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.
7.
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.
.
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.
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