EDITOR: B. SOMANATHAN
NAIR
1. INTRODUCTION
The
following are the mathematical operations performed on signals:
1. Addition
of signals.
2. Multiplication
of signals.
3. Differentiation
of signals.
4. Integration
of signals.
5. Amplitude
scaling and frequency scaling of signals.
6. Inversion
or reflection of signals.
7. Time
shifting of signals.
In
this blog, we discuss the scaling, inversion, and time-shifting operations of
signals.
2. SCALING OPERATION OF SIGNALS
Scaling operations can be performed on amplitude and
frequency of a signal.
2.1 AMPLITUDE SCALING
In
amplitude scaling, the amplitude of a given signal is multiplied by an integer
or fraction; multiplication by integer increases the given amplitude and multiplication
by fraction reduces it. For example,
consider a signal y(t) = f(t). Then we may express its
amplitude-scaled version as
y(t)
= Af(t) (1)
where
A is a constant of multiplication,
known as the amplitude scaling factor.
It may be noted that A can be greater
than, equal to, or less than 1.
2.2 TIME SCALING
In
time scaling, the period of a given signal is multiplied by an integer or a
fraction; multiplication by integer reduces the period and multiplication by
fraction increases it. Let y(t) = f(t) be a given signal; then its
time-scaled version is:
y(t) = f(at)
(2)
where
a is an integer or fraction, whose
value can be greater than, equal to, or less than 1.
3. INVERSION (REFLECTION)
Let x(t)
be a given signal in the positive time axis. Then the signal x(‒t) is called the reflection or inversion
of x(t).
4. TIME SHIFTING
In
many applications, we want a signal to be shifted time from its original
location in the left or right directions. For example, consider a signal x(t).
Then, the signal x(t ‒
T) will move x(t)
to the right by T units of time from
its original location. Similarly, the signal x(t + T) will move x(t) to the left by T units
of time.
ILLUSTRATIVE EXAMPLES
Example 1 (Inversion): Invert the pulse shown in Fig. 1.
Solution: The
inverted signal is shown in Fig. 2.
It can be seen that the whole pulse is inverted in time with respect to y axis as fulcrum.
Example 2 (Time Shifting): Figure 3 shows a signal x(t). Find the signal y(t)
= x(t − 2).
Solution: The
signal shown in Fig. 3 is
symmetrical with respect to the origin. When this signal is shifted to the
right by one unit of time, we obtain the waveform shown in Fig. 4.
Example 3: (Time Scaling): Figure 5 shows a triangular function x(t)
of base width 2 units of time and an amplitude of one unit. Plot the functions (a) y1(t) = x(t/2) and (b) y2(t) = x(2t).
Solution:
(a) y1(t) = x(t/2): In this case, the time gets
halved. Hence using basic theory, we find that the base width of the function
gets doubled; but the amplitude does not change and still remains at unity.
This is plotted in Fig. 6.
(b) y2(t) =
x(2t): In this case,
the time is multiplied by a factor of 2. Hence using basic theory, we find that
the base width of the function gets halved; but the amplitude does not change
and still remains at unity. This is plotted as shown in Fig. 7.
Example 4 (Amplitude Scaling): Using the data in Example 3, find y3(t) = 2x(2t).
Solution:
In this case, the amplitude of the function y2(t) = x(2t) becomes twice that shown in Fig. 7.
The result is shown in Fig. 8.
No comments:
Post a Comment