tu t
t


and also for t =0 you can sometimes
define it as
1
2
. At t =0, there is a discontinuity, that is it jumps from 0 to 1.
(Refer Slide Time: 07:04)
So this is your unit step function or the unit step signal. And another such important
signal is the unit impulse function.
(Refer Slide Time: 08:04)

5. And this is one of the most fundamental or key signals to understand the various
properties or the behavior of system.
(Refer Slide Time: 08:37)
Consider the following signal which is a pulse from
2

 to
2

. So the pulse has width
equals  and a height equals
1

. So the area under the pulse equals
1
1

  . So basically
we denote this pulse by ( )t . So there is a sequence of pulses, one pulse for each value
of . So we are considering a narrow pulse with width  from
2

 to
2

and of height
1

.
(Refer Slide Time: 10:48)

6. So basically what we have is for each pulse ( )t for every  the area under pulse equal
to 1. Now we define the pulse ( )t as
0
( ) lim ( )t t

 

 and this is your impulse function
or simply known as the impulse. As 0  , the width goes to 0 and height
1

 , but
the area under it still remains constant that is unity.
(Refer Slide Time: 11:48)
So basically what you can see is that the area under this that is  01( )
0
b
a
if a b
t dt
otherwise

 
 .
(Refer Slide Time: 12:35)

7. The impulse has several interesting properties as follows ( ) ( ) (0)x t t dt x


 that is if
you multiply the impulse function by any signal x(t) and integrate it, it picks the value of
the function x(t) at t = 0. Now 0 0( ) ( ) ( )x t t t dt x t


  .
(Refer Slide Time: 14:39)
(Refer Slide Time: 16:00)

8. Now 0( )t t  is nothing but basically the impulse shifted to t = t0. Some of the other
properties of the impulse function are
1
( ) ( )at t
a
  and for any signal x(t),
( ). ( ) (0). ( )x t t x t  .
(Refer Slide Time: 16:57)
Further, if you consider ( ) ( ) ( )x t d x t   


  and this is a very important property.
(Refer Slide Time: 18:13)

9. (Refer Slide Time: 18:58)
So we are going to set t t   which means t t   and d dt  . So the integral
simplifies as ( ) ( )x t t t dt


   . Here we use the property that ( ) ( )t t   .
(Refer Slide Time: 20:06)
So this becomes ( ) ( )x t t t dt


  , but ( ) ( )x t t t is nothing but
0
( )
t
x t t

 . So this is
equal to simply x(t).

10. (Refer Slide Time: 20:54)
So what we have as a result is that ( ) ( ) ( )x t d x t   


  and this property is known as
the sifting property of the impulse function or the sifting property of the delta function.
So it would be good to examine and understand the properties of the impulse function
because this arises very frequently in the analysis of signals and systems and it is very
important to both understand the behavior of systems and we will see several
applications of this impulse function as we go through the rest of this course. Let us
come to another interesting function that occurs frequently which is the complex
exponential.
(Refer Slide Time: 23:01)

11. The complex exponential function is defined as 0
( ) j t
x t e 
 or 02
( ) j f t
x t e 
 which is
0 0cos( ) sin( )t j t  where 1j   , is the imaginary number and if you can look at this,
0
( ) j t
x t e 
 and you have magnitude of x(t) which is equal to well
2 2
0 0( ) cos ( ) sin ( )x t t t   . Now 2 2
0 0cos ( ) sin ( ) 1t t   . So ( )x t is always 1,
that is a complex exponential always has unit magnitude and further it is a periodic
signal.
(Refer Slide Time: 24:27)
Here 0 02j f t j t
e e 
 where 0
0
2
f


 and this is an important relation. Here 0 is the
circular frequency and this is also known as the angular frequency and is in radians per
second, the unit of f0 frequency is Hertz.

12. (Refer Slide Time: 25:22)
This is a periodic signal and its period
0
1
T
f
 where 0
0
2
f


 . So the period, for instance,
if f0 equals 5Hz implies the period
1
0.2
5
T s  . So the period is basically the reciprocal
of the frequency.
(Refer Slide Time: 26:11)
A general complex exponential is defined as s j   and ( ) st
x t e which is basically
well, ( )
( ) (cos( ) sin( ))st j t t
x t e e e t j t  
 
    .

13. (Refer Slide Time: 26:54)
So what we have seen in this module is that we have seen other different classes of
signals such as energy and power signals etc., we have also seen some very commonly
arising and important continuous time signals such as the unit impulse or the unit step
function and we have also seen the complex exponential and the general complex
exponential signals. So we will stop here and continue with other aspects in the
subsequent modules. Thank you very much.