SIGNALS AND THEIR REPRESENTATION

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• SIGNALS AND THEIR REPRESENTATION
• Signals can be of two types
• Continuous
• (ii) Discrete
• Most of the signals are inherently

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• continuous but for use in computer they are
  discretised or sampled at regular intervals.
• Thus a voltage can be recorded continuously
  with the help of a pen-recorder but if we want
  to use

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• it for some computational purpose, we have to
  discretise it in magnitude as well as in time
  (Fig. 2.9 a, b).

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f (nT)
f (t)
0
0
t
T
2T
nT
t
Fig. 2.9(b)
Fig. 2.9(a)
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• The values sampled at regular intervals give some
  idea about the nature of variation of the
  original function f (t) with t. However, if the
  sampling interval is increased, then

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• many important informations about the signal will
  be lost.
• So for getting all the important information
  about the original continuous signal, it is
  desirable that

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• the sampling time is made very very small such
  that no peak or trough is missed out between two
  successive samples.
• This necessitates that we sample

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• the signal at a rate to trap a maximum or a
  minimum. This is ensured if the sampling
  frequency f s ? 2 fmax ,where fmax is the highest
  frequency present in the signal.

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• UNIT STEP
• When we switch on a constant voltage source to
  excite a circuit, the applied voltage is zero
  before the switching instant and remains constant
  after that (Fig. 2.10).

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t 0
R
V (t)

Vc
-
Vc
V (t)
L
t
0
Fig. 2.10
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• If the magnitude of the voltage is unity (or
  normalised to some scale to unity), it is called
  a unit step.
• Mathematically we write this as
• u (t) 1 t ? 0
• 0 otherwise

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• UNIT RAMP
• If a function increases continuously at a
  uniform rate, it is called a ramp function.
• r (t) k t t ? 0
• 0 otherwise
• When k 1, it is called a unit ramp.

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• UNIT IMPULSE
• When we apply a very large force for a very short
  duration such that neither the magnitude of the
  force nor the duration is measurable,

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• but their product is finite and measurable, it is
  an impulse. If the product is unity, it is a unit
  impulse.

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The product of f (t) and ?t is the area of the
thin rectangular strip. (Fig 2.11).
?


f (t)
t
? t
Fig. 2.11
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• We denote this function as ?(t) which is defined
  mathematically as
• Ideally ? (t) 0 at t 0 to ?
• and t 0- to - ?
• 1 at t 0

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• An approximation to this is a situation when a
  cricket ball is hit by a bat the time of
  contact is very small, and the force applied is
  very large, but the change in momentum is finite.

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• Signals can also be classified a deterministic
  or random. At this moment, we shall not
  consider random signals, and our studies will be
  restricted to deterministic signals.

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• A shifted signal f(t ?) is obtained when f(t)
  is shifted to the right by an interval ? in time.
  Mathematically this will be written as (Fig.
  2.12)
• f1 (t) f (t - ?) u (t - ?).

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f1 (t- ? )
f1 (t)
f (t)
0
t
t
?
Fig.2.12(a)
Fig.2.12(b)
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• This is multiplied by u (t - ?) to indicate that
  the function f (t) does not exist before t 0,
  and in electrical engg. we deal with such
  functions most of the time. Before switching a
  voltage at t 0, the circuit does not receive
  any excitation.

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• Thus an exponentially decaying voltage or
  current function, say 5e-3t, will be as shown
  by the solid curve and strictly speaking should
  be written as 5e-3t u(t) (see Fig. 2.13).

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Fig. 2.13
The function 5e-3t extends to the negative
region as shown by the dotted portion, upto - ?.
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• We can express any function mathematically by the
  sum of the above regular functions (defined upto
  t ?).
• For example, a rectangular function shown in
  Fig. 2.14 is given by (Fig 2.14)

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10
Fig. 2.14
3
t
3
10
f2 (t)
t

f1(t)
-10
t
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• f(t) 10, 0 ? t ? 3
• 0 otherwise
• f1 (t) f2(t)
• 10 u (t) 10 u (t-3)

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• Example
• Express a train of rectangular pulses in terms of
  regular functions.

V0
0
T1
T2
T1 T2
2T2
T1 2T2
t
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• The first pulse can be written as
• f1 (t) V0 u(t) u (t T1)
• The Second Pulse can be written as a shifted
  f1(t)
• as f2 (t) f1(t T2) u (t T2).
• similarly, f3 (t) f1(t-2T2) u (t-2T2)
• .. and so on.

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• Thus
• f (t) f1(t) f2(t) f3(t) .?
• V0 u (t) u (t-T1)
• u (t T2) (t T1-T2)
• u (t 2T2) u (t T1 2T2) . ?

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• Exercise
• Express the following functions mathematically
  in terms of regular continuous functions.
  (Fig.2.16)

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f (t)

f (t)
10
k
0
4
8
t
0
T
t
Fig 2.16 (ii)
f (t)
Fig 2.16 (i)
2
16
0
8
Fig 2.16 (iii)
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f (t)
10 sin 6 ? t
Fig 2. 16 (iv)
t
0
T
10
Fig 2.16 (v)
f (t)
0
3
4
t
1