CHAPTER 1 Signals and Basic Operations.

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Signals and Basic

Operations.

CHAPTER 1

2

Classification

of Signals

Operation

of the

Signal.

Elementary

Signals.

The objective of this chapter is to understand the signals and their classifications, basic operation of the signal.

Chapter Overview.

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1.1 What is a Signal ?

1.2 Classification of a Signals.

1.2.1 Continuous-Time and Discrete-Time Signals

1.2.2 Even and Odd Signals.

1.2.3 Periodic and Non-periodic Signals.

1.2.4 Deterministic and Random Signals.

1.2.5 Energy and Power Signals.

1.3 Basic Operation of the Signal.

1.4 Elementary Signals.

1.4.1 Exponential Signals.

1.4.2 Sinusoidal Signal.

1.4.3 Sinusoidal and Complex Exponential Signals.

1.4.4 Exponential Damped Sinusoidal Signals.

1.4.5 Step Function.

1.4.6 Impulse Function.

1.4.7 Ramped Function.

Signals and Systems.

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A common form of human communication;

(i) use of speech signal, face to face or telephone channel.

(ii) use of visual, signal taking the form of images of people or objects around us.

Real life example of signals;

(i) Doctor listening to the heartbeat, blood pressure and temperature of the patient. These indicate the state of health of the patient.

(ii) Daily fluctuations in the price of stock market will convey an information on the how the share for a company is doing.

(iii) Weather forecast provides information on the temperature, humidity, and the speed and direction of the prevailing wind.

1.1 What is a Signal ?

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By definition, signal is a function of one or more variable, which conveys information on the nature of a physical phenomenon.

A function of time representing a physical or mathematical

quantities.

e.g. : Velocity, acceleration of a car, voltage/current of a circuit.

An example of signal; the electrical activity of the heart recorded

with electrodes on the surface of the chest — the

electrocardiogram (ECG or EKG) in the figure below.

Cont’d…

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Cont’d…

Figure 1.1 (A) Left:

(a) Snapshot of Pathfinder exploring the surface of

Mars. (b) The 70-meter (230-foot) diameter antenna

located at Canberra, Australia. The surface of the 70-

meter reflector must remain accurate within a fraction

of the signal’s wavelength. (Courtesy of Jet Propulsion

Laboratory.)

Figure 1.1 (B)

Right: Perspectival view of Mount Shasta

(California), derived from a pair of

stereo radar images acquired from orbit

with the shuttle Imaging Radar (SIR-B).

(Courtesy of Jet Propulsion Laboratory.)

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There are five types of signals;

(i) Continuous-Time and Discrete-Time Signals

(ii) Even and Odd Signals.

(iii) Periodic and Non-periodic Signals.

(iv) Deterministic and Random Signals.

(v) Energy and Power Signals.

1.2 Classifications of a Signal.

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Continuous-Time (CT) Signals

Continuous-Time (CT) Signals are functions whose amplitude or value varies continuously with time, x(t).

The symbol t denotes time for continuous-time signal and (. ) used to denote continuous-time value quantities.

Example, speed of car, converting acoustic or light wave into electrical signal and microphone converts variation in sound pressure into correspond variation in voltage and current.

Figure 1.1: Continuous-Time Signal.

1.2.1 Continuous-Time and Discrete-Time Signals.

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Discrete-Time Signals

Discrete-Time Signals are function of discrete variable, i.e. they are defined only at discrete instants of time.

It is often derived from continuous-time signal by sampling at uniform rate. Ts denotes sampling period and n denotes integer.

The symbol n denotes time for discrete time signal and [. ] is used to denote discrete-value quantities.

Example: the value of stock at the end of the month.

Figure 1.3: Discrete-Time Signal.

( ) ,....2,1,0, == nnTxnx s

Cont’d…

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A continuous-time signal x(t) is said to be an even signal if

The signal x(t) is said to be an odd signal if

In summary, an even signal are symmetric about the vertical axis

(time origin) whereas an odd signal are antisymetric about the

origin.

Figure 1.4: Even Signal Figure 1.5: Odd Signal.

( ) ( ) tallfortxtx =−

( ) ( ) tallfortxtx −=−

1.2.2 Even and Odd Signals.

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Cont’d…

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Example 1.1: Even and Odd Signals.

Find the even and odd components of each of the following signals:

(a) x(t) = Cos(t) + Sin(t) + Cos(t)Sin(t)

(b) x(t) = 1 + t + 3t2 + 5t3 +9t4

Solution:(In Class)

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Periodic Signal.

A periodic signal x(t) is a function of time that satisfies the

condition

where T is a positive constant.

The smallest value of T that satisfy the definition is called a period.

Figure 1.6: Aperiodic Signal. Figure 1.7: Periodic Signal.

( ) ( ) ,tallforTtxtx +=

1.2.3 Periodic and Non-Periodic Signals.

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Deterministic Signal.

A deterministic signal is a signal that has no uncertainty with

respect to its value at any time.

The deterministic signal can be modeled as completely specified

function of time.

Figure 1.8: Deterministic Signal; Square Wave.

1.2.4 Deterministic and Random

Signals.

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Random Signal.

A random signal is a signal about which there is uncertainty

before it occurs. The signal may be viewed as belonging to an

ensemble or a group of signals which each signal in the ensemble

having a different waveform.

The signal amplitude fluctuates between positive and negative in a

randomly fashion.

Example; noise generated by amplifier of a radio or television.

Figure 1.9: Random Signal

Cont’d…

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1.2.5 Energy Signal and Power Signals.Energy Signal.

A signal is refer to energy signal if and only if the total energy

satisfy the condition;

Power Signal.

A signal is refer to as power signal if and only if the average

power satisfy the condition;

−

=

=1

0

21 N

n

nxN

P

−=

=n

nxE 2

E0

P0

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❑

Figure 1.10: Bounded and Unbounded Signal

1.2.6 Bounded and Unbounded Signals.

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1.3 Basic Operation of the Signals.1.3.1 Time Scaling.

1.3.2 Reflection and Folding.

1.3.3 Time Shifting.

1.3.4 Precedence Rule for Time Shifting and Time Scaling.

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Time scaling refers to the multiplication of the variable by a real

positive constant.

If a > 1 the signal y(t) is a compressed version of x(t).

If 0 < a < 1 the signal y(t) is an expanded version of x(t).

Example:

Figure 1.11: Time-scaling operation; continuous-time signal x(t),

(b) version of x(t) compressed by a factor of 2, and

(c) version of x(t) expanded by a factor of 2.

( ) ( )atxty =

1.3.1 Time Scaling.

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In the discrete time,

It is defined for integer value of k, k > 1. Figure below for k = 2,

sample for n = +-1,

Figure 1.12: Effect of time scaling on a discrete-time signal:

(a) discrete-time signal x[n] and (b) version of x[n] compressed by a factor of 2, with

some values of the original x[n] lost as a result of the compression.

,knxny =

Cont’d…

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Let x(t) denote a continuous-time signal and y(t) is the signal

obtained by replacing time t with –t;

y(t) is the signal represents a refracted version of x(t) about t = 0.

Two special cases for continuous and discrete-time signal;

(i) Even signal; x(-t) = x(t) an even signal is same as reflected

version.

(ii) Odd signal; x(-t) = -x(t) an odd signal is the negative of its

reflected version.

( ) ( )txty −=

1.3.2 Reflection and Folding.

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Example 1.2: Reflection.Given the triangular pulse x(t), find the reflected version of x(t) about

the amplitude axis (origin).

Solution:Replace the variable t with –t, so we get y(t) = x(-t) as in figure below.

Figure 1.13: Operation of reflection: (a) continuous-time signal x(t) and

(b) reflected version of x(t) about the origin

x(t) = 0 for t < -T1 and t > T2.

y(t) = 0 for t > T1 and t < -T2.

.

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A time shift delay or advances the signal in time by a time

interval +t0 or –t0, without changing its shape.

y(t) = x(t-t0)

If t0 > 0 the waveform of y(t) is obtained by shifting x(t)

toward the right, relative to the tie axis.

If t0 < 0, x(t) is shifted to the left.

Example:

Figure 1.14: Shift to the Left. Figure 1.15: Shift to the Right.

Q: How does the x(t) signal looks like?

1.3.3 Time Shifting.

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Example 1.3: Time Shifting.Given the rectangular pulse x(t) of unit amplitude and unit duration.

Find y(t)=x (t-2)

Solution:t0 is equal to 2 time units. Shift x(t) to the right by 2 time units.

Figure 1.16: Time-shifting operation:

(a) continuous-time signal in the form of a rectangular pulse of amplitude 1.0 and

duration 1.0, symmetric about the origin; and

(b) time-shifted version of x(t) by 2 time shifts.

.

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Time shifting operation is performed first on x(t), which results in

Time shift has replace t in x(t) by t - b.

Time scaling operation is performed on v(t), replacing t by at and

resulting in,

Example in real-life: Voice signal recorded on a tape recorder;

(a > 1) tape is played faster than the recording rate, resulted in

compression.

(a < 1) tape is played slower than the recording rate, resulted

in expansion.

1.3.4 Precedence Rule for Time Shifting and Time Scaling.

( ) ( )

( ) ( )batxty

atvty

−=

=

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Example 1.4: Continuous Signal. A CT signal is shown in Figure 1.17 below, sketch and label each of

this signal;

a) x(t -1)

b) x(2t)

c) x(-t)

Figure 1.17

-1 3

2

t

x(t)

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Solution:(a) x(t -1) (b) x(2t)

(c) x(-t)

-3 1

2

t

x(-t)

0 4

t

x(t-1)

2

-1/2 3/2

2

t

x(t)

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Example 1.5: Discrete Time Signal.

A discrete-time signal x[n] is shown below,

Sketch and label each of the following signal.

(a) x[n – 2] (b) x[2n]

(c.) x[-n+2] (d) x[-n]

x[n]

n

4

2

0 1 2 3

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(a) A discrete-time signal, x[n-2].

❑A delay by 2

4

2

0 1 2 3 4 5 n

x(n-2)

Cont’d…

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(b) A discrete-time signal, x[2n].

Down-sampling by a factor of 2.

4

2

0 1 2 3 n

x(2n)

Cont’d…

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(c) A discrete-time signal, x[-n+2].

Time reversal and shifting

4

2

-1 0 1 2 n

x(-n+2)

Cont’d…

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(d) A discrete-time signal, x[-n].

❑Time reversal

4

2

-3 -2 -1 0 1 n

x(-n)

Cont’d…

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In Class Exercises .A continuous-time signal x(t) is shown below, Sketch and label each

of the following signal

(a) x(t – 2) (b) x(2t) (c.) x(t/2) (d) x(-t)

x(t)

t

4

0 4

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1.4 Elementary Signals. There are many types of signals prominently used in the study of

signals and systems.





1.4.5 Ramp Function.

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A real exponential signal, is written as x(t) = Beat.

Where both B and a are real parameters. B is the amplitude of the

exponential signal measured at time t = 0.

(i) Decaying exponential, for which a < 0.

(ii) Growing exponential, for which a > 0.

Figure 1.18: (a) Decaying exponential form of continuous-time signal. (b)

Growing exponential form of continuous-time signal.

Figure 1.19: (a) Decaying exponential form of discrete-time signal.

(b) Growing exponential form of discrete-time signal.


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Continuous-Time.

Case a = 0: Constant signal x(t) =C.

Case a > 0: The exponential tends to infinity as t→infinity.

Case a > 0 Case a < 0

Case a < 0: The exponential tend to zero as t→infinity (here

C > 0).

Cont’d…

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Discrete-Time.

where B and a are real.

There are six cases to consider apart from a = 0.

Case 1 (a = 0): Constant signal x[n]=B.

Case 2 (a > 1): positive signal that grows exponentially.

Case 3 (0 < a < 1): The signal is positive and decays exponentially.

nBenx a=

Cont’d…

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Case 4 (a < 1): The signal alternates between positive and negative

values and grows exponentially.

Case 5 (a = -1): The signal alternates between +C and -C.

Case 6 (-1 < a <0): The signal alternates between positive and

negative values and decays exponentially.

Cont’d…

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A general form of sinusoidal signal is

where A is the amplitude, wo is the frequency in radian per

second, and q is the phase angle in radians.

Figure 1.20: Continuous-Time Sinusoidal signal A cos(ωt + Φ).

( ) ( )qw += tAtx ocos

1.4.2 Sinusoidal Signals.

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Discrete time version of sinusoidal signal, written as

Figure 1.21: Discrete-Time Sinusoidal Signal A cos(ωt + Φ).

Cont’d…

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Continuous time sinusoidal signals,

In the discrete time case,

( ) ( )

( ) ( ) tj

tj

BetASin

BetACos

tASintx

w

w

w

w

w

Im

Re

=+

=+

+=

1.4.3 Sinusoidal and Complex Exponential Signals.

( ) ( ) tj

tj

BenASin

BenACos

=+

=+

Im

Re

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Figure 1.22: Complex plane, showing eight points uniformly distributed on the

unit circle.

Cont’d…

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Multiplication of a sinusoidal signal by a real-value decaying

exponential signal result in an exponential damped sinusoidal

signal.

Where ASin(wt + ) is the continuous signal and e-at is the

exponential

Figure 1.23: Exponentially damped

sinusoidal signal Ae-at sin(ωt), with

A = 60 and α = 6.

Observe that in Figure 1.23, an increased in time t, the amplitude

of the sinusoidal oscillation decrease in an exponential fashion and

finally approaching zero for infinite time.

( ) ( ) 0,sin += − awa tAetx t


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The discrete-time version of the unit-step function is defined

by,

Figure 1.24: Discrete–time of Step Function of Unit Amplitude.

0

0

,0

,1

=n

nnu


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The continuous-time version of the unit-step function is defined

by,

Figure 1.25: Continuous-time of step function of unit amplitude.

The discontinuity exhibit at t = 0 and the value of u(t) changes

instantaneously from 0 to 1 when t = 0. That is the reason why u(0)

is undefined.

( )0

0

,0

,1

=t

ttu

Cont’d…

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The discrete-time version of the unit impulse is defined by,

Figure 1.26: Discrete-Time form of Impulse.

Figure 1.41 is a graphical description of the unit impulse d(t).

The continuous-time version of the unit impulse is defined by the

following pair,

The d(t) is also refer as the Dirac Delta function.

==

0,0

0,1

n

nnd

( ) 1

00

=

=

−

dtt

tforn

d

d


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Figure 1.27 is a graphical description of the continuous-time unit

impulse d(t).

Figure 1.27: (a) Evolution of a rectangular pulse of unit area into an impulse of unit

strength (i.e., unit impulse). (b) Graphical symbol for unit impulse. (c)

Representation of an impulse of strength a that results from allowing the duration

Δ of a rectangular pulse of area a to approach zero.

The duration of the pulse (t) decreased and its amplitude is

increased. The area under the pulse is maintained constant at unity.

Cont’d…

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Cont’d…

49

Institutive Impulse definition;

Application of unit impulse;

Impulse of current in time delivers a unit charge

instantaneous to the network.

Impulse of force in time delivers an instantaneous

momentum to a mechanical system.

Cont’d…

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The integral of the step function u(t) is a ramp function of unit

slope.

or

Figure 1.28: Ramp Function of Unite Slope.

The discrete-time version of the ramp function,

Figure 1.29: Discrete-Time Version of the Ramp Function.

( )

=

0,0

0,

t

tttr

( ) ( )ttutr =

=

0,0

0,

n

nnnr

1.4.7 Ramp Function.

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Successive Integration of Unit Impulse Function.

CHAPTER 1 Signals and Basic Operations.

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