sinusoidal signals

1. Principles of Signals and Systems
Prof. Aditya K. Jagannatham
Department of Electrical Engineering
Indian Institute of Technology, Kanpur
Lecture – 04
Real Exponential, Sinusoidal Signals, Basic Discrete Time Signals – Unit Impulse/
Complex Exponential
Keywords: Real Exponential, Sinusoidal Signals, Basic Discrete Time Signals – Unit
Impulse and Complex Exponential
Hello, welcome to another module in this massive open online course. So we are looking
at a classification of signals and let us look at yet another class of signals which are the
real exponential signals.
(Refer Slide Time: 00:29)
The real exponential signals are simply referred to as exponential signals similar to the
complex exponentials. The real exponential signal is of the form ( ) t
x t e
 and if 0  it
is increasing. At 0t  it is 1 and as , 0t
t e
   .

2. (Refer Slide Time: 01:36)
So t
e
, for any t is in fact a positive signal, similarly when 0  it is a decreasing signal.
(Refer Slide Time: 03:03)

3. (Refer Slide Time: 03:37)
So let me just draw it, that is, it is decreasing for 0  and as , 0t
t e
   .
(Refer Slide Time: 05:01)
The last class of signals which are also very important and one of the most fundamental
classes of signals is the sinusoidal signal simply a real sinusoidal signal. This
is 0( ) cos( )x t A t   , where A is the amplitude of the sinusoid, 0 is the angular
frequency or the radian frequency and  is the phase of the sinusoid. The frequency f0 as

4. we already seen is given as 0
2


and f0 is the fundamental frequency and we have also
seen that it can be represented as 0cos(2 )A f t  . The sinusoid is a periodic signal.
(Refer Slide Time: 06:21)
(Refer Slide Time: 07:12)
Now let us look at some basic discrete time signals which again occur frequently in the
analysis of systems. The unit step 1 0( )
0 0
nu n
n


and this is termed as the discrete unit
step function, which means it is defined only at discrete time instants.

5. (Refer Slide Time: 08:06)
(Refer Slide Time: 08:25)

6. (Refer Slide Time: 10:19)
Similarly we have the unit impulse function 1 0( )
0 0
nn
n
 

.
(Refer Slide Time: 10:46)
This also has some interesting properties, for instance ( ) 1
n
n


 and we can also have
the sifting property for this unit impulse function similar to what we had for the
continuous time scenario that is ( ) ( ) ( )
k
x k n k x n


  .

7. (Refer Slide Time: 11:27)
We can see that ( ) ( ) ( 1)n u n u n    where u(n) is the discrete time unit step signal and
( 1)u n  is the unit step signal shifted by 1. So this gives us the discrete time impulse
function. Similarly ( ) ( )
k
u n k


  , so this is again another representation of the unit step
function.
(Refer Slide Time: 13:22)

8. We have a discrete time complex exponential which is defined as ( )
j n
ox n e

 where
o
 is again the frequency, so 0 0( ) cos( ) sin( )
j n
ox n e n j n

     .
(Refer Slide Time: 14:35)
Now we need to know whether this discrete time complex exponential is periodic or not.
We have seen that the continuous time complex exponential is always periodic and the
fundamental period is given as
0
1
f
. So let us look at
2
0
j f n
e

where 0
0 2
f


 . So for
this to be periodic we have
2 2 ( ) 2 2
0 0 0 0.
j f n j f n N j f n j f N
e e e e
   
  .

9. (Refer Slide Time: 16:25)
So this is periodic if
2
0 1
j f N
e

 which means f0N must be equal to an integer. So
0f N k which means 0
k
f
N
 . So it must be a rational number. So the condition for
periodicity of the discrete time impulse is that 0
k
f
N
 which is a rational number.
(Refer Slide Time: 17:17)

10. Therefore,
j n
oe

is periodic only if 0
0 2
k
f
n

  or some
m
n
, which has to be a rational
number. So that is an interesting aspect of the discrete time complex exponential while
the complex exponential for the continuous time is always periodic.
(Refer Slide Time: 18:15)
So this basically completes the classification of signals where we have looked at a broad
set of signals which frequently arise from practice. There are several other signals and it
is not possible to characterize the entire set of signals, however, we have managed to
classify a fairly large set of frequently occurring signals in practice, the properties of
these signals are important to understand because these arise frequently in practice and
these are important to understand the principles of signals and principles of analysis of
signals and systems. So we will stop this module here and look at other aspects in the
subsequent modules. Thank you very much.