02-Elementary Signals

Instructor: Dr. Ramesh Kumar

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 Classification of Signals
We will restrict our attention to one-dimensional signals defined as single valued
functions of time. "Single valued" means that for every instant of time, there is a
unique value of the function. This value may be a real number, in which case we
speak of a real-valued signal, or it may be a complex number, in which case we
speak of a complex valued signal. In either case, the independent variable, namely,
time, is real valued.

1. Continuous Time and Discrete Time Signals 5. Energy and Power Signals

2. Deterministic and Non-deterministic Signals

3. Periodic and Aperiodic Signals

4. Even and Odd Signals 2

 Continuous-time signals (CTS)
• One way of classifying signals is based on how they are defined as a function of
time. In this context, a signal x(t) is said to be a continuous-time signal if it is
defined for all time t. Continuous Time and Discrete Time Signals

• Continuous-time signals arise naturally when a physical waveform such as an

acoustic wave or a light wave is converted into an electrical signal.

• The conversion is effected by means of a transducer; examples include the

microphone, which converts variations in sound pressure into corresponding
variations in voltage or current, and the photocell, which does the same for
variations in light intensity.

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 Continuous-time signals
− A value of signal exists at every instant of time

Independent variable

t
Independent variable

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 Discrete-time signals (DTS)
• Discrete-time signal is defined only at discrete instants of time. Thus, the
independent variable has discrete values only, which are usually uniformly
spaced.

• A discrete-time signal is often derived from a continuous-time signal by

sampling it at a uniform rate.

• Let Ts; denote the sampling period and n denote an integer that may assume
positive and negative values. Then sampling a continuous-time signal x(t) at time
t = nTs; yields a sample with the value x(nTs).
𝑥 𝑛 = x(nTs) n = 0, ± 1, ±2, ....

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 Discrete-time signals
• Consequently, a discrete-time signal is represented by the sequence of numbers
... , x[-2], x[-1], x[0], x[1], x[2], ... , which can take on a continuum of values.

• Such a sequence of numbers is referred to as a time series, written as

{x[n], n = 0, ±1, ±2, ... }, or simply x[n].

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 Discrete-time signals
− The value of signal exists only at equally spaced discrete
points in time

n
Independent variable

n
Independent variable

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 Notation
− A continuous-time signal is represented by enclosing the
independent variable (time) in parentheses ()
xt 

t
− A discrete-time signal is represented by enclosing the
independent variable (index) in square brackets []

xn

n 8
 Deterministic and Non-deterministic(Random) Signals

• A signal is said to be deterministic if there is no uncertainty with respect to its

value at any instant of time.

• Accordingly, we find that deterministic signals may be modeled as

completely specified functions of time.

• Signals which can be defined exactly by a mathematical formula are known

as deterministic signals.

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 Deterministic and Non-deterministic (Random) Signals
• A signal is said to be non-deterministic if there is uncertainty with respect to its
value at some instant of time.

• Non-deterministic signals are random in nature hence they are called

random signals.

• Random signals cannot be described by a mathematical equation.

• They are modelled in probabilistic terms.

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 Periodic signals and nonperiodic signals.
• A periodic signal x(t) is a function of time that satisfies the condition
x(t) = x(t + T) for all t,

• where T is a positive constant. Clearly, if this condition is satisfied for T = T0,

say, then it is also satisfied for T = 2 T0, 3 T0, 4 T0, ....

• The smallest value of T that satisfies the equation is called the fundamental
period of x(t). Accordingly, the fundamental period T defines the duration of one
complete cycle of x(t). The reciprocal of the fundamental period T is called the
fundamental frequency of the periodic signal x(t); it describes how frequently the
periodic signal x(t) repeats itself. We thus formally write
1
𝑓=
𝑇
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 Periodic signals and nonperiodic signals.
• The frequency f is measured in hertz (Hz), or cycles per second. The angular
frequency, measured in radians per second, is defined by
2𝜋
𝜔 = 2𝜋𝑓 =
𝑇
since there are 2𝜋 radians in one complete cycle. To simplify terminology, 𝜔 is
often referred to simply as the frequency.

Example of periodic signal: Square wave with amplitude A = 1 and period T = 0.2 s.
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Periodic and Non Periodic Signals

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 Periodic signals and nonperiodic signals.
Any signal x(t) for which no value of T satisfies the condition such that
x(t) ≠ x(t + T) for all t,
is called an aperiodic, or nonperiodic, signal.

Example of Aperiodic signal: Rectangular pulse of amplitude A and duration T1•

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Periodic and Non Periodic Signals

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Examples

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 Even and odd signals.
• A continuous-time signal x(t) is said to be an even signal if

x(-t) = x(t) for all t,

(a) Continuous-time signal x(t). (b) Representation of x(t) as a discrete-time

signal x[n].

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 Even and odd signals.
• A continuous-time signal x(t) is said to be an odd signal if

x(-t) = -x(t) for all t,

• In other words, even signals are symmetric about the vertical axis, or time origin,
whereas odd signals are antisymmetric about the time origin. Similar remarks
apply to discrete-time signals.

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 Even and odd signals.
• Example: Consider the signal
 t
sin , T  t  T
x(t )   T
 0, otherwise
Is the signal x(t) an even or an odd function of time t?
Solution: Replacing t with -t yields
  t    t 
sin    ,  T  t  T  sin   , T  t  T
x(t )    T   T    x t 
 0,  0,
 otherwise  otherwise

Hence, x(t) is an odd signal.

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 Even and odd signals.
• Suppose we are given an arbitrary signal x(t). We may develop an even-odd
decomposition of x(t) by applying the corresponding definitions. To that end, let
x(t) be expressed as the sum of two components xe(t) and xo(t) as follows:
x(t) = xe(t) + xo(t).
 Define xe(t) to be even and xo(t) to be odd; that is,
xe(-t) = xe(t) and xo(-t) = - xo(t)

 Putting t = -t in the expression for x(t), we may write

x(-t) = xe(-t) + xo(-t).
= xe(t) - x0(t).
 Solving for xe(t) and xo(t), we thus obtain
1
xe(t) = [x(t) + x(-t)], (1)
2
1
xo(t) = [x(t) - x(-t)] (2) 20
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 Even and odd signals
◆ Conjugate symmetric:
A complex-valued signal x(t) is said to be conjugate symmetric if
x(t )  x (t ) (1.6) Refer to
Let x(t )  a(t )  jb(t ) Fig. 1-13
Problem 1-2
x* (t )  a (t )  jb(t ) a( t )  a(t )
a(t )  jb(t )  a(t )  jb(t ) b( t )  b(t )

(1.7)

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 Even and odd signals.
• ANOTHER EXAMPLE OF EVEN AND ODD SIGNALS:
Find the even and odd components of the signal
x(t )  e cos  t 
2 t

Replacing t with -t in the expression for x(t) yields

x(t )  e cos  t 
2t

x(t )  e cos t
2t

Thus, applying Eqs. (1) and (2) to the problem at hand, we get

1 2t

x e(t )  2 e cos t  e cos t  cosh(2t )cos t
2t


1 2t

x e(t )  2 e cos t  e cos t   sinh(2t )cos t
2t

where cosh(2t) and sinh(2t) respectively denote the hyperbolic cosine and sine of
time t. 23
 Energy signals and power signals.
In electrical systems, a signal may represent a voltage or a current. Consider a
voltage v(t) developed across a resistor R, producing a current i(t). The
instantaneous power dissipated in this resistor is defined by
2

p (t )  v (t )
R
or, equivalently,
p (t )  Ri  t 
2

In both cases, the instantaneous power p(t) is proportional to the square of the
amplitude of the signal. Furthermore, for a resistance R of 1 ohm, Eqs. (1.12) and
(1.13) take on the same mathematical form. Accordingly, in signal analysis, it is
customary to define power in terms of a 1-ohm resistor, so that, regardless of
whether a given signal x(t) represents a voltage or a current, we may express the
p (t )  x  t 
2
instantaneous power of the signal as 24
 Energy signals and power signals.
The total energy of the continuous-time signal x(t) is ◆ Discrete-time case:
T
 Total energy of x[n]:
E  lim  x (t )dt   x 2 (t )dt
2
T
2
(1.15)
T   
E 
2
x 2 [ n] (1.18)
Time-averaged, or average, power is n 

1 T2 2 Average power of x[n]:

P  lim T x (t )dt (1.16)
T  T N
1
P  lim  x [n]
2 2
n  2 N
(1.19)
For periodic signal, the time-averaged power is n  N

1 T2 2 1 N 1
P  T x (t )dt
T 2
(1.17) P
N
 x [n]
n 0
2

(1.20)
★ Energy signal:
If and only if the total energy of the signal satisfies the condition
0 E 
★ Power signal:
If and only if the average power of the signal satisfies the condition
0 P
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 Energy signals and power signals

• x(t) ( or x[n]) is said to be an energy signal (or sequence) if and only if

0 < E < ∞, and P = 0.

• x(t) (or x[n]) is said to be a power signal (or sequence) if and only if 0 < P < ∞ ,
thus implying that E = ∞.

• Note that a periodic signal is a power signal if its energy content per period is
finite, and then the average power of this signal need only be calculated over a
period.

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 Energy signals and power signals

Solve: x(t)=Acos (𝜔0 𝑡 + 𝜃)

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