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1. Chapter Two: - Transmission of
Signals and Spectral Analysis:

2. 1.2
2.1. Classifications and properties of signals
A. Continuous-Time and Discrete-Time Signals:
A signal x(t) is a continuous-time signal if x(t) is a
continuous with t. If x(t) is defined at discrete times,
then x(t) is a discrete-time signal.
Since a discrete-time signal is defined at discrete
times, a discrete-time signal is often identified as a
sequence of numbers, denoted x[n], where n =
integer. Illustrations of a continuous-time signal x(t)
and of a discrete-time signal x[n] are shown in Fig.
1.

3. 1.3
2.1 cont’d
Fig. 1 Graphical representation of (a) continuous-time and (b)
discrete-time signals.

4. 1.4
2.1 cont’d
B. Analog and Digital Signals:
If a continuous-time signal x(t) take any value in the
continuous interval (a, b), where a may be - ꝏ and b
may be + ꝏ, then the continuous-time signal x(t) is
called an analog signal. If a discrete-time signal x[n]
can take on only a finite number of distinct values,
then we call this signal a digital signal.

5. 1.5
2.1 cont’d
where x,( t ) and x2( t ) are real signals and
C. Real and Complex Signals:
 A signal x(t) is a real signal if its value is a real number, and
a signal x(t) is a complex signal if its value is a complex
number. A general complex signal x(t) is a function of the
form
Note that in above equation t represents either a continuous
or a discrete variable.
1

6. 1.6
2.1 cont’d
D. Deterministic and Random Signals:
 Deterministic signals are those signals whose values
are completely specified for any given time. Thus, a
deterministic signal can be modeled by a known
function of time t .
Random signals are those signals that take random
values at any given time and must be characterized
statistically.

8. 1.8
2.1 cont’d
E. Even and Odd Signals: :
A signal x ( t ) is referred to as an even signal if.
A signal x ( t ) is referred to as an odd signal if.
 Geometrically, an even signal is symmetric about the vertical line t = 0.
And an odd signal is symmetric about the origin.
Some common examples of even signals include the cosine,
absolute value, and square functions.
Some common examples of odd signals include the sine, and Cube
fonctions.
.

9. Some Identities in trigonometry

10. 1.10
2.1 cont’d
Identify the following signals as even or odd signals.

11. 1.11
2.1 cont’d
F. Periodic and Non-periodic Signals:
 A signal x(t) is said to be periodic if it satisfies x(t) = x(t +T), for
all t and some constant T, T > 0.
 The quantity T is referred to as the period of the signal.
Two quantities closely related to the period are the frequency and
angular frequency, denoted as f and w, respectively, and defined as
A signal that is not periodic is said to be aperiodic.
Examples of periodic signals include the cosine and sine functions.

12. 1.12
2.1 cont’d
The period of a periodic signal is not unique. That is,
a signal that is periodic with period T is also periodic
with period NT, for every (strictly) positive integer N.
The smallest period with which a function is periodic
is called the fundamental period and its
corresponding frequency is called the fundamental
frequency.
Any continuous-time signal which is not periodic is
called a nonperiodic (or aperiodic ) signal.

13. 1.13
2.1 cont’d
Sum of periodic functions.
Let x1(t) and x2(t) be periodic signals with fundamental
periods T1 and T2, respectively. Then, the sum y(t) =
x1(t)+x2(t) is a periodic signal if and only if the ratio
T1/T2 is a rational number (i.e., the quotient of two
integers). Suppose that T1/T2 = q/r where q and r are
integers and co-prime (i.e., have no common factors),
then the fundamental period of y(t) is rT1 (or
equivalently, qT2, since rT1 = qT2).
Let T1=9 and T2=21; then (T1/T2)=(9/21)=3/7=(q/r)
Example:

14. 1.14
2.1 cont’d
G. Energy and Power Signals:
The energy E contained in the signal x(t) is given by
 signal with finite energy is said to be an energy signal.
 The average power P contained in the signal x(t) is given
by.
 A signal with (nonzero) finite average power is said to
be a power signal.
Energy= Power*time=(v2 /r)*t
Power= Energy/time

15. 1.15
2.1 cont’d
Unit-Step Function
 The unit-step function (also known as the Heaviside
function), denoted u(t), is defined as .
A plot of this function is shown below.

16. 1.16
2.1 cont’d
 Similarly, the shifted unit step function u(t - to) is
defined as.
A plot of this function is shown below.

17. 1.17
2.1 cont’d
Unit-Impulse Function
The unit-impulse function (also known as the Dirac delta
function or delta function), denoted δ(t), is defined by the
following two properties:
The total area under the impulse is equal to unity
Technically, ⸹
is not a function in the ordinary sense.
Rather, it is what is known as a generalized function.
Consequently, the ⸹
function sometimes behaves in
unusual ways.

18. 1.18
2.1 cont’d
 Graphically, the delta function is represented as
shown below.
The product of ⸹
(t) with any function of time, say s(t) depends only
upon the value of that function at t=0
Sifting property. For any continuous function x(t) and any
real constant to,
shifted impulse function

19. 1.19
2.1 cont’d
 The δ function also has the following properties:
 Equivalence property. For any continuous function x(t)
and any real constant to, where a is a nonzero real
constant.
Evaluate the following integrals

20. Evaluate the following integrals
The impulse occurs at t=1, which is
outside the range of integration
(a)
(b)
(c)

21. 1.21
2.1 cont’d
Complex Exponential Signals:
The complex exponential signal
is an important example of a complex signal. Using
Euler's formula, this signal can be defined as
Thus, x(t) is a complex signal whose real part is cos⍵ot
and imaginary part is sin⍵ot.
The fundamental period To of x(t) is given by
Note that x(t) is periodic for any value of ⍵o.

22. 1.22
2.1 cont’d
Exponentially increasing and decreasing sinusoidal signal

23. 1.23
2.1 cont’d
Sinusoidal Signals:
A continuous-time sinusoidal signal can be expressed as
where A is the amplitude (real), ⍵o, is the radian
frequency in radians per second, and θ is the phase angle
in radians. The sinusoidal signal x(t) is shown in Figure
below, and it is periodic with fundamental period
The reciprocal of the fundamental period To is called the
fundamental frequency fo:

24. 1.24
2.1 cont’d
 Using Euler's formula, the sinusoidal signal can be
expressed as
Example:
Write the following sinusoidal in terms of exponential function
We know that
We also know that,
This imply