classification of systems

1. Principles of Signals and Systems
Prof. Aditya K. Jagannatham
Department of Electrical Engineering
Indian Institute of Technology, Kanpur
Lecture – 05
Classifications of Systems – Memoryless and Casual/ Non-Casual Systems
Keywords: Memoryless Systems, Casual Systems and Non-Casual Systems
Hello, welcome to another module in this massive open online course. So we have
looked at the classification of signals and we have looked at various frequently occurring
signals. Let us now start discussing the fundamental aspect of this course which is
systems, their behavior and their properties.
(Refer Slide Time: 00:39)
So we are going to start looking at the classification of systems and its representation.

2. (Refer Slide Time: 01:33)
So a system representation is a mathematical model for an actual physical process or a
physical device or an object, it is an analytical model that precisely captures the input
output relationship of the system that is given an input signal what is the behavior of the
system or what the output that is given out by the system. So for any given input signal
this system model or system representation must be able to precisely characterize the
output.
(Refer Slide Time: 03:49)

3. And the system is presented mathematically as ( ) ( ( ))y t T x t where T basically captures
this physical process of the system, x(t) is the input signal and y(t) is the output signal
and this is a continuous time system. We can also think of it as a transformation or it is
also going to be an input output mapping.
(Refer Slide Time: 04:43)
(Refer Slide Time: 05:34)
In the same manner I can also define a discrete time system when the output is y(n) and
the input is x(n). I can similarly have ( ) ( ( ))y n T x n .

4. (Refer Slide Time: 06:00)
(Refer Slide Time: 06:51)
Now let us start with the classification of systems to better understand the properties of
various systems and the first important class of systems that we are going to look at is
systems with memory and memory less systems.

5. (Refer Slide Time: 07:23)
The systems can either have memory or they can be memoryless. Now a memoryless
system implies if output depends only on current input that is input at current time
instant, that is the past inputs or the past history of the signal has no bearing on the
output of the system. So this characterizes a memoryless system.
(Refer Slide Time: 08:55)
For instance, we have ( ) ( )y t Kx t where K is some constant. For instance in a circuit
we have ( ) ( )V t Ri t from the Ohm’s law, where V is the voltage, i is the current, R is

6. the resistance and the voltage at time instant t depends only on the current at time instant
t. So this is a classic example of a memoryless system.
(Refer Slide Time: 10:08)
Similarly, I can have for a discrete time signal, ( ) ( )y n Kx n where K is some constant
and this is also an example of a memoryless system.
(Refer Slide Time: 11:00)
.

7. On the other hand, systems with memory are opposite of naturally systems which have
memory. Here the output depends not only on the current input at the current time
instant, but also at the input at the past time instants. The output depends not only on
x(t), but also depends on the past values of x(t). So let us call this output y(t0) which
depends not only on x(t0), but also on x(t) for 0t t and this is known as a system with
memory.
(Refer Slide Time: 12:56)
For instance, if you again go back to the example of circuits you can have
1
( ) ( )
t
V t i d
C
 

  where V(t) is the voltage signal, ( )i  is the current signal and C is the
value of the capacitance and this is a system with memory, the capacitor has memory
because the voltage depends not only on the value of the current at the time instant t, but
also on the values of the current signal i(t) at the past time instants.

8. (Refer Slide Time: 15:07)
If you treat this as an input-output system where the input is current, output is voltage
then the present values of the voltage also depend on the past values of the current signal.
So the capacitor is a classic example of a system with memory. Let us now come to yet
another important property of a system that is causality that is causal and non- causal
systems.
(Refer Slide Time: 15:58)
Now a system is causal if output y(t0) depends only on x(t) for 0t t , that is output
depends only on past values of x(t) and does not depend on future values of x(t), so such

9. a system is known as a causal system and this property is known as causality . If a
system is not causal it means non- causal obviously that is the output also depends on
future values of the input signal.
(Refer Slide Time: 19:26)
(Refer Slide Time: 20:28)
For instance, let us go back to a simple system, ( ) ( )
t
y t x d 

  , this is a continuous time
causal system because y(t) depends on all x(t) for t  .

10. (Refer Slide Time: 21:20)
However on the other hand if you have a discrete time system such
that
1
( ) ( )
2 1
N
k N
y n x n k
N 
 

 , if you look at this, y(n) depends on x(n-N), x(n-N+1), so
on until x(n+N). So it also depends on the future values of x(n), hence this is non- causal.
Since the output depends not only on the past values, but also on the future values of x(n)
it is an example of a non- causal system.
(Refer Slide Time: 23:17)
So we are looking at the classification of systems and we will stop here and continue this
discussion in the subsequent lectures. Thank you very much.