EDITOR: B. SOMANATHAN
NAIR
3. TIME VARIANCE
In the previous blog, we
have discussed the properties of linearity and causality of systems. In this
blog, we discuss the properties of time variance and memory.
A system is said to be time
invariant if, for any change in the time factor associated with the system, the
system output (amplitude) remains constant, but a corresponding change occurs
in its time factor. The idea of time
variance can be illustrated with the help of actual physical examples.
ILLUSTRATIVE EXAMPLE 1: Test the time-variance condition of the system
governed by the equation
y(t)
= ax(t) (1)
Solution: To
test the time-variance of a given system, we adopt the following procedure.
First, we delay the input by a certain amount of time delay, say, T units. Then determine the output using
the equation governing the system. After this, we delay all the terms
representing time (i.e., t) by the
same delay, viz., T units. If the
end-results of the two procedures are the same, we say that the system is
time-invariant. If there is a difference between the two final expressions,
then the system is time-variant.
Proceeding as described above, we first delay the input x(t)
by T units to get x(t
‒ T) as the new input.
Substituting this value in (1), we obtain
y(t, T) = ax(t ‒ T) (2)
where we have assumed that
the output, after introducing the delay, becomes a function of both t and T.
Next, introduce a delay of T units, wherever the term T
appears in the equation of the system. In (1), t appears in both the x and
y terms. Following this direction we
write
y(t
‒ T) = ax(t ‒ T) (3)
We find that the
right-hand side of (2) and (3) are the same. Hence, we conclude that the system
governed by the equation y(t) = ax(t) is time-invariant.
ILLUSTRATIVE EXAMPLE 2: Test the time-variance of the system
governed by the equation y(t) = tx(t).
Solution: As before, we first delay the input x(t)
by T to get x(t ‒ T) as the new input. Substituting this
value in the given equation, we get
y(t,
T) = tx(t ‒ T) (1)
Now, change the term t to (t - T), wherever it appears in the equation.
Then, we get from the given equation, the relation
y(t ‒ T) = (t ‒ T)x(t ‒ T)
(2)
Inspection
of (1) and (2) show that their respective right-hand sides are different. We
therefore conclude that the system governed by the equation y(t) = tx(t)
is time variant. In fact, we can see
that every system, which contains t
as a multiplying factor, will invariably be time variant.
4. MEMORY
A system is said to be memoryless, if
the equation governing the system does not contain any time- delay element in
it. Such systems are also called static systems. If the equation governing the
system contains one or more delay terms in it, it is called a dynamic system or
a system with memory.
A system described by a continuous
time-domain equation and whose amplitude is varying at every instant can be
described only by using differential equations. Similarly, a system described
by a discrete time-domain equation, and whose amplitude is varying at every
instant, can be described only by using difference equations. Systems containing differential or difference equations
are therefore called dynamic systems or systems with memory.
ILLUSTRATIVE EXAMPLE 3: Test whether the system given by the relation y(t)
= Ax(t) is dynamic or not.
Solution: We find that the equation of the system does not
contain any delay term in it. Hence, the system is static or memoryless.
ILLUSTRATIVE EXAMPLE 4: Test whether the system governed by y(t) = Ax(t)+
by(t ‒ 1) is dynamic or not.
Solution: We find that the equation of the system contains the
delay term y(t ‒ 1) in it. Hence, the system is dynamic or one with memory.
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